Class 12

Odisha State Board BSE Odisha 9th Class Maths Solutions Geometry Chapter 4 କ୍ଷେତ୍ରଫଳ Ex 4 Textbook Exercise Questions and Answers.

BSE Odisha Class 9 Maths Solutions Geometry Chapter 4 କ୍ଷେତ୍ରଫଳ Ex 4

Question 1.
ନିମ୍ନଲିଖ୍ ଚିତ୍ରଗୁଡ଼ିକ ମଧ୍ୟରୁ କେଉଁଥ‌ିରେ ଚିତ୍ରିତ (shaded) ତ୍ରିଭୁଜର କ୍ଷେତ୍ରଫଳ, ଆୟତକ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳର ଅଧା ?

ସମାଧାନ:
(i), (ii) ଏବଂ (iv) ରେ ଚିତ୍ରିତ ତ୍ରିଭୁଜର କ୍ଷେତ୍ରଫଳ ଆୟତକ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳର ଅଧା

Question 2.
ଚିତ୍ରରେ ABCD ଓ DCEX ଦୁଇଟି ସାମାନ୍ତରିକ କ୍ଷେତ୍ର, AB = CF; B ଓ X ବିନ୍ଦୁ A ଓ E ର ମଧ୍ୟବର୍ତ୍ତୀ ହେଲେ,

(i) ନିମ୍ନଲିଖ ଉକ୍ତିଗୁଡ଼ିକ ମଧ୍ୟରୁ କେଉଁଗୁଡ଼ିକ ଠିକ୍ ଉକ୍ତି ?
(a) ABCD ଓ DCEX କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(b) ABCD ଓ CFEX କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(c) DCEX ଓ EFCB କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ । 
(d) DCEX ଓ CFEX କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
ସମାଧାନ:
(a) ଠିକ୍
(b) ଠିକ୍
(d) ଠିକ୍

(ii) ନିମ୍ନଲିଖ ଉକ୍ତିମାନଙ୍କରେ ଭୁଲ୍ ଥିଲେ ସଂଶୋଧନ କର ।
(a) Δ  XDC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ABCD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(b) Δ XCE ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BCFE କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(c) Δ BCE ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BCFE କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(d) Δ CEX ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ CEX କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(e) ABCD ର କ୍ଷେତ୍ରଫଳ = 2 x Δ CEX ର କ୍ଷେତ୍ରଫଳ ।
(f) BCEF ର କ୍ଷେତ୍ରଫଳ = 2 x Δ DCX ର କ୍ଷେତ୍ରଫଳ ।
ସମାଧାନ:
(b) Δ XCE ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) XEFC ର କ୍ଷେତ୍ରଫଳ ।
(c) Δ CXE ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × XDCE ର କ୍ଷେତ୍ରଫଳ ।
(f) DCEX ର କ୍ଷେତ୍ରଫଳ = 2 × Δ DCX ର କ୍ଷେତ୍ରଫଳ ।

Question 3.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ \(\overleftrightarrow{\mathbf{A F}}\|\overleftrightarrow{\mathbf{B G}}, \overleftrightarrow{\mathbf{A B}}\| \overleftrightarrow{\mathbf{D C}}, \overleftrightarrow{\mathbf{B E}} \| \overleftrightarrow{\mathbf{C F}} \text { ଓ } \overleftrightarrow{\mathbf{A C}} \| \overleftrightarrow{\mathbf{D G}}\) ।

(a) ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।
(i) ABCD କ୍ଷେତ୍ରସହ _________ ଓ _________ କ୍ଷେତ୍ରଦ୍ଵୟର କ୍ଷେତ୍ରଫଳ ସମାନ ।
(ii) Δ ABC କ୍ଷେତ୍ରସହ _________ ଓ _________ କ୍ଷେତ୍ରଦ୍ଵୟର କ୍ଷେତ୍ରଫଳ ସମାନ ।
ସମାଧାନ:
(i) EBCF ସାମାନ୍ତରିକ, ACGD ସାମାନ୍ତରିକ
(ii) Δ ACD, Δ DCG

(b) ପ୍ରମାଣ କର ଯେ :
(i) Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (ACGD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)
(ii) Δ ACD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (BCFE କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)
ସମାଧାନ:
(i) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = ACGD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ACGD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (ACGD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)

(ii) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = BCEF ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BCEF ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ Δ ACD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (BCEF ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)

(c) E ଯଦି AD ର ମଧ୍ୟବିନ୍ଦୁ ହୁଏ, ତେବେ ନିମ୍ନୋକ୍ତ କ୍ଷେତ୍ରମାନଙ୍କର କ୍ଷେତ୍ରଫଳ ମଧ୍ୟରେ ସମ୍ପର୍କ ନିର୍ଣ୍ଣୟ କର ।
(i) Δ ABC ଓ Δ BCF
(ii) Δ AEB ଓ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ABCD
(iii) Δ BCF ଓ BCFE କ୍ଷେତ୍ର
(iv) Δ DFC ଓ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର BCFE
(v) Δ ABE ଓ Δ DCF
ସମାଧାନ:
(i) Δ ABC ର କ୍ଷେତ୍ରଫଳ = Δ BCF ର କ୍ଷେତ୍ରଫଳ
(ଏକା ଭୂମି ଏବଂ ଏକା ସମାନ୍ତର ସରଳରେଖାଦ୍ଵୟ ମଧ୍ୟରେ ଅବସ୍ଥିତ ।)

(ii) Δ AEB ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{4}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
[Δ AEB ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (\(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)]

(iii) Δ BCF ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BCFE ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(∵ B͞F କଣ୍ଠ, ସାମାନ୍ତରିକ କ୍ଷେତ୍ରକୁ ଦୁଇଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।)

(iv) Δ DFC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{4}\) BCFE ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(∵ AD = EF ⇒ AE + DE = DE + DF ⇒ AE = DF)
∴ Δ AEB ର କ୍ଷେତ୍ରଫଳ = Δ DCF ର କ୍ଷେତ୍ରଫଳ 

(v) Δ ABE ର କ୍ଷେତ୍ରଫଳ = Δ DCF ର କ୍ଷେତ୍ରଫଳ ।
(∵ AD = EF ⇒ AD – ED = EF – ED ⇒ AE = DF)

Question 4.
ଚିତ୍ର (i) ଓ (ii) ରେ ଚିହ୍ନିତ ଅଂଶଦ୍ଵୟର କ୍ଷେତ୍ରଫଳ କାହିଁକି ସମାନ ?

ସମାଧାନ:
ଉତ୍ତର : ପ୍ରତ୍ୟେକ କ୍ଷେତ୍ରରେ ଚିହ୍ନିତ ଅଂଶମାନ ଗୋଟିଏ ଗୋଟିଏ ତ୍ରିଭୁଜ ।
ଚିତ୍ର (i) ରେ ଚିହ୍ନିତ ଅଂଶର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × 6 × 4 = 12 ବ.ମି. (∵ ତ୍ରିଭୁଜର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ଭୂମି × ଉଚ୍ଚତା)
ଏବଂ (ii) ରେ ଚିହ୍ନିତ ଅଂଶର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × 8 × 3 = 12 ବ.ମି.
ତେଣୁ ଉଭୟ ଚିହ୍ନିତ ଅଂଶର କ୍ଷେତ୍ରଫଳ ସମାନ ।

Question 5.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର । CX ⊥ AD, BY ⊥ \(\overrightarrow{\mathrm{CA}}\) ଏବଂ CZ ⊥ \(\overrightarrow{\mathrm{BA}}\), ନିମ୍ନଲିଖିତ ଉକ୍ତିମାନଙ୍କରୁ କେଉଁ ଭକ୍ତିଗୁଡ଼ିକ ଠିକ୍ ? କାରଣ ଦର୍ଶାଅ ।

(i) AD . CX = BZ . CZ
(ii) AD . CX = CY . BY
(iii) BZ . CZ = AC . BY
(iv) BC . CX = AB . CZ
(v) AB . CZ = 2AC . BY
ସମାଧାନ:
(i) AD . CX = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
BZ . CZ ≠ ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ (i) ଉକ୍ତିଟି ଭୁଲ୍ ଅଛି ।

(ii) AD . CX = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
CY . BY ≠ ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ (ii) ଉକ୍ତିଟି ଭୁଲ୍ ଅଛି ।

(iii) BZ . CZ ≠ ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
AC . BY = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ (iii) ଉକ୍ତିଟି ଭୁଲ୍ ଅଛି ।

(iv) BC . CX = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
AB . CZ = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ (iv) ଉକ୍ତିଟି ଠିକ୍ ଅଛି ।

(v) AB . CZ = ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
2AC . BY = 2 × ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ (v) ଉକ୍ତିଟି ଭୁଲ୍ ଅଛି ।

Question 6.
Δ ABC ରେ BC ଓ AC ବାହୁର ଦୈର୍ଘ୍ୟ ଯଥାକ୍ରମେ 16 ସେ.ମି. ଓ 12 ସେ.ମି. ।
Aରୁ BC ଉପରେ ପତିତ ଲମ୍ବର ଦୈର୍ଘ୍ୟ 9 ସେ.ମି ।
(i)  Δ ABC ର କ୍ଷେତ୍ରଫଳ ସ୍ଥିର କର ।
(ii) B ରୁ AC ଉପରେ ପତିତ ଲମ୍ବର ଦୈର୍ଘ୍ୟ ନିର୍ଣ୍ଣୟ କର ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ BC = 16 ସେ.ମି., AC = 12 ସେ.ମି. ।
AD ⊥ BC ଏବଂ BE ⊥ AC, AD = 9 ସେ.ମି. ।
ନିର୍ଦେୟ : (i) Δ ABC ର କ୍ଷେତ୍ରଫଳ (ii) BE ର ଦୈର୍ଘ୍ୟ

ଉତ୍ତର : (i) Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BC × AD = \(\frac{1}{2}\) × 16 × 9 = 72 ସେ.ମି.
(ii) Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) AC × DE
⇒ 72 = \(\frac{1}{2}\) × 12 × BE 
⇒ BE = \(\frac{72}{6}\) = 12 ବ. ସେ.ମି. ।

Question 7.
ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ AN ⊥ BC ଏବଂ AM ⊥ CD । BC = 25 ସେ.ମି., AN = 10 ସେ.ମି., CD = 15 ସେ.ମି. ହେଲେ,
(i) AM କେତେ ହେବ ସ୍ଥିର କର ।
(ii) Δ ABC ର କ୍ଷେତ୍ରଫଳ ସ୍ଥିର କର ।
(iii) Δ ADC ର କ୍ଷେତ୍ରଫଳ ସ୍ଥିର କର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ AM ଓ AN ଉତ୍ତତାଦ୍ବୟ ।
BC = 25 ସେ.ମି., AN = 10 ସେ.ମି., CD = 15 ସେ.ମି.

ନିର୍ମେୟ : (i) AM ର ଦୈର୍ଘ୍ୟ
(ii) Δ ABC ର କ୍ଷେତ୍ରଫଳ
(iii) Δ ADC ର କ୍ଷେତ୍ରଫଳ
[ ତ୍ରିଭୁଜର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × ଭୂମିର ଦୈର୍ଘ୍ୟ x ଉଚ୍ଚତା
ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = ଭୂମିର ଦୈର୍ଘ୍ୟ × ଭୂମି ପ୍ରତି ଅଙ୍କିତ ଉଚ୍ଚତା ]
ଉତ୍ତର : (i) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = BC × AN = CD × AM
⇒ BC × AN = CD × AM ⇒ 25 × 10 = 15 × AM
⇒ AM = \(\frac{25 \times 10}{15}\) = \(\frac{50}{3}\) ସେ.ମି. ବା 16 \(\frac{2}{3}\) ସେ.ମି.
(ii) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BC × AN = \(\frac{1}{2}\) × 25 × 10 = 125 ବର୍ଗ ସେ.ମି.
(iii) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) CD × AM = \(\frac{1}{2}\) × 15 × \(\frac{50}{3}\) = 125 ବ. ସେ.ମି.

Question 8.
ଚିତ୍ରରେ ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର । PQ || AD, XY || AB ହେଲେ, ପ୍ରମାଣ କର ଯେ
(i) POYB ଓ XOOD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(ii) AXYB ଓ APQD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(iii) PBCQ ଓ XYCD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ PQ || AD ଏବଂ XY || AB ।
ପ୍ରାମାଣ୍ୟ : (i) POYB ଓ XOOD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(ii) AXYB ଓ APQD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(iii) PBCQ ଓ XYCD କ୍ଷେତ୍ରଦ୍ଵୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।

ପ୍ରମାଣ : (i) AXOP ଓ OYCQ ଗୋଟିଏ ଗୋଟିଏ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ।
AO ଏବଂ OC ଯଥାକ୍ରମେ କ୍ଷେତ୍ରଗୁଡ଼ିକର କର୍ଣ୍ଣ । ∴ β = β ଏବଂ α = α₁
ପୁନଶ୍ଚ, ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ହେତୁ A ABCର କ୍ଷେତ୍ରଫଳ = A ADCର କ୍ଷେତ୍ରଫଳ
⇒ β₁ + POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ + α₁ = β + XOQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ + α
⇒ POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = XOQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ (∵ β = β₁ ଏବଂ α = α₁) (ପ୍ରମାଣିତ)

(ii) (i)ରୁ POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = XOQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ + APOX ର କ୍ଷେତ୍ରଫଳ
= XOOD ର କ୍ଷେତ୍ରଫଳ + APOX ର କ୍ଷେତ୍ରଫଳ
⇒ AXYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = APQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

(iii) (i)ରୁ ପ୍ରମାଣିତ POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = XOOD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ POYB କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ + OYCQ ର କ୍ଷେତ୍ରଫଳ
= XOQD ର କ୍ଷେତ୍ରଫଳ + OYCQ ର କ୍ଷେତ୍ରଫଳ
⇒ PBCQ ର କ୍ଷେତ୍ରଫଳ = XYCD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 9.
ଦତ୍ତ ମାନ ଅନୁଯାୟୀ ନିମ୍ନଲିଖତ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରଗୁଡ଼ିକର କ୍ଷେତ୍ରଫଳ ନିଶ୍ଚୟ କର – ଯାହାର,
(i) ଉଚ୍ଚତା 5 ସେ.ମି. ଓ ଭୂମିର ଦୈର୍ଘ୍ୟ 10 ସେ.ମି. ।
(ii) ଗୋଟିଏ ବାହୁର ଦୈର୍ଘ୍ୟ 18 ମି. ଓ ବିପରୀତ ସମାନ୍ତର ବାହୁଠାରୁ ତାହାର ଦୂରତା 7 ସେ.ମି. ।
(iii) ଭୂମିର ଦୈର୍ଘ୍ୟ 120 ଡେ.ମି. ଓ ତାହାର ବିପରୀତ କୌଣିକ ବିନ୍ଦୁରୁ ଅଙ୍କିତ ଲମ୍ବର ଦୈର୍ଘ୍ୟ 22 ଡେ.ମି. ।
ସମାଧାନ :
(i) ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = ଭୂମିର ଦୈର୍ଘ୍ୟ x ଉଚ୍ଚତା = (10 × 5) ବ. ସେ.ମି. = 50 ବ. ସେ.ମି. ।
(ii) ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = ବାହୁର ଦୈର୍ଘ୍ୟ × ବିପରୀତ ସମାନ୍ତର ବାହୁଠାରୁ ବାହୁର ଦୂରତା
= 18 ମି. × 7 ବ. ସେ.ମି. = (180 × 7) ବ. ସେ.ମି. = 1260 ବ. ସେ.ମି. ।
(iii) ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = ଭୂମିର ଦୈର୍ଘ୍ୟ × ବିପରୀତ କୌଣସି ବିନ୍ଦୁରୁ ଅଙ୍କିତ ଲମ୍ବର ଦୈର୍ଘ୍ୟ
= (120 × 22) ବ. ଡେ.ମି. = 2640 ବ. ଡେ.ମି. ।

Question 10.
ଚିତ୍ରରେ AP ⊥ BC, CQ ⊥ AB ଏବଂ XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ର । ନିମ୍ନ ଦତ୍ତମାନ ଅନୁଯାୟୀ Δ ABC ଓ XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ନିର୍ଣ୍ଣୟ କର ଏବଂ ଦର୍ଶାଅ ଯେ, Δ ABC ର କ୍ଷେତ୍ରଫଳ, XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳର ଅର୍ଦ୍ଧେକ ।
(i) BC = 16 ସେ.ମି., AP = 6 ସେ.ମି.
(ii) AB = 12 ସେ.ମି., CQ = 8 ସେ.ମି.

ସମାଧାନ:
(i) BC = 16 ସେ.ମି., AP = 6 ସେ.ମି.
∴ Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) BC . AP = \(\frac{1}{2}\) × 16 × 6 = 48 ବ. ସେ.ମି.
XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ର କ୍ଷେତ୍ରଫଳ = 16 × 6 = 96 ବ. ସେ.ମି.
ଏଠାରେ A ABCର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।  (ପ୍ରମାଣିତ)

(ii) XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = BC × AP = 16 × 6 = 96 ବ. ସେ.ମି.
∴ Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) AB × CQ = \(\frac{1}{2}\) × 12 × 8 = 48 ବ. ସେ.ମି.
ଏଠାରେ A ABCର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) XBCY ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।  (ପ୍ରମାଣିତ)

Question 11.
ଗୋଟିଏ ବର୍ଗକ୍ଷେତ୍ର ଓ ଗୋଟିଏ ରମ୍ବସ୍ ଏକା ଭୂମି ଉପରେ ଓ ତାହାର ଏକ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ; ସେମାନଙ୍କର କ୍ଷେତ୍ରଫଳ ସମାନ; ପ୍ରମାଣ କର ଯେ, ସେମାନେ ଏକ ସମାନ୍ତର ସରଳରେଖାଦ୍ଵୟ ମଧ୍ଯରେ ଅବସ୍ଥିତ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ବର୍ଗକ୍ଷେତ୍ର ଏବଂ EBCF ରମ୍ବସ୍ ଏକା ଭୂମି ଉପରେ ଅବସ୍ଥିତ ।
ପ୍ରାମାଣ୍ୟ : AF || BC
ଅଙ୍କନ : EM ⊥ BC ଅଙ୍କନ କର ।
ପ୍ରମାଣ : ABCD ବର୍ଗକ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = EBCF ରମ୍ବସ୍‌ର କ୍ଷେତ୍ରଫଳ ।
⇒ BC × AB = BC × EM
⇒ AB = EM
ପୁନଶ୍ଚ, AB ⊥ BM ଓ EM ⊥ BM
∴ AE || BM ⇒ AF || BC
 (ପ୍ରମାଣିତ)

Question 12.
Δ ABC ର BC ଉପରିସ୍ଥ D ଏପରି ଏକ ବିନ୍ଦୁ ଯେପରିକି BD = \(\frac{1}{2}\) DC । ପ୍ରମାଣ କର ଯେ,
Δ ABD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{3}\) × Δ ABC ର କ୍ଷେତ୍ରଫଳ ।
ସମାଧାନ:

ଦତ୍ତ : Δ ABC ର BC ଉପରିସ୍ଥ D ଏପରି ଏକ ବିନ୍ଦୁ ଯେପରିକି BD = \(\frac{1}{2}\) DC ।
ପ୍ରାମାଣ୍ୟ : Δ ABD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{3}\) × Δ ABC ର କ୍ଷେତ୍ରଫଳ ।
ଅଙ୍କନ : AM ⊥ BC ଅଙ୍କନ କର ।

Question 13.
ପ୍ରମାଣ କର ଯେ, କୌଣସି ତ୍ରିଭୁଜର ମଧ୍ୟମା ତାହାକୁ ଦୁଇ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ବିଭକ୍ତ କରେ ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ AD ମଧ୍ୟମା ।

ପ୍ରାମାଣ୍ୟ : Δ ABD ର କ୍ଷେତ୍ରଫଳ = Δ ADC ର କ୍ଷେତ୍ରଫଳ ।
ଅଙ୍କନ : A ବିନ୍ଦୁରୁ BC ପ୍ରତି AM ଲମ୍ବ ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ ABC ରେ AD ମଧ୍ୟମା ⇒ BD = CD 
Δ ABD ର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) BD . AM = CD . AM (∵ BD = CD)
ସେହିପରି Δ ADC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) CD · AM
∴ Δ ABD ର କ୍ଷେତ୍ରଫଳ = Δ ADC ର କ୍ଷେତ୍ରଫଳ ।

Question 14.
ପ୍ରମାଣ କର ଯେ, ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର ଗୋଟିଏ କର୍ଷ କ୍ଷେତ୍ରଟିକୁ ଦୁଇଗୋଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ବିଭକ୍ତ କରେ ।
ସମାଧାନ:

ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ AC ଏକ କଣ୍ଠ ।
ପ୍ରାମାଣ୍ୟ : B͞D କଣ୍ଠ ଅଙ୍କନ କରାଯାଉ ।
ଅଙ୍କନ : Δ ABC ର କ୍ଷେତ୍ରଫଳ = Δ ADC ର କ୍ଷେତ୍ରଫଳ
ପ୍ରମାଣ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର AC ଓ BD କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ ଜରନ୍ତ ।
ଅର୍ଥାତ୍ AO = OC ଏବଂ BO = DO
Δ ABC ରେ BO ମଧ୍ୟମା ।
ତ୍ରିଭୁଜର ମଧ୍ୟମା ତ୍ରିଭୁଜକୁ ଦୁଇଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।
Δ AOB ର କ୍ଷେତ୍ରଫଳ = Δ BOCର କ୍ଷେତ୍ରଫଳ
ସେହିପରି Δ BCD ରେ CO ମଧ୍ୟମା ।
⇒ Δ BOC ର କ୍ଷେତ୍ରଫଳ = Δ COD ର କ୍ଷେତ୍ରଫଳ
Δ ADC ରେ DO ମଧ୍ୟମା ।
⇒ Δ COD ର କ୍ଷେତ୍ରଫଳ = Δ AOD ର କ୍ଷେତ୍ରଫଳ
ପୁନଶ୍ଚ, Δ ABD ରେ AO ମଧ୍ୟମା
⇒ Δ AOD ର କ୍ଷେତ୍ରଫଳ = Δ AOB ର କ୍ଷେତ୍ରଫଳ
Δ AOB ର କ୍ଷେତ୍ରଫଳ = Δ BOC ର କ୍ଷେତ୍ରଫଳ = Δ COD ର କ୍ଷେତ୍ରଫଳ = Δ AOD ର କ୍ଷେତ୍ରଫଳ ।
Δ AOB ର କ୍ଷେତ୍ରଫଳ + Δ BOC ର କ୍ଷେତ୍ରଫଳ = Δ AOD ର କ୍ଷେତ୍ରଫଳ + Δ COD ର କ୍ଷେତ୍ରଫଳ
Δ ABC ର କ୍ଷେତ୍ରଫଳ = Δ ACD ର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 15.
ଚିତ୍ରରେ ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର । ପ୍ରମାଣ କରଯେ,
(i) ADQP ଓ BCQP କ୍ଷେତ୍ରତ୍ବୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(ii) Δ AOD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{4}\) ABCD
ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।

ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର AC ଓ BD କର୍ଣ୍ଣଦ୍ଵୟର ଛେଦବିନ୍ଦୁ O । 
PQ, AB ଓ DC କୁ ଯଥାକ୍ରମେ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରୁଛି ।
ପ୍ରାମାଣ୍ୟ : (i) ADQP କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = BCQP କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
(ii) Δ AOD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{4}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
ପ୍ରମାଣ : (i) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ A͞C ଓ BD କର୍ଣ୍ଣଦ୍ୱୟର ଛେଦବିନ୍ଦୁ O ।
Δ AOP ଓ Δ QOC ଦ୍ବୟରେ AO = CO, m∠AOP = m∠QOC (ପ୍ରତୀପ) ଏବଂ
m∠PAO = m∠OCQ (ଏକାନ୍ତର)
∴ Δ AOP ≅ Δ QOC (କୋ-ବା-କୋ ସର୍ବସମତା)
⇒ Δ AOP ର କ୍ଷେତ୍ରଫଳ = Δ QOC ର କ୍ଷେତ୍ରଫଳ … (i)
ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର AC (କର୍ଣ୍ଣ) ।
∴ Δ ADC ର କ୍ଷେତ୍ରଫଳ = Δ ABC ର କ୍ଷେତ୍ରଫଳ
⇒ Δ ADC ର କ୍ଷେତ୍ରଫଳ – Δ QOC ର କ୍ଷେତ୍ରଫଳ + Δ AOP ର କ୍ଷେତ୍ରଫଳ
= Δ ABC ର କ୍ଷେତ୍ରଫଳ – Δ AOP ର କ୍ଷେତ୍ରଫଳ + Δ QOC ର କ୍ଷେତ୍ରଫଳ
(∵ Δ AOP ର କ୍ଷେତ୍ରଫଳ = Δ QOC ର କ୍ଷେତ୍ରଫଳ)
⇒ ADOP କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = BCQP କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।

(ii) Δ AOD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) × A ADC ର କ୍ଷେତ୍ରଫଳ (∵ AO = \(\frac{1}{2}\) AD)
= \(\frac{1}{2}\) × \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{4}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 16.
ABCD ଏକ ଟ୍ରାପିଜିୟମ୍, ଏହାର \(\stackrel{\leftrightarrow}{\mathrm{AB}}\) || \(\stackrel{\leftrightarrow}{\mathrm{DC}}\); ପ୍ରମାଣ କର ଯେ,
(i) Δ ADC ଓ Δ BDC ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
(ii) Δ ADB ଓ Δ ACB ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।

ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍‌ରେ \(\stackrel{\leftrightarrow}{\mathrm{AB}}\) || \(\stackrel{\leftrightarrow}{\mathrm{DC}}\) ।
ପ୍ରାମାଣ୍ୟ : (i) Δ ADCର କ୍ଷେତ୍ରଫଳ = Δ BDC ର କ୍ଷେତ୍ରଫଳ
(ii) Δ ADBର କ୍ଷେତ୍ରଫଳ = Δ ACB ର କ୍ଷେତ୍ରଫଳ
ପ୍ରମାଣ : (i) \(\stackrel{\leftrightarrow}{\mathrm{AB}}\) || \(\stackrel{\leftrightarrow}{\mathrm{DC}}\) ଏବଂ Δ ADC ଦ୍ବୟ ଓ Δ BDC ଏକ ଭୂମି D͞C ବିଶିଷ୍ଟ ।
⇒ Δ ADC ର କ୍ଷେତ୍ରଫଳ = Δ BDC ର କ୍ଷେତ୍ରଫଳ ।
(ii) \(\stackrel{\leftrightarrow}{\mathrm{AB}}\) || \(\stackrel{\leftrightarrow}{\mathrm{DC}}\) ଏବଂ Δ ADB ଓ Δ ACB ଦ୍ଵୟ ଏକା ଭୂମି AB ବିଶିଷ୍ଟ ।
⇒ Δ ADB ର କ୍ଷେତ୍ରଫଳ = Δ ACB ର କ୍ଷେତ୍ରଫଳ ।

Question 17.
Δ ABC ର E ଓ F ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ
(i) ଦର୍ଶାଅ ଯେ, EBCF ଏକ ଟ୍ରାପିଜିୟମ୍ ।
(ii) Δ ABCର କ୍ଷେତ୍ରଫଳ 50 ବ. ସେ.ମି. ହେଲେ, ଦର୍ଶାଅ ଯେ, EBCF ଟ୍ରାପିଜିୟମ୍‌ର କ୍ଷେତ୍ରଫଳ 37.5 ବ. ସେ.ମି. ।
ସମାଧାନ:
(i) ଦତ୍ତ : Δ ABC ରେ E ଓ F ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : EBCF ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ପ୍ରମାଣ : Δ ABC ରେ AB ର ମଧ୍ୟବିନ୍ଦୁ E ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ F ।
⇒ EF || BC
∴ EBCF ଏକ ଟ୍ରାପିଜିୟମ୍ ।

(ii) Δ ABC ରେ E, AB ର ମଧ୍ୟବିନ୍ଦୁ ଓ F, ACର ମଧ୍ୟବିନ୍ଦୁ ।
⇒ EF || BC ଏବଂ EF = \(\frac{1}{2}\) BC = BD = CD
EFDB କ୍ଷେତ୍ରରେ EF || BD ଓ EF = BD ।
∴ EFDB ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।

EFDB ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ ED କର୍ଣ୍ଣ EFDB ସାମାନ୍ତରିକ କ୍ଷେତ୍ରକୁ ଦୁଇ ସମାନ କ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।
⇒ Δ EBD ର କ୍ଷେତ୍ରଫଳ = Δ EFD ର କ୍ଷେତ୍ରଫଳ ।
∴ ସେହିପରି ପ୍ରମାଣ କରାଯାଇପାରେ ଯେ, Δ AEF ର କ୍ଷେତ୍ରଫଳ = Δ EBD ର କ୍ଷେତ୍ରଫଳ
= Δ EFD ର କ୍ଷେତ୍ରଫଳ = Δ FDC ର କ୍ଷେତ୍ରଫଳ ।
Δ ABC ର କ୍ଷେତ୍ରଫଳ = 50 ବର୍ଗ ସେ.ମି. ।
Δ AEF ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{4}\) × 50 = 12.5 ବର୍ଗ ସେ.ମି. ।
∴ EBCF ଟ୍ରାପିଜିୟମ୍‌ର କ୍ଷେତ୍ରଫଳ = Δ ABC ର କ୍ଷେତ୍ରଫଳ – Δ AFF ର କ୍ଷେତ୍ରଫଳ
= 50 ବର୍ଗ ସେ.ମି. – 12.5 ବର୍ଗ ସେ.ମି. = 37.5 ବର୍ଗ ସେ.ମି. ।

Question 18.
Δ ABC ର E ଓ F ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ । C͞E ଓ B͞F ର ଛେଦବିନ୍ଦୁ O ହେଲେ,
ଦର୍ଶାଅ ଯେ Δ OBC ର କ୍ଷେତ୍ରଫଳ = AEOF ଚତୁର୍ଭୁଜର କ୍ଷେତ୍ରଫଳ ।
ସମାଧାନ:
ଦତ୍ତ :  Δ ABC ର E ଓ F ଯଥାକ୍ରମେ A͞B ଓ A͞C ର ମଧ୍ୟବିନ୍ଦୁ । C͞E ଓ B͞F ର ଛେଦବିନ୍ଦୁ O।
ପ୍ରାମାଣ୍ୟ : Δ OBC ର କ୍ଷେତ୍ରଫଳ  =  AEOF ଚତୁର୍ଭୁଜର କ୍ଷେତ୍ରଫଳ

ପ୍ରମାଣ :  Δ ABC ରେ E ଓ F ଯଥାକ୍ରମେ A͞B ଓ A͞C ର ମଧ୍ୟବିନ୍ଦୁ ।
Δ ABC ରେ B͞F ମଧ୍ୟମା ।
Δ ABF ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ
[∵ ମଧ୍ୟମା ତ୍ରିଭୁଜକୁ ଦୁଇ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ବିଭକ୍ତ କରେ ।]
Δ ABC ରେ C͞E ମଧ୍ୟମା ।
Δ BCE ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ
⇒ Δ ABF ର କ୍ଷେତ୍ରଫଳ = Δ BCE ର କ୍ଷେତ୍ରଫଳ
⇒ AEOF ଚତୁର୍ଭୁଜର କ୍ଷେତ୍ରଫଳ + Δ BOE ର କ୍ଷେତ୍ରଫଳ
= Δ OBC ର କ୍ଷେତ୍ରଫଳ + Δ BOE ର କ୍ଷେତ୍ରଫଳ
⇒ AEOF ଚତୁର୍ଭୁଜର କ୍ଷେତ୍ରଫଳ = Δ OBC ର କ୍ଷେତ୍ରଫଳ (ପ୍ରମାଣିତ)

Question 19.
ଦର୍ଶାଅ ଯେ ଗୋଟିଏ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କର୍ଣ୍ଣଦ୍ଵୟ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରକୁ ଚାରିଗୋଟି ସମକ୍ଷେତ୍ରଫଳ
ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର । ଏହାର କଣ୍ଠଦ୍ଵୟ ପରସ୍ପରକୁ O ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ।
ପ୍ରାମାଣ୍ୟ : Δ AOB ର କ୍ଷେତ୍ରଫଳ = Δ BOC ର
କ୍ଷେତ୍ରଫଳ = Δ COD ର
କ୍ଷେତ୍ରଫଳ = Δ AOD ର କ୍ଷେତ୍ରଫଳ ।
ଅର୍ଥାତ୍ କର୍ଣ୍ଣଦ୍ୱୟ କ୍ଷେତ୍ରଟିକୁ ଚାରିଗୋଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ବ ତ୍ରିଭୁଜରେ ପରିଣତ କରନ୍ତି ।

ପ୍ରମାଣ : ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର A͞C ଓ BD କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମନ୍ଦିଖଣ୍ଡ କରନ୍ତି । 
ଅର୍ଥାତ୍ AO = OC ଏବଂ BO = DO 
Δ ABC ରେ B͞O ମଧ୍ୟମା ।
[ତ୍ରିଭୁଜର ମଧ୍ଯମା, ତ୍ରିଭୁଜକୁ ଦୁଇଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ବ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।]
⇒ Δ AOB ର କ୍ଷେତ୍ରଫଳ = A BOC ର କ୍ଷେତ୍ରଫଳ … (i) 
ସେହିପରି Δ BCD ରେ C͞O ମଧ୍ୟମା ।
⇒ Δ BOC ର କ୍ଷେତ୍ରଫଳ = Δ COD ର କ୍ଷେତ୍ରଫଳ … (ii)
Δ ADC ରେ D͞O ମଧ୍ୟମା ।
⇒ Δ COD ର କ୍ଷେତ୍ରଫଳ = Δ AOD ର କ୍ଷେତ୍ରଫଳ … (iii)
ପୁନଶ୍ଚ, Δ ABD ରେ A͞O ମଧ୍ୟମା । 
⇒ Δ AODର କ୍ଷେତ୍ରଫଳ = Δ AOBର କ୍ଷେତ୍ରଫଳ … (iv)
∴ (i), (ii), (iii) (iv)ରୁ Δ AOBର କ୍ଷେତ୍ରଫଳ = Δ BOCର  କ୍ଷେତ୍ରଫଳ = Δ CODର କ୍ଷେତ୍ରଫଳ = Δ AODର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 20.
କ୍ଷେତ୍ରଫଳ ସମ୍ବନ୍ଧୀୟ ଉପପାଦ୍ୟ ପ୍ରୟୋଗ କରି ପ୍ରମାଣ କର ଯେ,
(i) ତ୍ରିଭୁଜର ଯେକୌଣସି ଦୁଇ ବାହୁର ମଧ୍ୟବିନ୍ଦୁକୁ ଯୋଗ କରୁଥିବା ସରଳରେଖା ତୃତୀୟ ବାହୁ ସଙ୍ଗେ ସମାନ୍ତର ।
(ii) ଟ୍ରାପିଜିୟମ୍‌ର ଅସମାନ୍ତର ବାହୁଦ୍ୱୟର ମଧ୍ୟବିନ୍ଦୁକୁ ଯୋଗ କରୁଥିବା ସରଳରେଖା ସମାନ୍ତର ବାହୁ ସହ ସମାନ୍ତର ।
ସମାଧାନ:
(i) ଦତ୍ତ : Δ ABC ରେ D ଓ E ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : DE || B͞C
ଅଙ୍କନ : DC ଓ B͞E ଅଙ୍କନ କର ।
ପ୍ରମାଣ : AEB ରେ E͞D ମଧ୍ୟମା ।
∴ Δ ADE ର କ୍ଷେତ୍ରଫଳ = Δ DBE ର କ୍ଷେତ୍ରଫଳ
[∵ ମଧ୍ୟମା ତ୍ରିଭୁଜକୁ ଦୁଇ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ବିଭକ୍ତ କରେ ।]

ସେହିପରି Δ ADC ରେ DE ମଧ୍ୟମା ।
∴ Δ ADE ର କ୍ଷେତ୍ରଫଳ = Δ ECD ର କ୍ଷେତ୍ରଫଳ ।
∴ Δ DBE ର କ୍ଷେତ୍ରଫଳ = Δ ECD ର କ୍ଷେତ୍ରଫଳ
କିନ୍ତୁ ଏମାନେ ଏକା ଭୂମି DE ବିଶିଷ୍ଟ । ତେଣୁ ତ୍ରିଭୁଜଦ୍ଵୟ ଏକା ସମାନ୍ତର ସରଳରେଖାଦ୍ବୟ ମଧ୍ୟରେ ଅବସ୍ଥିତ ।
ଅର୍ଥାତ୍ DE || BC ।

(ii) ଦତ୍ତ : ABCD ଟ୍ରାପିଜିୟମ୍‌ରେ AD || BC । M ଓ N ଯଥାକ୍ରମେ ଅସମାନ୍ତର ବାହୁଦ୍ୱୟ AB ଓ DC ର ମଧ୍ୟବିନ୍ଦୁ । 
ପ୍ରାମାଣ୍ୟ : MN || AD

ଅଙ୍କନ : \(\overrightarrow{\mathrm{DM}}\), \(\overrightarrow{\mathrm{CB}}\) କୁ P ବିନ୍ଦୁରେ ଛେଦ କରୁ ।
ପ୍ରମାଣ : Δ ADM ଏବଂ Δ PBM ଦ୍ବୟରେ
AM = MB (ଦତ୍ତ), m∠AMD = m∠PMB (ପ୍ରତୀପ)
ଏବଂ m∠ADM = m∠MPB (ଏକାନ୍ତର)
∴ Δ ADM ≅ Δ PBM (କୋ-କୋ-ବା)
MD = PM
⇒ M, P͞D ର ମଧ୍ୟବିନ୍ଦୁ ।
ବର୍ତ୍ତମାନ Δ DPC ରେ M ଓ N ଯଥାକ୍ରମେ DP ଓ D͞C ର ମଧ୍ୟବିନ୍ଦୁ ।
MN || PC
⇒ MN || AD (∵ AD || BC) (ପ୍ରମାଣିତ)

ବିକଳ୍ପ ପ୍ରଣାଳୀ :
ଦତ୍ତ : ABCD ଟ୍ରାପିଜିୟମ୍‌ରେ AD || BC । AB ଓ DC ର ମଧ୍ୟବିନ୍ଦୁ ଯଥାକ୍ରମେ M ଓ N ।

ପ୍ରାମାଣ୍ୟ : MN || AD ।
ଅଙ୍କନ : AC ଓ BD ଅଙ୍କନ କର ।
ପ୍ରମାଣ : AD || BC ଏବଂ Δ ABC ଓ Δ DBC ଦ୍ଵୟ ଏକା ଭୂମି BC ବିଶିଷ୍ଟ ।
⇒ Δ ABC ର କ୍ଷେତ୍ରଫଳ = Δ DB Cର କ୍ଷେତ୍ରଫଳ ।
⇒ \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ DBC ର କ୍ଷେତ୍ରଫଳ ।
Δ BMC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ (C͞M ମଧ୍ୟମା ହେତୁ)
ସେହିପରି Δ BNC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ DBC ର କ୍ଷେତ୍ରଫଳ ( BY ମଧ୍ୟମା ହେତୁ)
Δ BMC ର କ୍ଷେତ୍ରଫଳ = Δ BNCର କ୍ଷେତ୍ରଫଳ ।
Δ BMC ଓ Δ BNC ଦ୍ଵୟ BC ର ଏକ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ ।
⇒ MN || BC
BC || AD (ଦତ୍ତ)
⇒ MN || AD (ପ୍ରମାଣିତ)

Question 21.
P, ABCD ସାମାନ୍ତରିକ ଚିତ୍ରର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ହେଲେ, ପ୍ରମାଣ କର ଯେ
Δ APB ର କ୍ଷେତ୍ରଫଳ + Δ CDP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ।
‘P’ ଏହାର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।

ପ୍ରାମାଣ୍ୟ : Δ ABP ର କ୍ଷେତ୍ରଫଳ + Δ CDP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
ଅଙ୍କନ : P ବିନ୍ଦୁ ମଧ୍ୟଦେଇ AB ସହ ସମାନ୍ତର କରି ଏକ ରେଖାଖଣ୍ଡ MN ଅଙ୍କନ କର, ଯାହା AD ଓ BC କୁ ଯଥାକ୍ରମେ M ଓ N ବିନ୍ଦୁରେ ଛେଦ କରିବ ।
ପ୍ରମାଣ : ABNM ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ଓ ABP ଦ୍ଵୟ ଏକା ଭୂମି ଏବଂ ଏକା ସମାନ୍ତର ସରଳରେଖାଦ୍ବୟ ମଧ୍ୟରେ ଅବସ୍ଥିତ ।
∴ Δ ABP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ABNM ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
ସେହପରି Δ CDP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) MDCN ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ Δ ABP ର କ୍ଷେତ୍ରଫଳ + Δ CDP ର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) (ABNM ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ + MDCN ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ)
= \(\frac{1}{2}\) ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 22.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ ଥ‌ିବା Δ ABCରେ AB = AC, BC ଉପରିସ୍ଥ P କୌଣସି ଏକ ବିନ୍ଦୁ । PX ⊥ AB, PY ⊥ AC ଓ  CQ ⊥ AB ହେଲେ ପ୍ରମାଣ କର ଯେ, PX + PY = CQ ।

ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ AB = AC; BC ଉପରିସ୍ଥ P ଏକ ବିନ୍ଦୁ ।
PX ⊥ AB, PY ⊥ AC ଓ  CQ ⊥ AB
ପ୍ରାମାଣ୍ୟ : PX + PY = CQ
ଅଙ୍କନ : A͞P ଅଙ୍କନ କର ।

ପ୍ରମାଣ : Δ ABP ର କ୍ଷେତ୍ରଫଳ + Δ APC ର କ୍ଷେତ୍ରଫଳ = Δ ABC ର କ୍ଷେତ୍ରଫଳ
⇒ \(\frac{1}{2}\) AB . PX + \(\frac{1}{2}\) AC . PY = \(\frac{1}{2}\) AB . CQ
⇒ \(\frac{1}{2}\)AB . PX + \(\frac{1}{2}\) AB . PY = \(\frac{1}{2}\) AB . CQ (∵ AB = AC)
⇒ \(\frac{1}{2}\) AB (PX + PY) = \(\frac{1}{2}\) AB . CQ
⇒ PX + PY = CQ (ପ୍ରମାଣିତ)

Question 23.
Δ ABC ଏକ ସମବାହୁ ତ୍ରିଭୁଜ; O ଏହାର ଅନ୍ତଃସ୍ଥ ଏକ ବିନ୍ଦୁ; OX, OY ଓ OZ ଯଥାକ୍ରମେ ତ୍ରିଭୁଜର ବାହୁମାନଙ୍କ ପ୍ରତି ଲମ୍ବ । ପ୍ରମାଣ କର ଯେ, OX + OY + OZ = ତ୍ରିଭୁଜର ଉଚ୍ଚତା ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ B = BC = AC । ‘O’ ଏହାର ଅନ୍ତଃସ୍ଥ ଏକ ବିନ୍ଦୁ ।
OX ⊥ BC, OY ⊥ AC, OZ ⊥ AB
ଏବଂ AM ⊥ BC ।

ପ୍ରାମାଣ୍ୟ : OX + OY + OZ = AM
ଅଙ୍କନ : OA, OB, OC ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ OBC ର କ୍ଷେତ୍ରଫଳ + Δ OAC ର କ୍ଷେତ୍ରଫଳ + Δ OAB ର କ୍ଷେତ୍ରଫଳ = Δ ABC ର କ୍ଷେତ୍ରଫଳ
⇒ \(\frac{1}{2}\) BC . OX + \(\frac{1}{2}\) AC . OY + \(\frac{1}{2}\) AB . OZ = \(\frac{1}{2}\) BC . AM
⇒ \(\frac{1}{2}\) BC . OX + \(\frac{1}{2}\) BC . OY + \(\frac{1}{2}\) BC . OZ = \(\frac{1}{2}\) BC . AM
⇒ \(\frac{1}{2}\) BC (OX + OY + OZ) = \(\frac{1}{2}\) BC . AM (∵ AB = BC = AC)
⇒ OX + OY + OZ = AM

Question 24.
ଦର୍ଶାଅ ଯେ, ଗୋଟିଏ ରମ୍ବସ୍‌ର କ୍ଷେତ୍ରଫଳ, ଏହାର କର୍ଣ୍ଣଦ୍ୱୟର ଦୈର୍ଘ୍ୟର ଗୁଣଫଳର ଅର୍ଦ୍ଧେକ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ରମ୍ବସ୍‌ରେ AC ଓ BD ଏହାର ଦୁଇଟି କଣ୍ଠ ।
ସେମାନଙ୍କର ଛେଦବିନ୍ଦୁ O ।

ପ୍ରାମାଣ୍ୟ : ରମ୍ବସ୍‌ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) AC . BD
ପ୍ରମାଣ : ABCD ରମ୍ବସ୍‌ର କ୍ଷେତ୍ରଫଳ
= Δ AOD ର କ୍ଷେତ୍ରଫଳ + Δ AOB ର କ୍ଷେତ୍ରଫଳ + Δ BOC ର କ୍ଷେତ୍ରଫଳ + Δ DOC ର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) AO . OD + \(\frac{1}{2}\) AO . BO + \(\frac{1}{2}\) BO . OD + \(\frac{1}{2}\) OC . OD
= \(\frac{1}{2}\) AO . (OD + BO) + \(\frac{1}{2}\) OC . (BO + OD) = \(\frac{1}{2}\) AO . BD + \(\frac{1}{2}\) OC . BD
= \(\frac{1}{2}\) BD (AO + OC) = \(\frac{1}{2}\) AC . BD

ବିକଳ୍ପ ପ୍ରଣାଳୀ :
ABCD ରମ୍ବସ୍‌ର କ୍ଷେତ୍ରଫଳ = Δ ABD ର କ୍ଷେତ୍ରଫଳ + Δ BCD ର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) BD . AO + \(\frac{1}{2}\) BD . CO
= \(\frac{1}{2}\) BD (AO + OC) = \(\frac{1}{2}\) BD . AC

Question 25.
Δ ABC ର AD ମଧ୍ୟମା ଉପରେ X ଯେକୌଣସି ଏକ ବିନ୍ଦୁ ହେଲେ ପ୍ରମାଣ କର ଯେ, Δ ABX ଓ Δ ACX ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
ସମାଧାନ:
ଦତ୍ତ : Δ AB Cର AD ମଧ୍ୟମା । AD ମଧ୍ୟମା ଉପରେ ‘X’ ଯେକୌଣସି ଏକ ବିନ୍ଦୁ ।

ପ୍ରାମାଣ୍ୟ : Δ ABX ର କ୍ଷେତ୍ରଫଳ = Δ ACX ର କ୍ଷେତ୍ରଫଳ
ପ୍ରମାଣ : Δ ABC ର AD ମଧ୍ୟମା ।
⇒ Δ ABC ର କ୍ଷେତ୍ରଫଳ = Δ ADC ର କ୍ଷେତ୍ରଫଳ
ସେହିପରି Δ ABD ରେ XD ମଧ୍ୟମା ।
⇒ Δ BXD ର କ୍ଷେତ୍ରଫଳ = Δ CXD ର କ୍ଷେତ୍ରଫଳ
∴ Δ ABD ର କ୍ଷେତ୍ରଫଳ – Δ BXD ର କ୍ଷେତ୍ରଫଳ
= Δ ADC ର କ୍ଷେତ୍ରଫଳ – Δ CXD ର କ୍ଷେତ୍ରଫଳ ।
⇒ Δ ABX ର କ୍ଷେତ୍ରଫଳ = Δ ACX ର କ୍ଷେତ୍ରଫଳ ।  (ପ୍ରମାଣିତ)

Question 26.
Δ ABC ର BC ବାହୁ ଉପରେ P ଯେକୌଣସି ଏକ ବିନ୍ଦୁ, AP ର ମଧ୍ୟବିନ୍ଦୁ X ହେଲେ ପ୍ରମାଣ କର ଯେ,
Δ XBC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (Δ ABC ର କ୍ଷେତ୍ରଫଳ) ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର BC ବାହୁ ଉପରେ P ଯେକୌଣସି ଏକ ବିନ୍ଦୁ ।
A͞P ର ମଧ୍ୟବିନ୍ଦୁ X ।

ପ୍ରାମାଣ୍ୟ : Δ XBC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ
ପ୍ରମାଣ : Δ ABP ର BX ର ମଧ୍ୟମା ।
⇒ Δ BXP ର କ୍ଷେତ୍ରଫଳ = Δ ABX ର କ୍ଷେତ୍ରଫଳ
⇒ Δ BXP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABP ର କ୍ଷେତ୍ରଫଳ
ସେହିପରି Δ ACP ର CX ମଧ୍ୟମା ।
⇒ Δ CPX ର କ୍ଷେତ୍ରଫଳ = Δ ACX ର କ୍ଷେତ୍ରଫଳ
⇒ Δ CPX ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ACP ର କ୍ଷେତ୍ରଫଳ
∴ Δ BXP ର କ୍ଷେତ୍ରଫଳ = Δ CXP ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (Δ ABP ର କ୍ଷେତ୍ରଫଳ + Δ ACP ର କ୍ଷେତ୍ରଫଳ)
⇒ Δ XBC ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 27.
ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର; P ଓ Q ଯଥାକ୍ରମେ AB ଓ DC ର ମଧ୍ୟବିନ୍ଦୁ । ପ୍ରମାଣ କର ଯେ, PBOD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳର ଅଧା ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ।
P ଓ Q ଯଥାକ୍ରମେ AB ଓ DC ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : PBQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) ABCD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ ।

ଅଙ୍କନ : P͞Q ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ POD ର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) APQD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
[∵ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କଣ୍ଠ, କ୍ଷେତ୍ରକୁ ଦୁଇ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ ।]
ସେହିପରି Δ PBQର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) PBCQ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
⇒ Δ POD ର କ୍ଷେତ୍ରଫଳ + Δ PBQ ର କ୍ଷେତ୍ରଫଳ
= \(\frac{1}{2}\) Δ APQD ର କ୍ଷେତ୍ରଫଳ + \(\frac{1}{2}\) PBCQ ର କ୍ଷେତ୍ରଫଳ
⇒ PBOD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) (APQD ର କ୍ଷେତ୍ରଫଳ + PBCQ ର କ୍ଷେତ୍ରଫଳ)
⇒ PBOD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) ABCD କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 28.
ପ୍ରମାଣ କର ଯେ, କୌଣସି ତ୍ରିଭୁଜର ବାହୁମାନଙ୍କର ମଧ୍ୟବିନ୍ଦୁଗୁଡ଼ିକୁ ଯୋଗ କରୁଥିବା ରେଖାଖଣ୍ଡତ୍ରୟ ତ୍ରିଭୁଜଟିକୁ ଚାରୋଟି
ସମାନ କ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ବିଭକ୍ତ କରନ୍ତି ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ D, E ଓ F ଯଥାକ୍ରମେ AB, BC ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : DE, EF, DF ରେଖାଖଣ୍ଡତ୍ରୟ ତ୍ରିଭୁଜଟିକୁ
ଚାରିଗୋଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରିବ ।

ପ୍ରମାଣ : D ଓ F ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ । ⇒ DE || BC
ସେହିପରି EF || AB ଏବଂ DE || AC ।
∴ ADEF, DBEF ଏବଂ CEDF ଗୋଟିଏ ଗୋଟିଏ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ।
ADEF ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର D͞F କର୍ଣ୍ଣ ।
⇒ Δ ADF ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
ସେହିପରି ପ୍ରମାଣ କରାଯାଇ ପାରେ ଯେ, Δ BDE ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
ଏବଂ Δ CEF ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
∴ Δ ADF ର କ୍ଷେତ୍ରଫଳ = Δ BDE ର କ୍ଷେତ୍ରଫଳ = Δ CEF ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 29.
ABCD ଚତର୍ଭୁଜର A͞C ଓ BD କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ‘O’ ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । AO = CO ହେଲେ, ପ୍ରମାଣ
କର ଯେ Δ ABD ଓ Δ BCD ଦ୍ବୟ ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତର୍ଭୁଜର AC ଓ BD କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ O
ଏବଂ AO = CO ।

ପ୍ରାମାଣ୍ୟ : Δ ABD ର କ୍ଷେତ୍ରଫଳ = Δ BCD ର କ୍ଷେତ୍ରଫଳ ।
ପ୍ରମାଣ : Δ ABC ର BO ମଧ୍ୟମା । (‘. AO = OC)
∴ Δ ABO ର କ୍ଷେତ୍ରଫଳ = Δ BOCର କ୍ଷେତ୍ରଫଳ … (i)
ସେହିପରି Δ ADC ର DO ମଧ୍ୟମା । (∵ AO = OC)
∴ Δ ADO ର କ୍ଷେତ୍ରଫଳ = Δ DOC ର କ୍ଷେତ୍ରଫଳ … (ii)
(i) ଓ (ii) ରୁ Δ ABO ର କ୍ଷେତ୍ରଫଳ + Δ ADO ର କ୍ଷେତ୍ରଫଳ = Δ BOC ର କ୍ଷେତ୍ରଫଳ + Δ DOC ର କ୍ଷେତ୍ରଫଳ
∴ Δ ABD ର କ୍ଷେତ୍ରଫଳ = Δ BCD ର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Question 30.
D, E ଓ F ଯଥାକ୍ରମେ Δ ABC ର AB, BC ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ । ଦର୍ଶାଅ ଯେ,
(i) FDEC ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ଏବଂ
(ii) FDEC ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ D, E ଓ F ଯଥାକ୍ରମେ AB, BC ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : (i) FDEC ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ii) FDEC ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ ।

ପ୍ରମାଣ : D ଓ F ଯଥାକ୍ରମେ AB ଓ AC ର ମଧ୍ୟବିନ୍ଦୁ ।
DF || BC ⇒ DF || EC
ସେହିପରି ପ୍ରମାଣ କରାଯାଇପାରେ ଯେ DE || CF ।
∴ DECF ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । (ପ୍ରମାଣିତ)
ସେହିପରି ADEF, DBEF ଗୋଟିଏ ଗୋଟିଏ ସାମାନ୍ତରିକ କ୍ଷେତ୍ର ।
DECF ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର EF କର୍ଣ୍ଣ ।
[∵ ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କର୍ଣ୍ଣ, କ୍ଷେତ୍ରକୁ ଦୁଇଟି ସମକ୍ଷେତ୍ରଫଳ ବିଶିଷ୍ଟ ତ୍ରିଭୁଜରେ ପରିଣତ କରେ]
⇒ Δ DEF ର କ୍ଷେତ୍ରଫଳ = Δ CEF ର କ୍ଷେତ୍ରଫଳ
ସେହିପରି ପ୍ରମାଣ କରାଯାଇ ପାରେ ଯେ
= Δ BDE ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
ଏବଂ Δ ADF ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
∴ Δ ADF ର କ୍ଷେତ୍ରଫଳ = Δ BDE ର କ୍ଷେତ୍ରଫଳ = Δ CEF ର କ୍ଷେତ୍ରଫଳ = Δ DEF ର କ୍ଷେତ୍ରଫଳ
Δ ADF ର କ୍ଷେତ୍ରଫଳ + Δ BDE ର କ୍ଷେତ୍ରଫଳ + Δ CEF ର କ୍ଷେତ୍ରଫଳ + Δ DEF ର କ୍ଷେତ୍ରଫଳ
= 2 × Δ DEF ର କ୍ଷେତ୍ରଫଳ + 2 × Δ CEF ର କ୍ଷେତ୍ରଫଳ
⇒ Δ ABC ର କ୍ଷେତ୍ରଫଳ = 2 (Δ DEF ର କ୍ଷେତ୍ରଫଳ + Δ CEF ର କ୍ଷେତ୍ରଫଳ)
⇒ Δ ABC ର କ୍ଷେତ୍ରଫଳ = 2 × FDEC ସାମାନ୍ତରିକ କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ
∴ FDEC କ୍ଷେତ୍ରର କ୍ଷେତ୍ରଫଳ = \(\frac{1}{2}\) Δ ABC ର କ୍ଷେତ୍ରଫଳ । (ପ୍ରମାଣିତ)

Odisha State Board CHSE Odisha Class 12 Invitation to English 1 Solutions Poem 1 Daffodils Textbook Exercise Questions and Answers.

Class 12th Invitation English Poem Chapter 1 Daffodils Question Answers CHSE Odisha

Daffodils Class 12 Questions and Answers

Think it out

Question 1.
When did the poet see the daffodils?
Answer:
The poet saw the daffodils when he was moving about aimlessly.

Question 2.
Where did the poet see the daffodils?
Answer:
The poet saw the daffodils under the trees and beside the lake.

Question 3.
Fill in the blanks to describe the idea of stanza 1: The poet was __________ in the English Countryside. He saw thousands of __________ fluttering and dancing beneath _________ and beside __________. The daffodils appeared to be ___________ in the strong breeze.
Answer:
The poet was wandering in the English Countryside. He saw thousands of golden daffodils fluttering and dancing beneath the trees and beside the lake. The daffodils appeared to be dancing in the strong breeze.

Question 4.
What does the poet compare the daffodils with?
Answer:
The poet compares the daffodils with the stars on the Milkyway.

Question 5.
What resemblance does he find between the stars and the daffodils?
Answer:
The resemblance the poet finds between the stars and the daffodils is one of countlessness.

Question 6.
What does the poet say about the number of flowers?
Answer:
The poet says that he saw ten thousand flowers at a glance.

Question 7.
Where were the flowers?
Answer:
The flowers were beside the lake.

Question 8.
Which of the two danced more sprightly – the waves or the daffodils?
Answer:
The daffodils danced more sprightly.

Question 9.
How does the poet feel while looking at the daffodils?
Answer:
While looking at the daffodils, the poet’s happiness knows no bounds. The site is a gay inspiration to him.

Question 10.
What happens to the poet when he lies on his couch?
Answer:
When he lies in his couch in a thoughtful or thoughtless mood, the lively picture of the sprightly dancing daffodils flashed upon his inward eye and fills his entire being with joy.

Question 11.
Mention the two moods of the poet?
Answer:
The two moods of the poet are vacant (free from thought) mood and pensive (thoughtful) mood.

Question 12.
What does the poet feel when he remembers the sight of the daffodils?
Answer:
When he remembers the sight of the daffodils, it fills his entire being with joy. His heart begins to dance with the flowers. In short, the sight of daffodils proves to be a source not only of immediate pleasure but also Of lasting joy.

Question 13.
When does the poet write the poem – beside or off the lake?
Answer:
The poet writes the poem off the lake.

Question 14.
Do you find a rhyme scheme in the poem? The rhyming scheme of the first stanza is a b a b (a – ‘cloud’ and ‘crowd’; b – ‘hills’ and ‘daffodils’), ending with a rhyming couplet cc (c – ‘trees’ and breeze’). Is the rhyme scheme similar in the other three stanzas or do you find any variation?
Answer:
The rhyme scheme in the other three stanzas is similar.

Question 15.
How many times is the word “dance” repeated in this poem? In which line does it show the happiness and liveliness of the flowers?
Answer:
The word ‘dance’ is repeated four times in this poem. The line “Tossing their heads in sprightly dance” shows the happiness and liveliness of the flowers.

Question 16.
In which line does it create a sense of harmonious relationship between the daffodils and the waves?
Answer:
The line – ‘The waves beside them danced’ creates a sense of the harmonious relationship between the daffodils and the waves.

Question 17.
In which line does this harmonious relationship include the poet himself?
Answer:
The line ‘In such a jocund company’ establishes this harmonious relationship that includes the poet himself.

Question 18.
What figures of speech do you find in the poem?
Answer:
We find the figures of speech such as ‘metaphors’ and ‘similes’ in the poem.

Question 19.
‘Simile’ is a figure of speech that makes an explicit comparison between two unlike things by using ‘like’, ‘as’, etc. For example, in ‘I wandered lonely as a cloud’, as the loneliness of the poet resembles the loneliness of the cloud that is floating high in the sky, the figure of speech used is a simile. What other example of a simile do you find in the poem?
Answer:
The other example of a simile we find in the poem is “continuous as the stars that shine”.

Question 20.
‘Metaphor’ is a figure of speech that makes an implicit comparison between two, unlike things. In ‘What wealth the show to me had brought’, the poet imagines the happiness brought to him by the beautiful scene of the flowers as “wealth”. Does he use a metaphor here?
Answer:
Yes, he uses a metaphor here.

Question 21.
“Ten thousand saw I at a glance” – is it an exaggeration? Will you call it a ‘hyperbole’?
Answer:
It is an exaggeration. We call it a “hyperbole”.

Question 22.
What figure of speech does the poet use in “They stretched in never-ending line.”?
Answer:
The poet uses “hyperbole” in “They stretched in never-ending line

CHSE Odisha Class 12 English Daffodils Important Questions and Answers

I. Multiple-Choice Questions (MCQs) with Answers

Question 1.
The poem ‘Daffodils’ is composed by ___________?
(A) William Wordsworth
(B) John Keats
(C) P.B. Shelly
(D) None of the above
Answer:
(A) William Wordsworth

Question 2.
William Wordsworth was born in _________?
(A) 1770
(B) 1780
(C) 1775
(D) 1772
Answer:
(A) 1770

Question 3.
Does this poem incorporate the ideas and aspects essential to _____________?
(A) romantic poetry
(B) metaphysical poetry
(C) spiritual poetry
(D) None of these
Answer:
(A) romantic poetry

Question 4.
‘I wandered lonely as a cloud’. Here I refers to ______________?
(A) the reader
(B) the cloud
(C) the daffodils
(D) the poet
Answer:
(D) the poet

Question 5.
‘Continuous as the stars that shine and twinkle on the milky way. In the above lines, the daffodils have been compared to ____________?
(A) the clouds
(B) the other flowers
(C) the stars
(D) milky way
Answer:
(C) the stars

Question 6.
The poet saw the daffodils _____________?
(A) along the margin of a bay
(B) by the side of a pool
(C) by the riverside
(D) in a garden
Answer:
(A) along the margin of a bay

Question 7.
‘A poet could not but be gay in such a jocund company’. Here ‘jocund company’ refers to the ____________?
(A) Mends of the poet
(B) waves
(C) daffodils
(D) stars
Answer:
(C) daffodils

Question 8.
“What wealth the show to me had brought.” By ‘wealth’ the poet means _____________?
(A) happiness
(B) good
(C) pleasure
(D) money
Answer:
(A) happiness

Question 9.
‘A host of golden daffodils,’ Here ‘A host’ means ______________?
(A) a few
(B) one who entertains
(C) a guest
(D) a large number
Answer:
(D) a large number

Question 10.
‘Bliss’it means ____________?
(A) great love
(B) great joy
(C) great blessing
(D) great loneliness
Answer:
(B) great joy

Question 11.
‘They stretched in never-ending line along the margin of a bay” Here ‘they’ refers to _____________?
(A) the stars
(B) the daffodils
(C) the clouds
(D) the hills
Answer:
(B) the daffodils

Question 12.
The daffodils were growing?
(A) along the margin of a bay
(B) by the side of a pool
(C) by the riverside
(D) in a garden
Answer:
(A) along the margin of a bay

Question 13.
The poet saw _____________ daffodils?
(A) a few
(B) white
(C) blue
(D) a large number of
Answer:
(D) a large number of

Question 14.
When the poet was looking at the daffodils?
(A) he felt sad to seé them
(B) he was totally lost in their beauty
(C) he thought they were ugly
(D) he felt like dancing
Answer:
(B) he was totally lost in their beauty

Question 15.
“Never-ending line” means _________?
(A) a curved line
(B) a short line
(C) a continuous line
(D) line of stars
Answer:
(C) a continuous line

Question 16.
‘1.’en thousand saw I at a glance tossing their heads in sprightly dance.” Here ‘sprightly dance’ means ____________?
(A) a slow dance
(B) a lively dance
(C) dance of the spirits
(D) a religious dance
Answer:
(B) a lively dance

Question 17.
“Solitude” means ____________?
(A) being alone
(B) being together
(C) being in trouble
(D) being sad
Answer:
(A) being alone

Question 18.
“Glee” means __________?
(A) joy
(B) sadness
(C) gloom
(D) happiness
Answer:
(D) happiness

Question 19.
“A poet could not but be gay” means __________?
(A) the poet could not sleep
(B) the poet was very sad
(C) the poet was happy
(D) the poet could not be happy
Answer:
(C) the poet was happy

Question 20.
William Wordsworth was an __________ poet?
(A) Irish
(B) English
(C) Scottish
(D) Welsh
Answer:
(B) English

Question 21.
William Wordsworth was a poet of _____________?
(A) nature
(B) romance
(C) beauty
(D) loves
Answer:
(A) nature

Question 22.
William Wordsworth died in ____________?
(A) 1770
(B) 1772
(C) 1850
(D) 1852
Answer:
(C) 1850

Question 23.
William Wordsworth was a poet of natural objects and _____________ people?
(A) country
(B) common
(C) aristocratic
(D) gentry
Answer:
(A) country

Question 24.
Wordsworth was honored as _____________?
(A) romantic poet
(B) England poet laureate
(C) common people’s poet
(D) pathçtic poet
Answer:
(B) England poet laureate

Question 25.
What did the poet see?
(A) a lot of clouds
(B) a lonely cloud
(C) a host of golden daffodils
(D) both (B) and (C)
Answer:
(C) a host of golden daffodils

Question 26.
Where did the poet see the daffodils?
(A) beside the lake
(B) near the tree
(C) in a big field
(D) on the lap of the hill
Answer:
(A) beside the lake

Question 27.
What were the daffodils doing?
(A) dancing and singing
(B) fluttering and dancing in the breeze
(C) shinning with a happy glance
(D) dancing in the water
Answer:
(B) fluttering and dancing in the breeze

Question 28.
Whom does the poet compare with the daffodils?
(A) star
(B) tree
(C) cloud
(D) lake
Answer:
(A) star

Question 29.
Where do the stars shine?
(A)on the sky
(B) on the milky way
(C) in the water
(D) top of daffodils
Answer:
(B) on the milky way

Question 30.
How many daffodils did the poet see at a glance?
(A) five thousand
(B) ten thousand
(C) fifteen thousand
(D) none of these
Answer:
(B) ten thousand

Question 31.
Where are the daffodils fluttering and dancing?
(A) in the sea
(B) in the lake
(C) in the ocean
(D) in the breeze
Answer:
(D) in the breeze

Question 32.
Who was dancing like the daffodils?
(A) rivers
(B) seas
(C) waves
(D) lakes
Answer:
(C) waves

Question 33.
The poem ‘Daffodils’ can be called a ___________?
(A) nature poem
(B) heroic poem
(C) psychoanalytical poem
(D) None of these
Answer:
(A) nature poem

Question 34.
“A state of loneliness” is called __________?
(A) pensive
(B) vacant
(C) solitude
(D) pleasure
Answer:
(C) solitude

Question 35.
What does the poet mean by “little thought” when he says “I gazed and gazed-but little thought”?
(A) He had thought a little
(B) He had not thought
(C) He had some thought
(D) He had a small thought
Answer:
(B) He had not thought

Question 36.
How did the daffodils surpass the waves?
(A) In their cheerfulness and brightness
(B) In their sprightly dance
(C) With their shine and number
(D) In their beauty and joyfulness
Answer:
(A) In their cheerfulness and brightness

Question 37.
Who was/were wandering lonely?
(A) A cloud
(B) The poet
(C) The daffodils
(D) The stars
Answer:
(B) The poet

Question 38.
To whom does the poet compare himself?
(A) The daffodils
(B) The cloud
(C) The stars
(D) The lake
Answer:
(B) The cloud

Question 39.
The poet has made two comparisons in the poem “Daffodils”. What are they?
(A) Himself with a cloud, daffodils with the stars
(B) Happiness with thè daffodils, stars with daffodils
(C) Himself with the daffodils, daffodils with the stars
(D) Himself with the stars, daffodils with the stars
Answer:
(A) Himself with a cloud, daffodils with the stars

Question 40.
“I wandered lonely as a cloud”. This statement is an example of a _____________?
(A) Simile
(B) Metaphor
(C) Personification
(D) Hyperbole
Answer:
(A) Simile

Question 41.
1 gazed and gazed suggests that the poet?
(A) took time to see all the flowers
(B) had no other work
(C) was lonely
(D) was enchanted
Answer:
(D) was enchanted

Question 42.
Whom does the poet personify in the poem “Dáffodils”?
(A) The daffodils
(B) The cloud
(C) The stars
(D) The lake and trees
Answer:
(A) The daffodils

Question 43.
How does the poet personify the daffodils?
(A)By calling them a company
(B) By saying they are fluttering and dancing
(C) By referring to them as a host
(D) By keeping them in his memory
Answer:
(B) By saying they are fluttering and dancing

Question 44.
Which of the following is an exaggerated phrase/statement?
(A) A host of golden daffodils
(B) Ten thousand saw I at a glance
(C) The waves beside them dance
(D) I wandered lonely as a cloud
Answer:
(B) Ten thousand saw I at a glance

Question 45.
Which of the following means “Moving to and for”?
(A) Dancing
(B) Sprightly
(C) Fluttering
(D) Tossing
Answer:
(D) Tossing

Question 46.
Which of the following is the opposite of “Vacant”?
(A) Plaintive
(B) Pensive
(C) Gleeful
(D) Jocund
Answer:
(B) Pensive

Question 47.
What is the “inward eye” the poet mentions in the poem “Daffodils”?
(A) Inner eye
(B) Mind’s eye
(C) Heart’s eye
(D) Closed eyes
Answer:
(B) Mind’s eye

Question 48.
How many times is the word “dance” repeated in this poem?
(A) Two
(B) Three
(C) Four
(D) Five
Answer:
(C) Four

II. Short Type Questions with Answers

Question 1.
Where did the poet see the daffodils and what does the poet compare the daffodils with?
Answer:
The poet saw the daffodils when he was moving about aimlessly. He saw the daffodils under the trees and beside the lake. The poet compares the daffodils with the stars on the Milkyway.

Question 2.
What resemblance does he find between the stars and the daffodils and how I does the poet feel while looking at the daffodils?
Answer:
The resemblance the poet finds between the stars and the daffodils is one of countlessness. While looking at the daffodils, the poet’s happiness knows no bounds. The site is a gay inspiration to him.

Question 3.
What happens to the poet when he lies on his couch?
Answer:
When he lies in his couch in a thoughtful or thoughtless mood, the lively picture of the sprightly dancing daffodils flashed Upon his inward eye and fills his entire being with joy.

Question 4.
What does the poet feel when he remembers the sight of the daffodils?
Answer:
When he remembers the sight of the daffodils, it fills his entire being with joy. His heart begins to dance with the flowers. In short, the sight of daffodils proves to be a source not only of immediate pleasure but also of lasting joy.

Question 5.
What is the similarity between the stars and the daffodils?
Answer:
The stars twinkle continuously in a milky way being innumerable and the daffodils, in the same way, danced and swayed in the breeze as if having no end.

Question 6.
What impact did the dancing daffodils have on the poet?
Answer:
The poet was deeply enchanted by the beautiful dancing daffodils. It was as if a great wealth for the poet. He was happy with the jocund company of the flowers.

Question 7.
How did a jocund company’ impact the poet?
Answer:
The expression ‘jocund company’ refers to the merry association of the waves and the daffodils dancing in joy. While gazing at the daffodils, the poet was beside himself with joy. The waves and the daffodils produced a cheerful effect on the poet in their company.

Question 8.
Where does a cloud float?
Answer:
The poet wanders all alone like a piece of cloud floating high over valley and hills.

Question 9.
‘The poet has described the motion of the daffodils.’ Quote the words to support your answer?
Answer:
The motion or the movement of the daffodils has been reflected in their “fluttering” and “dancing” to the tune of the breeze blowing.

Question 10.
Quote the words that give an instance of alliteration?
Answer:
“Alliteration” is a type of comparison. The far-stretching daffodils appear to the poet be continuous like the stars that shine in the night sky.

Question 11.
Which figure of speech does the poet use and why? Give an example?
Answer:
In the figure of speech, the poet has used ‘hyperbole’ for exaggeration and effect. He has tried to show the plentitude (profuse) of the flowers using the expression “ten thousand”.

Detailed Summaries and Glossary

Stanza – 1
I wandered………………………………………………………………………the breeze.
At the beginning, the poet is not only lonely, but also in a state of wandering, that is, moving about aimlessly as a floating cloud. All on a sudden, he sees a crowd – a whole bank of beautiful daffodils, quivering and ‘dancing in the breeze’ by the side of the lake, ‘beneath the trees’. The simile that compares the poet’s state to that of a cloud reinforces the very idea of an aimless drift.

ସାରମର୍ମ :
କବିତାର ଆଦ୍ୟଭାଗର ବର୍ଣ୍ଣନାନୁଯାୟୀ କବି କେବଳ ଏକାକୀ ନାହାନ୍ତି, ବରଂ ସେ ଏକ ଭାସମାନ ବାଦଲ ଭଳି ଲକ୍ଷ୍ୟହୀନ ଭାବେ ଘୂରି ବୁଲୁଛନ୍ତି । ହଠାତ୍ ଏକ ହ୍ରଦକୂଳରେ ଅନେକ ଡାଫୋଡ଼ିଲ୍ ହଲି ଦୋହଲି ପବନରେ ନାଚୁଥିବାର ସେ ଦେଖିବାକୁ ପାଇଛନ୍ତି । କବିଙ୍କ ମାନସିକ ଅବସ୍ଥାକୁ ଭାସମାନ ବାଦଲ ସହିତ ତୁଳନା କରାଯାଇଥ‌ିବାରୁ କବି ଯେ ଲକ୍ଷ୍ୟହୀନ ଭାବେ ଭ୍ରମଣ କରୁଛନ୍ତି ତାହା ବୁଝାପଡ଼ୁଛି ।

Glossary
lonely : The poet was not exactly lonely, for his sister Dorothy, was with him. The word ‘lonely’ refers to the state of the poet’s mind. This word helps to create a
beautiful atmosphere.
cloud : ବାଦଲ
floats : ଭାସିବା
vales : ଭ୍ୟାଲେସ୍
daffodils : bell-shaped flowers of golden yellow colour, with narrow leaves, which bloom ¡n early . spring usually by the side of lakes ଏକ ପ୍ରକାର ସ୍ଥଳପଦ୍ମ ଜାତୀୟ ଫୁ ଲ
a host of : a large number of – ବହୁ ସଙ୍ଖ୍ୟକ
beside : at the side of (ପାଖରେ)
lake : the poet here refers to lake Ullswater on the borders of Cumberland and west Moonland – (ହ୍ରଦ)
beneath : ତଳେ
fluttering : quivering
breeze : ପବନ

Stanza – 2
Continuous as ………………………………………………………. sprightly dance.
In this stanza, in the poet’s ‘inward eye’, his imagination moves from earth to heaven and discovers a similarity between the daffodils and the stars. The beautiful spectacle of the crowd of dancing daffodils reminds the poet of the luminous stars in the sky. The flowers by the shore of the lake seem as countless as the stars. He catches sight of ‘ten thousand at a glance’, moving their heads in a gay and lively dance.

ସାରମର୍ମ :
ଏହି ପଦରେ କବିଙ୍କର ଅନ୍ତଃଦୃଷ୍ଟିରେ ତାଙ୍କ କଳ୍ପନାଶକ୍ତି ପୃଥ‌ିବୀପୃଷ୍ଠରୁ ଆକାଶକୁ ଗତିଶୀଳ ହୋଇଛି ଏବଂ ସେ ଡାଫୋଡ଼ିଲ୍‌ ଏବଂ ତାରକାପୁଞ୍ଜ ମଧ୍ୟରେ ସାମଞ୍ଜସ୍ୟ ଆବିଷ୍କାର କରିଛନ୍ତି । ଅଗଣିତ ନୃତ୍ୟମାନ ଡାଫୋଡ଼ିଲ୍‌ ତାଙ୍କୁ ଆକାଶର ଉଜ୍ଜ୍ଵଳ ତାରକାରାଜି କଥା ମନେପକାଇ ଦେଇଛି । ହ୍ରଦକୂଳର ଫୁଲ୍‌ଗୁଡ଼ିକ ତାଙ୍କୁ ଅଗଣିତ ତାରକା ଭଳି ମନେ ହୋଇଛି । ଏକାଥରକେ ସେ ଦଶହଜାରସଂଖ୍ୟକ ଡାଫୋଡ଼ିଲ୍‌ ମୁଣ୍ଡ ହଲାଇ ଜୀବନ୍ତଭାବେ ନୃତ୍ୟ କରୁଥିବାର ସେ ଦେଖାରି ଛନ୍ତି ।

Glossary
twinkle : shine
milkway : the broad, luminous band of stars encircling the sky (ଛାୟାପଥ)
never-ending : endless (ସୀମୀହୀନ)
margin : border(ସୀମୀ)
bay : here refers to a lake (ହୀନ)
ten thousand : many. not to be taken seriously
ten thousand at a glance : Here we come across an example of hyperbole, a figure of speech, rather rare in Wordsworth, purposeful exaggeration. The line of flowers is imagined to be stretching almost into infinity, through the use of phrase like ‘ten thousand’, etc.
glance : ବାହାପ
tossing : moving (ଟସ୍ ମାରିବା)
sprightly : lively (ସ୍ପଷ୍ଟ ଭାବରେ)
Tossing dance : moving their heads in gay and lively dance (ସେହି ଫୁଲ ଗୁଡ଼ିକ)

Stanza – 3
The waves ……………………………………………………………………..had brought.
The waves besides the flowers are dancing, but the mirth of daffodils is far greater than that of the waves. The entire atmosphere is one of joy and this delight sinks deep into the poet’s heart. His eyes are fixed on the spectacular sight. The poet fails to realize that the beautiful scene is going to be a source of joy for him in the future also. In short, the show of the flowers brings great ‘wealth’ to the poet,because the poet’s imagination changes them, again and again, into something precious and permanently lovable.

ସାରମର୍ମ :
ଫୁଲଗୁଡ଼ିକ ଭଳି ଲହରୀମାଳା ମଧ୍ଯ ନୃତ୍ୟ କରୁଛନ୍ତି, ମାତ୍ର ଡାଫୋଡ଼ିଲ୍‌ର ପ୍ରଫୁଲ୍ଲତା ଲହରୀମାଳାର ପ୍ରଫୁଲ୍ଲତାଠାରୁ ଢେର ବେଶି । ସମଗ୍ର ପରିବେଶ ଆନନ୍ଦପ୍ରଦ ରହିଛି ଏବଂ ସେହି ଆନନ୍ଦ କବିଙ୍କ ହୃଦୟକନ୍ଦରକୁ ବିଗଳିତ କରିଛି । ସେହି ଚମତ୍କାର ଦୃଶ୍ୟ ଉପରେ କବିଙ୍କ ଆଖିଯୋଡ଼ିକ ଲାଖିଯାଇଛି । ସେହି ମନୋଲୋଭା ଦୃଶ୍ୟ ଯେ ଭବିଷ୍ୟତରେ ତାଙ୍କ ପାଇଁ ଆନନ୍ଦର ଉତ୍ସ ହେବାକୁ ଯାଉଛି ବୋଲି କବି ଅନୁଭବ କରିପାରି ନାହାନ୍ତି । ସଂକ୍ଷେପରେ କହିଲେ, ଫୁଲଗୁଡ଼ିକର ଦୃଶ୍ୟ କବିଙ୍କ ପାଇଁ ସମ୍ପଦ ଆଣିଦେଇଛି କାରଣ କବିଙ୍କ ଚିନ୍ତାଧାରା ସେଗୁଡ଼ିକୁ ବେଳକୁବେଳ କିଛି ମହାର୍ଘ ଓ ପରମ ଆନନ୍ଦକାରୀ ବସ୍ତୁରେ ରୂପାନ୍ତରିତ କରିବାରେ ଲାଗିଛି ।

Glossary
nationalistic : promoting nationalism
out-did : surpassed (ଅତିକ୍ରମ କଲା |)
sparkling : shining (ଉଜ୍ଜ୍ୱଳ)
Out-did …. waves : The dancing of the daffodils seemed even more spontaneous and cheerful than that of the waves. (ଅଧିକ ଆନନ୍ଦପ୍ରଦ ଥିଲା)
glee : mirth (ଆନନ୍ଦ)
A poet …. gay : The poet identified himself with the daffodils. The sight might not have appealed to ordinary folk, but to a poet, it was a gay inspiration. (କବିଙ୍କ ପାଇଁ ପ୍ରେରଣାର ଉତ୍ସ ପାଲଟିଥିଲା )
gay : light-hearted and carefree (ଜଞ୍ଜାଳଶୂନ୍ୟ)
jocund : joyful (ଅତ୍ୟନ୍ତ ଖୁସୀ)
gazed-and gazed : The poet’s eyes were fixed on the beautiful sight of dancing daffodils ( ଆଖୁ ଲାଖ୍ ରହିଥୁଲା)
little thought : no thought ( ଭାବନା ନ ଥିଲା)
show : sight (ଦୃଶ୍ୟ) The poet here refers to the beauty created by the dancing waves and daffodils.

Stanza – 4
For oft, ………………………………………………………the daffodils.
This stanza describes the wealth of delight the ‘show’ has bestowed on the poet. The sight of the golden daffodils becomes a thing of the past. Time flies by. In later years, when he lies on his couch in a thoughtful mood, the lovely picture of the sprightly daffodils flashes upon his imagination and fills his entire being with joy. Thus the lovely sight proves to be a source not only of immediate pleasure but of lasting joy as well. This experience brings about a profound change in the poet’s mode of perception as well as a deeper spiritual life.

ସାରମର୍ମ :
ଏହି ପଦଟି ସୁନ୍ଦର ଡାଫୋଡ଼ିଲ୍‌ଗୁଡ଼ିକର ଦୃଶ୍ୟ କବିଙ୍କୁ ପ୍ରଦାନ କରିଥିବା ଆନନ୍ଦରୂପକ ସମ୍ପଦ ବିଷୟରେ ବର୍ଣ୍ଣନା କରିଛି । ସୁନେଲୀ ଡାଫୋଡ଼ିଲ୍‌ଗୁଡ଼ିକର ଦୃଶ୍ୟ ଅତୀତ ପାଲଟି ଯାଇଛି । ସମୟ ଗଡ଼ିଚାଲିଛି । ପରବର୍ତ୍ତୀ କାଳରେ କବି ଯେତେବେଳେ ଦୁଃଖଦ ମନରେ ଖଟିଆରେ ଗଡ଼ି ପଡ଼ିଛନ୍ତି, ସେତେବେଳ ସତେଜ ଡାଫୋଡ଼ିଲ୍‌ଗୁଡ଼ିକର ମନୋଲୋଭା ଦୃଶ୍ୟ ତାଙ୍କ କଳ୍ପଦୃଷ୍ଟିରେ ଭାସିଯାଇଛି ଓ ତାଙ୍କର ସାରା ଶରୀରରେ ଭରିଦେଇଛି ଆନନ୍ଦର ଶିହରଣ । ଏହିପରି ସେହି ସୁନ୍ଦର ଦୃଶ୍ୟ କେବଳ ତାତ୍‌କ୍ଷଣିକ ଆନନ୍ଦର ଉତ୍ସ ନୁହେଁ ବରଂ ପରମାନନ୍ଦର ଉତ୍ସ ଭାବେ ପ୍ରମାଣିତ ହୋଇଛି । ଏହି ଅନୁଭବ କବିଙ୍କର ଦୃଷ୍ଟିଭଙ୍ଗୀ ଓ ଆଧ୍ୟାତ୍ମିକ ଜୀବନଶୈଳୀ ଉପରେ ଗଭୀର ପ୍ରଭାବ ପକାଇଛି ।

Glossary
Oft : often (ଅନେକରିବା)
Vacant : not thinking of anything in particular (ଶୂନ୍ୟ)
Pensive : sad and thoughtful (ଦୁଃଖ ଏବଂ ଚିନ୍ତାପୂର୍ଣ୍ଣ)
flash upon : ଚମକି ଉଠିବା
inward eye: here it means the mind that contemplates imagination not the eye proper that only sees (ଆଭ୍ୟନ୍ତରୀଣ ଚକ୍ଷୁ)
bliss : great joy (ସୁଖ)
solitude: the state of loneliness (ଏକାକୀ)
They flash …. solitude : The poet was in a happy mood in the jocund company of the waves and flowers. That was the experience of the ‘outer eyes’ but that of the ‘inward eye’ was more joyful. The feeling of sympathy with the waves and the flowers and to the breeze in their glee and the Wordsworthian emotion recollected in tranquillity expressed ‘in themselves’ — A. C. Bradley
bliss of solitude : Solitude is a recurrent theme in Wordsworth’s poetry. Here it means ‘a supreme delight that comes out of loneliness. Coleridge says that ‘inward eye’ should be reserved for higher uses i.e. mental and spiritual delight. He adds that the thoughts and images in this poem are too great for the subject. vide : Biographia Literania. The last stanza is also reminiscent of Wordsworth’s The Solita?Reaper The music in mv heart I bore Long after it was heard no more

Introducing the Poet
William Wordsworth (1770-1850) is perhaps the best-known of all the Romantic poets. He was born in the Lake District, in Cumberland, his love of the English Lakes never left him and remained to the end a major influence in all he wrote. Wordsworth launched his poetic career in a company with S.T. Coleridge; they jointly published Lyrical Ballads. He was one of the greatest poets of the country and of natural life. As a nature poet and a poet willing and able to see the man against a backdrop of nature, he has no equal. He excels at taking one particular moment of experience and conveying it richly with a hint of moral comment. His greatest contribution to English literature was A Return to Nature. His idea of nature is related to the concept of unity, the idea of poetic power, and even his moral beliefs. Nature is seen in many aspects of Wordsworth’s work. At its deepest in ‘The Prelude’, ‘Tintern Abbey’, Nature is seen as possessing a definite mystical bond with man’s spirit. Nature in its moral aspects is seen in poems such as ‘Tintern Abbey’ and especially ‘The Prelude’. Wordsworth became a master of all forms of poetry: narrative verse, ballads, lyrical poems, sonnets, odes, and elegies. He was a conscious, deliberate poet. He had a fine mastery of language. To him, “Poetry is the spontaneous overflow of powerful feelings: it takes its origin from emotion recollected in tranquillity.”

About the Poem
The title ‘Daffodils ’ is likely to suggest a poem in worshipful praise of the flowers so named. The poem, composed in 1807, is an instance of emotion recollected in tranquillity. ‘Daffodils’ describes the beauty and power of the flowers. More significantly, it is a vivid dramatic record of the poet’s encounter with a charming spectacle of nature. On careful reading, the poem turns out to be a beautiful, dramatic lyric on the poet’s spiritual conversion in the company of Nature. The fact that Wordsworth achieves his purpose in ‘Daffodils’ through the use of simple, spontaneous, yet sweet and forceful language bears witness to his poetic greatness and excellence.

Summary
The poem gives vent to a personal experience. In one of his wanderings, the poet happened to catch sight of a host of golden daffodils fluttering and dancing in the breeze by the side of a lake. They also reminded him of the stars at night in brightness and multitude. They were ‘tossing their heads in sprightly dance’. The waves were dancing too, but the glee of the lovely flowers was far greater than that of the waves. The entire atmosphere was one of joy and this joy swelled the poet’s heart. The sight had also another and a more profound effect on the mind of the poet. In later years, when he was in a pensive mood, the lively picture of the dancing daffodils flashed upon his ‘inward eye’ and filled his mood of solitude with immense joy.

ସାରାଂଶ:
କବିତାଟି ଏକ ବ୍ୟକ୍ତିଗତ ଅନୁଭୂତିକୁ ଭାବପୂର୍ଣ୍ଣଭାବେ ପରିପ୍ରକାଶ କରିଛି । ଥରେ ବୁଲୁବୁଲୁ କବି ୱାର୍ଡସ୍ୱର୍ଥ ଏକ ହ୍ରଦକୂଳରେ ଗୁଡ଼ିଏ ସୁନେଲି ରଙ୍ଗର ଡାଫୋଡ଼ିଲ୍‌ ଫୁଲର ସମ୍ଭାର ପବନ ଲହରୀରେ ଦୋଳି ଖେଳୁଥ‌ିବାର ଦେଖ‌ିବାକୁ ପାଇଲେ । ସେହି ଫୁଲଗୁଡ଼ିକ କବିଙ୍କୁ ଉଜ୍ଜଳ ରାତ୍ରିରେ ଆକାଶରେ ଚିକିମିକି କରୁଥିବା ଅସଂଖ୍ୟ ତାରକାମାନଙ୍କୁ
ମନେପକାଇ ଦେଇଥିଲା । ସେହି ଫୁଲଗୁଡ଼ିକ ମୁଣ୍ଡ ହଲାଇ ଜୀବନ୍ତ ନୃତ୍ୟ କଲାଭଳି ପ୍ରତୀୟମାନ ହେଉଥିଲା । ଲହରୀମାଳା ଡେଇଁ ଡେଇଁ ନୃତ୍ୟ କରୁଥିଲେ ହେଁ ଏହି ସୁନ୍ଦର ଫୁଲଗୁଡ଼ିକର ଆନନ୍ଦ ତା’ଠାରୁ ଅନେକ ଗୁଣରେ ଅଧିକ ଥିଲା । ସମଗ୍ର ପରିବେଶ ଆନନ୍ଦବିଭୋର ଥିଲା ଏବଂ ସେହି ଆନନ୍ଦ କବିଙ୍କ ହୃଦୟକୁ ଉଦ୍‌ବେଳିତ କରିଥିଲା । ସେହି ଦୃଶ୍ୟ କବିଙ୍କ ମନରେ ଆଉ ଏକ ଗଭୀର ପ୍ରଭାବ ପକାଇଥିଲା । ପରବର୍ତ୍ତୀ ଜୀବନରେ କବିଙ୍କ ମନ ଯେତେବେଳେ ଦୁଃଖରେ ଭରିଯାଇଥିଲା, ନୃତ୍ୟରତା ଡାଫୋଡ଼ିଲ ଫୁଲଗୁଡ଼ିକର ସୁନ୍ଦର ଦୃଶ୍ୟ ତାଙ୍କ ଅନ୍ତଃଚକ୍ଷୁ ସମ୍ମୁଖରେ ଭାସିଯାଇଥିଲା ଏବଂ
ତାଙ୍କ ନିରୋଳା ମନରେ ଆନନ୍ଦ ଭରି ଦେଇଥିଲା ।

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Odisha State Board BSE Odisha 9th Class Maths Solutions Geometry Chapter 2 ତ୍ରିଭୁଜମାନଙ୍କ ସର୍ବସମତା Ex 2(b) Textbook Exercise Questions and Answers.

BSE Odisha Class 9 Maths Solutions Geometry Chapter 2 ତ୍ରିଭୁଜମାନଙ୍କ ସର୍ବସମତା Ex 2(b)

Question 1.
ନିମ୍ନ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।
(a) Δ ABC ରେ m∠A = 40°, m∠B = 75° ହେଲେ, ତ୍ରିଭୁଜର ବୃହତ୍ତମ ଏବଂ କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁମାନ ସ୍ଥିର କର ।
ସମାଧାନ:
m∠A = 40°, m∠B = 75°
⇒ Δ ABCର m∠C = (180 – 40 – 75)° = 65°
m∠B > m∠C > m∠A
⇒ AC > AB > BC
∴ ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ବାହୁ AC ଓ କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ବାହୁ BC ।

(b) Δ ABC ରେ m∠A = 110°, m∠B = 20° ହେଲେ, ତ୍ରିଭୁଜର କେଉଁ ବାହୁଟି କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ?
ସମାଧାନ:
Δ ABC ରେ m∠A = 110°, m∠B = 20° ∴ m∠C = (180 – 110 – 20)° = 50°
m∠A > m∠C > m∠B
⇒ BC > AB > AC
∴ AC କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ବାହୁ ।

(c) Δ ABC ରେ m∠B = 90° ହେଲେ, ତ୍ରିଭୁଜର କେଉଁ ବାହୁଟି ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ?
ସମାଧାନ:
Δ ABC ରେ m∠B = 90° ।
⇒ m∠C < 90° ଓ m∠A < 90°
∴ ∠B ର ସମ୍ମୁଖୀନ ବାହୁ AC ବାହୁଟି ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ।

(d) Δ ABC ରେ m∠A = m∠B + m∠C ହେଲେ, ତ୍ରିଭୁଜର ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ କେଉଁଟି ?
ସମାଧାନ:
Δ ABC ରେ m∠A = m∠B + m∠C
ଆମେ କଣ୍ଙ m∠A + m∠B + m∠C = 180°
⇒ m∠A = m∠B + m∠C = 90°
∴ ∠A ର ସମ୍ମୁଖୀନ ବାହୁ BC ର ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ବାହୁ ।

(e) Δ ABC ରେ m∠A = 40°, m∠B = 50° । ବାହୁଗୁଡ଼ିକର ଦୈର୍ଘ୍ୟର ଉକ୍ରମରେ ସଜାଇ ଲେଖ ।
ସମାଧାନ:
Δ ABC ରେ m∠A = 40°, m∠B = 50° ⇒ m∠C = 180° – 40° – 50° = 90°
∴ m∠C > m∠B > m∠A
⇒ AB > AC > BC

Question 2.
ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।
(a) ତ୍ରିଭୁଜର ଦୁଇ ବାହୁର ଦୈର୍ଘ୍ୟର ସମଷ୍ଟି, ଏହାର ତୃତୀୟ ବାହୁର ଦୈର୍ଘ୍ୟଠାରୁ ________ ।
ସମାଧାନ:
ବୃହତ୍ତର

(b) ତ୍ରିଭୁଜର ଦୁଇ ବାହୁର ଦୈର୍ଘ୍ୟର ଅନ୍ତର, ଏହାର ତୃତୀୟ ବାହୁର ଦୈର୍ଘ୍ୟଠାରୁ ________ ।
ସମାଧାନ:
କ୍ଷୁଦ୍ରତର

(c) ତ୍ରିଭୁଜର ଉଚ୍ଚତା ତ୍ରୟର ଦୈର୍ଘ୍ୟର ସମଷ୍ଟି, ଏହାର ପରିସୀମାଠାରୁ ________ ।
ସମାଧାନ:
କ୍ଷୁଦ୍ରତର

(d) ତ୍ରିଭୁଜର ପରିସୀମା, ଏହାର ମଧ୍ୟମାତ୍ରୟର ସମଷ୍ଟିଠାରୁ ________ ।
ସମାଧାନ:
ବୃହତ୍ତର

(e) ତ୍ରିଭୁଜର ଶୀର୍ଷବିନ୍ଦୁରୁ ଭୂମିପ୍ରତି ଅଙ୍କିତ ଲମ୍ବର ଦୈର୍ଘ୍ୟ ଏହାର ଅନ୍ୟ ଦୁଇ ବାହୁମାନଙ୍କର ଦୈର୍ଘ୍ୟଠାରୁ ________ ।
ସମାଧାନ:
କ୍ଷୁଦ୍ରତର

Question 3.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ m∠CBD > m∠BCE ହେଲେ, ଦର୍ଶାଅ ଯେ AB > AC ।
ସମାଧାନ:
ଦତ୍ତ : ଚିତ୍ରରେ m∠CBD > m∠BCE
ପ୍ରାମାଣ୍ୟ : AB > AC
ପ୍ରମାଣ : m∠CBD > m∠BCE
⇒ m∠A + m∠ACB > m∠A + m∠ABC
(∵ ତ୍ରିଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ଅନ୍ତଃସ୍ଥ ଦୂରବର୍ତ୍ତୀ କୋଣଦ୍ୱୟର ପରିମାଣର ସମଷ୍ଟି ସହ ସମାନ)
⇒ m∠ACB > m∠ABC ⇒ AB > AC
(ପ୍ରମାଣିତ)

Question 4.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ PQ = PR । ଦର୍ଶାଅ ଯେ PS > PQ ।
ସମାଧାନ:
ଦତ୍ତ : ଚିତ୍ରରେ PQ = PR
ପ୍ରାମାଣ୍ୟ : PS > PQ
ପ୍ରମାଣ : PQ = PR ⇒ m∠PRQ = m∠PQR … (i)
m∠PQR > m∠PSQ
(∵ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ଅନ୍ତଃସ୍ଥ ଦୂରବର୍ତ୍ତୀ କୋଣର ପରିମାଣଠାରୁ ବୃହତ୍ତର)
⇒ m∠PQR >m∠PSR ⇒ m∠PRQ > m∠PSR (∵ m∠PRQ = m∠PQR)
⇒ m∠PRS > m∠PSR ⇒ PS > PR
⇒ PS > PQ (∵ PQ = PR)

Question 5.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ A͞D, ∠A ର ସମଦ୍ବିଖଣ୍ଡକ ହେଲେ, ଦର୍ଶାଅ ଯେ (i) AB > BD (ii) AC > CD ।
ସମାଧାନ:
ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ A͞D, ∠A ର ସମଦ୍ବିଖଣ୍ଡକ ।
ପ୍ରାମାଣ୍ୟ : (i) AB > BD ଏବଂ (ii) AC > CD
ପ୍ରମାଣ : Δ ADC ରେ m∠ADB > m∠CAD
m∠ADB > m∠BAD (∵ m∠BAD = m∠CAD)
AB > BD … (i)
ପୁନଶ୍ଚ, Δ ABD ରେ m∠ADC > m∠BAD
m∠ADC > m∠CAD (∵ m∠CAD = m∠BAD)
AC > CD … (ii)
(ପ୍ରମାଣିତ)

Question 6.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ PR > PQ ଏବଂ P͞S, ∠P ର ସମଦ୍ବିଖଣ୍ଡକ । ଦର୍ଶାଅ ଯେ x > y ।
ସମାଧାନ:
ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ PR > PQ । P͞S, ∠P ର ସମଦ୍ଵିଖଣ୍ଡକ ।
ପ୍ରାମାଣ୍ୟ : x > y
ପ୍ରମାଣ : PQR ରେ PR > PQ ⇒ m∠POS > m∠PRS … (i)
APQS ରେ ବହିଃସ୍ଥ m∠PSR = m∠PQS + m∠QPS
APSR ରେ ବହିଃସ୍ଥ m∠PSQ = m∠PRS + m∠RPS
କିନ୍ତୁ (i) ରୁ m∠PQS > m∠PRS
⇒ m∠PQS + m∠QPS > m∠PRS + m∠RPS (∵ m∠QPS = m∠RPS)
⇒ m∠PSR > m∠PSQ ⇒ x > y
(ପ୍ରମାଣିତ)

Question 7.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ PQ > PR, \(\overrightarrow{\mathrm{QS}}\) ଏବଂ \(\overrightarrow{\mathrm{RS}}\) ଯଥାକ୍ରମେ ∠Q ଓ ∠R ର ସମଦ୍ବିଖଣ୍ଡକ । ଦର୍ଶାଅ ଯେ SQ > SR ।
ସମାଧାନ:
ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ PQ > PR ।
\(\overrightarrow{\mathrm{QS}}\) ଏବଂ \(\overrightarrow{\mathrm{RS}}\) ଯଥାକ୍ରମେ ∠Q ଓ ∠R ର ସମଦ୍ବିଖଣ୍ଡକ ।
ପ୍ରାମାଣ୍ୟ : SQ > SR
ପ୍ରମାଣ : PQ > PR ⇒ m∠PRQ > m∠PQR
\(\frac{1}{2}\)m∠PRQ > \(\frac{1}{2}\)m∠PQR ⇒ m∠SRQ > m∠SQR
⇒ SQ > SR
(ପ୍ରମାଣିତ)

Question 8.
ଦର୍ଶାଅ ଯେ, ସମକୋଣୀ ତ୍ରିଭୁଜର କର୍ଣ୍ଣ ତ୍ରିଭୁଜର ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ ।
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ m∠B = 90° ।
ପ୍ରାମାଣ୍ୟ : A͞C, ତ୍ରିଭୁଜର ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ ।
ପ୍ରମାଣ : m∠ABC = 90°
⇒ m∠BAC +m∠ACB = 90°
∴ m∠ABC > m∠BAC ⇒ AC > BC … (i)
ପୁନଣ୍ଚ, m∠ABC > m∠ACB ⇒ AC > AB … (ii)
∴ (i) ଓ (ii) ରୁ A͞C ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ ।
(ପ୍ରମାଣିତ)

Question 9.
PQRS ଚତୁର୍ଭୁଜରେ P͞S ଓ Q͞R ଯଥାକ୍ରମେ ଚତୁର୍ଭୁଜର ବୃହତ୍ତମ ଏବଂ କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ । ପ୍ରମାଣ କର ଯେ,
(i) m∠PQR > m∠PSR
(ii) m∠QRS > m∠SPQ ଏବଂ
(iii) m∠P + m∠S < m∠Q + m∠R
ସମାଧାନ:
ଦତ୍ତ : PQRS ଚତୁର୍ଭୁଜରେ P͞S ଓ Q͞R ଯଥାକ୍ରମେ ଚତୁର୍ଭୁଜର ବୃହତ୍ତମ ଏବଂ କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ ।
ପ୍ରାମାଣ୍ୟ : (i) m∠PQR > m∠PSR
(ii) m∠QRS > m∠SPQ ଏବଂ
(iii) m∠P + m∠S < m∠Q + m∠R
ଅଙ୍କନ : S͞Q ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ PSQ ରେ m∠PQS > m∠PSQ
(:: P͞S ବୃହତ୍ତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ)
Δ QSR ରେ m∠SQR > m∠QSR
(: Q͞R କ୍ଷୁଦ୍ରତମ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ବାହୁ)
⇒ m∠PQS + m∠SQR > m∠PSQ + m∠QSR
⇒ m∠PQR > m∠PSR … (i)
ସେହିପରି P͞R ଅଙ୍କନ କରି ପ୍ରମାଣ କରାଯାଇପାରେ ଯେ, 
m∠QRS > m∠SPQ … (ii)
(i) ଓ (ii) ରୁ m∠PQR + m∠QRS > m∠PSR + m∠SPQ
⇒ m∠Q + m∠R > m∠S + m∠P
⇒ m∠S + m∠P < m∠Q + m∠R … (iii)
(ପ୍ରମାଣିତ)

Question 10.
Δ ABC ର AD, BE ଓ CF ଉଚ୍ଚତାତ୍ରେୟ । ପ୍ରମାଣ କର ଯେ,
(i) AB + AC > 2AD
(ii) AB + BC + AC > AD + BE + CF
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର A͞D ⊥ B͞C, BE ⊥ AC ଏବଂ CF ⊥ AB
ପ୍ରାମାଣ୍ୟ : (i) AB + AC > 2AD
(ii) AB + BC + AC > AD + BE + CF
ପ୍ରମାଣ : Δ ABD ରେ AB > AD ଏବଂ Δ ADC ରେ AC > AD
⇒ AB + AC > 2AD … (i)
ସେହିପରି ପ୍ରମାଣ କରାଯାଇ ପାରେ ଯେ, AC + BC > 2CF ଏବଂ AB + BC > 2BE
∴ AB + AC + AC + BC + AB + BC > 2AD + 2CF + 2BE
⇒ 2(AB + AC + BC) > 2(AD + BE + CF)
⇒ AB + AC + BC > AD + BE + CF … (ii)
(ପ୍ରମାଣିତ)

Question 11.
Δ ABC ର AD, BE ଏବଂ CF ମଧ୍ଯମାତ୍ରୟ । ପ୍ରମାଣ କର ଯେ,
(i) AB + AC > 2AD
ସମାଧାନ:
ଦତ୍ତ : (i) Δ ABC ରେ AD ମଧ୍ୟମା ।
ପ୍ରାମାଣ୍ୟ : AB + AC > 2AD
ଅଙ୍କନ : \(\overrightarrow{\mathrm{AD}}\) ଉପରେ M ଏପରି ଏକ ବିନ୍ଦୁ ଯେପରିକି AD = DM ହେବ । CM ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ ABD ଓ Δ CDM ଦ୍ବୟରେ BD = CD (ଦତ୍ତ),
AD = DM (ଅଙ୍କନ) ଏବଂ m∠ADB = m∠CDM (ପ୍ରତୀପ)
Δ ABD ≅ Δ CDM => AB = CM … (i)
ବର୍ଭମାନ Δ ACM ରେ AC + CM > AM
AC + CM > 2AD => AC + AB > 2AD … (i) ରୁ
(ପ୍ରମାଣିତ)

(ii) AB + AC + BC > AD + BE + CF
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର AD, BE ଓ CF ମଧ୍ଯମାତ୍ରୟ ।
ପ୍ରାମାଣ୍ୟ : AB+ AC + BC > AD + BE + CF 
ପ୍ରମାଣ : (i) ରେ ପ୍ରମାଣିତ ଯେ, AB + AC > 2AD
ସେହିପରି ପ୍ରମାଣ କରାଯାଇପାରେ, 
AB + BC > 2BE ଏବଂ BC + AC > 2BF
∴ AB + AC + AB + BC + BC + AC > 2AD + 2BE + 2CF
⇒ 2(AB + AC + BC) > 2(AD + BE + CF)
⇒ AB + AC + BC > AD + BE + CF … (ii)
(ପ୍ରମାଣିତ)

Question 12.
Δ ABC ର O ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ହେଲେ, ପ୍ରମାଣ କର ଯେ
(i) BO + CO < AB + AC
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର O ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : BO + CO < AB + AC
ଅଙ୍କନ : \(\overrightarrow{\mathrm{BO}}\), A͞C କୁ M ବିନ୍ଦୁରେ ଛେଦ କରୁ ।
ପ୍ରମାଣ : Δ MOC ରେ OM + MC > CO, Δ ABM ରେ AB + AM > BM
∴ AB + AM + MC + OM > CO + BM ⇒ AB + AC + OM > CO + BO + OM
⇒ AB + AC > CO + BO ⇒ BO + CO < AB + AC … (i)
(ପ୍ରମାଣିତ)

(ii) AO + BO + CO < AB + AC + BC ଏବଂ
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର O ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : AO + BO + CO < AB + AC + BC
ପ୍ରମାଣ : (i) ରେ ପ୍ରମାଣିତ BO + CO < AB + AC
ସେହିପରି ପ୍ରମାଣ କରାଯାଇପାରେ,
AO + CO < AB + BC ଏବଂ AO + BO < AC + BC
⇒ BO + CO + AO + CO + AO + BO < AB + AC + AB + BC + AC + BC
⇒ 2(AO + BO + CO) < 2(AB + AC + BC)
⇒ AO + BO + CO < AB + AC + BC … (ii)
(ପ୍ରମାଣିତ)

(iii) AO + BO + CO > \(\frac{1}{2}\)(AB + AC + BC)
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ର O ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : AO + BO + CO > \(\frac{1}{2}\)(AB + AC + BC)
ପ୍ରମାଣ : Δ AOB ରେ AO + BO > AB, Δ BOC ରେ BO + CO > BC
Δ AOC ରେ AO + CO > AC
∴ AO + BO + BO + CO + AO + CO > AB + BC + AC
⇒ 2(AO + BO + CO) > AB + BC + AC
⇒ AO + BO + CO > \(\frac{1}{2}\)(AB + AC + BC) … (iii)
(ପ୍ରମାଣିତ)

Question 13.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ Δ ABC ରେ AB > AC ଏବଂ AD = AC । ପ୍ରମାଣ କର ଯେ,

(i) m∠ACD = \(\frac{1}{2}\) (m∠B + m∠C)
(ii) m∠BCD = \(\frac{1}{2}\) (m∠C – m∠B)
ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ AB > AC ଏବଂ AD = AC ।
ପ୍ରାମାଣ୍ୟ : (i) m∠ACD = \(\frac{1}{2}\) (m∠B + m∠C)
(ii) m∠BCD = \(\frac{1}{2}\) (m∠C – m∠B)
ପ୍ରମାଣ : (i) Δ ADC ରେ AD = AC
⇒ m∠ADC = m∠ACD ⇒ 2m∠ADC = 2m∠ACD
⇒ m∠ADC + m∠ADC = 2m∠ACD
⇒ m∠ADC + m∠DBC + m∠DCB = 2m∠ACD
⇒ (m∠ACD + m∠DCB) + m∠DBC = 2m∠ACD (∵ m∠ADC = m∠ACD)
⇒ m∠C + m∠B = 2m∠ACD
⇒ m∠ACD = \(\frac{1}{2}\) (m∠C + m∠B) … (i) (ପ୍ରମାଣିତ)

(ii) ପୁନଣ୍ଚ, m∠BCD = m∠ACB – m∠ACD
⇒ 2m∠BCD = 2m∠ACB – 2m∠ACD
⇒ 2m∠BCD = 2m∠ACB – m∠ACD – m∠ACD = 2m∠ACB – m∠ACD – m∠ADC
= 2m∠ACB – m∠ACD – (m∠DBC + m∠DCB)
= 2m∠ACB – m∠ACD – m∠DBC – m∠DCB
= 2m∠ACB – (m∠ACD + m∠DCB) – m∠DBC
= 2m∠ACB – m∠ACB – m∠DBC = m∠ACB – m∠DBC = m∠C – ∠B
m∠BCD = \(\frac{1}{2}\) (m∠C – m∠B) … (ii) (ପ୍ରମାଣିତ)

Question 14.
ABCD ଚତୁର୍ଭୁଜରେ ପ୍ରମାଣ କର ଯେ,

(i) AB + BC + CD > AD
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତୁର୍ଭୁଜ
ପ୍ରାମାଣ୍ୟ : AB + BC + CD > AD
ପ୍ରମାଣ : Δ ABC ରେ AB + BC > AC … (1)
Δ ADC ରେ AC + CD > AD … (2)
(1) ଓ (2) କୁ ଯୋଗକଲେ AB + BC + AC + CD > AC + AD
⇒ AB + BC + CD > AD (ଉଭୟ ପାର୍ଶ୍ଵରୁ AC ବାଦଦେଲେ) (ପ୍ରମାଣିତ)

(ii) AB + BC + CD + AD > AC + BD
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ଚତୁର୍ଭୁଜ
ପ୍ରାମାଣ୍ୟ : AB + BC + CD + AD > AC + BD
ପ୍ରମାଣ : Δ ABC ରେ AB + BC > AC … (1)
Δ BDC ରେ BC + CD > BD … (2)
Δ ADC ରେ AD + CD > AC … (3)
Δ ADB ରେ AD + AB > BD … (4)
( 1), (2), (3) ଓ (4) କୁ ଯୋଗକଲେ
AB + BC + BC + CD + AD + CD + AD + AB > AC + AC + BD + BD
⇒ 2(AB + BC + CD + AD) > 2(AC + BD)
⇒ AB + BC + CD + AD > AC + BD (ପ୍ରମାଣିତ)

(ii) AB + BC + CD + AD > AC + BD
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ଚତୁର୍ଭୁଜ
ପ୍ରାମାଣ୍ୟ : AB + BC + CD + AD > 2AC
ପ୍ରମାଣ : ABC Δରେ AB + BC > AC … (1)
ସେହିପରି ADC Δରେ AD + CD > AC … (2)
(1) ଓ (2) କୁ ଯୋଗକଲେ AB + BC + CD + AD > AC + AC
⇒ AB + BC + CD + AD > 2AC (ପ୍ରମାଣିତ)

 

Question 15.
Δ ABC ରେ AC > AB ଏବଂ A͞D ତ୍ରିଭୁଜର ମଧ୍ୟମା ହେଲେ ପ୍ରମାଣ କର ଯେ m∠BAD > m∠CAD ।

ସମାଧାନ:
ଦତ୍ତ : Δ ABC ରେ AC > AB । A͞D, Δ ABC ର ଏକ ମଧ୍ୟମା ।
ପ୍ରାମାଣ୍ୟ : m∠BAD > m∠CAD
ଅଙ୍କନ : \(\overrightarrow{\mathrm{AD}}\) ଉପରେ M ଏପରି ଏକ ବିନ୍ଦୁ ଯେପରିକି AD = DM ।
CM ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ ABD ଓ Δ CMD ଦ୍ବୟରେ BD = CD,
AD = DM ଏବଂ m∠ADB = m∠CDM (ପ୍ରତୀପ)
∴ Δ ABD ≅ ACMD ⇒ AB = CM ଏବଂ m∠CMD = m∠BAD
∴ Δ ACM ରେ AC > CM (∵ AC > AB ଦତ୍ତ)
⇒ m∠CMD > m∠CAD
⇒ m∠BAD > m∠CAD (∵ m∠CMD = m∠BAD) (ପ୍ରମାଣିତ)

Question 16.
ABCD ଚତୁର୍ଭୁଜର ‘O’ ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ (କର୍ଣ୍ଣଦ୍ୱୟର ଛେଦବିନ୍ଦୁ ଭିନ୍ନ) ହେଲେ ପ୍ରମାଣ କର ଯେ,

(i) 2(OA + OB + OC + OD) > AB + BC + CD + AD ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ଚତୁର୍ଭୁଜର ଅନ୍ତର୍ଦେଶରେ ‘O’ ଏକ ବିନ୍ଦୁ । ଯାହା କର୍ଣ୍ଣଦ୍ଵୟର ଛେଦବିନ୍ଦୁ ନୁହେଁ ।
ପ୍ରାମାଣ୍ୟ : 2(OA + OB + OC + OD) > AB + BC + CD + AD
ପ୍ରମାଣ : Δ AOB ରେ OA + OB > AB … (i)
Δ AAD ରେ OA + OD > AD … (ii)
Δ ADC ରେ OD + OC > CD … (iii)
Δ ABC ରେ OB + OC > BC … (iv)
(i), (ii), (iii) ଓ (iv) କୁ ଯୋଗକଲେ
OA + OB + OA + OD + OD + OC + OB + OC > AB + AD + CD + BC
⇒ 2(OA + OB + OC + OD) > AB + BC + CD + AD (ପ୍ରମାଣିତ)

(ii) OA + OB + OC + OD > AC + BD
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତୁର୍ଭୁଜରେ ‘O’ ଏକ ଅନ୍ତସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : OA + OB + OC + OD > AC + BD
ପ୍ରମାଣ : Δ AOC ରେ OA + OC > AC … (1)
Δ BOD ରେ OB + OD > BD … (2)
(1) ଓ (2) କୁ ଯୋଗକଲେ
OA + OC + OB + OD > AC + BD
OA + OB + OC + OD > AC + BD (ପ୍ରମାଣିତ)

Question 17.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ m∠PAX = m∠QAY ହେଲେ ଦର୍ଶାଅ ଯେ, PA +AQ < PB + BQ ।

ସମାଧାନ:

ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ m∠PAX = m∠QAY ।
ପ୍ରାମାଣ୍ୟ : PA + AQ < PB + BQ
ଅଙ୍କନ : \(\overrightarrow{\mathrm{PA}}\) ଉପରେ R ଏକ ବିନ୍ଦୁ ନିଅ ଯେପରିକି P – A – R ଓ AQ = AR ହେବ ।
ପ୍ରମାଣ : m∠PAX = m∠BAR (ପ୍ରତୀପ କୋଣ)
m∠PAX = m∠QAY (ଦତ୍ତ)
⇒ m∠QAY = m∠BAR
∴ Δ ABQ ଓ Δ ABR ମଧ୍ୟରେ
AQ = AR (ଅଙ୍କନ)
m∠QAY = m∠BAR (ପ୍ରମାଣିତ)
A͞B ସଧାରଣ ବିନ୍ଦୁ
Δ ABQ ≅ Δ ABR (ବା-କୋ-ବା ସର୍ବସମତା)
⇒ BQ = BR
Δ PBR ରେ PR < PB + BR
⇒ PA + AR < PB + BR
⇒ PA + AQ < PB + BQ (∵ AR = AQ ଓ BR = BQ) (ପ୍ରମାଣିତ)

Question 17.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ  AB = AC ହେଲେ ଦର୍ଶାଅ ଯେ, AF > AE ।

ସମାଧାନ:
ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ  AB = AC
ପ୍ରାମାଣ୍ୟ : AF > AE
ପ୍ରମାଣ : Δ ABC ରେ AB = AC (ଦତ୍ତ)
m∠ABC = m∠ACB
Δ DFC ରେ ବତ୍ହିଃସ୍ଥ m∠FCB > m∠CFD
⇒ m∠ACB > m∠CFD ⇒ m∠ACB > m∠AFE … (i)
[7 m∠AFE = m∠CFD (ପ୍ରତୀପ)]
⇒ m∠ABC > m∠AFE (∵ m∠ABC = m∠ACB)
ପୁନଶ୍ଚ ବତ୍ହିଃସ୍ଥ m∠AEF > m∠ABC
m∠AEF > m∠ACB (∵ m∠ABC = m∠ACB)
m∠AEF > m∠AFE [∵ (i) ରୁ]
⇒ AF > AE (ପ୍ରମାଣିତ)

Odisha State Board BSE Odisha 9th Class Maths Solutions Geometry Chapter 3 ଚତୁର୍ଭୁଜ Ex 3(b) Textbook Exercise Questions and Answers.

BSE Odisha Class 9 Maths Solutions Geometry Chapter 3 ଚତୁର୍ଭୁଜ Ex 3(b)

Question 1.
ନିମ୍ନଲିଖ ଉକ୍ତିଗୁଡ଼ିକ ଭୁଲ୍ କି ଠିକ୍ ଲେଖ ।
(a) ଚତୁର୍ଭୁଜର ଚାରୋଟି ବାହୁ ସର୍ବସମ ହେଲେ, ତାହା ଏକ ବର୍ଗଚିତ୍ର ।
ସମାଧାନ:
ଭୁଲ

(b) ପ୍ରତ୍ୟେକ ରମ୍ବସ୍ ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଠିକ୍

(c) ପ୍ରତ୍ୟେକ ସାମାନ୍ତରିକ ଚିତ୍ର ଏକ ରମ୍ବସ୍ ।
ସମାଧାନ:
ଭୁଲ

(d) ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ ସନ୍ନିହିତ ବାହୁର ଦୈର୍ଘ୍ୟ ସମାନ ହେଲେ, ତାହା ଏକ ରମ୍ବସ୍ ।
ସମାଧାନ:
ଠିକ୍

(e) ରମ୍ବସ୍‌ର କର୍ଣ୍ଣଦ୍ଵୟ ସର୍ବସମ ।
ସମାଧାନ:
ଭୁଲ

(f) ଗୋଟିଏ ଆୟତଚିତ୍ରର କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରନ୍ତି ।
ସମାଧାନ:
ଠିକ୍

(g) ଗୋଟିଏ ରମ୍ବସ୍‌ର ଗୋଟିଏ କୋଣର ପରିମାଣ 90° ହେଲେ, ତାହା ଏକ ବର୍ଗଚିତ୍ର ।
ସମାଧାନ:
ଠିକ୍

(h) ଗୋଟିଏ ବର୍ଗଚିତ୍ରର କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମକୋଣରେ ସମର୍ଦ୍ଦିଖଣ୍ଡ କରନ୍ତି ।
ସମାଧାନ:
ଠିକ୍

(i) ଯଦି ଏକ ଚତୁର୍ଭୁଜର କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପର ପ୍ରତି ଲମ୍ବ ହୁଅନ୍ତି, ତେବେ ଚତୁର୍ଭୁଜଟି ଏକ ବର୍ଗଚିତ୍ର ।
ସମାଧାନ:
ଭୁଲ

(j) ପ୍ରତ୍ୟେକ ସାମାନ୍ତରିକ ଚିତ୍ର ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ସମାଧାନ:
ଠିକ୍

(k) ପ୍ରତ୍ୟେକ ବର୍ଗଚିତ୍ର ଏକ ଆୟତଚିତ୍ର ।
ସମାଧାନ:
ଠିକ୍

(l) ରମ୍ବସ୍ ଏକ ବର୍ଗଚିତ୍ର ।
ସମାଧାନ:
ଭୁଲ

Question 2.
ନିମ୍ନ ଚିତ୍ରଗୁଡ଼ିକୁ ଦେଖ୍ ‘x’ ର ମୂଲ୍ୟ ସ୍ଥିର କର ।

ସମାଧାନ:
ସାମାନ୍ତରିକ ଚିତ୍ର:
m∠DAB + m∠ABC = 180°
⇒ 75° + m∠ABD + m∠DBC = 180°
⇒ 75° + m∠ABD + 60° = 180°
⇒ m∠ABD = 180° – 135° = 45°
ବର୍ତ୍ତମାନ x = 45° (∵ m∠ABD = m∠CDB)

ରମ୍ବସ୍:
m∠ABC + m∠BCD = 180°
⇒ m∠BCD = 180° – m∠ABC = 180° – 120° = 60°
କିନ୍ତୁ Δ ABC ରେ m∠BAC + m∠BCA
= 180° – 120° = 60°
⇒ m∠BCA = \(\frac{60^{\circ}}{2}\) = 30° (∵ m∠BAC = m∠BCA)
m∠BCD = 60°
⇒ m∠BCA + m∠ACD = 60°
⇒ 30° + x = 60°
⇒ x = 30°

ଅ‍।ୟତଚିତ୍ର:
m∠BAC + m∠ACB = 90°
⇒ m∠ACB = 90° – m∠BAC = 90° – 32° = 58°
କିନ୍ତୁ m∠OBC = m∠OCB = 58° (∵ OB = OC)
⇒ x = 58°

ବର୍ଗଚିତ୍ର:
m∠DOC = m∠POB = 85° କିନ୍ତୁ m∠OBP = 45°
Δ APB ରେ m∠POB + m∠OBP = m∠OPA
⇒ 85° + 45° = x
⇒ x = 130°

Question 3.
ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।
(a) ________ ର କର୍ଣ୍ଣଦ୍ଵୟ ସର୍ବସମ ଏବଂ ପରସ୍ପର ପ୍ରତି ଲମ୍ବ ।
ସମାଧାନ:
ବର୍ଗଚିତ୍ର

(b) ABCD ଚତୁର୍ଭୁଜରେ ∠A ଓ ∠B ପରସ୍ପର ପରିପୂରକ ହେଲେ, ଚତୁର୍ଭୁଜଟି ________ ।
ସମାଧାନ:
ଟ୍ରାପିଜିୟମ୍

(c) ଗୋଟିଏ ରମ୍ବସ୍‌ର କଣ୍ଠଦ୍ଵୟ ସର୍ବସମ ହେଲେ, ରମ୍ବସ୍‌ ________ ।
ସମାଧାନ:
ବର୍ଗଚିତ୍ର

(d) ABCD ଚତୁର୍ଭୁଜର AB = CD, A͞B || C͞D ହେଲେ, ଚତୁର୍ଭୁଜଟି ________ ।
ସମାଧାନ:
ସାମାନ୍ତରିକ ଚିତ୍ର

(e) ABCD ଚତୁର୍ଭୁଜର AB = BC ଏବଂ AC = BD ଏବଂ ∠B ଏକ ସମକୋଣ ହେଲେ ଚତୁର୍ଭୁଜଟି ________ ।
ସମାଧାନ:
ବର୍ଗଚିତ୍ର

(f) ଗୋଟିଏ ରମ୍ବସ୍‌ର ଗୋଟିଏ କୋଣର ପରିମାଣ 90° ହେଲେ ରମ୍ବସ୍‌ ________ ।
ସମାଧାନ:
ବର୍ଗଚିତ୍ର

(g) ABCD ଚତୁର୍ଭୁଜର A͞C ଓ B͞D କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମସ୍ଵିଖଣ୍ଡ କରନ୍ତି ଏବଂ m∠A = 90° ହେଲେ, ଚତୁର୍ଭୁଜଟି ________ ।
ସମାଧାନ:
ଅ‍।ୟତଚିତ୍ର

(h) ABCD ଚତୁର୍ଭୁଜର A͞C ଓ B͞D କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ୱିଖଣ୍ଡ କରନ୍ତି ଏବଂ AC ≅ BD ହେଲେ, ଚତୁର୍ଭୁଜଟି ________ ।
ସମାଧାନ:
ଅ‍।ୟତଚିତ୍ର

Question 4.
(i) ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ m∠B = (x + 30°) 3 m∠C = (2x – 60°) ହେଲେ m∠A କେତେ ?
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ରରେ m∠B = (x + 30°), m∠C = (2x – 60°) ।
ନିର୍ଦେୟ : m∠A ର ପରିମାଣ
ଉତ୍ତର : ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ m∠B+ m∠C = 180°
⇒ (x + 30) + (2x – 60) = 180°
⇒ 3x – 30° = 180°
⇒ 3x = 210°
⇒ x = 70°
∴ m∠A = m∠C = 2x – 60° = 2 × 70° – 60° = 80°

(ii) ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । ∠A ଓ ∠B ର ସମଦ୍ଵିଖଣ୍ଡକଦ୍ବୟ ପରସ୍ପରକୁ P ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । ∠APB ର ପରିମାଣ କେତେ ?
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । ∠A ଓ ∠B ର ସମଦ୍ଵିଖଣ୍ଡକ ଯଥାକ୍ରମେ \(\overrightarrow{\mathrm{AP}}\) ଓ \(\overrightarrow{\mathrm{BP}}\)
ପରସ୍ପରକୁ P ବିନ୍ଦୁରେ ଛେଦ କରୁଛନ୍ତି ।
ନିର୍ମେୟ : ∠APB ର ପରିମାଣ ।
ଡତ୍ତର : m∠A + m∠B = 180°
⇒ \(\frac{1}{2}\) m∠A + \(\frac{1}{2}\) m∠B = \(\frac{1}{2}\) × 180°
⇒ m∠PAB + m∠PBA = 90°
Δ APB ରେ m∠PAB + m∠PBA + m∠APB = 180°
[ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ ସନ୍ନିହିତ କୋଣଦ୍ଵୟର ପରିମାଣର ସମଷ୍ଟି 180° ।]
⇒ 90° + m∠APB = 180°
(∵ m∠PAB + m∠PBA = 90°)
⇒ m∠APB = 180° – 90° = 90°

(iii) ଗୋଟିଏ ରମ୍ବସର କ୍ଷୁଦ୍ରତର କର୍ପୂର ଦୈର୍ଘ୍ୟ, ଏହାର ଏକ ବାହୁର ଦୈର୍ଘ୍ୟ ସହ ସମାନ ହେଲେ, ରମ୍ବସ୍‌ର ବୃହତ୍ତର କୋଣର ପରିମାଣ କେତେ ?
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ରମ୍ବସ୍‌ରେ A͞C କ୍ଷୁଦ୍ରତର କର୍ଣ୍ଣ ଓ BC = AC ।
ନିଶ୍ଚେୟ : ∠BCD ର ପରିମାଣ ।
ଉତ୍ତର : ABCD ରମ୍ବସ୍‌ରେ AB = BC । କିନ୍ତୁ BC= AC (ଦତ୍ତ)
∴ AB = BC = AC
⇒ Δ ABC ସମବାହୁ
⇒ m∠ACB = 60°
ସେହିପରି Δ ACD ସମବାହୁ
⇒ m ∠ACD = 60°
∴ m∠BCD=m∠ACB + m∠ACD = 60° + 60° = 120°

(iv) ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇଟି କ୍ରମିକ ଶୀର୍ଷରେ ଉତ୍ପନ୍ନ କୋଣମାନଙ୍କର ପରିମାଣର ଅନୁପାତ 2 : 3 ହେଲେ, ବୃହତ୍ତର କୋଣର ପରିମାଣ କେତେ ?
ସମାଧାନ:
ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇଟି କ୍ରମିକ ଶୀର୍ଷରେ ଉତ୍ପନ୍ନ କୋଣମାନଙ୍କର ପରିମାଣର ଅନୁପାତ 2 : 3 ।
ଦତ୍ତ : ବୃହତ୍ତର କୋଣର ପରିମାଣ ।
ନିଶ୍ଚେୟ : ମନେକର କୋଣଦ୍ଵୟର ପରିମାଣ 2x ଏବଂ 3x ।
ଉତ୍ତର : ସାମାନ୍ତରିକ ଚିତ୍ରରେ 2x + 3x = 180°
[ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ ସନ୍ନିହିତ କୋଣଦ୍ଵୟର ପରିମାଣର ସମଷ୍ଟି 180° ।]
ସାମାନ୍ତରିକ ଚିତ୍ରରେ 2x + 3x = 180°
⇒ 5x = 180°
⇒ x = 36°
∴ ବୃହତ୍ତମ କୋଣର ପରିମାଣ = 3x = 3 x 36° = 108°

(v) ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ରର ଗୋଟିଏ କୋଣର ପରିମାଣ ଏହାର ଏକ ସନ୍ନିହିତ କୋଣର \(\frac{4}{5}\) ହେଲେ, ସନ୍ନିହିତ କୋଣର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
ଦତ୍ତ : ସାମାନ୍ତରିକ ଚିତ୍ରର ଗୋଟିଏ କୋଣର ପରିମାଣ ଅନ୍ୟ ସନ୍ନିହିତ କୋଣର ପରିମାଣ \(\frac{4}{5}\) ଅଂଶ ।
ନିଶ୍ଚେୟ : ସନ୍ନିହିତ କୋଣଦ୍ଵୟର ପରିମାଣ ।
ଉତ୍ତର : ମନେକର ସନ୍ନିହିତ କୋଣଦ୍ୱୟର ପରିମାଣ x ଏବଂ \(\frac{4 \mathrm{x}^{\circ}}{5}\) ।
ସାମାନ୍ତରିକ ଚିତ୍ରରେ ସନ୍ନିହିତ କୋଣଦ୍ୱୟର ପରିମାଣର ସମଷ୍ଟି 180° ।
x + \(\frac{4 \mathrm{x}^{\circ}}{5}\) = 180° 
⇒ 5x + 4x = 900
⇒ 9x = 900
⇒ x = 100
∴ ଗୋଟିଏ କୋଣର ପରିମାଣ 100 ଏବଂ ଅନ୍ୟ ସନ୍ନିହିତ କୋଣର ପରିମାଣ = 180° – 100° = 80° ।

Question 5.
(i) ABCD ଏକ ଉତ୍ତଳ ଚତୁର୍ଭୁଜ । ଏଥ‌ିରେ ∠B, ∠C, ∠D ର ପରିମାଣ ଯଥାକ୍ରମେ ∠A ର ପରିମାଣର ଦୁଇଗୁଣ, ତିନିଗୁଣ, ଚାରିଗୁଣ ହେଲେ ଦର୍ଶାଅ ଯେ, ଏହା ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ଚତୁର୍ଭୁଜ । m∠B = 2m∠A, m∠C = 3m∠A ଏବଂ m∠D = 4m∠A
ପ୍ରାମାଣ୍ୟ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ପ୍ରମାଣ : ମନେକର m∠A = x ତେବେ m∠B = 2x, m∠C = 3x ଏବଂ m∠D = 4x
ABCD ଟ୍ରାପିଜିମ୍ବସ୍‌ରେ m∠A + m∠B + m∠C + m∠D = 360°
⇒ x + 2x + 3x + 4x = 360°
⇒ 10x = 360°
⇒ x = 36°
∴ m∠A = 36°, m∠B = 2 x 36° = 72°,
m∠C = 3 × 36° = 108° ଏବଂ m∠D = 4 × 36° = 144°
ଲଷ୍ୟକର ଏଠାରେ m∠B + m∠C = 72° + 108° = 180°
∴ A͞B || D͞C
⇒ ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ । (ପ୍ରମାଣିତ)

(ii) ABCD ଚତୁର୍ଭୁଜରେ ∠A ଓ ∠B ର ସମତ୍ତିଖଣ୍ଡକ ପରସ୍ପରକୁ ‘O’ ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ଏବଂ ∠AOB ଏକ ସମକୋଣ ହେଲେ ପ୍ରମାଣ କର ଯେ, ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତୁର୍ଭୁଜରେ ∠A ଓ ∠B ର ସମଦ୍ବିଖଣ୍ଡକ ପରସ୍ପରକୁ ‘O’ ବିନ୍ଦୁରେ ଛେଦ କରୁଛନ୍ତି ।
m∠AOB = 90°
ପ୍ରାମାଣ୍ୟ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ପ୍ରମାଣ : Δ AOB ରେ m∠OAB + m∠ABO + m∠AOB = 180°
⇒ m∠OAB + m∠ABO = 90°
[∵ m∠AOB = 90° (ଦତ୍ତ)]
⇒ 2m∠OAB + 2m∠ABO = 2 × 90°
⇒ m∠A + m∠B = 180°
⇒ A͞D || B͞C
⇒ ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
(ପ୍ରମାଣିତ)

(iii) ABCD ଚତୁର୍ଭୁଜରେ ∠ADC ଏକ ସମକୋଣ, m∠BAC = m∠ACB = 45° ଏବଂ AD = DC ହେଲେ ପ୍ରମାଣ କରି ଯେ, ଏହା ଏକ ବର୍ଗଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ତୁର୍ଭୁଜରେ m∠ADC = 90°, AD = DC, m∠BAC = m∠ACB = 45° ।
ପ୍ରାମାଣ୍ୟ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ପ୍ରମାଣ : Δ ADC ରେ m∠ADC = 90° ଏବଂ AD = DC
⇒ m∠DAC = m∠DCA => m∠DAC = m∠DCA = 45°
Δ ADC ଓ Δ ABC ଦ୍ଠୟରେ m∠DAC = m∠BAC
m∠DCA = m∠ACB ଏବଂ A͞C ସାଧାରଣ ।
∴ Δ ADC ≅ Δ ABC
⇒ AD = AB ଏବଂ DC = BC କିନ୍ତୁ ଦତ୍ତ AD = DC
∴ AD = AB = DC = BC
⇒ ABCD ଏକ ବର୍ଗଚିତ୍ର । (∵ m∠D = 90°)
(ପ୍ରମାଣିତ)

(iv) ABCD ଚତୁର୍ଭୁଜରେ AD = BC = 3 ସେ.ମି., CD = 8 ସେ.ମି. । AB ୟପରେ E ଓ F ଦୁଇଟି ବିନ୍ଦୁ,
m∠BCF = m∠BFC = m∠AED = m∠ADE = 45° ହେଲେ ପ୍ରମାଣ କର ଯେ, ABCD ଏକ ଆୟତଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତୁର୍ଭୁଜରେ AD = BC = 3 ସେ.ମି., CD = 8 ସେ.ମି., EF = 2 ସେ.ମି. ଏବଂ
m∠BCF = m∠BFC = m∠AED = m∠ADE = 45°
ପ୍ରାମାଣ୍ୟ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ ।
ପ୍ରମାଣ : Δ ADE ରେ m∠ADE = m∠AED = 45°
⇒ AD = AE ଏବଂ m∠A = 90°
∴ AE = 3 ସେ.ମି.
ସେହିପରି Δ FBC ରେ m∠B = 90° ଏବଂ BC = BF = 3 ସେ.ମି.
∴ AB = AE + EF + BF = 3 + 2 + 3 = 8 ସେ.ମି.
ଏଠାରେ AD = BC (ଦତ୍ତ)
AB = DC ଏବଂ m∠A = m∠B = 90°
∴ ABCD ଏକ ଆୟତଚିତ୍ର ।
(ପ୍ରମାଣିତ)

(v) ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । ଯଦି AB = 2AD ଏବଂ P, C͞D ର ମଧ୍ୟବିନ୍ଦୁ ହୁଏ ତେବେ ଦର୍ଶାଅ ଯେ,m∠APB = 90° ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ କ୍ଷେତ୍ରରେ AB = 2AD, P, D͞C ର ମଧ୍ୟବିନ୍ଦୁ 
ପ୍ରାମାଣ୍ୟ : m∠APB = 90°
ପ୍ରମାଣ : Δ ADP ରେ AD = DP
(∵ 2AD = AB = CD ଏବଂ P, C͞D ର ମଧ୍ୟବିନ୍ଦୁ)
m∠DAP = m∠APD = α (ମନେକର)
ସେହିପରି Δ PBC ରେ PC = BC (∵ AD = BC = DP = PC)
m∠PBC = m∠BPC = β (ମନେକର)
Δ ADP ରେ m∠D =180° – 2α (∵ m∠D + m∠PAD + m∠APD = 180°)
ଓ Δ BPC ରେ m∠C = 180° – 2β
କିନ୍ତୁ m∠D + m∠C = 180° (∵ A͞D || B͞C)
180° – 2α + 180° – 2β = 180°
⇒ 2α + 2β = 180°
⇒ α + β = 90°
m∠APD + m∠BPC = 90°
⇒ m∠APB = 90°
(ପ୍ରମାଣିତ)

Question 6.
ABCD ଚତୁର୍ଭୁଜରେ m∠ABD = m∠BDC ଏବଂ m∠ADB = m∠CBD ହେଲେ ପ୍ରମାଣ କରି ଯେ, ଏହା ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ଏବଂ Δ ABC = Δ ADC ।
ସମାଧାନ:
ଦତ୍ତ : m∠ABD = m∠BDC ଏବଂ m∠ADB = m∠CBD ।
ପ୍ରାମାଣ୍ୟ : (i) ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ii) Δ ABC ≅ Δ ADC
ପ୍ରମାଣ : Δ ABD ଓ Δ BDC ରେ m∠ABD = m∠BDC,
m∠ADB = m∠DBC ଏବଂ B͞D ସାଧାରଣ ବାହୁ
⇒ Δ ABD ≅ Δ ADC ⇒ m∠A = m∠C … (i)
ପୁନଶ୍ଚ, m∠ABD + m∠DBC = m∠BDC + m∠ADB
⇒ m∠B = m∠D … (ii)
(i) ଓ (ii) ରୁ ପାଇଲେ, ABCD ଚତୁର୍ଭୁଜର ବିପରୀତ କୋଣମାନଙ୍କର ପରିମାଣ ସମାନ । ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
∴ ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ପ୍ରମାଣିତ)

Question 7.
ABCD ଚତୁର୍ଭୁଜରେ BC || AD । A͞C ଓ B͞D ଯଥାକ୍ରମେ ∠BAD ଓ ∠CDA କୁ ସମର୍ଦ୍ଦିଖଣ୍ଡ କରୁଥିଲେ, ପ୍ରମାଣ
କର ଯେ AB = BC = CD ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଚତୁର୍ଭୁଜରେ BC || AD । A͞C ଓ B͞D ଯଥାକ୍ରମେ ∠BAD ଓ ∠CDA କୁ ସମର୍ଦ୍ଦିଖଣ୍ଡ ଜରେ
ପ୍ରାମାଣ୍ୟ : AB = BC = CD
ପ୍ରମାଣ : m∠BAC = m∠CAD (∵ A͞C, ∠BAD ର ସମର୍ଦ୍ଦିଖଣ୍ଡ)
BC || AD
⇒ m∠CAD = m∠BCA (ଏକାନ୍ତର କୋଣ)
⇒ m∠BAC = m∠BCA
⇒ AB = BC … (i)
ସେହିପରି m∠BDC = m∠BDA (ଦତ୍ତ)
BC || AD ⇒ m∠CBD = m∠BDA (ଏକାନ୍ତର କୋଣ)
⇒ m∠BDC = m∠CBD
⇒ BC = CD … (ii)
(i) ଓ (ii) ରୁ ପାଇଲେ, AB = BC = CD
(ପ୍ରମାଣିତ)

Question 8.
ABCD ସାମାନ୍ତରିକ ଚିତ୍ର । ∠A ଓ ∠C ର ସମଦ୍ବିଖଣ୍ଡକ ଯଥାକ୍ରମେ \(\overrightarrow{\mathrm{AP}}\) ଓ \(\overrightarrow{\mathrm{CQ}}\) । ଏମାନେ ଯଦି D͞C ଓ A͞B କୁ ଯଥାକ୍ରମେ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ପ୍ରମାଣ କର ଯେ, APCQ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । ∠A ଓ ∠C ର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମି ଯଥାକ୍ରମେ DC ଓ AB କୁ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ।
ପ୍ରାମାଣ୍ୟ : APCQ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ପ୍ରମାଣ : m∠A = m∠C
⇒ \(\frac{1}{2}\)m∠A = \(\frac{1}{2}\)m∠C
⇒ m∠BCQ = m∠DAP ଏବଂ m∠PCQ = m∠PAQ … (i)
⇒ m∠BCQ + m∠B = m∠DAP + m∠D (∵ m∠B = m∠D)
⇒ m∠AQC = m∠APC … (ii)
(i) ଓ (ii) ରୁ ପାଇବା APCQ ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(∵ ବିପରୀତ କୋଣମାନଙ୍କର ପରିମାଣ ସମାନ ।)
(ପ୍ରମାଣିତ)

Question 9.
ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ M ଓ N ଯଥାକ୍ରମେ D͞C ଓ A͞B ର ମଧ୍ୟବିନ୍ଦୁ । ପ୍ରମାଣ କର ଯେ,
(i) MCBN ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର 
(ii) DMBN ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର ଏବଂ
(iii) D͞B ଓ M͞N ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରନ୍ତି ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ M ଓ N ଯଥାକ୍ରମେ D͞C ଓ A͞B ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : (i) MCBN ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ii) DMBN ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(iii) D͞B ଓ M͞N ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରନ୍ତି ।
ପ୍ରମାଣ : MCBN ସାମାନ୍ତରିକ ଚିତ୍ରରେ N͞B || M͞C (∵ A͞B || D͞C)
NB = MC (∵ AB = DC ଏବଂ \(\frac{1}{2}\)AB = \(\frac{1}{2}\)DC)
∴ MCBN ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । … (i) (ପ୍ରମାଣିତ)
DMBN ଚିତ୍ରରେ B͞M || D͞N (∵ A͞B || D͞C)
NB = DM (∵ AB = DC ଏବଂ \(\frac{1}{2}\)AB = \(\frac{1}{2}\)DC)
∴ DMBN ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । … (ii) (ପ୍ରମାଣିତ)
DMBN ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ହେତୁ B͞D ଓ N͞M କଣ୍ଠଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରିବେ । … (iii) (ପ୍ରମାଣିତ)

Question 10.
ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ A͞C ଓ B͞D କଣ୍ଠଦ୍ଵୟ ପରସ୍ପରକୁ ‘O’ ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । DO ର ମଧ୍ୟବିନ୍ଦୁ X ଓ BO ର ମଧ୍ୟବିନ୍ଦୁ Y ହେଲେ, ପ୍ରମାଣ କର ଯେ AXCY ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ A͞C ଓ B͞D କଣ୍ଠଦ୍ଵୟର ଛେଦବିନ୍ଦୁ ‘O’ । X ଓ Y ଯଥାକ୍ରମେ DO ଏବଂ B͞O ର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : AXCY ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ପ୍ରମାଣ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
AO = CO ଏବଂ DO = BO
AO = CO ଏବଂ \(\frac{1}{2}\)DO = \(\frac{1}{2}\)BO
AYCX ଏକ ଚତୁର୍ଭୁଜ ଯାହାର କଣ୍ଠଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରନ୍ତି ।
AXCY ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ପ୍ରମାଣିତ)

Question 11.
ABCD ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର । A͞C ଉପରେ K, L ଦୁଇଟି ବିନ୍ଦୁ ଯେପରିକି AK = CL । ପ୍ରମାଣ କର ଯେ, DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ ଚିତ୍ର । A͞C ଉପରେ K, L ଦୁଇଟି ବିନ୍ଦୁ ଯେପରିକି AK = CL ।
ପ୍ରାମାଣ୍ୟ : DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ଅଙ୍କନ : B͞D ଅଙ୍କନ କର । A͞C ଓ B͞D ର ଛେଦବିନ୍ଦୁ O ହେଉ ।
ପ୍ରମାଣ : ABCD ସାମାନ୍ତରିକ ଚିତ୍ର ।
⇒BO = DO ଏବଂ AO = CO
⇒BO = DO ଏବଂ AO – AK = CO – CL
⇒BO = DO ଏବଂ KO = LO
⇒ DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ପ୍ରମାଣିତ)

Question 12.
ABCD ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର । B͞D ଉପରେ P ଓ Q ଦୁଇଟି ବିନ୍ଦୁ ଯେପରିକି AP || CQ । ପ୍ରମାଣ କର ଯେ, DP = BQ ଏବଂ APCQ ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ର । B͞D ଉପରେ P ଓ Q ଦୁଇଟି ବିନ୍ଦୁ ଯେପରିକି A͞P || C͞Q ।
ପ୍ରାମାଣ୍ୟ : (i) DP = BQ
(ii) APCQ ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ପ୍ରମାଣ : A͞P || C͞Q, B͞D ଛେଦକ
∴ m∠APQ = m∠PQC
⇒ m∠APD = m∠BQC … (i)
Δ ADP ଓ Δ BQC ରେ
AD = BC (ସାମାନ୍ତରିକ ଚିତ୍ରର ବିପରୀତ ବାହୁ)
m∠ADP = m∠QBC, ( AD || BC )
m∠APD = m∠BQC [(i) ରୁ ପ୍ରମାଣିତ]
Δ ADP ≅ Δ BQC
DP = BQ ଏବଂ AP = QC … (ii)
⇒ AP || QC (ଦତ୍ତ) ଏବଂ AP = QC,
⇒ APCQ ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ପ୍ରମାଣିତ)

Question 13.
ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ DK ⊥ AC, BL ⊥ AC ଏବଂ K ଓ I ଯଥାକ୍ରମେ ଲମ୍ବନ୍ବୟର ପାଦବିନ୍ଦୁ । ପ୍ରମାଣ କର ଯେ, DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ DK ⊥ AC, BL ⊥ AC ।
ପ୍ରାମାଣ୍ୟ : DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
ଅଙ୍କନ : D͞B ଅଙ୍କନ କର ।
ପ୍ରମାଣ : Δ ADK ଓ Δ BCL ଦ୍ଠୟରେ AD = BC,
ଏବଂ m∠DKA = m∠BLC
Δ ADK ≅ Δ BCL
⇒ AK = CL
ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ DO = BO ଏବଂ AO = CO
⇒ DO = BO ଏବଂ AO – AK = CO – CL
⇒ DO = BO ଏବଂ KO = OL
⇒ DKBL ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(ପ୍ରମାଣିତ)

Question 14.
ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । A͞D ଉପରେ P ଏକ ବିନ୍ଦୁ ଯେପରିକି DC = DP, \(\overrightarrow{\mathrm{CP}}\) ଓ \(\overrightarrow{\mathrm{BA}}\) ପରସ୍ପରକୁ Q ବିନ୍ଦୁରେ ଛେଦ କରୁଥିଲେ, ପ୍ରମାଣ କର ଯେ
(i) AQ = AP
(ii) BC = BQ
(iii) AD = CD + AQ
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । A͞D ଉପରେ P ଏକ ବିନ୍ଦୁ ଯେପରିକି DC = DP । \(\overrightarrow{\mathrm{CP}}\) ଓ \(\overrightarrow{\mathrm{BA}}\) ପରସ୍ପରକୁ Q ।
ପ୍ରାମାଣ୍ୟ : (i) AQ = AP
(ii) BC = BQ
(iii) AD = CD + AQ

ପ୍ରମାଣ : Δ PCD ରେ DC = DP
m∠DCP = m∠DPC
କିନ୍ତୁ m∠DPC = m∠QPA (ପ୍ରତୀପ)
ଏବଂ m∠DCP = m∠AQP (ଏକାନ୍ତର) (∵ B͞Q || C͞D, Q͞C ଛେଦକ)
m∠DPC = m∠DCP ତେଣ୍ଙ m∠QPA = m∠AQP
⇒ AQ = AP … (i) (ପ୍ରମାଣିତ)
କିନ୍ତୁ m∠DCP = m∠AQP (ଏକାନ୍ତର)
ପୁନଶ୍ଚ, m∠DPC = m∠PCB (ଏକାନ୍ତର)
∠DCP = m∠DPC ତେଣ୍ଙ m∠AQP = m∠PCB
⇒ BC = BQ … (ii) (ପ୍ରମାଣିତ)
ପୂର୍ବ ରୁ ପ୍ରମାଣିତ CD = PD ଏବଂ AQ = AP
CD + AQ = PD + AP
⇒ CD + AQ = AD … (iii) (ପ୍ରମାଣିତ)

Question 15.
ABCD ସାମାନ୍ତରିକ ଚିତ୍ରରେ D͞C ବାହୁ ଉପରେ X ଏକ ବିନ୍ଦୁ ଯେପରିକି AD = AX । ପ୍ରମାଣ କର ଯେ, m∠XAB = m∠ABC ଏବଂ AC = BX ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । D͞C ଉପରେ X ଏକ ବିନ୍ଦୁ ଯେପରିକି AD = AX ।
ପ୍ରାମାଣ୍ୟ : (i) m∠XAB = m∠ABC
(ii) AC = BX

ପ୍ରମାଣ : AD = AX
⇒ m∠ADX = m∠AXD
କିନ୍ତୁ m∠ADX = m∠ABC (ସାମାନ୍ତରିକ ଚିତ୍ରର ବିପରୀତ କୋଣ)
m∠AXD = m∠XAB (ଏକାନ୍ତର)
∴ m∠XAB = m∠ABC … (i) (∵ m∠ADX = m∠AXD) (ପ୍ରମାଣିତ)
Δ AXB ଓ Δ BAC ଦ୍ଠୟରେ m∠XAB = m∠ABC,
∴ A͞B ସାଧାରଣ ଏବଂ AX = BC (∵ AX = AD = BC)
Δ AXB ≅ Δ BAC
⇒ BX = AC … (ii) (ପ୍ରମାଣିତ)

Question 16.
ପ୍ରମାଣ କର ଯେ, ସାମାନ୍ତରିକ ଚିତ୍ରର କୋଣମାନଙ୍କର ସମଦ୍ବିଖଣ୍ଡକ ରେଖାମାନଙ୍କ ଦ୍ବାରା ଗଠିତ ଚତୁର୍ଭୁଜଟି ଏକ ଅ‍।ୟତଚିତ୍ର ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର । ∠A ଓ ∠B ର ସମଦ୍ଵିଖଣ୍ଡକ ରଶ୍ମିଦ୍ବୟର ଛେଦବିନ୍ଦୁ M, ∠B ଓ ∠C ର
ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମିଦ୍ବୟର ଛେଦବିନ୍ଦୁ N, ∠C ଓ ∠D ର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମିଦ୍ବୟର ଛେଦବିନ୍ଦୁ P ଏବଂ ∠D ଓ ∠Aର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମିଦ୍ଵୟର ଛେଦବିନ୍ଦୁ Q ।
ପ୍ରାମାଣ୍ୟ : MNPQ ଏକ ଅ‍।ୟତଚିତ୍ର ।

ପ୍ରମାଣ : m∠A + m∠B = 180°
⇒ \(\frac{1}{2}\)m∠A + \(\frac{1}{2}\)m∠B = 180°
⇒ m∠MAB + m∠MBA = 90°
⇒ m∠AMB = 90°
⇒ m∠AMB = m∠QMN = 90° (ପ୍ରତୀପ କୋଣ)
ସେହିପରି ପ୍ରମାଣ କରାଯାଇପାରେ ଯେ, m∠QPN = 90°, m∠MQP= 90° ଏବଂ m∠MNP = 90°
∴ MNPQ ଏକ ଅ‍।ୟତଚିତ୍ର । (ପ୍ରମାଣିତ)

Question 17.
ପ୍ରମାଣ କର ଯେ, ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ରର କର୍ଣ୍ଣଦ୍ୱୟର ଛେଦବିନ୍ଦୁ ମଧ୍ୟଦେଇ ଅଙ୍କିତ ଓ ସାମାନ୍ତରିକ ଚିତ୍ରର ବାହୁମାନଙ୍କ ଦ୍ଵାରା ସୀମାବଦ୍ଧ ରେଖାଖଣ୍ଡ କର୍ତ୍ତୃମାନଙ୍କ ଛେଦବିନ୍ଦୁଠାରେ ସମଦ୍ବିଖଣ୍ଡିତ ହୁଏ ।
ସମାଧାନ:
ଦତ୍ତ : ABCD ଏକ ସାମାନ୍ତରିକ ଚିତ୍ରରେ A͞C ଓ B͞D କର୍ଣ୍ଣଦ୍ୱୟର ଛେଦବିନ୍ଦୁ O।
O ବିନ୍ଦୁ ଦେଇ ଅଙ୍କିତ ଏକ ରେଖା A͞B ଓ CD କୁ M ଓ N ବିନ୍ଦୁରେ ଛେଦକରୁଛି ।

ପ୍ରାମାଣ୍ୟ : MO = ON
ପ୍ରମାଣ : Δ BMO ଏବଂ Δ DNO ଦ୍ବୟରେ
m∠MOB = m∠NOD (ପ୍ରତୀପ)
m∠MBO = m∠NDO (ଏକାନ୍ତର),
BO = CO (ସାମାନ୍ତରିକ ଚିତ୍ରର କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କରନ୍ତି)
∴ Δ BMO ≅ Δ DNO (∵ କୋ-ବା-କୋ ସର୍ବସମତା)
⇒ MO = ON (ପ୍ରମାଣିତ)

Question 18.
ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ R͞P || A͞B, RQ || BC ଏବଂ PQ || AC ହେଲେ ଦର୍ଶାଅ ଯେ BC = \(\frac{1}{2}\)QR ।
ସମାଧାନ:
ଦତ୍ତ : ପାର୍ଶ୍ୱସ୍ଥ ଚିତ୍ରରେ RP || AB, RQ || BC ଏବଂ PQ || AC ।
ପ୍ରାମାଣ୍ୟ : BC = \(\frac{1}{2}\)QR

ପ୍ରମାଣ : RQ || BC, RP || AB
⇒ RABC ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର
⇒ BC = RA
ସେହିପରି RQ || BC, PQ || AC
⇒ AQBC ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର
⇒ BC = AQ
⇒ 2BC = RA + AQ
⇒ 2BC = RQ
⇒ BC = \(\frac{1}{2}\)QR (ପ୍ରମାଣିତ)

Odisha State Board BSE Odisha 9th Class Maths Solutions Geometry Chapter 3 ଚତୁର୍ଭୁଜ Ex 3(a) Textbook Exercise Questions and Answers.

BSE Odisha Class 9 Maths Solutions Geometry Chapter 3 ଚତୁର୍ଭୁଜ Ex 3(a)

Question 1.
ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।
(i) ଗୋଟିଏ ଉତ୍ତଳ ଚତୁର୍ଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି ________ ।
ସମାଧାନ:
360°

(ii) ଗୋଟିଏ ପଞ୍ଚଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି ________ ।
ସମାଧାନ:
540°

(iii) ଗୋଟିଏ ଅଷ୍ଟଭୁଜର ବହିଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି ________ ।
ସମାଧାନ:
360°

(iv) ଗୋଟିଏ ସୁଷମ ଷଡ଼ଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ ________ ।
ସମାଧାନ:
120°

(v) ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ 45° ହେଲେ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ________ ।
ସମାଧାନ:
8

(vi) ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ 150° ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ________ ।
ସମାଧାନ:
12

(vii) ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି 1440° ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ________ ।
ସମାଧାନ:
10

(viii) ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ୨ ହେଲେ, ଏହାର ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ________ ।
ସମାଧାନ:
40°

(ix) n ସଂଖ୍ୟକ ବାହୁ ବିଶିଷ୍ଟ ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ ________ ।
ସମାଧାନ:
\(\frac{2 n-4}{n}\) × 90°

(x) n ସଂଖ୍ୟକ ବାହୁ ବିଶିଷ୍ଟ ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ________ ।
ସମାଧାନ:
\(\frac{360^{\circ}}{n}\)

Question 2.
(i) ଗୋଟିଏ ଚତୁର୍ଭୁଜର କୋଣମାନଙ୍କର ଅନୁପାତ 2 : 3 : 4 : 6 ହେଲେ, ସେମାନଙ୍କର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
ମନେକର ଚତୁର୍ଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣ 2x°, 3x°, 4x° ଏବଂ 6x° ।
∴ 2x° + 3x° + 4x° + 6x° = 360° ⇒ 15x  = 360° ⇒ x = 24
∴ 2x° = 2 × 24 = 48°, 3x° = 3 × 24 = 72°,
4x° = 4 × 24 = 96° ଏବଂ 6x° = 6 × 24 = 144°
∴ ଚତୁର୍ଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣ 48°, 72°, 96° ଓ 144° ।

(ii) ଏକ ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ କ୍ରମିକ କୋଣ ମଧ୍ୟରୁ ଗୋଟିକର ପରିମାଣ ଅନ୍ୟର ପରିମାଣର \(\frac{3}{2}\) ଗୁଣ ହେଲେ କୋଣଗୁଡ଼ିକର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ ସନ୍ନିହିତ କୋଣଦ୍ଵୟର ପରିମାଣର ସମଷ୍ଟି 180° ।
ମନେକର ସାମାନ୍ତରିକ ଚିତ୍ରର ଗୋଟିଏ କୋଣର ପରିମାଣ x ଏବଂ ପ୍ରଶ୍ନନୁସାରେ ଅନ୍ୟଟିର ପରିମାଣ \(\frac{3 x^{\circ}}{2}\) ।
∴ x° + \(\frac{3 x^{\circ}}{2}\) = 180°
⇒ \(\frac{5 x^{\circ}}{2}\) = 180° ⇒ x = 72°
ଗୋଟିଏ କୋଣର ପରିମାଣ 72° ଏବଂ ଅନ୍ୟ କୋଣର ପରିମାଣ = \(\frac{3 x^{\circ}}{2}\) × 72 = 108°
∴ ସାମାନ୍ତରିକ ଚିତ୍ରର କୋଣଗୁଡ଼ିକର ପରିମାଣ 72°, 108°, 72° ଓ 108° ।

(iii) ଗୋଟିଏ ପଞ୍ଚଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣର ଅନୁପାତ 2 : 3 : 4: 5 : 6 ହେଲେ ବୃହତ୍ତମ କୋଣର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
n ବାହୁ ବିଶିଷ୍ଟ ଏକ ବହୁଭୂଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି (2n – 4) ସମକୋଣ ।
ଗୋଟିଏ ପଞ୍ଚଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି = (2 × 5 – 4) × 90° = 540°
ମନେକର ପଞ୍ଚଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣ 2x°, 3x°, 4x°, 5x° ଏବଂ 6x° ।
∴ 2x° + 3x° + 4x° + 5x° + 6x° = 540° → 20x = 540 = x = 27
∴ ବୃହତ୍ତମ କୋଣର ପରିମାଣ = 6x = 6 × 27 = 162°

(iv) ଗୋଟିଏ ଉତ୍ତଳ ଚତୁର୍ଭୁଜର ଦୁଇଟି କୋଣ ସମକୋଣ ଏବଂ ଅନ୍ୟ କୋଣମାନଙ୍କର ପରିମାଣ ପ୍ରତ୍ୟେକ 120° ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ସ୍ଥିର କର ।
ସମାଧାନ:
ମନେକର ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା n । 
∴ ଏହାର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କ ସଂଖ୍ୟା n ।
ପ୍ରଶାନୁସାରେ, 90° + 90° + (n – 2) 120° = (2n – 4) × 90°
n ସଂଖ୍ୟକ ବାହୁବିଶିଷ୍ଟ ବହୁଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି (2n – 4) ସମକୋଣ
ଏଠାରେ ଲକ୍ଷ୍ୟକର, ବହୁଭୁଜର n ସଂଖ୍ୟକ କୋଣ ମଧ୍ୟରୁ ଦୁଇଟି କୋଣ ସମକୋଣ ଏବଂ ଅନ୍ୟ ସମସ୍ତ କୋଣମାନ (n – 2 ସଂଖ୍ୟକ) ପ୍ରତ୍ୟେକେ 120° ।
⇒ 180 + (n – 2) 120 = (2n – 4) x 90
⇒ 18 + (n – 2) 12 = (2n – 4) 9
⇒ 18 + 12n – 24 = 18n – 36
⇒ 12n – 6 = 18n – 36
⇒  6n = 30 ⇒ n = 5
∴ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା 5 ।

(v) ଗୋଟିଏ ପଞ୍ଚଭୁଜର କୋଣମାନଙ୍କର ପରିମାଣର x°, (x − 10°), (x − 20°), (2x – 40°), (2x – 90°) ହେଲେ ‘x’ ର ମାନ ସ୍ଥିର କର ।
ସମାଧାନ:
n ଭୁଜ ବିଶିଷ୍ଟ ବହୁଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି (2n – 4) ସମକୋଣ । 
∴ ଗୋଟିଏ ପଞ୍ଚଭୁଜର ଅନ୍ତଃସ୍ଥ କୋଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି = (2 × 5 – 4) × 90° = 540°
⇒ 7x – 160 = 540 ⇒ 7x = 700 ⇒ x = 100

(vi) ଗୋଟିଏ ଅଷ୍ଟଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି ଏବଂ ବହିଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି ସ୍ଥିର କର ।
ସମାଧାନ:
ଗୋଟିଏ ଅଷ୍ଟଭୁଜର ଅନ୍ତଃ ସ୍ଥ କୋଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି = (2 × 8 – 4) × 90° = 1080°
ଏବଂ ବହିଃସ୍ଥ କୋଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି = 360°

(vii) ଗୋଟିଏ ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ କ୍ରମିକ କୋଣଦ୍ୱୟର ପରିମାଣର ଅନୁପାତ 2 : 3 ହେଲେ ସାମାନ୍ତରିକ ଚିତ୍ରର ଅନ୍ୟ କୋଣମାନଙ୍କର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
ସାମାନ୍ତରିକ ଚିତ୍ରର ଦୁଇ ସନ୍ନିହିତ କୋଣଦ୍ଵୟର ପରିମାଣର ସମଷ୍ଟି 180° ।
ମନେକର ଦୁଇ ସନ୍ନିହିତ କୋଣଦ୍ୱୟର ପରିମାଣ ଯଥାକ୍ରମେ 2x° ଏବଂ 3x° ।
∴ 2x + 3x = 180°
⇒ 5x = 180° ⇒ x = 36
∴ କୋଣମାନଙ୍କର ପରିମାଣ 2x° = 2 × 36 = 72° ଓ 3x° = 3 × 36 = 108°
∴ ସାମାନ୍ତରିକ ଚିତ୍ରର କୋଣମାନଙ୍କର ପରିମାଣ 72°, 108°, 72° ଏବଂ 108° ।

Question 3.
ଦର୍ଶାଅ ଯେ, ଗୋଟିଏ ସୁଷମ ଅଷ୍ଟଭୁଜର ଗୋଟିଏ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ ଏହାର ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣର ତିନିଗୁଣ ।
ସମାଧାନ:
ସୁଷମ ବହୁଭୁଜ (n ବାହୁ ବିଶିଷ୍ଟ) ର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ =  \(\frac{2 n-4}{n}\) ସମକୋଣ ।
ଗୋଟିଏ ସୁଷମ ଅଷ୍ଟଭୁଜର ଗୋଟିଏ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{2×8-4}{8}\) × 90° = \(\frac{12}{8}\) × 90 = 135°
ସୁଷମ ଅଷ୍ଟଭୁଜର ଗୋଟିଏ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = \(=\frac{360^{\circ}}{n}\)
ସୁଷମ ଅଷ୍ଟଭୁଜର ଗୋଟିଏ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = \(=\frac{360^{\circ}}{8}\) = 45°
ଏଠାରେ ଲକ୍ଷ୍ୟକର, 45° x 3 = 135° ହେତୁ ଅଷ୍ଟଭୁଜର ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣର ତିନିଗୁଣ, ଏହାର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ ସହ ସମାନ ।

Question 4.
ABCDE ଗୋଟିଏ ସୁଷମ ପଞ୍ଚଭୁଜ ହେଲେ, Δ BED ର ପ୍ରତ୍ୟେକ କୋଣର ପରିମାଣ ସ୍ଥିର କର ।
ସମାଧାନ:
ABCDE ଗୋଟିଏ ସୁଷମ ପଞ୍ଚଭୁଜ ।
ସୁଷମ ବହୁଭୂଜ (n ବାହୁ ବିଶିଷ୍ଟ)ର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{2 n-4}{n}\) ସମକୋଣ ।
ଏହାର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{2×5-4}{5}\) × 90° = 108°
Δ ABE ରେ m∠A = 108°, m∠ABE = m∠AEB = \(\frac{72}{2}\) = 36°
∴ Δ BED ରେ m∠BED = m∠AED – m∠AEB
= 108° – 36° = 72°
ସେହିପରି m∠BDE = 72°
∴ Δ BDE ରେ m∠EBD = 180° – (72° + 72°) = 36°
∴ Δ BDE ର କୋଣଦ୍ୱୟର ପରିମାଣ 36°, 72°, 72°।

Question 5.
ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ ଏବଂ ବହିଃସ୍ଥ କୋଣର ପରିମାଣର ଅନୁପାତ 5 : 1 ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ସ୍ଥିର କର ।
ସମାଧାନ:
ମନେକର ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = 5x°
ଓ ପ୍ରତ୍ୟେକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = x°

⇒ n= 10 + 2
⇒ n= 12
∴ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା 12 ।

Question 6.
n ସଂଖ୍ୟକ ବାହୁ ବିଶିଷ୍ଟ ଏକ ବହୁଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ଓ (n + 2) ସଂଖ୍ୟକ ବାହୁ ବିଶିଷ୍ଟ ଏକ ବହୁଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ ମଧ୍ଯରେ ଅନ୍ତର 9° ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ସ୍ଥିର କର ।
ସମାଧାନ:
n-ବାହୁ ବିଶିଷ୍ଟ ବହୁଭୁଜର ଏକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{360^{\circ}}{n}\)

⇒ n² + 2n – 80 = 0
⇒ n² + 10n – 8n – 80 = 0
⇒ n(n + 10) – 8(n + 10) = 0
⇒ (n + 10)(n – 8) = 0
⇒ n = -10 କିମ୍ଚ‍।
ଏଠାରେ n = 8 ଗ୍ରହଣୀୟ କିନ୍ତୁ n = -10 ଗ୍ରହଣୀୟ ନୁହେଁ । 
∴ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା 8 ।

Question 7.
ଗୋଟିଏ ସୁଷମ ବହୁଭୁଜର ଏକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ 120° ହେଲେ, ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା ସ୍ଥିର କର ।
ସମାଧାନ:
ମନେକର ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା n ।
ପ୍ରଶ୍ନନୁସାରେ, \(\frac{2 n-4}{n}\) × 90° = 120°
ସୁଷମ ବହୁଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{2 n-4}{n}\) ସମକୋଣ ।
⇒ (2n – 4) 9 = 12n
⇒ 18n – 36 = 12n
⇒ 6n = 36
⇒ n = 6
∴ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା 6 ।

ବିକଳ୍ପ ସମାଧାନ :
ସୁଷମ ବହୁଭୁଜର ଏକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ 120° ।
ସୁଷମ ବହୁଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = 180° – 120° = 60° ।
n-ବାହୁ ବିଶିଷ୍ଟ ବହୁଭୁଜର ଏକ ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{360^{\circ}}{n}\)
⇒ \(\frac{360^{\circ}}{n}\) = 60° 
⇒ 60° n = 360° 
⇒ n =\(\frac{360^{\circ}}{60}\) = 6
∴ ବହୁଭୁଜର ବାହୁ ସଂଖ୍ୟା 6 ।

Question 8.
(n – 1) ସଂଖ୍ୟକ ଏବଂ (n + 2) ସଂଖ୍ୟକ ସୁଷମ ଚତୁର୍ଭୁଜର ବହିଃସ୍ଥ ବହୁଭୁଜର ବହିଃସ୍ଥ କୌଣଦ୍ୱୟର ଅନ୍ତର 6° ହେଲେ, ଦର୍ଶାଅ ଯେ, ‘n’ ର ମାନ 13 ହେବ ।
ସମାଧାନ:
n ସଂଖ୍ୟକ ବାହୁ ବିଶିଷ୍ଟ ଏକ ସୁଷମ ବହୁଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{360^{\circ}}{n}\)

⇒ n² + n – 2 = 180
⇒ n² + n – 182 = 0
⇒ n² + 14n – 13n – 182 = 0
⇒ n (n + 14) – 13(n + 14) = 0
⇒ (n + 14) (n- 13) = 0
⇒ n + 14 = 0 ବା n – 13 = 0
⇒ n = -14   ଅସମ୍ଭବ ∴ n = 13

Question 9.
ଗୋଟିଏ ପଞ୍ଚଭୁଜର ଗୋଟିଏ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ 140 । ଅନ୍ୟ କୋଣଗୁଡ଼ିକର ପରମାଣର ଅନୁପାତ 1 : 2 : 3 : 4 ହେଲେ ଦର୍ଶାଅ ଯେ, ବୃହତ୍ତମ କୋଣର ପରିମାଣ 160° ।
ସମାଧାନ:
n ବାହୁ ବିଶିଷ୍ଟ ଏକ ବହୁଭୁଜର ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣର ସମଷ୍ଟି (2n – 4) ସମକୋଣ ।
ଗୋଟିଏ ପଞ୍ଚଭୁଜର ଅନ୍ତଃସ୍ଥ କୌଣମାନଙ୍କର ପରିମାଣର ସମଷ୍ଟି = (2 × 5 – 4) × 90 = 540° 
ପଞ୍ଚଭୁଜର ଗୋଟିଏ କୋଣର ପରିମାଣ = 140°
ମନେକର ଅନ୍ୟ 4ଟି କୋଣର ପରିମାଣ x°, 2x°, 3x° ଓ 4x° ।
ପ୍ରଶ୍ନନୁସାରେ, x + 2x° + 3x° + 4x° + 140° = 540°
⇒ 10x = 400 ⇒ x = 40°
∴ ବୃହତ୍ତମ କୋଣର ପରିମାଣ = 4x° = 4 × 40° = 160°

Question 10.
ABCDE ଗୋଟିଏ ସୁଷମ ପଞ୍ଚଭୁଜର AD, ∠CDE କୁ ଦୁଇଭାଗ କରୁଥିଲେ, ଦର୍ଶାଅ ଯେ m∠ADE : m∠ADC = 1 : 2 ।
ସମାଧାନ:
ଦତ୍ତ : ABCDE ଗୋଟିଏ ସୁଷମ ପଞ୍ଚଭୁଜର A͞D, ∠CDE କୁ ଦୁଇଭାଗ କରେ ।
ପ୍ରାମାଣ୍ୟ : m∠ADE : m∠ADC = 1 : 2 ।
ପ୍ରମାଣ : ସୁଷମ ପଞ୍ଚଭୁଜର ପ୍ରତ୍ୟେକ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ = \(\frac{2 \times 5-4}{5}\) × 90 = \(\frac{6}{5}\) × 90° = 108°
AE = ED
⇒ m∠EAD = m∠EDA
m∠AED = 108° = m∠EDC
AAED ରେ m∠EAD = m∠ADE = \(\frac{180^{\circ}-108^{\circ}}{2}\) = \(\frac{72^{\circ}}{2}\) = 36°
m∠ADC = m∠EDC – m∠ADE = 108° – 36° = 72°
∴ m∠ADE : m∠ADC = 36° : 12°
⇒ m∠ADE : m∠ADC = 1 : 2
(ପ୍ରମାଣିତ)

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 2 Solutions Poem 1 Indian Children Speak Textbook Exercise Questions and Answers.

Class 12th Alternative English Poem Chapter 1 Indian Children Speak Question Answers CHSE Odisha

Indian Children Speak Class 12 Questions and Answers

Pre-Reading Activity:
Very often we fall to understand each other. This happens especially when we are prejudiced against each other. To understand the other we need to develop a positive attitude. How would you react if you are described as just the opposite of what really you are? What can you do to clear the misunderstanding? Now read the poem below to see how the speaker tries to clear one such misunderstanding of some white people about the Red Indian children.

Questions For Discussion:
Question 1.
When you read the poem, you come across such names as Pansy, Delores, Ramon and Joe Henry. How can you describe them together.
Answer:
They are all Red Indian children.

Question 2.
Throughout the poem the phrase ‘people said’ has been repeated. Who are these people?
Answer:
These people are the white people.

Question 3.
What does moon-coloured dress refer to?
Answer:
It is bright, yellowish-white coloured dress.

Question 4.
Are the Indian children really dumb? Give reasons for your answer.
Answer:
The Indian children are really not dumb. It is because the narrator says, ‘clearly, I hear Delores answer/yes, the sunset is so good,I think God is throwing /Abright show I around the shoulders of the sky.’

Question 5.
Who do you think are rude – the white people or the Indians? Why do you think so?
Answer:
The white people were rude because they say Indian children are heard to reach, they are silent, they are dumb, they have no affection, they don’t seem very bright and they never take other in.

Question 6.
What is the speaker’s attitude towards the Indian children?
Answer:
The speaker is sympathetic and co-operative with the Indian children. His sense of sympathy stands as a sharp contrast with the uncompromising apathy of white men.

Question 7.
How many voices do you hear in this poem? Whose are they?
Answer:
There are three voices – the voice of the speaker the voice of white men and the voice of the Indian children in this poem.

Question 8.
The poem begins with ‘people said’. But towards the end of the poem the speaker says – ‘I have forgotten the idle words that people said’. Does this suggest a transition of mood and attitude in the speaker? Explain
Answer:
This certainly suggests a transition of mood and attitude in the speaker. It is because shedding the age old hackneyed bitter expression of the white man on the Indian children, the speaker switches over to another mood and mind.

Question 9.
What does the speaker convey in the last three lines of the poem?
Answer:
The last three lines are the concluding lines which convey that the speaker has personally parted with the indecent attitude towards the Indian children. He keeps in store in his heart to slip into the heart of Indian Lands and wants for that time to come.

Question 10.
Do you think the Indian children’s view of the world is different from that of the white people’s? How so?
Answer:
The Indian children’s view of the world is certainly different from that of the white people’s. The white people’s view is quite detrimental, command and selfish.

Question 11.
Is the speaker in the poem an American, Indian or a white American? How do you know?
Answer:
The speaker in the poem happens to be an American who makes it explicit in the line of the poem. ‘AndI steeped into the heart ofIndianLand’. This statement proves that the poet is an on-Indian.

Composition:
Question 1.
The speaker in the poem is not one of the ‘Indian children’. When then does the poem bear the title ‘Indian children speak’? Examine the appropriateness of the title of the poem.
Answer:
The title of the poem reads ‘Indian children speak’. The very title is suggestive of the voice of the Indian children although the poet himself is not one of the Indian children. The Indian children are the focus point of the poem. They are the pivotal characters around whom the whole edifice of the poem revolver. The speaker in the poem is not one of the ‘Indian children’. But the poem bears the title ‘Indian children speak’. The speaker is dissociated from the Indian children and makes an impersonal approachonbehalfoftheIndian children whom he likes and wishes to become apart of them He also desires to forget the idle words the people said and wants to treasure the day when the day when the iron doors swung wide so that he would slip into the heart of Indian land. All these speaks volumes of the question in contest.

Question 2.
There is an undercurrent of irony throughout the poem. Discuss how.
Iron refers to _______________________________________
Now discuss how there is an under current of irony in the poem?
Answer:
There is an under current of irony throughout the poem Irony refers to the expression of one’s meaning by saying something which is the direct opposite of one’s thoughts in order to make one’s remarks forceful.

Activity On Poem Completion:

Fill in the gaps appropriately with the lines given below the text of the incomplete poem ‘Nurses Song’.

Nurses’ Song
When the voices of children are heard on the green
_______________________________________
My hear is at rest without my breast
And everything else is still
Then come home, my children the sun is gone down.
And the dews of the night arise
Come, come, leave off play and let us away
_______________________________________
_______________________________________
And we cannot go to sleep
Besides, in the sky the little birds fly.
_______________________________________
The little ones leaped and shouted and laughed

Missing Lines:
(i) And the hills are all covered up with sleep.
(ii) And laughing is heard on the hill.
(iii) And, then go home to bed.
(iv) No, no, let us play for it yet day.
(v) Till the morning appears in the skies.

Answer:
When the voices of children are heard on the green
And laughing is heard on the hill.
My hear is at rest without by breast
And everything else is still
Then come home, my children the sun is gone down.
And the dews of the night arise
Come, come, leave off play and let us away
And the hills are all coverup with sleep.
And, then go home to bed.
And we cannot go to sleep
Besides, in the sky the little birds fly.
Till the morning appears in the skies
Well, well go and play till the fight fades away.
‘No, no, letus play, for it is yet day’
The little ones leaped and shouted and laughed
And all the hills echoed.

Indian Children Speak Summary in English

It is said that Indian children are hard to teach. They should not be expected to talk. One day a short and fat little boy said that the moon had gone all the way with him the previous night. It is said that Indian children are very silent and their works are ‘yes’ or ‘no’. But the ragged pansy confused softly and said that his dress was old but at night the moon was kind when he wore a beautiful moon, colored dress. It is again said that Indian children are dumb. They hardly make any replay. He clearly hears Delores answer. The sunset is so good that he thinks God is throwing a bright shawl around the shoulders in the sky. However, it is also said that Indian children have no affection. They just don’t care for anyone. Then he feels that Ramon’s hand and hears him whisper. A wild animal races in me since his mother sleeps under the ground. Whether it will always run and run. It is also said that Indian children are rude. They don’t seem very bright. Then he remembers Joe Henery’sremark. The tree is hanging down her head because the sun is staring at her. White people always stare. They do not know it is not polite. It is said that Indian children never take people in. One usually always stands outside their thoughts. He has forgotten the idea words that people said but tree sure the day when iron doors swang wide and he supports into the heart of Indian Land.

Analytical Outlines:

• It is said that Indian children are hard to teach.
• They should not be expected to talk.
• One day a short and fat little boy said something.
• He said about the moon
• He said that it had gone all the way with him.
• It happened in the previous right.
• Again people say something about Indian children.
• It is said that Indian children are Very silent.
• Their works are ‘yes’ or ‘no’
• But the ragged pansy confided softly.
• He said that his dress was old.
• But at night the moon is kind.
• Because he wore a beautiful moon-colored dress at night.
• It isagain said something by thepeople.
• They say Indian children are dumb.
• They hardly make any reply.
• He clearly hears Delores answer.
• The sunset is extremely good.
• He thinks God is throwing a bright shawl around the shoulders in the sky.
• However, people say something negatively.
• They say that Indian children have no affection.
• They just don’t care for anyone.
• Then he feels the Ramon’s hand.
• He also hears him whisper.
• A wild animal raises in me.
• His mother sleeps under the ground.
• Whether it will run and run.
• It is also said that Indian children are rude.
• They don’t seem very bright.
• Then he remembers something:
• It is Joe Henry’s remark.
• The trees is hanging down her head.
• Because, the sun is staring at her.
• While people always stare.
• They do not know it is not polite.
• It is said that Indian children never take people in.
• One usually always stands outside their thoughts.
• He has forgotten the idle words people say.
• He treasures the day when iron doors swung wide.
• He supports into the heart of Indian Land.

Meaning Of Difficult Words:
stubby – short and thick
ragged – (with clothes) tom (here, refers to someone wearing rags)
dumb – speechless, mute, here stupid, unintelligent
shawl – loose square cloth worn over the shoulders or head by women.
confided – told a secret.

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Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 4 Text C: The Mushroom of Death Textbook Activity Questions and Answers.

Class 12th Alternative English Chapter 12 The Mushroom of Death Question Answers CHSE Odisha

The Mushroom of Death Class 12 Questions and Answers

Activity -11

Vocabulary:
Find out words from the passage, which mean more or less the following:
(i) to think deeply for a long time (3).
(ii) to spread something so that it will influence a lot of people (4)
(in) a written list of all the objects in a particular place (5)
(iv) Seen to be true of valid (explanation, argument, or statement) (6)
(v) Change in the genetic structure (12)
(vi) a similar action or decision in the past (15)

Answer:
(i) to think deeply for a long time-contemplate
(ii) to spread something so that it will influence a lot of people – propagate
(iii) a written list of all the objects in a particular place – inventory
(iv) Seen to be true of valid (explanation, argument, or statement) -plausible
(v) Change in the genetic structure -genetic mutations.
(vi) a similar action or decision in the past – historical precedent

Activity-12

Link Words:
Choose the correct alternatives and rewrite the sentences after removing the brackets.
(a) I understand your point of view. (However / Although), I don’t agree to it.
(b) Ramesh had lived in this village for 20 years (Even though / Nevertheless) the villagers still considered him an outsider.
(c) He has lived next door to us for a year, (yet / however), we hardly even seen him.
(d) I walked up the stairs cautiously. (Even / though) I nearly slipped twice.
(e) He has refused entry to the USA. (Though / Instead) he was forced to return to India instead.
(f) Someone of his poems was published in newspapers and magazines (so that / consequently) he thought of himself as an established poet.

Answer:
(a) I understand your point of view. However, I don’t agree to it.
(b) Ramesh had lived in this village for 20 years. Nevertheless, the villagers still considered him an outsider.
(c) He has lived next door to us for a year, yet, we hardly even seen him.
(d) I walked up the stairs cautiously even I nearly slipped twice.
(e) He has refused entry to the USA. He was forced to return to India instead.
(f) Someone of his poems was published in newspapers and magazines so he thought of himself as an established poet.

Activity – 15

Cohesive Devices: Link Words
Predict whether the following words present an addition, a result, or a contrast.
(a) Those events have taught us a great deal about nuclear war’s potential physical and biological impact. But ………………….(1)
(b) The light ……………………… causes thermal effects that depend upon the thermal energy incident on the skin of man. Also ………………………. (4)
(c) Authorised unclassified estimates indicate that world arsenals contain about 50,000 weapons, although ………………………… (5)
(d) The impact of the Hiroshima bomb was geographically limited thus ………………………… (2)
(e) The consequences can be a kind that was not even contemplated till recently, that is ………………………. (3).
(f) The fireball spreads out to affirm the distinctive mushroom could and ………………………… (5)

Answer:
(a) Those events have taught us a great deal about nuclear war’s potential physical and biological impact. But it must be remembered that the cities of Hiroshima and Nagasaki experienced only a single explosion each of a weapon much smaller in field than many of those stockpiled in world nuclear arsenals today, (addition)
(b) The light ………………… causes thermal effects that depend upon the thermal energy incident on the skin of man. Also …………….. (addition)
(c) Authorised unclassified estimates indicate that world arsenals contain about 50,000 weapons, although …………………… (contrast)
(d) The impact ofthe Hiroshima bomb was geographically limited thus …………………… (result)
(e) The consequences can be a kind that was not even contemplated till recently, that is …………………….. (addition)
(f) The fireball spreads out of firm the distinctive mushroom could and ……………………. (addition)

Activity-14(A)

Say which tense is the following sentences in:
(1) Those events have taught us a great deal ____________.
(2) The bomb dropped in Hiroshima released ____________.
(3) All the energy of the explosion is added ____________.
(4) If the explosion occurs close ____________.
(5) It may be better ____________.
(6) As the fireball rises ____________ it spreads ____________.
(7) He became an engineer.
(8) She played well
(9) She took my photograph.

Answer:
(1) Present tense
(2) Past tense
(3) Present tense
(4) Present tense
(5) Present tense
(6) Present tense
(7) Past tense
(8) Past tense
(9) Past tense

Extra Activity – (Miscellaneous)
(c) Directions: Pick up synonyms of the words from the words which follow:

Question 1.
Composure:
(a) Assumed attitude
(b) Liberty or musical
(c) Restlessness
(d) Work tranquility
Answer:
(d) Work tranquility

Question 2.
Implicate:
(a) to insult
(b) doubt
(c) involve
(d) make clear
Answer:
(c) involve

Question 3.
Concert:
(a) agreement
(b) beauty
(c) power
(d) yielding
Answer:
(c) power

Question 4.
Mitigate:
(a) to heal
(b) soften
(c) pardon
(d) send on a mission
Answer:
(b) soften

Question 5.
Buoyant:
(a) childlike
(b) brisk
(c) sturdy
(d) light-hearted
Answer:
(c) sturdy

Question 6.
Unalloyed:
(a) not connected
(b) calm
(c) absolute and complete
(d) inferior
Answer:
(c) absolute and complete

Question 7.
Blandishment
(a) slanders
(b) flattering speech
(c) thieveries
(d) immaturities
Answer:
(b) flattering speech

Question 8.
Propulsive:
(a) explosive
(b) disgusting
(c) impatient
(d) impelling to action
Answer:
(d) impelling to action

Question 9.
Athorart:
(a) crosswise
(b) following
(c) flattened out
(d) just ahead
Answer:
(a) crosswise

Question 10.
Flagging:
(a) becoming afraid
(b) hesitation
(c) growing weak
(d) limping
Answer:
(c) growing weak

Question 11.
Intransigence:
(a) power
(b) bitter criticism
(c) obstinate unwillingness to agree
(d) great anger
Answer:
(c) obstinate unwillingness to agree

Question 12.
Rectify:
(a) to command
(b) destroy
(c) correct
(d) build
Answer:
(c) correct

Question 13.
Incitement:
(a) timmil
(b) calm
(c) stimulus
(d) noise
Answer:
(c) stimulus

Question 14.
Devoid:
(a) evasive
(b) hopeless
(c) lacking
(d) stupid
Answer:
(c) lacking

Question 15.
Resolved:
(a) dummerised
(b) dispelled
(c) strengthened
(d) tonglad
Answer:
(b) dispelled

Question 16.
Privy:
(a) dishonest
(b) caution
(c) secretly aware
(d) quiet
Answer:
(c) secretly aware

Question 17.
Differentiation:
(a) distinction
(b) caution or grounds of difference
(c) argument
(d) quiet
Answer:
(a) distinction

Question 18.
Condon:
(a) pile of logs
(b) smokeless gun powder
(c) line of people as a guard
(d) heavy clock
Answer:
(c) line of people as a guard

Question 19.
Pilfer:
(a) to gossip
(b) steal
(c) trifle
(d) loiter
Answer:
(b) steal

Question 20.
Lore:
(a) sentiment
(b) body of tradition
(c) suspicion
(d) fabestories
Answer:
(b) body of tradition

Question 21.
Baleful:
(a) harmful
(b) kind
(c) happy
(d) dark
Answer:
(a) harmful

Question 22.
Hallowed:
(a) old
(b) decayed
(c) sacred
(d) mellowed
Answer:
(c) sacred

Question 23.
Liar:
(a) landowner
(b) evil glance
(c) den
(d) trap
Answer:
(c) den

Question 24.
Bridle:
(a) to bow
(b) insult
(c) show anger
(d) trap
Answer:
(c) show anger

Question 25.
Slothful:
(a) flit
(b) stubborn
(c) lazy
(d) ignorant
Answer:
(c) lazy

Question 26.
Shift:
(a) to manage
(b) show
(c) slide
(d) drag one’s feet
Answer:
(a) to manage

Question 27.
Gruesome:
(a) dark
(b) rude
(c) painful
(d) ghostly
Answer:
(d) ghostly

Question 28.
Be token:
(a) to be a sign of
(b) invite
(c) threaten
(d) enrich
Answer:
(a) to be a sign of

Question 29.
Last:
(a) unless
(b) but
(c) for fear
(d) enrich
Answer:
(a) unless

Question 30.
Requite:
(a) to repay
(b) demand
(c) complete
(d) need
Answer:
(a) to repay

Question 31.
Mite:
(a) precious stone
(b) small object
(c) strength
(d) probability
Answer:
(b) small object

Question 32.
Cite:
(a) to memorize
(b) use clearly
(c) point out with a figure
(d) quote
Answer:
(d) quote

Question 33.
Satellite:
(a) sparkling
(b) ruler
(c) gem
(d) servile attention
Answer:
(d) servile attention

Question 34.
Respite:
(a) breath
(b) fatigue
(c) ill will
(d) interval of rest
Answer:
(d) interval of rest

Question 35.
Incite:
(a) to cut of
(b) perceive the inner nature of the thing
(c) arouse or stair up
(d) commence
Answer:
(c) arouse or stair up

TEST – II
Directions

Pick up synonyms of the words from die list of words that follow every word:

Question 1.
Parasite:
(a) disease
(b) a loss of motion
(c) a hanger on
(d) an insect
Answer:
(c) a hanger on

Question 2.
Rite:
(a) solemn activity
(b) justice
(c) straitness
(d) a cleaning
Answer:
(a) solemn activity

Question 3.
Apposite:
(a) appropriate
(b) highly unpleasant
(c) fulish
(d) painful
Answer:
(a) appropriate

Question 4.
Chafe:
(a) torudicile
(b) to fret and fume
(c) to cheat
(d) to etch
Answer:
(b) to fret and fume

Question 5.
Bald:
(a) broad
(b) rash
(c) unadorned
(d) insulting
Answer:
(c) unadorned

Question 6.
Clean:
(a) to get bit by bit
(b) speak
(c) to discover
(d) to polish
Answer:
(a) to get bit by bit

Question 7.
Shard:
(a) part of a plough
(b) swindle
(c) fragment
(d) layer of earth
Answer:
(c) fragment

Question 8.
Barge:
(a) to thrush forward
(b) to brag
(c) to smell
(d) to oppose
Answer:
(a) to thrush forward

Question 9.
Claim:
(a) care
(b) fortress
(c) well
(d) heap of stone
Answer:
(d) heap of stone

Question 10.
Wrought:
(a) made or fashioned
(b) broken
(c) complicated
(d) strengthened
Answer:
(a) made or fashioned

Question 11.
Drab:
(a) dull or colorless
(b) tired
(c) discouraged
(d) shabby
Answer:
(a) dull or colorless

Question 12.
Err:
(a) to weaver
(b) to make a mistake
(c) to delay
(d) to become confused
Answer:
(b) to make a mistake

Question 13.
Lode:
(a) weight
(b) discouragement
(c) power
(d) vein of
Answer:
(d) vein of

Question 14.
Cadge:
(a) to be cautious
(b) to sponge
(c) to make a reservation
(d) to snatch
Answer:
(b) to sponge

Question 15.
Irk:
(a) to scold
(b) to make a werry fall
(c) to urge
(d) to annoy
Answer:
(d) to annoy

Question 16.
Butt:
(a) blunt ness
(b) stupidity
(c) target
(d) support
Answer:
(c) target

Question 17.
Wield:
(a) to throw
(b) to use with full effect
(c) to grap
(d) to cut
Answer:
(b) to use with full effect

Question 18.
Wreck:
(a) to twist
(b) to inflict
(c) to emit an unpleasant odor
(d) cadence
Answer:
(c) to emit an unpleasant odor

Question 19.
Lilt:
(a) laughter
(b) physical beauty
(c) hopefulness
(d) cadence
Answer:
(d) cadence

Question 20.
Wrath:
(a) anger
(b) garland of flower
(c) phantom
(d) halo
Answer:
(c) phantom

Question 21.
Chaff
(a) banter
(b) grist
(c) abrasion
(d) comfort
Answer:
(c) abrasion

Question 22.
Crypt:
(a) puzzle
(b) silence
(c) brevity
(d) vault
Answer:
(d) vault

Question 23.
Tilt:
(a) cultivated land
(b) dispute
(c) balance
(d) point of view
Answer:
(d) point of view

Question 24.
Perturb:
(a) to upset
(b) to cause doubt
(c) to burden
(d) to test
Answer:
(c) to burden

Question 25.
Usurp:
(a) to yield
(b) to cause doubt
(c) to burden
(d) to test
Answer:
(c) to burden

Question 26.
Recriminate:
(a) to resist authority
(b) to accuse in return
(c) to respect an illegal act
(d) to restate
Answer:
(b) to accuse in return

Question 27.
Ensconce:
(a) to surround
(b) promote
(c) honor
(d) to settle comfortably
Answer:
(d) to settle comfortably

Question 28.
Elude:
(a) to evade
(b) to omit or leave out
(c) to make mention of
(d) to deceive
Answer:
(a) to evade

Question 29.
Rifle:
(a) to disturb
(b) to shoot
(c) to seize
(d) to plunder or ransack
Answer:
(d) to plunder or ransack

Question 30.
Mollify:
(a) to irritate
(b) to appease
(c) to amuse
(d) to limit the meaning of
Answer:
(b) to appease

Question 31.
Recoup:
(a) to recover
(b) to trap
(c) to strengthen
(d) to shuffle
Answer:
(a) to recover

Question 32.
Substantiate:
(a) to weaken
(b) to substitute
(c) to verify
(d) to make wealthy
Answer:
(c) to verify

Question 33.
Solicit:
(a) to command
(b) to worry
(c) to sympathise with
(d) to ask for
Answer:
(d) to ask for

Question 34.
Embroil:
(a) to anger
(b) to involve in the discussion
(c) to encompass
(d) to bring to boiling point
Answer:
(b) to involve in the discussion

Question 35.
Envisage:
(a) to face
(b) to seek
(c) to understand
(d) to foresee in imagination
Answer:
(d) to foresee in imagination

TEST – III

Question 1.
Compound:
(a) to emphasize
(b) to confuse
(c) to put together
(d) to compress
Answer:
(c) to put together

Question 2.
Beguile:
(a) to charm
(b) to become shy
(c) to fetter
(d) to smile at
Answer:
(a) to charm

Question 3.
Slaken:
(a) to grow weary
(b) to hampen
(c) to become less active
(d) to quentch
Answer:
(c) to become less active

Question 4.
Submerge:
(a) to walk on
(b) to sink
(c) to appear
(d) to join together
Answer:
(b) to sink

Question 5.
Replenish:
(a) to spread around
(b) to fulfill
(c) to give up
(d) to provide a new supply for
Answer:
(d) to provide a new supply for

Question 6.
Convulse:
(a) to shake violently
(b) to restrict
(c) to befuddle
(d) to impel
Answer:
(a) to shake violently

Question 7.
Placade:
(a) tofettenout
(b) to pacify
(c) to annoy
(d) to make secure
Answer:
(b) to pacify

Question 8.
Ingratiate:
(a) to make ungrateful
(b) to force one’s way in
(c) to place oneself in a favorable position
Answer:
(c) to place oneself in a favorable position

Question 9.
Augury:
(a) dispute
(b) alter
(c) place of refuse
(d) omen
Answer:
(d) omen

Question 10.
Flagrant:
(a) widely scattered
(b) poisonous
(c) scandalous
(d) absurd
Answer:
(c) scandalous

Question 11.
Ferret:
(a) to search
(b) to trap
(c) to hide
(d) to flee
Answer:
(a) to search

Question 12.
Impediment:
(a) opposition
(b) told
(c) obstruction
(d) disparagement
Answer:
(c) obstruction

Question 13.
Nomenclature:
(a) adoption of a pen name
(b) system of names
(c) parliamentary rule
(d) history of names
Answer:
(b) system of names

Question 14.
Cumulative:
(a) serious
(b) swollen
(c) rich
(d) steadily increasing
Answer:
(d) steadily increasing

Question 15.
Pedantic:
(a) hanging
(b) making a needless display of leaming
(c) ignorant
(d) solemn
Answer:
(b) making a needless display of leaming

Question 16.
Disparate:
(a) radically different
(b) discouraged
(c) reckless
(d) stingy
Answer:
(a) radically different

Question 17.
Regime:
(a) order of procedure
(b) system of government
(c) recipe for cooking
(d) peacefulness
Answer:
(b) system of government

Question 18.
Inimical:
(a) favorable
(b) unique
(c) unfriendly
(d) wicked
Answer:
(c) unfriendly

Question 19.
Deplete:
(a) to flatten
(b) to conquer
(c) to finish
(d) to exhaust
Answer:
(d) to exhaust

Question 20.
Despensation:
(a) distribution
(b) dismissal
(c) surrender of power
(d) delaying
Answer:
(a) distribution

Question 21.
Circuitous:
(a) surrounded
(b) dizzy
(c) round-about
(d) deceptive
Answer:
(c) round-about

Question 22.
Scintilla
(a) Knsal
(b) trace
(c) veil
(d) brilliant surface
Answer:
(d) brilliant surface

Question 23.
Conversant:
(a) well-mannered
(b) talkative
(c) argumentative
(d) familiar
Answer:
(d) familiar

Question 24.
Villify:
(a) to lie
(b) to prove
(c) to defame
(d) to defraud
Answer:
(c) to defame

Question 25.
Noxious:
(a) dark
(b) injurious
(c) hateful
(d) evil-smelling
Answer:
(b) injurious

Question 26.
Cursory:
(a) informal
(b) penetrating
(c) angry
(d) rapid and superficial
Answer:
(d) rapid and superficial

Question 27.
Actuate:
(a) to explain
(b) to put in action
(c) to furnish proof
(d) to prepare a financial statement
Answer:
(b) to put in action

Question 28.
Flaceid:
(a) weak
(b) pale
(c) dull
(d) scared
Answer:
(b) pale

Question 29.
Dire:
(a) severe
(b) wicked
(c) dreadful
(d) hopeless
Answer:
(c) dreadful

Question 30.
Sequestered:
(a) quiet
(b) shady
(c) safe
(d) secluded
Answer:
(d) secluded

Question 31.
Inconceivable:
(a) unimportant
(b) unthinkable
(c) improbably
(d) inconsequential
Answer:
(b) unthinkable

Question 32.
Inopportune:
(a) untimely
(b) not instant
(c) unreasonable
(d) leisurely
Answer:
(a) untimely

Question 33.
Tactless:
(a) considerable
(b) sharp
(c) pertains to the origin of touch
(d) strong
Answer:
(c) pertains to the origin of touch

Question 34.
Inconclusive:
(a) not apparent
(b) not decisive
(c) positive
(d) unanswerable
Answer:
(b) not decisive

Question 35.
Disputation:
(a) controversy
(b) formal enquiry
(c) dissertation
(d) distribution
Answer:
(a) controversy

TEST-IV

Question 1.
Benign:
(a) radiant
(b) religion
(c) kindly
(d) hopeful
Answer:
(c) kindly

Question 2.
Dictum:
(a) enunciation
(b) law
(c) autocratic ruler
Answer:
(b) law

Question 3.
Appurtenance:
(a) accessory
(b) apt retort
(c) personal characteristic
(d) insult
Answer:
(a) accessory

Question 4.
Asperity:
(a) Ambition
(b) eagerness
(c) promptness
(d) harshness
Answer:
(d) harshness

Question 5.
Cogent:
(a) brief
(b) wise
(c) convincing
(d) mathematical term
Answer:
(c) convincing

Question 6.
Feline:
(a) delicate
(b) catlike
(c) very feminine
(d) slack
Answer:
(b) catlike

Question 7.
Sibilant:
(a) talkative
(b) secret
(c) soft
(d) hissing
Answer:
(d) hissing

Question 8.
Jocose:
(a) merry
(b) fat
(c) clumsy
(d) foolish
Answer:
(a) merry

Question 9.
Mendacious:
(a) bitter
(b) beggarly
(c) boastful
(d) untrustful
Answer:
(d) untrustful

Question 10.
Capitulate:
(a) to emphasize
(b) to rush
(c) to surrender
(d) to overturn
Answer:
(c) to surrender

Question 11.
Recapitulate:
(a) to recover property
(b) to sum up
(c) to repeat oneself tiresomely
(d) to surrender again
Answer:
(b) to sum up

Question 12.
Celerity:
(a) grace
(b) fame
(c) slipperiness
(d) speed
Answer:
(b) fame

Question 13.
Head:
(a) to pay attention
(b) to team
(c) to hesitate
(d) to be positive
Answer:
(a) to pay attention

Question 14.
Rack:
(a) to fleece
(b) to pile up
(c) to torture
(d) to shatter
Answer:
(c) to torture

Question 15.
Squib:
(a) young pigeon
(b) pm point
(c) feather
(d) brief with paragraph
Answer:
(d) brief with paragraph

Question 16.
Bak:
(a) to luxuriate
(b) to be modest
(c) to lie down
(d) to moisten
Answer:
(c) to lie down

Question 17.
Coy:
(a) dainty
(b) glamorous
(c) petish
(d) demure
Answer:
(d) demure

Question 18.
Blurt:
(a) effusive description
(b) impulsive utterance
(c) splash of color
(d) stain
Answer:
(b) impulsive utterance

Question 19.
Want:
(a) need
(b) wish
(c) habit
(d) refusal
Answer:
(c) habit

Question 20.
Refex share:
(a) to splice
(b) to split apart
(c) to unload
(d) to brace
Answer:
(b) to split apart

Question 21.
Pore:
(a) to perspire
(b) to read carefully
(c) to look serious
(d) to rain hard
Answer:
(b) to read carefully

Question 22.
Tome:
(a) large book
(b) mausoleum
(c) echo
(d) aulted roof
Answer:
(a) large book

Question 23.
Marie:
(a) grit
(b) stone
(c) gloom
(d) smudge
Answer:
(c) gloom

Question 24.
Drain:
(a) to shift
(c) to emaciate
(b) to stretch
(d) to exhaust
Answer:
(d) to exhaust

Question 25.
Feint:
(a) to challenge
(b) to make a sham
(c) to withdraw
(d) to grow weak
Answer:
(b) to make a sham

Question 26.
Brawl:
(a) to shout
(b) to cry
(c) to quarrel noisily
(d) to revolt
Answer:
(c) to quarrel noisily

Question 27.
Crime:
(a) frost
(b) dirt
(c) lubricant
(d) grain to be grown
Answer:
(b) dirt

Question 28.
Gad
(a) to stare
(b) to tease
(c) to rush about
(d) to criticize
Answer:
(c) to rush about

Question 29.
Shade:
(a) to secret
(b) amount
(c) privacy
(d) slight difference
Answer:
(a) to secret

Question 30.
Shidge:
(a) soft mud
(b) menial worker
(c) slattern
(d) bookish
Answer:
(a) soft mud

Question 31.
Scrimp:
(a) to shrivel
(b) to be frugal
(c) to be selfish
(d) to be fussy
Answer:
(b) to be frugal

Question 32.
Drub:
(a) to bounce
(b) to leaf
(c) to beat
(d) to be stupid
Answer:
(c) to beat

Question 33.
Dross
(a) lustre
(b) dull surface
(c) mental depression
(d) impurity
Answer:
(d) impurity

Question 34.
Straff
(a) to discipline
(b) to bombard
(c) to rub
(d) to slice
Answer:
(d) to slice

Question 35.
Wend:
(a) to direct one’s course
(b) to wander
(c) to weave
(d) to sloop
Answer:
(a) to direct one’s course

TEST-V

Question 1.
Blunt:
(a) abrupt manner
(b) direct insult
(c) mainshock
(d) retarded shock
Answer:
(c) mainshock

Question 2.
Prime:
(a) to supply with facts
(b) to begin
(c) to assist
(d) to strut
Answer:
(a) to supply with facts

Question 3.
Bode:
(a) to dwell
(b) to foreshadow
(c) to endure
(d) to wait
Answer:
(b) to foreshadow

Question 4.
Wrest:
(a) to grapple with an opponent
(b) to twist into a distorted shape
(c) to compiler
(d) to stretch forcibly
Answer:
(d) to stretch forcibly

Question 5.
Frond:
(a) decorative border
(b) palm leaf
(c) thick branch
(d) prong
Answer:
(b) palm leaf

Question 6.
Mite:
(a) to come up to or touch
(b) to make suitably
(c) to allot
(d) to challenge
Answer:
(b) to make suitably

Question 7.
Flay:
(a) to whip
(b) to spread out
(c) to splice together
(d) to strip off the skin
Answer:
(b) to spread out

Question 8.
Tend:
(a) to sympathize
(b) to incline
(c) to delay
(d) to offer
Answer:
(b) to incline

Question 9.
Pert:
(a) hide
(b) wealth
(c) track of a wild animal
(d) equipment
Answer:
(b) wealth

Question 10.
Tant:
(a) stingy
(b) hard
(c) secretive
(d) tightly drawn
Answer:
(d) tightly drawn

Question 11.
Track:
(a) climb
(b) to travel by wagon
(c) to deceive
(d) to carry
Answer:
(a) climb

Question 12.
Design:
(a) to condescend
(b) to pretend
(c) to disparage
(d) to refuse
Answer:
(a) to condescend

Question 13.
Spume:
(a) spray
(b) anger
(c) foam
(d) noise
Answer:
(c) foam

Question 14.
Effectuate:
(a) to accomplish
(b) begin
(c) practice
(d) end
Answer:
(a) to accomplish

Question 15.
Perceptive:
(a) wise
(b) alert
(c) discerning
(d) precise
Answer:
(c) discerning

Question 16.
Syndrome:
(a) council
(b) combination of symptoms
(c) fetish
(d) monopoly
Answer:
(b) combination of symptoms

Question 17.
Fastidious:
(a) literal
(b) clear
(c) discrete
(d) fussy
Answer:
(d) fussy

Question 18.
Apotheosis:
(a) revelation
(b) pithy saying
(c) perfect example
(d) rhetorical address
Answer:
(c) perfect example

Question 19.
Pristine:
(a) beautiful
(b) prudish
(c) shining
(d) original
Answer:
(d) original

Question 20.
Forbearance:
(a) patience
(b) foresight
(c) stubbornness
(d) inherited traits
Answer:
(a) patience

Question 21.
Coercive:
(a) stick
(b) compelling
(c) persuasive
(d) complaining
Answer:
(b) compelling

Question 22.
Hybrid:
(a) pure
(b) carefully selected
(c) mixed
(d) hardy
Answer:
(c) mixed

Question 23.
Sully:
(a) to ridicule
(b) leap forth
(c) deceive
(d) tarnish
Answer:
(d) tarnish

Question 24.
Blatant:
(a) conceited
(b) unpleasantly noise
(c) brutal
(d) openly hostile
Answer:
(c) brutal

Question 25.
Peregrination:
(a) land measurement
(b) uncertainty
(c) travel
(d) scheme
Answer:
(b) uncertainty

Question 26.
Oblogay:
(a) abusive language
(b) state of being of forgotten
(c) discussion
(d) burial rite
Answer:
(a) abusive language

Question 27.
Mettle:
(a) mood
(b) courage
(c) sternness
(d) belligerence
Answer:
(b) courage

Question 28.
Infraction:
(a) small portion
(b) collision
(c) oversight
(d) violation of law
Answer:
(d) violation of law

Section – C
In Text – A you are exposed to a futuristic view of the scientific and technological world that is likely to emerge by 2050. But will the world survive so long? If your answer is in negative what possible threats do you apprehend?
(i) __________________
(ii) __________________
(iii) __________________
Discuss the possible threats in consultation with others. Now read Amalendu Bandopadhyay’s, ‘The Mushroom of Death’ and find out what dangers, the writer thinks to lie ahead for humanity.

The Mushroom of Death Summary in English

Summary:
The bombing of Hiroshima and Nagasaki has taught us the potential physical and biological impact of a nuclear war. The bomb dropped on Hiroshima released energy equalling 20 kilotons of chemical explosives. Now, the question is what will happen if many modem nuclear weapons are exploded? It is clear that the consequences can be of kind that smoke from massive nuclear-ignited urban fires can cause a worldwide disruption in the planet’s weather and climate. The effects of an air burst will form an extremely strong shock wave that propagates outward rendering the air luminous and creating a fireball in the immediate vicinity of the burst.

If the explosion takes place close to the surface, there will be a shock wave coupled to the ground and a crater can be dug in the ground. Gamma rays and neutrons also release from an air burst. While detonating a nuclear weapon, it releases heat of about tens of millions of degrees Celsius into the nearby air. Even buildings of heavy construction will collapse. Scientists are of the view that the smoke produced by the burning of cities in after match of a nuclear war may significantly affect the earth’s climate for long periods of time. There will be a substantial decrease in precipitation.

Analytical Outlines

• There was a bombing on Hiroshima and Nagasaki.
• It has brought us the potential physical impact.
• It has also brought us a biological impact.
• It was the impact of nuclear war.
• The bomb was dropped on Hiroshima.
• It released high energy.
• It was equal to 20 kilotons of chemical explosives.
• Then the writer asked a very powerful question.
• What will happen if many modem nuclear weapons are exploded?
• It is clear that the consequences will be far more severe than in 1945.
• The consequences can be of such a kind.
• The smoke from massive nuclear-ignited urban fires.
• It can cause a worldwide disruption.
• It can cause it in the planet’s weather and climate.
• It is the effect of an air burst.
• It will form an extremely strong shock wave.
• It propagates outward rendering of the air luminous.

• It creates a fireball.
• The fireball is created in the immediate vicinity of the burst.
• When the explosion takes place close to the surface.
• There will be a shock wave coupled to the ground.
• The aerator can be dug in the ground.
• Gamma rays are released from an air burst.
• Neutrons are also released from this air burst.
• A nuclear weapon is detonated.
• It releases heavy heat.
• It is about tens of millions of degrees Celcius.
• It releases it into the nearby am
• Even buildings of heavy construction will collapse.
• Scientists provide opinions about it.
• They talk about the smoke produced by burning cities.
• This burning is in the aftermath of a nuclear war.
• It may significantly affect the earth’s climate.
• It will affect it for a long period of time.
• There will be a substantial decrease in the precipitation

Meanings Of Difficult Words

arsenals – stores of weapons.
deployment – organizing troops and equipment for immediate action.
incident on – something that occurs in connection with something else.
gamma rays – high-frequency rays emitted from a radioactive atom.
inventory – lit of articles.
unclassified – no longer secret.
megaton – one million tons.
buoyant – capable of keeping an object afloat.
precipitation – condensation in the atmosphere as rain, snow or hait

aftermath – a situation resulting from an important event.
consequences – results, aftermaths
obtain – get
devastating – terrible, horrible, dangerous
vicinity – in the nearby area,
initial – beginning, at the outset.
lethal – deadly, life-killing, dangerous.
plausible – evident, having proofs.
thermal – relating to heat or temperature
precipitation – rainfall.

Read More:

• Text A: The Year 2050-Reflections of a Futurist Question Answer
• Text B: Powershift Question Answer
• Text C: The Mushroom of Death Question Answer

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 4 Text B: Powershift Textbook Activity Questions and Answers.

Class 12th Alternative English Chapter 11 Powershift Question Answers CHSE Odisha

Powershift Class 12 Questions and Answers

Activity-5:

Vocabulary:
Find the words from the passage which more or less mean the following:
(i) Shocking or amazing(1)
(ii) Careful watching a suspect (2)
(ii) boring and uninteresting (9)
(iv) to include something in a larger group(10)
(v) a group of three similar things(18)

Answer:
(i) Shocking or amazing – astounding
(ii) Careful watching a suspect – surveillance technologies
(iii) boring and uninteresting-tedious repetition
(iv) to include something in larger group-subsume information.
(v) a group of three similar things – the Patriarch of a family President of a company. Prime Minister of a nation.

Activity-6

Writer’s Bias:
‘Our best computers are still stone-axe primitive’. The italicized metaphor emphasizes the obsoleteness and inadequacy of our best computers. Can you find any other such expression in the passage which have a metaphysical meaning? Do you such expressions convey the writer’s bias?

Expression	Meaning	Any bias of the writer

Answer:

Expression	Meaning	Any bias of the writer
Stone-axe primitive	obsoleteness and inadequacy of our best computer	undeveloped condition of our computers.
‘dumb’ weapons	uncaring for	violent, abortive
power-seekers	people seeking power	power hungry, selfish

Activity – 7:

In many old-fashioned Grammar books, you may find some objection to the use of one-line paragraphs. Are there any one-line paragraphs in this passage? Why does the writer use them? Is he justified in using such small paragraphs?

Answer:
Yes, there are some one-line paragraphs in this passage. They are:

1. Besides its great flexibility, knowledge has other important characteristics that make it fundamentally different from lesser sources of power in tomorrow’s world. (Paragraph – 12)
2. Today in the first changing affluent nations, despite all inequalities of income and wealth, the coming struggle for power will increasingly turn into a struggle over the distribution of and access to knowledge. (Paragraph -12).
   The writer makes use of such one line paragraph in order to put forth his own views in the midst of a general discussion. He is quite justified to have used such a paragraph system.

Activity – 8:

Note-Making:
In the first section of the passage, we came to know that-
1. Force and wealth of knowledge.
2. Knowledge —> power.
3. Force and wealth g knowledge.
That is to say, force and wealth depend on knowledge, is the source of the highest quality power and knowledge is the most important ingredient of force and wealth. Now, make point notes on Sections 2 and 3 using such symbols wherever possible.

Answer:
Weapons and surveillance technologies & knowledge
Knowledge —> computers.
Non-facts and disputed facts a power conflicts in society. False and lies / ‘true’ facts and scientific ‘laws’ —> knowledge.

Activity-9:

Answer the questions as briefly as you can:-
Question (i).
How does Toffler establish that knowledge is the most important ingredient of force and wealth?
Answer:
The military which rests on force uses computerized knowledge. The advanced economy can not for thirty seconds without (knowledge) computers.

Question (ii).
How does he distinguish among ‘data’ information and ‘knowledge’?
Answer:
‘Data’ means more or less connected ‘facts’, ‘information’ refers to data that have been fitted into categories and classification schemes or patterns; ‘knowledge’ means information that has been further refined into more general statements.

Question (iii).
Why is the second section named ‘facts’, ‘lies’, and ‘truth’?
Answer:
Facts, lies, and truth are the things around which the second section has been centered. The whole section is a reflection of most things.

Question (iv).
How is it that knowledge is the most democratic source of power?
Answer:
Knowledge is the key to all kinds of things. It encompasses the means of communication that shapes the message that flows through them. Unlike bullets or budgers, knowledge itself gets used up. The revolutionary characteristics of knowledge that it can be grasped by the weak and poor as well make it the most democratic source of power.

Question (v).
What does Toffler mean by the concept of power-tria? (Paragraph -18)
Answer:
The power tria-patriarch of a family, president of a company, and Prime Minister of a nation wants to control the quantity, quality, and distribution of knowledge within his or her domain.

Question (iv).
Look at the introduction and the conclusion and say briefly, how they relate to the body paragraph of the exports.
Answer:
The introduction starts with knowledge of the part which had never imagined such an explosion of knowledge in the modem world. The conclusion tells that the modem knowledge has reached such a zenith that its provision is not too far. The control of knowledge has become the most important aspect. The introduction and conclusion of the passage are just like prefixes and suffixes ofthe body. The body passage is just an advancement between the introduction and the conclusion.

Extra Activity – 9(A)

Find words in Text – B and use them first as nouns and then as verbs in sentences of your own.

mind	shape
speed	dispute
rule	play
Advance	form
run	help
make	use
turn	process
concern	race
force	control

Answer:

mud	(N) You can’t change his mind. (V) Mind your own business.
speed	(N) We should not drive cars in a high speeds. (V) You should speed up your writing.
rule	(N) The students must obey the rules of the college. (V) Akbar ruled for a long time
advance	(N) The employee asked for an advance. (V) The army advanced forward.
run	(N) The player could not make a good run. (V) I can run five kilometers.
make	(N) The watch is a foreign make. (V) I can’t make a fire now.
turn	(N) His turn came last. (V) All his efforts turned into failure.
concern	(N) Power has become the principal concern of politics. (V) This book concerns human liberty.
force	(N) He exerted force to extract money. (V) Don’t force me to do this.
shape	(N) The shape of the globe is round. (V) Education shapes human personality.
dispute	(N) The two brothers are in dispute for land division. (V) We should not dispute for a piece of land.
play	(N) The play was very attractive. (V) He plays cricket every day.
form	(N) I want an admission form. (V) We have decided to form a club.
help	(N) I want your help to do this work. (V) I can help you in this matter.
use	(N) You should know the process use of a computer. (V) I always use a ballpoint pen.
process	(N) This is not the process of preparing coffee. (V) He processed everything for the meeting.
race	(N) He participated in the 100 mtrs. race. (V) He raced along the street to catch a thief.
control	(N) Our government provides much stress on fund control. (V) We should always control our anger.

Section – B

There is a popular saying ‘knowledge is power’. How can it be true? Give examples to explain this equation. Now, read Alvin Toffler’s ‘Power – shift’. You may find some of your points mentioned. You must focus your attention on the way Toffler presents those points while reading the text.

Power-Shift Summary in English

Summary:

From satellites to submarines, modem weapons are constructed of information-rich electronic components. Today’s fighter plane is a flying computer. Even ‘dumb; computers are manufactured with the help of supersmart computers with electronic chips. The military, to choose a single example uses computerized knowledge – ‘expert systems’ – in missile defense. The Pentagon’s Defence Advanced Research Projects Agency (D ARPA) has set as a long-range goal the design of a system that can make ‘one million logical inferences per second’.

Logic inference, and epistemology – is short, brain work by humans and machine is today’s precondition for military power. It has so become that the advanced economy could not run for thirty seconds without computers. Therefore, knowledge is not only the source of the highest quality power but also the most important ingredient of force and wealth. Put differently knowledge has gone from being an adjunct of money power and muscle power to bringing their very essence.

It is, in feet, the ultimate amplified which is the key to the power shift that lies ahead. There are as many definitions of knowledge as there are people who regard themselves as knowledgeable. Matters grow worse when words like signs, symbols, and imagery are given highly technical meanings. To make things simple and escape from these definitional quicksands, even at the expense of vigor, the term knowledge will be expanded in the pages ahead.

However, besides its great flexibility, knowledge has other important characteristics that make it fundamentally different from the lesser sources of power in tomorrow’s world. Thus force, for all practical concerns, is finite. There is a limit to how much force can be employed before we destroy what we wish to capture to defend knowledge, in principle different and infinitely expandable. Knowledge is also inherently different from both muscle and money because one gun can not be used simultaneously by two people. But by contrast, both ofthe men can use the same knowledge either for or against each other, and in that very process, we may even produce still more knowledge.

Unlike bullets or budgers, knowledge itself does not get used up. This alone tells us that the rules ofthe knowledge power game are sharply different from the precepts relied on by those who use force or money to accomplish their will. In fact, today, in the first changing affluent nations despite of all inequalities of income and wealth the coming struggle for power will increasingly turn into a struggle over the distribution of and access to knowledge. The control of knowledge has become the most important necessity which can save humanity.

Analytical Outlines

• It may be a satellite.
• Even it may be a submarine.
• The modem weapons are constructed of information-rich electronic components.
• Today’s fighter plane is a flying computer.
• Even we may consider the ‘dumb’ computers.
• They are manufactured with the help of super-smart computers.
• They are manufactured with electronic chips.
• Let us consider one burning example.
• It is that the military uses computerized knowledge.
• They use ‘expert systems’.
• They use it in missile defense.

• The Pentagon’s Defence Advanced Research Projects Agents have designed such a system.
• This system can make one million logical inferences per second.
• This logic inference is epistemology.
• Its use has increased military power.
• The advanced economy can’t run without computer knowledge.
• It becomes the source of the highest quality power.
• It also becomes the most important ingredient of force and wealth.
• It becomes an adjunct of money power and muscle power.
• In fact, it is the ultimate amplifier.
• It is really the ray to the power shift.
• There can be as many definitions of knowledge as there are people.
• Actually, matters grow worse.
• Words like signs, symbols, and imagery are given highly technical meanings.
• However, we can provide a simple definition of knowledge.
• Hence, the term knowledge will be given an expanded meaning.
• Actually, knowledge is greatly flexible in its meaning.
• It has other important characterization too.
• It makes it fundamentally different from other lesser sources of power.
• Thus, force is finite for all practical concerns.
• The employment force depends upon the wish of capture or defense.
• Knowledge is also inherently different from both muscle and money.
• Because one gun can not be simultaneously used by two people.
• But, by contrast, both of men can use the same knowledge either for or against each other.
• Through this process, We may even produce still more knowledge.
• Unlike bullets or budgers, knowledge itself does not get used up.
• The rules of knowledge are different from those who use force or money to accomplish their will.
• In feet, today, the nations are fatty changing.
• Even if they are having inequalities of income or wealth.
• They are in greater struggle in the distribution of knowledge.
• The control of knowledge has become the most important necessity.
• Because it can save humanity.

Meanings Of Difficult Words:

congeal – thickening of a liquid.
Pentagon – Headquarters of the US Department of Defence.
cliche(s) – A frequently used idea that has lost effectiveness.
chasm – a very deep creek (in rock, earth, or ice)
erus – the most important part of a problem.
sweeping – moving rapidly, quick movements
genius – a talented mind, a person having fabulous intelligence
astounding – amazing, surprising, wonderful
technologies – technologies that make their master’s mere servants.
supersmart – doing things very smartly even more smartly than expected.
epistemology – theory of knowledge
diverse – different, not the same of similar
maldistribution- uneven, distribution of wealth.
affluent – rich, abundant, plenty, having a lot.
abuse – misuse, wrong use of something
threat – danger, jeopardy

Read More:

• Text A: The Year 2050-Reflections of a Futurist Question Answer
• Text B: Powershift Question Answer
• Text C: The Mushroom of Death Question Answer

Odisha State Board CHSE Odisha Class 12 Alternative English Solutions Grammar Clauses Exercise Questions and Answers.

CHSE Odisha Class 12 Alternative English Grammar Clauses

Observe the following sentences:
1. She knows where you live. (Knows what?)
2. She knows the place you live (Which place?)
3. She will reach where you live (Shall reach where?)

The clauses, where you live in sentences No. 1 is object of the verb ‘knows’. Therefore, it is noun clause. The clause where you live in sentence No 2. qualifies the noun ‘place’. Therefore, it is an adjective clause. The clause where you live in sentence No. 3 modifies the verb ‘will reach’. Therefore, it is an adverb clause. It is important to note that the same clause (where you live) may be a noun clause, adjective clause and an Adverb clause in different sentences according to its function. Therefore, we cannot state the kind of a clause without finding its function.

Definition:
Those parts of a Sentence which have subjects and predicates are called clauses. There are as many clauses in a sentences as there are Finite Verbs.

Kinds Of Clauses:
Clauses can be classified as the following three types. Such as:
(a) Co-ordinate Clauses.
(b) Subordinate Clauses.
(c) Principal Clauses

A. Co-ordinate Clauses:
1. Observe the following sentences:
2. Rim set or you will lose the race.
3. He ran fast but (he) lost the race.
The above sentences are joined by co-ordinate conjunctions e.g., ‘and’, ‘or’ and ‘but’. They are the examples of co-ordinate clauses. Some more co-ordinate conjunctions are, not only… but also, either… or, neither… nor, or else, otherwise, as well as, for, therefore, both…. and etc. They also join co-ordinate clauses.

Kinds of connections between two co-ordinate clauses.
(a) Copulative
Examples:
1. Gandhi was not only a good leader, he was also a reformer.
2. She cannot sing, nor can she dance.
3. She as well as her parents is stupid.
4. I took my lunch packet and boarder the bus.
In the above sentences, the italicised words (co-ordinate conjunctions) simply couple together two sentences.

(b) Alternative:
Examples:
1. Either you or your sister is naughty.
2. Neither a borrower nor a lender be.
3. Obey your teachers or you will repent.
4. Walk fast, you will not catch the bus.
In the above sentences, the italicised words (Co-ordinate disjunctions) simply offer a choice between the clauses disjointed in meaning.

(c) Adversative:
Examples:
1. She is intelligent but slow-working.
2. She ran last, you she missed the train.
3. Iam week, however, I shall carry your box.
4. Everybody, cursed her, neverthless, she did not come round.
In the above sentences, the italicised words (co-ordinate conjunction) show contrast and are opposite in meaning.

(d) Illative:
Examples:
1. She was not regular in her classes, therefore she was expelled from the college.
2. Her father is poor, so he can not. .
3. He missed the bus, for he did not run fast.
In the above sentences, the italicized words (co-ordinate conjunctions) jo in two clauses wherein the second clause draws inferences from the first clause.

Also obwerve the following sentences:
1. He cursed his parents which (and this) was wrong.
2. She went to Agra, where (and there) she saw the Taj Mahal.
3. Then he called on the Principal, who (and he) promised him to help.
In the above sentences, the co-ordinate clauses begin conjunction (which / where / who) are used in a conjunctive sense. Therefore, they introduce co-ordinate clauses and form a compound sentences.

Analysis Of Compound Sentences
Definition

Analysis is the process of breaking up a sentence into its compound parts.
Process of Analysis of a compound sentence.
(i) Pick out all the finite verbs to ascertain the number of clauses.
(ii) Break up the sentence into clauses.
(iii) Write the clauses in full (by supplying the missing verb or subject)

Model Solutions

Example -1
He is strong but he is dull.
Analysis:
(i) He is strong Principal Clause.
(ii) He is dull (Co-ordinate Clause) Coordinate to (ii) Connective ‘but’

Example – 2
He was stupid, therefore, he was punished.
Analysis:
(i) He was stupid (Principal Clause)
(ii) He punished (Co-ordinate Clause) Coordiante to (i) Connective ‘therefore’

Example -3
I have bought a bicycle, which has proved a white elephant to me.
Analysis:
(i) I have bought a bicycle Principal Clauses
(ii) It has proved a white elephant to me (Co-ordiante Clause) Coordiante to (i) Connective ‘which’

Example – 4
You can fool some of the people all of the times and all of the people some of the times, but you cannot fool all the people all the time.
Analysis:
(i) You can fool some of the people all of the times Principle Clause.
(ii) You cannot fool all the people all the time (Co-ordinate Clause) Coordinate to (ii) and (iii) Connective ‘but’.

Exercises For Practice
I. Add suitable Co-ordinate Clauses in the sentences below:
1. She was proud, therefore ____________
2. She has no hope of recovery, nevertheless ____________
3. Ring up the Fire Brigade at once, otherwise ____________
4. She is both rich and ____________
5. He was hungry still (get) ____________
6. The patient’s condition was thinking, nevertheless ____________
7. It is very hot today, so ____________
8. I offered her money, but ____________

2. Combine the following pairs of sentences by using coordinate conjunctions.
1. She met Ram. He gave her this message.
2. I saw the scenery of the garden. It raised my spirits.
3. She cursed my relatives. It made my blood boil.
4. She went to Allahabad. She got a job there.
5. They studied till late at night. Then they went to bed.
6. She generous. Her sister is parsimonious.
7. She had no recommendation. She managed to get the job.
8. He deserved the prize. He worked.

3. Analyse the following sentences into clauses.
1. She ran very fast, so she started gasping.
2. He is ill and cannot study, yet the attends his classes.
3. She reached the platform, when (and then) the train was about to steam off.
4. I started for the city where (and there) I intended to rent a room.
5. Neither a lender, not a borrower can be good.
6. He is dishonest, so he is insolvent.
7. It rained but the programme was not cancelled.
8. He is weak, however, he will get through.
9. Talents differ, and all is well and wisely put.
10. I am ill, therefore, I cannot escort you.

B. Subordinate Clauses (Complex Sentences)
(I) The Noun Clause:
A noun claused may be:

(a) Subj ect to TVansitive Verb
Observe the following sentences:
1. That God exists everywhere is true.
2. Why the old lady cursed him is known to me,
3. When my father will return is uncertain.
4. How she has got this job is an open secret.
In the above sentences the italicised words are the Noun Clauses.

Definition
A Noun Clause always performs the functions of a noun and answers the questions “what”? The above sentences can be broken, (disjoined) into clauses as follows:

1. It is true. (Principal Clause).
God exists everywhere. (Subordinate / Noun Clause)
Conjunction That

2. It is known to me (Principal Clause)
the old lady cursed him. (Subordinate / Noun Clause)
Conjunction Why

3. It is uncertain (Principal Clause)
My father will return (Subordinate / Noun Clause)
Conjunction When

4. It is an open secret (Principal Clause)
She has got this job. (Subordinate / Noun Clause)
Conjunction How

To find the Noun Clause, we should ask questions like
1. What is true?
2. What is known to me?
3. What is uncertain?
4. What is an open crat?
The answer to the above questions will locate the subordinate ‘is’, ‘is’, known, ‘is’ and ‘is’ respectively.

(b) Object to a Transitive Verb
Observe the following sentences:
1. The bagger asked me ifl could help him.
2. Everybody known why you are late.
3. The teacher said that hard work is the key to success.
4. She asked me ifl would lend her a hundred rupees.
The italicised words in the above sentences are Noun Clauses Connectives (if / why / that) and they are object to the verbs asked, knows, said and asked respectively.

(c) Complement to an Incomplete verb
Observe the: following sentences:
1. It seems that she is very selfish.
2. My opinion is that we should quit this place.
3. He found that his cash was missing.
4. Everybody felt that the old man would not recover.
The clauses of the above sentences are Noun Clauses. Because they answer the question ‘what’? They are joined by the connectives (that). They serve as complement to the verbs (seems, is found and felt) preceding them.

(d) Object to a preposition
Observe the following sentences:
1. There is no truth, what she says.
2. Iam surprised at what step she has taken.
3. Don’t crave for what you cannot achieve.
4. You must stick to what you have promised.
The italicised words in the above sentences are Noun Clauses. They serve as objects to the prepositions (in / at / for / to) preceding them.

(e) Object to a participle
Observe the following sentences:
1. Hoping that I will see her, I visited her house.
2. Hearing that he was ill, I rang up to house.
3. Fearing that the wolf would like the sheep, the shepherd boy began to cry.
4. Seeing that the bear had gone away, the boy climbed down the tree.
The italicised words in the above sentences are Norm Clauses. They serve as objects to the participles (Hoping / Hearing / Fearing / Seeing) preceding them.

(f) Object to an infinite
Observe the following sentences:
1. I want to know what help you expect from me.
2. The girl was made to tell where she had stayed for the night.
3. He was shocked to learn that his father had met with a serious accident.
4. I want to ascertain whether you would accompany me.
The italicised words in the above sentences are Noun Clauses. They serve as objects to the infinite (to know / to tell / to learn / to ascertain) preceding them.

(g) In Apposition to a Noun or a Pronoun
Observe the following sentences:
1. The saying that pride health a fall is true.
2. Then came the news that Mahatma Gandhi was shot dead.
3. It is quite certain that she is not at home.
4. The idea that man is a humble tool in the hands of destiny seems to be true. The italicized words in the above sentences are Noun Clauses. They stand in Apposition to a Noun (saying /news /idea) or Pronoun (it) preceding them.

The following connective words begin the Noun clauses:
(a) The Conjunction ‘that as-
1. He thought that he was right.
2. Iam sure that she would write a letter to me.
Sometimes the conjunction ‘that ’ is omitted but its meaning is implied as:
1. She brought she was mistaken.
2. I am sure that you would stand first.

(b) The Interrogative or Relative Words as:
1. That is what he means.
2. Tell me why you disobeyed your teachers.
3. I know where you go every night.
4. How she manages for household is very astonishing.

(c) The Interrogative or Relative Pronouns as:
1. I can’t say whose house it is.
2. Can you guess who is wandering in the street ?

(d) The Conjunction: ‘If /weather’as:
1. I asked him if (whether) he had packed his luggage.
2. She asked me if (whether) I would teach her.

Exercise For Practice: (Solved)

Analysis of some model sentences:
Study the following model sentences analysed below:

1. Why he abused me is not clear.
2. The begger asked me if I could help him.
3. Hearing that he was ill, I went to see him.
4. You must stick to what you have said.
5. I want to know what she has done.
6. It seems that she is very lucky.
7. It is true that she has been kindnapped.
8. It is a well-known saying that uneasy lies the head that wears the crown.

Answer:
Analysis of the given sentences:
1. Why he abused me is not clear.
(a) (It) is not clear.
Kind – Main clause
Function-It is required to frame a sentence.

(b) Why he abused me.
Kind – Noun Clause
Function – It is necessary to frame a sentence.

2. The beggar asked me if I could help him.
(a) The beggar asked me
Kind – Main Clause
Function – It is necessary to frame a sentence.

(b) If I could help him.
Kind – Noun Clause
Function – Object to the verb ‘asked’

3. Hearing that he was III, I went to see him.
(a) Hearing, I want to see him
Kind – Main Clause
Function – It is necessary to frame a sentence.

(b) that he was ill.
Kind – Main clause
Function – Object to the participle – ‘Hearing’

4. You must stick to what you have said.
(a) You must stick to
Kind – Main Clause
Function – It is necessary to frame a sentence.

(b) What you have said
Kind – Noun Clause
Function – Object to the preposition – ‘to’

5. I want to know what she has done.
(a) I want to know.
Kind – Main Clause
Function – It is necessary to frame a sentence.

(b) What she has done.
Kind – Noun Clause
Function – Object to the infinite ‘to know’

6. It seems that she is very lucky.
(a) It seems
Kind – Main Clause
Function – It is necessary to frame a sentence

(b) that she is very lucky
Kind – Noun clause .
Function – Complement to the verb – ‘seems’

7. It is true that she has been kindnapped.
(a) It is true
Kind – Main Clause
Function – It is necessary to frame a sentence

(b) that she has been kidnapped
Kind – Noun Clause
Function – Apposition to the pronoun ‘it’.

8. It is a well-known saying that uneasy lies the head that wears the crown.
(a) It is a well known saying
Kind – Main clause
Function – It is necessary to frame a sentence.

(b) that necessary lies the head that wears the crown.
Kind – Noun Clause
Function – Apposition to the noun – ‘saying’

Exercise For Practice: (Unsolved)

1. Complete the following sentences inserting Noun Clauses:
1. ____________ is possible.
2. ____________ is a wonder.
3. He promised that ____________.
4. He cannot rely on ____________
5. She is ready to pay ____________
6. The fact is ____________
7 ____________ is quite clear.
8. Poor Suniti ate ____________
9. The ide ____________ seems to be true.
10. Hoping ____________ I appeared at the interview.

2. Locate the Noun Clauses in the following sentences:
1. Do you know who came to my house yesterday?
2. What she says is hundred percent true.
3. His belief was that his daughter would return.
4. Tell me where she is putting up.
5. She returned saying that she would take revenge on me.
6. Thinking that Hari is poor. I lent him to rupees.
7. Life is what we make it.
8. Her wish is that she may win a lottery.
9. No one knows who she is.
10. I want to know how fer Puri from Bhubaneswar.

3. Analyse the following sentences into clauses giving the function of each sub ordinate clause:
1. How long will she stay in Lucknow is not known.
2. Ask her if she can accompany you.
3. I don’t know why she committed suicide.
4. I agreed to what he said.
5. There is no sense in what she says.
6. She wants to know why you entered her private room.
7. It pained me to know that Hari’s daughter had eloped with an out-caste.
8. It is believed that truth always triumphs.
9. That you should cheat me hurts me.
10. It seems that Priya is very smart.

(II) Adjective Clause:
Definition:

The adjective clause performs the functions of an adjective to quality a noun or pronoun of the main clause.
Observe the following sentences:
1. This is the old man who stumbled against a stone.
2. The elephant is an animal which has tusks.
3. She is the girl whose husband divorced her.
4. This is the place where my friend lives.

Analysis of Adjective Clauses:
1. This is the old a stone.
Clause (a) This is the old man.
Kind – Main Clause.
Function-It is the main part of a sentence.

Clause (b) Who stumbled against a stone.
Kind – Adjective clause
Function – Qualifying – “Old man”.

2. The elephant tusks.
Clause (a) The elephant is an animal.
Kind – Main Clause
Function – The main part of the sentence

Clause (b) Which is tusks.
Kind – Adjective Clause
Function – Qualifying – “elephant”

3. She is divorced her.
Clause (a) She is the girl.
Kind – Main Clause
Function – Main part of a sentence.

Clause (b) whose husband divorced her.
Kind-Adjective Clause
Function – Qualifying – ‘girl’

4. This is ____________ lives.
Clause (a) This is the place.
Kind – Main Clause
Function – The main part of a sentence.

Clause (b) Where my friend lives.
Kind-Adjective clause
Function – Qualifying – “place”

The relative pronouns (‘who, which and whose’) join the adjective clauses to the Principal Clause in sentences No. 1,2 and 3 above. Relative Adverb (‘where’) also joins the adjective clause to the Principal Clause in sentence No. 4 above. Sometimes an Adjective Clause is introduced by ‘but’ which is equivalent to a Relative Pronoun followed by not’ ….as:

1. There was not a woman who shed tears at the bride’s departure.
Or,
There was not a woman who did not shed tears at the bride’s departure.

2. There are a few of us who love our motherland.
Or,
There are few of us who don’t love our motherland.

3. There is none in the neighborhood who was not prepared to help her.
Or,
There was none in the neighborhood who was not prepared to help her.

Exercise For Practice: (Unsolved)

I. Complete the following sentences inserting adjective clauses:
1. Sindhi is the only girl ____________
2. The greedy farmer killed the goose ____________
3. This is not such a book ____________
4. Do you know the time ____________
5. The dog ____________ has been shot dead.
6. The bullet ____________ has not yet been cast.
7. ____________ A fox gave the hounds a capital run.
8. The man ____________ is like an animal.
9. Such stories ____________ are very romantic.
10. He ____________ himself falls into it.

2. Locate the Adjective Clauses in the following sentences:
1. The man who is holding the flag is my brother.
2. All that glitters is not gold.
3. The place where the accident had taken place is near Bhubaneswar.
4. The reason why she failed is clear.
5. This is the same car as my father has purchased last year.
6. God helps those who help themselves.
7. This is the time when you should work hard.
8. This is the book that I had presented to my grand father.
9. They also serve who only stand and wait.
10. We all admire a man who helps others.

3. Analyse the following sentences into clauses giving the function of each subordinate clause:
1. This is the place where Indira Gandhi was associated.
2. Childhood is the eye when seeds of character are sown.
3. This is Mamata whose father is a officer in the Navy.
4. This is the bull which has eaten all the paddy.
5. This is the from (that) my elder sister gave (had given) me.
6. This is the school where I had my education.
7. Here comes the man you are looking for.
8. Those whom the Gods love die young.
9. Tomorrow is the day when we shall go on picnic.
10. Blessed is the whom the neighbour praise.

(III) The Adverb Clause
Definition:

The Adverb Clause performs the function of an adverb. It can modify a verb, an adjective or another adverb.

(a) Time-denoting Adverbial Clauses:
Observe the following sentences:
1. All stood up when the President came.
2. What here till I do not come back.
3. She sang while I danced.
4. The doctor had reached there before the patient died.
5. As the hot air cools, the ballon come down.
The italicised words in the above sentences are Adverb Clauses. Their introducing words (‘when, ill, while, before, as’) are time denoting adverbs.
Some other time – denoting adverbs are: after, since, as soon as, whenever as long as, so long as etc.

(b) Place – denoting Adverbial Clauses:
Observe the following sentences.
1. She studies where I study.
2. Live whenever you desire.
3. She returned whence (from where) she had marrie.
4. The soul has reached where from it might not return.
5. The ship sailed whither the wind took her.
The clauses printed in italics in the above sentences point to the place where the action of the main clause take place. They are adverbal clauses and serve as adverbs of places.

(c) Manner – denoting Adverbial clauses:
Observe the following sentences.
1. Try to finish it as she has shown you.
2. He run as if he were frightened.
3. She behaved as though she were annoyed.
4. I did according as I was directed.
The clauses printed in italics in the above sentences point to the manner in which the action of the main clause is done. They are adverbial clauses and serve as adverbs of manners.

(d) Reason or cause-denoting Adverbal Clauses:
Observe the following sentences:
1. As she has been laid up with fever, she cannot take our class.
2. She cannot solve this sum, because she is dull in mathematics.
3. Since you recommend him, I am approaching him.
4. I regret that I could not see you on the appointed day.
5. Now that the sun has set, we should return home.
The clauses printed in italics in the above sentences point to the reason behind the action expressed in the main clause. They are adverbial clauses and serve as adverbs of reasons / cause.

(e) Condition – denoting Adverbial Clauses:
Observe the following sentences:
1. We cannot get first division, unless we burn midnight oil.
2. I will lend you the required money provided that you promise me to return it in time.
3. I cannot led you in if you do not show me your identity card.
4. In case you do not return the library books in time, you will be fined.

The clauses printed in italics in the above sentences point to the condition behind the action in / of the main clause is done. They are adverbial clauses and serve as adverbs of condition. It is important to note that the condition denoting adverb (which introduces the adverbial clause of condition) is sometimes omitted; as

1. Should she came to me, I shall bring her round.
Or,
(If she comes to me, I shall bring her round.)

2. Supposing he falls, he cannot execute his studies.
Or,
(Ifhe fails, he can’t execute his studies)

3. Had you worked hard, you would have got first division.
Or,
(If you had worked hard, you would have got first division).

(f) Extend – denoting Adverbial clauses:
Observe the following sentences:
1. So far as I know, she is a dullard.
2. I cannot say how far I am correct.
3. There was water and water as far as I could see.
4. Can you tell me how long you will accompany me?
The clauses printed in italics in the above sentences point to the extent of the action (fact) mentioned in the main clause. They are adverbal clauses and serve as adverbs of extent.

(g) Comparison-showing Adverbial Clauses:
Observe the following sentences.
1. She is pretty as she is wise.
2. I like him to less than you (do).
3. Lila is cleaver than Shila is.
4. The aeroplane flies fester than railway train can run.
The clauses printed in italics in the above sentences point to the comparison of degrees of a quality in the main clause. They are adverbial clauses and serve as adverbs of comparison.

(h) Result of Effect – denoting Adverbial Clauses:
Observe the following sentences:
1. Run fast so that you may not be late.
2. She ate to much that she fell asleep.
3. He ran so much that he got tired.
4. So bravely did they fight that the enemies retreated.
The clauses printed in italics in the above sentences point to the result ofthe action expressed in the main clause. They are adverbal clauses and serve as adverbs of result / effect.

(I) Contrast or Concession-denoting Adverbial Clauses:
Observe the following sentences: .
1. He is miserly though he is rich.
2. We must go although it is raining.
3. Whatever you may say, I don’t believe a word of it.
4. Even if she apologises, I shall not visit her house.
The clauses printed in italics in the above sentences point to the action expressed in the main clause. They are adverbial clauses and serve as adverbs of contrast / concession.

(J)Purpose – denoting Adverbial Clauses:
Observe the following sentences:
1. She works hard, so that she may get a scholarship.
2. Keep awake – lest somebody should get down with your luggage.
3. You eat that you may live.
4. I went to the post office in order that I might post the letter.
The clauses printed in italics in the above sentences point to the purpose behind the action expressed in the main clause. They are adverbial clauses and serve as adverbs of purpose.

Important Point About Adverbial Clauses:
Some Grammarian treat, the Extent denoting Adverbial Clauses as per with the manner – denoting adverbial clauses and proportion – denoting adverbial clauses.

Subordinate Conjunction	Introduce Adverbial Clause of
when	time
whenever	time
after	time
before	time
while	time
as long as	time
as soon as	time
tin	time
since	time
where	place
wherever	place
whence	place
whither	place
where from	place
that	purpose
in order that	purpose
lest	purpose
so that	purpose
for	cause / reason
because	cause / reason
since	cause / reason
as	cause / reason
that	cause / reason
incase	condition
if	condition
unless	condition
on the condition	condition
provided that	condition
so… that	result / effect
so	result / effect
such	result / effect
such that	result / effect
As as	comparison
So as	comparison
than	comparison
such as	comparison
no less than	comparison
even if	concession/contrast
however	concession/contrast
whatever	concession/contrast
though	concession/contrast
although	concession/contrast
as	manner
as if	manner
as though	manner
The the	extent

Analysis Of Adverbial Clauses
Exercise For Practice (Solved)

Analyse the following sentences into clauses giving the kind and function of each subordinate (Adverbial) Clause:
1. Never talk while you are driving a car.
2. Keep the purse where you can collect it.
3. You may join an institute as you like.
4. She Med because she neglected her studies.
5. She can not finish the paper unless she increases her speed of writing.
6. I do not know how far your statement is true.
7. He is a short-sighted as he is short – statured.
8. He is so old that he cannot climb up the hill.
9. He is dishonest though he is rich.
10. We take exercise so that we may become strong.
Answer:
1. Never talk while you are driving a car.
Clause (a) Never talk
Kind – Main clause
Function – Main part of the sentences.

Clause (b) While you are driving a car.
Kind – Adverb clause
Function-showing-‘time’

2. Keep the purse where you can collect it.
Clause (a) Keep the purse
Kind – Main clause.
Function – Main part of the sentence.

Clause (b) where you can collect it.
Kind – Adverb clause
Function-showing-‘place’

3. You may join an institute as you like.
Clause (a) You may join
Kind – Main clause
Function – Main part of the sentence.

Clause (b) an institute as you like
Kind – Adverb clause
Function – showing – ‘manner ’.

4. She failed because she neglected her studies.
Clause (a) She failed
Kind – Main clause
Function – Main part of the sentence.

Clause (b) because she neglected her studies.
Kind-Adverb clause
Function – showing – ‘reason’.

5. She can not finish the paper unless she increases her speed of writing.
Clause (a) She can not finish the paper.
Kind – Main clause
Function – Main part of the sentence.

Clause (b) unless she increases her speed of writing.
Kind – Adverb clause
Function – showing – ‘condition’.

6. I do not know how far your statement is true.
Clause (a) I do not know.
Kind – Main clause
Function-Main part of the sentence.

Clause (b) how fer your statement is true
Kind-Adverb clause
Function – showing – ‘extent’.

7. He is a short-sighted as he is short – statured.
Clause (a) He is short – sighted
Kind – Main clause
Function – Main part of the sentence

Clause (b) as he is short – statured
Kind – Adverb clause
Function – showing – ‘composition’.

8. He is so old that he cannot climb up the hill.
Clause (a) He is so old
Kind – Main clause
Function – Main part of the sentence

Clause (b) that he can’t climb up the hill.
Kind – Adverb clause
Function – showing – ‘result’.

9. He is dishonest though he is rich.
Clause (a) He is dishonest
Kind – Main clause
Function – Main part of the sentence

Clause (b) through he is rich
Kind – Adverb clause
Function – showing – ‘contras’.

10. We take exercise so that we may become strong.
Clause (a) We take exercise
Kind – Main clause
Function – Main part of the sentence

Clause (b) so that we may become strong.
Kind – Adverb clause
Function – showing – ‘purpose’.

Exercise For Practice (Unsolved)

1. Complete the following sentences inserting Adverb Clauses:
1 ____________ there is hope.
2. ____________ The thiefhid himself
3. I worked hard ____________
4 ____________ he took his umbrella with him.
5. ____________I should never have come.
6. She is as kind a woman ____________
7. Puspa was as kind ____________
8. She behaved ____________

2. Locate the Adverb Clauses in the following sentences:
1. Since she came here, she is unwell.
2. I shall stay wherever you stay.
3. Iam happy that Rupali has got the gold medal.
4. She cannot understand this feet because she is dull minded.
5. Walk carefully least you should fell down.
6. He is so poor that he cannot buy costly medicines.
7. She behaved in such a manner that we were irritated.
8. She talks as if she will betray you.
9. As far as I think she will betray you.
10. Iam hard up these days, all the same, I shall help you.

3. Analyse the following sentences into clauses giving the function of each sub¬ordinate clause:
1. The patient had died before the doctor arrived.
2. The maid-servant went whither I sent her.
3. She works hard so that she may stand first.
4. Bhagat Singh died that the nation might live.
5. Work hard in order that you may win a scholarship.
6. He is so old that he cannot run fast.
7. You are not as tall as your younger sister.
8. She did according as she was ordered.
9. So far as I know Nirupama is an enchantress.
10. I waited for her till the sun set.

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 3 Text B: Some Samples of Advertisement Textbook Activity Questions and Answers.

Class 12th Alternative English Chapter 8 Some Samples of Advertisement Question Answers CHSE Odisha

Some Samples of Advertisement Class 12 Questions and Answers

Activity -1
Make sentences using the following from Text – B (ii)

tense	withstand
like-minded	rehearse
hold	satisfaction
alternative	picture
practise	career

Answer:
tense: The situation is very tense.
like-minded: The matter should be discussed among like-minded people.
hold: The statement does not hold true.
alternative: There is not other alternative than doing it.
practise: Practise yoga and keep healthy.
withstand: I cannot withstand his insolence.
rehearse: They rehearsed the drama before staging it.
satisfaction: His work is not up to my satisfaction.
picture: He gave a clear picture of the situation.
career : He should form your career at the right earnest.

Extra Activity – 1(A)

I. (i) Infinitives (used as nouns):
1. To teach humanity is the aim of this school
2. Other children are told off to help them.
3. The real aim of this school is to teach humanity.
Notice that the to – infinitive in sentence 1 is used as the subject, in sentence 2, it is used as the object and in sentence 3, it is used as the complement. We can say that the infinitives in these sentences above are used as nouns.

Fill in the blanks with ‘to – infinitive’ as necessary:
1. __________ is human, divine.
2. I promise __________ you in these crises.
3. __________ before you speak is always wise.
4. I want ___________ the place where the accident occurred.
5. Don’t forget __________ the door when you go out.
6. __________ your voice again was so pleasant.
7. Your first duty is __________ your motherland.
8. She appears __________ a clever girl.
9. His aim in life is __________ a college teacher.
10. Anita wanted __________ photography.
Answer:
1. To err is human, to forgive is divine.
2. I promise to help you in these crises.
3. To think before you speak is always wise.
4. I want to see the place where the accident occurred.
5. Don’t forget to lock the door when you go out.
6. To hear your voice again was so pleasant.
7. Your first duty is to serve your motherland.
8. She appears to be a clever girl.
9. His aim in life is to become a college teacher.
10. Anita wanted to read photography.

II. Derive nouns from the following verbs:

suspend	weigh
breathe	grow
indicate	build
hospitalise	require
exist	propose

Answer:

Verbs	their noun forms
suspend	suspension
breathe	breath
indicate	indication
hospitalise	hospitalisation
exist	existence
weigh	weight
grow	growth
build	building
require	requirement
propose	proposal

III. Make sentence with the following using them both as nouns as well as verbs:

poison	plan
damage	cause
rated	wear
increase	ban
benefit	show

Answer:
poison :
(N) She committed suicide by taking poison.
(V) She has poisoned your mind against me.

damage :
(N) The accident has caused a lot of damage.
(V) Corruption has damaged his personality.

rated :
(N) The rate of interest at present is very long.
(V) Sachin is rated as the best batsman.

increase :
(N) India has a very alarming increase in population.
(V) The price of petroleum product has increased.

benefit :
(N) This project promises no benefit.
(V) The price of petroleum product has increased.

plan :
(N) You should chalk out a plan.
(V) He is planning to visit France in November.

cause :
(N) One must read the theory of cause and effect.
(V) Ravan’s pride caused his fall.

wear :
(N) He deals in footwear.
(V) Players must wear white clothes.

ban :
(N) Government has put a ban on this film.
(V) Government have banned staging of the play.

show :
(N) It was a ground show.
(V) The wrinkles on his profiles show his age.

Extra Activity – 1(B)

Give antonyms of the following:

dirty	growth
low	risen
false	stringent
increasing	private
efficiency	majority

Answer:

Words	their antonyms
dirty	clean
low	high
false	true
increasing	decreasing
efficiency	inefficiency
growth	decay
stringent	lenient
private	public
majority	minority

II. Derive adjectives from the following:

regret	ornament
precision	seclusion
transition	commerce
response	attention
delicacy	decoration

Answer:

Words	their adjectives
regret	regretful
precision	precise
transition	transitional
response	responsive
delicacy	delicate
ornament	ornamental
seclusion	secluded
commerce	commercial
attention	attentive
decoration	decorative

III. Turn the following sentences into passive:
1. They have repaired the road.
2. Someone stole my bicycle last night.
3. No one knows his address.
4. I can carry a thin box.
5. They will give you a good salary.
6. You must post the letter now.
7. You must obey the rules of the road.
8 . My pen needs filling.
9. They laughed at her.
10. Do you know this man?
11. The news surprised to alL
12. He gave him what he wanted.
13. Switch on the lights.
14. She showed the visitor the new baby.
15. The doctor advised the patient to take rest.

Answer:
1. The road has been repaired.
2. My bicycle was stolen last night.
3. His address is known to none.
4. This box can be carried by me.
5. You will be given a good salary.
6. The letters must be posted now.
7. The rules of the road must be obeyed.
8. My pen needs to be filled.
9. She was laughed at.
10. Is this man known to you?
11. We were all surprised at the news.
12. He was given what he wanted.
13. Let the lights be switched on.
14. The new baby was shown to the visitor.
15. The doctor advised the patient that rest should be taken or the patient was advised to take rest.

IV. Give antonyms of the following:

closeness	genuine
modem	availability
extravagant	occupy
responsible	impressive
remember	appropriate

Answer:

Words	their antonyms
closeness	remoteness
modern	ancient
extravagant	frugal
responsible	irresponsible
remember	forget
genuine	false
availability	nonavailability
occupy	vacant
impressive	unimpressive
appropriate	inappropriate

Section – B

Focussing Question:
Look at the advertisement opposite and decide what its main idea is. Choose from the list below:
(a) Lila Hotels do their best make their guests feel and home.
(b) Lila Hotels mostly cater for businessmen/women.
(c) The guests in Lila Hotels will find all the facilities they require in the building itself.
(d) Lila Hotels take great care in looking after business women as well as businessmen.

Some Samples of Advertisement Summary in English

“What I’m really trying to say is that they treat me like a person. A rather overused phrase, I agree but other business women will know what I mean. If Fm in the restaurant there’s none of that over effusive welcome followed by a table behind a pilar or near the kitchen door. I don’t have to take my briefcase into the bar either, to prove all I want is a drink. When I go to my room, there are some little extra touches that make me feel specially welcome. It’s not simply the softer decor. Lila have thoughtfully provided a hair dryer and make up mirror, things I appreciate away from home.

And they can even come up with an iron or a pair of tights at a moments notice. So, I always stay at Lila Hotel whenever I can. I like their friendly and business-like attitude towards me. As speaking as a woman, you can’t say fairer then that.” Lila Hotels International. (Nobody works harder to make your stay better) Tell him your date of birth, your educational qualification and why you want to join us. He will send you booklets to give you a far longer picture, picture of the life and if you like, put you in touch with people who can tell you more about the career.

Analytical Outlines:

• The writer says that they treat him like a person
• The business women will know it better.
• He was in a restaurant.
• There was no over-effusive welcome.
• It was followed by a table.
• The table was behind a pillar near to the kitchen door.
• He had not taken his briefcase to the bar.
• He also wanted a drink.
• He went to his room
• There were some extra touches.
• This made him feel specially welcome.
• It was not simply the softer decor.
• Lila had thoughtfully provided a hair dryer.
• She had only provided one make up mirror.
• He appreciates these things away from home.
• They can come up with an iron or a pair of tights immediately.
• For this reason, he always stays at Lila Hotel.
• Whenever he goes there.
• He likes their friendly attitude.
• He likes their business-like attitude towards him
• That Hotel is actually, much more fairer than other.
• The Hotel is Lila Hotel International

Meaning Of Difficult Words
treat – to consider, to behave, to handle, to manage
restaurant – refreshment room
over-effusive – over-extensive
tank – here, fighting vehicle
roar – to make a loud
crisis – turning point, moment of danger of suspense
erupt – to break out
scour – to go along, to cleanse
remote – separate, indirect
guess – anticipation hope
tense – serious
mobile – easily moved, movable
prevail – to succeed, to be current, to predominate
confront-free, encounter

Read More: 

• Text A: How to Write a Winning ‘Resume’ Question Answer
• Text B: Some Samples of Advertisement Question Answer
• Text C: On the Education of a Man of Business Question Answer

Odisha State Board CHSE Odisha Class 12 Approaches to English Book 1 Solutions Unit 3 Text C: On the Education of a Man of Business Textbook Activity Questions and Answers.

Class 12th Alternative English Chapter 9 On the Education of a Man of Business Question Answers CHSE Odisha

On the Education of a Man of Business Class 12 Questions and Answers

Activity-10
Comprehension
Answer the following questions in a sentence of two:

Question (a)
How is ‘university’ in a course of study helpful to a man of business?
Answer:
‘University” in a course of study is helpful to a man of business because, it makes the mind agile but gives a variety of information. Such a system will make him acquainted with many modes of thought, with various classes of facts, and will enable him to understand men better.

Question (b)
What should a young person read during the transitional period from school to bis affairs of the world? ‘
Answer:
Ayoungmanshouldreadbooksthatmiximaginationandphilosophythatisbooksof the Bacoimian style during the transitional period from school to his entry into the affairs of the world.

Question (c)
How is a ‘ready man’ different from a ‘full man’?
Answer:
A ‘ready man’ is a practical man ready with facts and information but a ‘full man’ sustain bookish knowledge without practical utility.

Question(d)
How can a young man be trained to be methodical?
Answer:
Ayoung man can be trained to be methodical through letting him employ himself in making digests, arranging and classifying materials, writing narratives and in deciding upon conflicting evidence.

Question (e)
Why should a man of business bie allowed to repeat some apt expressions which a . learned man should avoid?
Answer:
A man ofbusiness should be allowed to repeat some apt expressions which a learned man should avoid because avoidance of such repetitions maybe carried too far in all kinds of writing. In literature, one is seldom brought to account for misleading people but in business one may soon be called upon to pay the penalty for having avoided the world which would exactly have expressed one’s meaning.

Question (f)
How can the sense of responsibility help in a man develop his personality?
Answer:
Aman of business must develop a sense of responsibility. He must believe in the power and validity of truth and in all he does or says should be anxious to express as much truth as possible.

Activity – 11
Grammar Review

We often use ‘should’ or ‘ought to ’ interchangeably with little difference of meaning. But there acts places where one is preferred to the other. ‘Should’ is preferred when an outside authority rather than the speaker himself recommends it. In case of power failure the computer should be switched off. ‘Should’ is also preferred when we give advice with ‘I’. I should avoid your company if I were you. Now, use should or ought to be in the blank spaces. In some places you can use either of them.

(a) This bottle ____________ be kept out of reach of children.
(b) If I were feeling ill, I ____________ stay at home today.
(c) I think you ____________ have listened to him, if I were you. It could have helped you.
(d) According to the instruction printed on the bottle, it ____________ be refrigerated after opening.
(e) The application you, sent ____________ includes the details of your past experience.
Answer:
(a) This bottle should be kept out of reach of children.
(b) If I were feeling ill, I should stay at home today.
(c) I think you ought to have listened to him, if I were you. It could have helped you.
(d) According to the instruction printed on the bottle, it should be refrigerated after opening.
(e) The application you, sent ought to includes the details of your past experience

Activity-12
Grammar Review
Complete the sentence with must or have/has to:

(a) The patient ____________ have at least eight hours sleep at night. He has got a longs problem and he ____________ give up smoking.
(b) That’s in reality a good news, I ____________ phone my friend Kim.
(c) I always sleep through the alarm clock. My Dad ____________ wake me up every morning.
(d) ‘Can we meet tomorrow evening?’ Sorry, no I ____________ go to the dentist at 7 O’clock. ’
(e) To get to Bangalore. I ____________ borrow money from my sister.

Answer:
(a) The patient must have a least eight hours sleep at night. He has got a longs problem and he has to give up smoking.
(b) That’s in reality good news, I have to phone my friend Kim
(c) I always sleep through the alarm clock. My Dad must wake me up every morning.
(d) ‘Can we meet tomorrow evening?’ Sorry, no I have to go to the dentist at 7 O’clock.
(e) To get to Bangalore. I have to borrow money from my sister.

Activity-15

The following passage was originally in 6 paragraphs. But all of them have been combined into one. You are to find out the places where new paragraphs begin and mark them with ( ) Unconsciousness is a state of where the person appears to be in deep sleep from which he/she can not be awoken. The individual does not respond to any external stimuli like sprinking cold water on the feet and for that matter even painful ones like piercing with a pin tingling a nerve etc.

This insensible state is brought about by some interference in the normal functioning of the brain and the nervous system Unconsciousness when partial is called stupor and when complete is termed as Coma. In cases of stupor, the individual can be roused with difficulty, but the eyelids can not be opened due to resistance by the individual In coma, however, although there is not response. When an individual is being called, the lids can be opened without any resistance.

The usual cases of unconsciousness include fainting, sun in the blood supply to the train, because of fright, unexpected good or bad news etc. People held up in stuffy places like elevators often faint. A sudden fell in blood pressure can also cause feinting. The individual appears pale becomes weak and slow, breathing becomes shallow and the skin turns cold and clammy. Excessive summer heat can make an individual faint. Prolonged exposure to sun may also caused sun stroke which starts with headache, vomiting, dizziness, cramps or dryness ofthe throat.

Conclusion commonly results in unconsciousness. Direct injury to the brain caused by either a blow on the head or a fell from a height etc. may result in short while in mild cases. Concussion and compression result in stupor or come in more serious cases. An individual could suddenly become unconscious due to a heart attack. The initial signs are vomiting, profiise sweating and pain in the left sided of the chest.

Answer:
Unconsciousness …………………… nervous system
Unconsciousness when partial ………………………
The usual cases of …………………….. an individual feint.
Prolonged ……………………. exposure in unconsciousness.
Direct injury …………………. in more serious cases.
An individual ……………. left side of the chest.

Activity -16
Cohesive Devices

Choose correct alternatives from the given choices to fill the blanks 1 – 5 in the following passage.
The problem of deep-sea pollution can only be solved by international corporation
(1) the problem of pollution and coastal degradation of our own shoreline in our special problem which we must be conscious of and tackle by ourselves. Pollution only means dirt
(2) ‘matter in the wrong place.’Getting late or becoming ignorant about it all can be disastrous. There are types and degrees of pollution, but even a slight amount can affect natural and necessary function and movements we have seen that pollution near the coast and in astuaries and creeks affects the breeding of fish, thus reducing their numbers in the deep sea. But heavy pollution,
(3) that of Mahim Creek Mumbai also kills the coastal vegetation like mangroves which is responsible for holding the sand and run in place and consequently for the health of coastal areas. Costal and estuarine lands are often extremely fertile, the nutrients washed down in rivers often ‘pileup’ in
(4) flat marshy areas, making the soil reach
(5) the lagoons are the fish nurseries. Good vegetation cover is one way of dealing with polluted water in such areas, for the plants absorb must of the waste matters.
1. However / though / although / but
2. On the other hand / in other words / whereas / broadly speaking.
3. Mainly / similar to / such as / specially.
4. And/when/where/while.

Answer:
1. but,
2. in other hands
3. such as
4. many
5. while

Section – C
Suppose you are going to take some major decisions in your life now. You have to choose a career and make preparation of your future life. What will be your three important considerations while choosing a career ?’
List them below:
(i) ____________
(ii) ____________
(iii) ____________
Whatever career you choose, the following essay, ‘On the Education of Man of Business’ with its insightful observations can light up the path of your life.

On the Education of a Man of Business Summary in English

A man of business should be closely brought up in the habit of reasoning. The study of geometry is hardly better for him. Any university course of study designed for him makes his mind agile and gives a variety of information. This system will make him grow acquainted with many modes of thought with various classes of facts and will enable him to understand men better. His youth time may be well spent by the study of metaphysical nature. A breath and a tone may be given to a man’s mode of thinking. It will afterward be of signal use to him in the business of everyday life. Some works transit from the school to the world.

These are particularly needed in a system of educational studies remote from real life. Such works tend to give the students interest in common things about him which he has scarcely even been called upon to feel They display how imagination and philosophy can be woven into practical wisdom. However, our student is not intended to become a learned man, a man of business not a full man but a ready man He must be taught to arrange and express what he knows, for this purpose let him employ himself in making digests arranging and classifying materials, writing narratives and in deciding upon this conflicting evidence.

All these exercises require a method. A method is developed from rule beginnings. There is hardly any degree of toil for which he would not be compensated by such a result. The student of business should begin soon to cultivate fluency in writing. Fluency does not mean the flow of words, but a habit of expressing his thoughts with accuracy with brevity, and readiness. Moreover, in the style of the man of business, nothing is to be aimed at but plainness and precision.

A close repetition of the same word for the same thing need not be avoided. In literature, however, you are seldom brought to account for misleading people, but in business, you may soon be called upon to pay the penalty for having shunned the word which should exactly have expressed the meaning. A consummate man of business should be able to fix his attention on details and be ready to give every kind of argument a hearing.

This will not encumber him for he must have been practiced beforehand in the exercise of his intellect and there they remain in a shapeless heap, another possessed of method can arrange what he has collected, but such a man by the aid of principles goes further and bulbs with his materials. In feet, in addition to a stout heart, he should have the patient temperament and a vigorous but disciplined imagination and then he will plan boldly and with a large extent of view, execute calmly and not be a stretching act of his hand for things not yet within his grasp. He will let opportunities grow before his eyes until they are ripped to be seized.

He will think steadily over possible failure, in order to provide a remedy or a retreat, there will be the strength of repose about him. He must have a deep sense of responsibility. He must believe in the power and vitality of truth and in all he does or says, should be anxious to express as much truth as possible. His feelings of responsibility and love of truth will almost inevitably endow him with diligence accuracy and discreteness these commonplace required for a good man of business.

Analytical Outlines

• A man of business should be closely brought up in the habit or reasoning.
• He should study geometry.
• It is hardly better for him.
• The university course of study is designed for him.
• It makes his mind agile.
• It gives varieties of information.
• This system acquaints him with many mode of thought.
• It acquaints him with various classes of facts.
• It will enable him to understand men better.
• His youth time may be spent by the study of metaphysical nature.
• A breath and a tone may be given to a man’s mode of thinking.
• It will be a good signal for him.
• It will help him in the business of everyday life.
• Some works transit from the school to the world.
• These are actually needed in educational studies.
• These are remote from real life.
• Such works provide the students interest in common things.
• He has scarcely been called upon to feel these.
• They display how imagination and philosophy can be woven.
• They can be woven into practical wisdom.
• Out student is not intended to be a learned man.
• Neither is he intended to be a business man.
• Nor is he intended to be a foil man.
• But he is intended to be a ready man.
• He must be taught to arrange what he knows.
• He must be taught to express what he knows.
• We should let him to digest arranging of materials.
• We should let him to classify these materials.
• We should allow him to writing narratives.
• We should allow him to decide upon this conflicting evidence.
• All these exercises require method.
• Method is developed from rule beginnings.
• There is hardly and degree of toil for.
• He would not be compensated by such a result.
• The student of business should begin to cultivate a fluency in writing.
• Fluency does not mean flow of words.
• But it means a habit of expressing his thoughts.
• It must be with accuracy.
• It must be with brevity.
• It must be also with readiness.
• Nothing much is required for the style of man of business.
• But it requires plainness.
• Again it also requires precision.
• A close repetition of the same word of something should not be avoided.
• In literature, we seldom use it.
• We consider here as misleading the people.
• But is business we should not be shunned.
• Otherwise, we have to pay penalty for it.
• Because it must express the meaning exactly.
• A consummate man of business must fix his attention on details.
• He should be ready to give every kind of argument a hearing.
• This will not encumber him
• As he has practised it before hand.
• It is actually exercise of his intellect.
• He should be strong in principles.
• He can collect materials together.
• There they remain a shapeless heap.
• He can also arrange the collected material
• But still then, he will go further by the aid of principles.
• The principles he builds with his materials.
• In feet, he should be having a stout heart.
• Again, he should have patient temperament.
• It should be vigorous.
• But there should be disciplined imagination.
• So that he will plan boldly.
• He will plan with large extent of view.
• He will execute it calmly.
• He should not stretch out for his out of his grasp.
• He will let opportunities grow before his eyes.
• Until they are riped to be seized.
• He will think steadily over possible failure.
• So that, he can provide a remedy or a retreat for it.
• There will be the strength ofrepose about him.
• He must have a deep sense of responsibility.
• He must believe in the power and vitality of truth.
• All his doing should be anxious to express much truth.
• His feeling of responsibility and loved oftruth will endow with him diligence.
• It will endow him with accuracy.
• It will also endow him with discreteness.
• All these common things are required for a good man of business.

Meanings Of Difficult Words
agi – active, nimble
variety – diversity di&rence
information – instruction, intimation
Mode – type, kind
metaphysical – of the science of being and knowledge meta means beyond, physic means earth.
transit – change, pass from one to other
remote – separate, indirect
tend – to look after, to incline, to conduce
scarcely – hardly rarely, seldom
display – to exhibit, show
conflicting – ling, colliding
evidence – clearness, obviousness, testimony, proof; witness, indication
oil – hard struggle
compensate – to make amends for
cultivate – to till to produce, to devote attention to
fluency – volubility
accuracy – exactness, correctness
brevity – briefiiess, consciousness
consummate – to perfect, to accomplish, to relish
encumber – to imped the motion of to hamper
vigorous – energetic, of vital power

Read More: 

• Text A: How to Write a Winning ‘Resume’ Question Answer
• Text B: Some Samples of Advertisement Question Answer
• Text C: On the Education of a Man of Business Question Answer