## CHSE Odisha Class 11 Math Notes Chapter 5 Principles of Mathematical Induction

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 5 Principles of Mathematical Induction will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 5 Principles of Mathematical Induction

Principles Of Mathematical Induction

(i) Principle – 1
Let P(n) is a statement , n ∈ Z

Step – 1: Verification step:
verify that P(1) is true.

Step – 2: Induction step – 1:
Assume that P(k) is true for any arbitrary k ∈ N.

Step – 3: Induction step – 2:
prove that P(k+1) is true using step – 1 and step – 2

Step – 4: Conclusion Step:
If P(k+1) is true then take a conclusion that by Principle of mathematical induction P(n) is true for all n ∈ N.

(ii) Principle – 2
Let P(n) be a statement, n ∈ N.

Step – 1: Verification step:
verify the P(1) is true

Step – 2: Induction step – 1:
Assume that P(2), P(3),….. P(k) is true.

Step – 3: Induction step – 2:
Using step – 1 and step – 2 prove that P(k + 1) is true.

Step – 4: Conclusion step:
If P(k + 1) is true, then take a conclusion that by Principle of mathematical induction P(n) is true for all n ∈ N.

## CHSE Odisha Class 11 Math Notes Chapter 4 Trigonometric Functions

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 4 Trigonometric Functions will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 4 Trigonometric Functions

Angle:
If A, B, and C are three non-collinear points, then ∠ABC = $$\overrightarrow{\mathrm{BA}} \cup \overrightarrow{\mathrm{BC}}$$

$$\overrightarrow{\mathrm{BA}}$$ is the initial side, $$\overrightarrow{\mathrm{BC}}$$ is called the terminal side and B is called the vertex of the angle.

Positive and negative angles:
If the direction of rotation is anti-clockwise then the angle is positive and if the direction of rotation is clockwise then the angle is negative.

Measure of an angle:
(a) Sexagesimal system or English System (Degree measure):
1 degree = 1° = $$\left(\frac{1}{360}\right) \text { th }$$ of revolution from initial side to terminal side.

• One revolution = 360°
• 1° = 60′ (sixty minute)
• 1′ = 60” (sixty seconds)

One radian = 1c = The angle at the centre of the circle by an arc where the arc length equals to

Note:
(i) θ = $$\frac{l}{r}=\frac{\text { arc }}{\text { radius }}$$ where θ is an radian.
(ii) θ in radian is created as a real number.

(c) Relation between Degree and radian measure:

• We can convert radian to degree or degree to radian by using the identity.
$$\frac{\mathrm{D}}{180}=\frac{\mathrm{R}}{\pi}$$ where D is the degree measure and R is the radian measure of an angle.

Trigonometry Functions:

(i) Sign of trigonometry functions:

(ii) $$\begin{array}{cccc} \text { Add } & \text { Sugar } & \text { To } & \text { Coffee } \\ \downarrow & \downarrow & \downarrow & \downarrow \\ \text { all }+ & \sin + & \tan + & \cos + \end{array}$$

(iii) Periodicity of trigonometry functions:

 Trigonometric function Period sin x 2π cos x 2π tan x π cot x π sec x 2π cosec x 2π sin2 x or cos2 x π |sin x| or |cos x| π

Trigonometric functions of some standard angles

Fundamental trigonometric identities:
(a) sin θ = $$\frac{1}{{cosec} \theta}$$
(b) cos θ = $$\frac{1}{{sec} \theta}$$
(c) tan θ = $$\frac{1}{{cot} \theta}$$
(d) sin2 θ + cos2 θ = 1
(e) sec2 θ – tan2 θ = 1
(f) cosec2 θ – cot2 θ = 1
(g) sin (-θ) = -sin (θ)
(h) cos (-θ) = -cos (θ)

Trigonometric functions of allied angles:
(a) sin $$\left((2 n+1) \frac{\pi}{2} \pm \theta\right)$$ = (±) cos θ choose + or – in (±) by using ASTC rule

(b) cos $$\left((2 n+1) \frac{\pi}{2} \pm \theta\right)$$ = (±) sin θ choose + or – in (±) by using ASIC rule
Similar technique can be used for other trigonometric functions.

(c) sin (nπ ± θ) = (±) sin θ
cos (nπ ± θ) = (±) cos θ
tan (nπ ± θ) = (±) tan θ
choose + or – in (±) by using ASTC rule.

Sum And Difference Formulae:
(a) sin(A + B) = sin A . cos b + cos A . sin B

(b) sin(A – B) = sin A . cos B – cos A . sin B

(c) cos(A + B) = sin A . cos B – cos A . sin B

(d) cos(A – B) = sin A . cos B + cos A . sin B

(e) tan(A + B) = $$\frac{\tan A+\tan B}{1-\tan A \cdot \tan B}$$

(f) tan(A – B) = $$\frac{\tan A-\tan B}{1+\tan A \cdot \tan B}$$

(g) cot(A + B) = $$\frac{\cot A \cdot \cot B-1}{\cot A+\cot B}$$

(h) cot(A – B) = $$\frac{\cot A \cdot \cot B+1}{\cot A-\cot B}$$

(i) sin(A + B) + sin (A – B) = 2 sin A . cos B

(j) sin (A + B) – sin(A – B) = 2 cos A . sin B

(k) cos(A + B) + cos(A – B) = 2 cos A . cos B

(l) cos(A + B) – cos(A – B) = -2sin A . sin B

(m) sin(A + B) sin(A – B) = sin2 A – sin2 B = cos2 B – cos2 A

(n) cos(A + B) cos(A – B) = cos2 A – sin2 B = cos2 B – sin2 A

(o) sin 2A = 2 sin A cos A = $$\frac{2 \tan \mathrm{A}}{1+\tan ^2 \mathrm{~A}}$$

(p) cos 2A = cos2 A – sin2 A
= 2 cos2 A – 1
= 1 – 2 sin2 A
= $$\frac{1-\tan ^2 \mathrm{~A}}{1+\tan ^2 \mathrm{~A}}$$

(q) tan 2A = $$\frac{2 \tan A}{1-\tan ^2 A}$$

(r) tan(A + B + C) = $$\frac{\tan A+\tan B+\tan C-\tan n A \cdot \tan B \cdot \tan C}{1-\tan A \cdot \tan B-\tan B \cdot \tan C-\tan C \cdot \tan A}$$

(s) sin 3A = 3 sin A – 4 sin3 A
= 4 sin A sin($$\frac{\pi}{3}$$ – A) sin($$\frac{\pi}{3}$$ + A)

(t) cos 3A = 4 cos3 A – 3 cos A
= 4 cos A cos($$\frac{\pi}{3}$$ – A) cos($$\frac{\pi}{3}$$ + A)

(u) tan 3A = $$\frac{3 \tan A-\tan ^3 A}{1-3 \tan ^2 A}$$
= tan A. tan($$\frac{\pi}{3}$$ – A) tan($$\frac{\pi}{3}$$ + A)

Sum or Difference → Product:
(a) sin A + sin B = 2 sin($$\frac{A+B}{2}$$) cos($$\frac{A-B}{2}$$)
(b) sin A – sin B = 2 cos($$\frac{A+B}{2}$$) cos($$\frac{A-B}{2}$$)
(c) cos A + cos B = 2 cos($$\frac{A+B}{2}$$) cos($$\frac{A-B}{2}$$)
(d) cos A – cos B = -2 sin($$\frac{A+B}{2}$$) sin($$\frac{A-B}{2}$$)

Submultiple Arguments:
(a)

(b) 2 sin2 $$\frac{\theta}{2}$$ = 1 – cos θ
2 cos2 $$\frac{\theta}{2}$$ = 1 + cos θ

(c) tan $$\frac{\theta}{2}$$ = $$\frac{\sin \theta}{1+\cos \theta}=\frac{1-\cos \theta}{\sin \theta}$$

(d) sin θ = 3 sin $$\frac{\theta}{2}$$ – 4 sin3 $$\frac{\theta}{2}$$
cos θ = 4 cos3 $$\frac{\theta}{2}$$ – 3 cos $$\frac{\theta}{2}$$

(e) tan θ = $$\frac{3 \tan \frac{\theta}{2}-\tan ^3 \frac{\theta}{2}}{1-3 \tan ^2 \frac{\theta}{2}}$$

Trigonometric Equations:
(a) Equation involving trigonometric equations of unknown angles are called trigonometric function.
(b) Principle solution: The solution ‘x’ of a trigonometric equation is said to be a principle solution if x ∈ (0, 2π)
(c) The solution considered over the entire set R are called the general solution.
(d) General solution of some standard trigonometric equations.

• sin x = 0 ⇒ x = nπ, n ∈ Z
• cos x = 0 ⇒ x = (2n + 1) $$\frac{\pi}{2}$$, n ∈ Z
• tan x = 0 ⇒ x = nπ, n ∈ Z
• sin x = sin α ⇒ x = nπ + (-1)n, n ∈ Z
• cos x = cos α ⇒ x = 2nπ ± α, n ∈ Z
• tan x = tan α ⇒ x = nπ + α, n ∈ Z
• $$\left.\begin{array}{l} \sin ^2 x=\sin ^2 \alpha \\ \cos ^2 x=\cos ^2 \alpha \\ \tan ^2 x=\tan ^2 \alpha \end{array}\right]$$ ⇒ x = nπ ± α
• $$\left.\begin{array}{l} \cos x=\cos \alpha \\ \text { and } \sin x=\sin \alpha \end{array}\right]$$ ⇒ x = nπ ± α, n ∈ Z

Sine Formula:
In any Δ ABC, $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ or, $$\frac{\sin \mathrm{A}}{a}=\frac{\sin \mathrm{B}}{b}=\frac{\sin \mathrm{C}}{c}$$ = 2R
∴ a = 2R sin A, b = 2R sin B and c = 2R sin C
Also, sin A = $$\frac{a}{2R}$$, sin B = $$\frac{b}{2R}$$ and sin c = $$\frac{c}{2R}$$

Cosine fromulae:
In any Δ ABC,
(i) a2 = b2 + c2 – 2bc cos A
(ii) b2 = c2 + a2 – 2ca cos B
(iii) c2 = a2 + b2 – 2ab cos C
or, → cos A = $$\frac{b^2+c^2-a^2}{2 b c}$$
→ cos B = $$\frac{c^2+a^2-a^2}{2 c a}$$
→ cos C = $$\frac{a^2+b^2-c^2}{2 a b}$$

Projection formulae:
In any Δ ABC,
(i) a = b sin C + c sin B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A

Tangent formulae (Napier’s Analogy);
In any Δ ABC

Area of Triangle (Heron’s formulae):
(i) Area of triangle ABC

(ii) Heron’s formulae:
In any Δ ABC Let 2S = a + b + c
Area of Δ ABC = Δ = $$\sqrt{s(s-a)(s-b)(s-c)}$$
Δ = $$\frac{1}{2}$$ bc sin A = $$\frac{1}{2}$$ ca sin B
= $$\frac{1}{2}$$ ab sin C, Δ = $$\frac{abc}{4R}$$

Semi-Angle Formulae:

## BSE Odisha 10th Class Maths Solutions Algebra Chapter 4 ସମ୍ଭାବ୍ୟତା Ex 4(b)

Odisha State Board BSE Odisha 10th Class Maths Solutions Algebra Chapter 4 ସମ୍ଭାବ୍ୟତା Ex 4(b) Textbook Exercise Questions and Answers.

## BSE Odisha Class 10 Maths Solutions Algebra Chapter 4 ସମ୍ଭାବ୍ୟତା Ex 4(b)

Question 1.
ନିମ୍ନଲିଖ ଉକ୍ତି ମଧ୍ୟରୁ କେଉଁଟି ଠିକ୍ ଦର୍ଶାଅ।
(i) ଘଟଣାଟି ϕ ହେଲେ ଏହାର ସମ୍ଭାବ୍ୟତା ଶୂନ ।
(ii) ଘଟଣା E = S, ଯେଉଁଠାରେ S (Sample Space) ତେବେ P(E) < 1।
(iii) ଗୋଟିଏ ମୁଦ୍ରାକୁ ଥରେ ଟସ୍ କଲେ Sample Spaceର ଉପାଦାନ ସଂଖ୍ୟା 4 ଅଟେ।
(iv) ‘Probability’ ଶବ୍ଦରୁ ଗୋଟିଏ ଅକ୍ଷର ‘i’ ବାଛିବାର ସମ୍ଭାବ୍ୟତା $$\frac{2}{11}$$।
(v) E1 ଓ E2 (E1 E2 ⊂ S) ପରସ୍ପର ବର୍ହିଭୁକ୍ତ ଘଟଣା ଦ୍ଵୟର ସମ୍ଭାବ୍ୟତାର ଯୋଗଫଳ 1 ।
(vi) ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଏକ ସଙ୍ଗେ ଦୁଇ ଥର ଗଡ଼ାଇଲେ ଲବ୍‌ଧ ସାମ୍ପଲ ସେସ୍‌ର ଉପାଦାନ ସଂଖ୍ୟା 36 ।
(vi) ଗୋଟିଏ ମୁଦ୍ରାକୁ 3 ଥର ଟସ୍ କଲେ ଲବ୍‌ଧ ସାମ୍ପଲ ସ୍ପେସ୍‌ରେ ବିଦ୍ୟମାନ ଉପାଦାନମାନଙ୍କ ସଂଖ୍ୟା
32 = 9।
(viii) ଗୋଟିଏ sample spaceର E1 ଏବଂ E2 ଦ୍ଵୟ ବହିର୍ଭୁକ୍ତ ଘଟଣା ହେଲେ
P(E1 ∪ E2) = P (E1) + P(E2)।
(ix) ଥରେ ମୁଦ୍ରାକୁ ଟସ୍ କଲେ E1 = {H} ଘଟଣାଟିର ପରିପୂରକ ଘଟଣାଟି E2 = {H, T}।
ଉ –
ଠିକରକ୍ତି: (i), (iv), (vi) ଓ (viii)

Question 2.
ଏକ ପରୀକ୍ଷଣରେ E1, E2, E3 ଏବଂ E4 ଚାରିଗୋଟି ବହିର୍ଭୁକ୍ତ ଘଟଣା । ଏଠାରେ (E1 ∪ E2 ∪ E3 ∪ E4) ନିଶ୍ଚିତ ରୂପେ ଘଟୁଥିବା ଘଟଣା । ଦତ୍ତ ଘଟଣାଗୁଡ଼ିକ ସମ ସମ୍ଭାବ୍ୟତା ବିଶିଷ୍ଟ ହେଲେ ପ୍ରତ୍ୟେକର ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
ସମାଧାନ:
ପରୀକ୍ଷଣରେ E1, 2, E3, E4 ଚାରୋଟି ବର୍ହିଭୁକ୍ତ ଘଟଣା ଅର୍ଥାତ୍ E1 ∩ E2 ∩ E3 ∩ E4 = ϕ
(E1 ∪ E2 ∪ E3 ∪ E4) ଏକ ନିଶ୍ଚିତ ଘଟଣା ହେତୁ ଏହାର ସମ୍ଭାବ୍ୟତା 1।
P (E1 ∪ E2 ∪ E3 ∪ E4) = P(E1) + P(E2) + P(E3) + P(E4)
⇒ 1 = P (E1) + P(E2) + P(E3) + P(E4)
⇒ P(E1) = P (E2) = P (E3) = P (E4) = $$\frac{1}{4}$$
କାରଣ ଘଟଣାଗୁଡିକ ସମ ସମ୍ଭାବ୍ୟତାବିଶିଷ୍ଟ ।

Question 3.
ଗୋଟିଏ ଲୁଡୁଗୋଟି ଥରେ ଗଡ଼ାଇ ଦିଆଗଲା । ତେବେ ନିମ୍ନଲିଖ୍ ଘଟଣାମାନଙ୍କ ସମ୍ଭାବ୍ୟତା ସ୍ଥିର କର ।
(i) ଫଳ ≤ 3
(ii) ଫଳ < 3
(iii) ଫଳ ≤ 4
(iv) ଫଳ < 6 (v) ଫଳ ≤ 6 (vi) ଫଳ > 6
ସମାଧାନ:
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଇ ଦିଆଗଲା ।

(i) ଫଳ ≤ 3 ଏକ ଘଟଣା = E1 ∴ E1 = {1, 2, 3} ଏବଂ |E1| = 3
ଦତ୍ତ ପ୍ରଶ୍ନରେ ସାମ୍ପଲ ସେଟ୍ |S| = = 6 ଓ | E1| = 3
∴ ଫଳ ≤ 3ର ସମ୍ଭାବ୍ୟତା P(E1) = $$\frac{\left|E_1\right|}{|S|}=\frac{3}{6}=\frac{1}{2}$$

(ii) ଫଳ < 3 ଏକ ଘଟଣା E2 ∴ E2 = {1, 2} ⇒ |E2| = 2
∴ ଫଳ < 3ର ସମ୍ଭାବ୍ୟତା P(E2) = $$\frac{\left|E_2\right|}{|S|}=\frac{2}{6}=\frac{1}{3}$$

(iii) ଫଳ ≤ 4 ଏକ ଘଟଣା E3 ∴ E3 = {1, 2, 3, 4} ⇒ |E3| = 4
∴ ଫଳ ≤ 4ର ସମ୍ଭାବ୍ୟତା P(E3) = $$\frac{\left|E_3\right|}{|S|}=\frac{2}{6}=\frac{1}{3}$$

(iv) ଫଳ < 6 ଏକ ଘଟଣା E4 ∴ E4 = {1, 2, 3, 4, 5} ⇒ |E4| = 5
∴ ଫଳ < 6 ର ସମ୍ଭାବ୍ୟତା P(E4) = $$\frac{\left|E_4\right|}{|S|}=\frac{5}{6}$$

(v) ଫଳ ≤ 6 ଏକ ଘଟଣା E5
∴ E5 = {1, 2, 3, 4, 5, 6} ⇒ |E5| = 6
∴ ଫଳ ≤ 6 ର ସମ୍ଭାବ୍ୟତା P(E5) = $$\frac{\left|E_5\right|}{|S|}=\frac{6}{6}=1$$
ବି.ଦ୍ର. : ଫଳ ≤ 6 ଘଟଣାଟି ଏକ ନିଶ୍ଚିତ ଘଟଣା ହେତୁ ସମ୍ଭାବ୍ୟତା l ହେବ ।

(vi) ଫଳ > 6 ଏକ ଘଟଣା E6
∴ E = ϕ ⇒ |E6| = 0
∴ ଫଳ > 6ରେ ସମ୍ଭାବ୍ୟତା P(E6) = $$\frac{0}{6}=0$$
ବି.ଦ୍ର. : ଫଳ > 6 ଏକ ଅନିଶ୍ଚିତ ଘଟଣା ହେତୁ ସମ୍ଭାବ୍ୟତା 0 ହେବ ।

Question 4.
ଗୋଟିଏ ଜାର୍‌ରେ 5 ଗୋଟି ନାଲି, 6 ଗୋଟି ସବୁଜ ଏବଂ 4 ଗୋଟି ନୀଳ ମାର୍ବଲ ରହିଛି । ଜାରୁରୁ ଯଦୃଚ୍ଛା ଗୋଟିଏ ସବୁଜ ମାର୍ବଲ୍ ବାହାର କରିବାର ସମ୍ଭାବ୍ୟତା ସ୍ଥିର କର ।
ସମାଧାନ:
ଗୋଟିଏ ଜାର୍‌ରେ 5ଟି ନାଲି, ଟି ସବୁଜ ଓ 4 ଗୋଟି ନୀଳ ମାର୍ବଲ ଅଛି ।
ସବୁଜ ମାର୍ବଲ ସଂଖ୍ୟା = 6
ସମୁଦାୟ ମାର୍ବଲ ସଂଖ୍ୟା = 5 + 6 + 4 = 15
ଗୋଟିଏ ସବୁଜ ମାର୍ବଲ ବାହାର କରିବାର ସମ୍ଭାବ୍ୟତା

Question 5.
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଢ଼ାଗଲା । ଯଦି E ଘଟଣାଟି ‘ଫଳ ଏକ ଯୁଗ୍ମ ସଂଖ୍ୟା’’କୁ ସୂଚାଏ ତେବେ E ଘଟଣାଟି ଘଟିବାର ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
ସମାଧାନ:
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଗଲା ।
ଏହାର Sample space, S = {1, 2, 3, 4, 5, 6} ⇒ |S|= 6
‘‘ଫଳ ଏକ ଯୁଗ୍ମ ସଂଖ୍ୟା’’ ଏକ ଘଟଣା = E ∴ E = {2, 4, 6} = |E|= 3
‘‘ଫଳ ଏକ ଯୁଗ୍ମ ସଂଖ୍ୟା’’ର ସମ୍ଭାବ୍ୟତା P(E) = $$\frac{|E|}{|S|}$$ = $$\frac{3}{6}$$ = $$\frac{1}{2}$$

Question 6.
ଗୋଟିଏ ଲୁଡୁ ଗୋଟିକୁ ଥରେ ଗଡ଼ାଇଲେ ‘‘ଫଳ ଏକ ଅଯୁଗ୍ମ ସଂଖ୍ୟା’’କୁ ସୂଚାଉଥବା ଘଟଣାଟି ଘଟିବାର ସମ୍ଭାବ୍ୟତା କେତେ ?
ସମାଧାନ:
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଗଲା ।
ଏଠାରେ Sample space S = {1, 2, 3, 4, 5, 6} = |S| = 6
“ଫଳ ଏକ ଅଯୁଗ୍ମ ସଂଖ୍ୟା” ଏକ ଘଟଣା = E, |E| = 3
∴ ଏହାର ସମ୍ଭାବ୍ୟତା P(E) = $$\frac{3}{6}$$ = $$\frac{1}{2}$$

Question 7.
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଗଲା । ଯଦି ‘‘ଫଳ ≤ 5’’କୁ ସୂଚାଉ ଥ‌ିବା ଘଟଣା E ହୁଏ, ତେବେ ଉକ୍ତ ଘଟଣାଟି ଘଟିବାର ସମ୍ଭାବ୍ୟତା କେତେ?
ସମାଧାନ:
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଗଲା ।
ଏଠାରେ Sample space, S = {1, 2, 3, 4, 5, 6} ⇒ |S| = 6
‘‘ଫଳ ≤ 5 ଏକ ଘଟଣା’’ = E
∴ |E| = {1, 2, 3, 4, 5} ⇒ |E|= 5
∴ ଏହାର ସମ୍ଭାବ୍ୟତା P(E) = $$\frac{|E|}{|S|}=\frac{5}{6}$$

Question 8.
ଗୋଟିଏ ମୁଦ୍ରାକୁ 2 ଥର ଟସ୍ କରାଗଲେ ନିମ୍ନଲିଖ୍ ଘଟଣାଗୁଡ଼ିକୁ ସ୍ଥିର କରି ସେମାନଙ୍କ ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
(i) ଅତି କମ୍‌ରେ ଗୋଟିଏ H;
(ii) ଫଳରେ କେବଳ T ରହିବା;
(i) ଫଳରେ ଅତି ବେଶିରେ ଗୋଟିଏ H ରହିବା ଓ
(iv) ଫଳରେ H ନ ରହିବା
ଗୋଟିଏ ମୁଦ୍ରାକୁ 2 ଥର ଟସ୍ କଲେ Sample space S = {HH, HT, TH, TT} ଏବଂ | S | = 4
(i) ମନେକର ଅତି କମ୍‌ରେ ଗୋଟିଏ H ଆସିବାର ଏକ ଘଟଣା =
∴ E1 = {HH, HT, TH} ⇒ |E1| = 3
ଅତି କମ୍‌ରେ ଗୋଟିଏ H ଆସିବାର ସମ୍ଭାବ୍ୟତା P(E) = $$\frac{\left|E_1\right|}{|S|}=\frac{3}{4}$$

(ii) ଫଳରେ କେବଳ T ଆସିବା ଏକ ଘଟଣା = E2
∴ E2 = {TT} ⇒ |E2| = 1
∴ P(E2) = $$\frac{\left|E_2\right|}{|S|}=\frac{1}{4}$$

(iii) ଫଳରେ ଅତିବେଶିରେ ଗୋଟିଏ H ରହିବା ଏକ ଘଟଣା = E
∴ E3 = {HH, TH, TT} ⇒ |E3| = 3
∴ P(E3) = $$\frac{\left|E_3\right|}{|S|}=\frac{3}{4}$$

(iv) ଫଳରେ H ନରହିବା ଏକ ଘଟଣା E4
∴ E4 = {TT} ⇒ |E4| = 1
∴ P(E4) = $$\frac{\left|E_4\right|}{|S|}=\frac{1}{4}$$

Question 9.
ଗୋଟିଏ ମୁଦ୍ରାକୁ 3 ଥର ଟସ୍ କରାଗଲା । ସାମ୍ପଲ ସ୍ପେସ୍ଟ ଲେଖ ଓ ନିମ୍ନଲିଖ୍ ଘଟଣାମାନଙ୍କ ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
(i) ଫଳରେ କେବଳ T ରହିବା
(ii) ଫଳରେ ଅତି କମ୍ରେ ଦୁଇଟି H ଥ‌ିବା
(iii) ଫଳରେ ଅତି ବେଶିରେ ଦୁଇଟି T ରହିବା
(iv) ଫଳରେ କେବଳ H କିମ୍ବା କେବଳ T ଥ‌ିବା ଓ
(v) କୌଣସି ଫଳରେ T ନ ଥ‌ିବା
ସମାଧାନ :
ଗୋଟିଏ ମୁଦ୍ରାକୁ 3 ଥର ଟସ୍ କରାଗଲା ।
ଏଠାରେ Sample space S = {HHH, HTH, HHT, HTT, TTT, TTH, THT, THH) ଏବଂ | S|=8
(i) ଫଳରେ କେବଳ ‘T ରହିବା ଏକ ଘଟଣା = E1
∴ E1 = {TTT} ଏଠାରେ |E1|= 1
∴ P(E1) = $$\frac{\left|E_1\right|}{|S|}=\frac{1}{8}$$

(ii) ଫଳରେ ଅତି କମ୍ରେ ଦୁଇଟି H ଥିବା ଏକ ଘଟଣା = F
∴ F = {HTH, HHT, THH, HHH} ⇒ |F|= 1
∴ P(F) = $$\frac{|F|}{|S|}=\frac{4}{8}=\frac{1}{2}$$

(iii) ଫଳରେ ଅତି ବେଶିରେ ଦୁଇଟି T ଥିବା ଏକ ଘଟଣା = A
∴ A = {HHH, HHT, HTH, HTT,THH, THT, TTH} = |A| = 7
P(F) = $$\frac{|A|}{|S|}=\frac{7}{8}$$

(iv) ଫଳରେ କେବଳ H ଥ‌ିବା ଏକ ଘଟଣା E = {HHH}, ⇒ |E| = 1
P(E) = $$\frac{|E|}{|S|}=\frac{1}{8}$$
ସେହିପରି F କେବଳ T ଥିବା ଏକ ଘଟଣା । P(F) =
ଫଳରେ କେବଳ H ଥ‌ିବା କିମ୍ବା କେବଳ T ଥ‌ିବା ଘଟଣାଟି E ∪ F
P(E ∪ F) = P(E) + P(F) = $$\frac{1}{8}+\frac{1}{8}=\frac{2}{8}=\frac{1}{4}$$
(∵ E ଓ F ଘଟଣାଦ୍ଵୟ ପରସ୍ପର ବହିଃର୍ଭୁକ୍ତ)

(v) କୌଣସି ଫଳରେ T ନଥିବା ଏକ ଘଟଣା = E
∴ E = {HHH} = |E|=1
∴ P(E) = $$\frac{|E|}{|S|}=\frac{1}{8}$$

Question 10.
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଦୁଇଥର ଗଡ଼ାଇ ଦିଆଯିବାରେ ନିମ୍ନଲିଖତ ଫଳ ଲବ୍‌ଧ ହେବାର ସମ୍ଭାବ୍ୟତା ସ୍ଥିର କର ।
(i) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ = 6,
(ii) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ = 4,
(iii) ସଂଖ୍ୟା ଦୁଇଟିରୁ ପ୍ରତ୍ୟେକଟି ଗୋଟିଏ ଗୋଟିଏ ବର୍ଗ ସଂଖ୍ୟା,
(iv) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ 2 10,
(v) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ < 6 ଓ
(vi) ପ୍ରଥମ ସଂଖ୍ୟାଟି ଅଯୁଗ୍ମ ଓ ଦ୍ବିତୀୟଟି 61
ସମାଧାନ :
(i) ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ 2 ଥର ଗଡ଼ାଇଲେ Sample space ସଂଖ୍ୟା |S| = 6² = 36 ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ 6 ଆସିବା ଏକ ଘଟଣା = E ∴ E = {15, 51, 24, 42, 33} |E| = 5 ∴ P(E) = $$\frac{|E|}{|S|}=\frac{5}{36}$$

(ii) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ = 4 ଏକ ଘଟଣା T, ∴ T = {13, 31, 22} |T| = 5 ∴ P(T) = $$\frac{|T|}{|S|}=\frac{3}{36}=\frac{1}{12}$$

(iii) ଦୁଇଟି ସଂଖ୍ୟାରୁ ପ୍ରତ୍ୟେକଟି ଗୋଟିଏ ଗୋଟିଏ ବର୍ଗ ସଂଖ୍ୟା ଏକ ଘଟଣା = F ∴ F = {11, 44} |F| = 2 ∴ P(F) = $$\frac{|F|}{|S|}=\frac{2}{36}=\frac{1}{18}$$ (iv) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ > 10 ଏକ ଘଟଣା = E
∴ E = {46, 64, 55, 56, 65, 66} = |E| = 6
∴ P(E) = $$\frac{|E|}{|S|}=\frac{6}{36}=\frac{1}{6}$$

(v) ସଂଖ୍ୟା ଦୁଇଟିର ଯୋଗଫଳ < 6 ଏକ ଘଟଣା = |E |
∴ E= (11, 12, 13, 14, 22, 23, 32, 41, 31, 21} |E| = 10
P(E) = $$\frac{|E|}{|S|}=\frac{10}{36}=\frac{5}{18}$$

(vi) ପ୍ରଥମ ସଂଖ୍ୟାଟି ଅଯୁଗ୍ମ ଓ 2ୟ ସଂଖ୍ୟାଟି 6 ଏକ ଘଟଣା ।
∴ E = {16, 36, 56) = |E| = 3
∴ P(E) = $$\frac{|E|}{|S|}=\frac{3}{36}=\frac{1}{12}$$

Question 11.
ଏକ ପରୀକ୍ଷଣରେ ପରସ୍ପର ବର୍ହିଭୁକ୍ତ ଦୁଇଟି ଘଟଣା E1 ଓ E2 ଏପରିକି P(E1) = 2P(E2) ଓ P(E1) + P(E2) = 0.9 । ତେବେ E1 ∪ E2 ଘଟଣା ତଥା E1, ଘଟଣାର ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
ସମାଧାନ :
ଏକ ପରୀକ୍ଷଣରେ ପରସ୍ପର ବର୍ହିଭୁକ୍ତ ଦୁଇଟି ଘଟଣା E1 ଓ E2
P(E1) = 2P (E2) P(E1) + P(E2) = 0.9
∴ P(E1) + P(E2) = 0.9 2P(E2) +P(E2) = 0.9
= 3P(E2) = 0.9 = P(E2) = 0.9 = 0.3 P(E1) = 2P (E2) = 2 × 0.3 = 0.6
∴ P(E1 ∪ E2) = P(E1) + P(E2) = 0.6 + 0.3 = 0.9.
P(E) = 0.6

Question 12.
ଯଦି E, ଓ E, ଏପରି ଦୁଇଟି ଘଟଣା ଯେଉଁଠାରେ P(E1) = $$\frac{5}{8}$$, P(E2) = $$\frac{2}{8}$$ ଓ P(E1 ∩ E2) = $$\frac{1}{8}$$ ତେବେ ନିମ୍ନଲିଖତଗୁଡ଼ିକ ସ୍ଥିର କର ।
(i) P(E1 ∪ E2)
(ii) P(E1’)
(iii) P(E2’)
(iv) P(E’1 ∪ E’2)
ସମାଧାନ :
E, ଓ E, ଏପରି ଦୁଇଟି ଘଟଣା ଯେଉଁଠାରେ P(E1) = $$\frac{5}{8}$$, P(E2) = $$\frac{2}{8}$$
P(E1 ∩ E2) = $$\frac{1}{8}$$
(i) P(E1 ∪ E2) = P(E1) + P(E2) – P(E1 ∩ E2) = $$\frac{5}{8}+\frac{2}{8}-\frac{1}{8}=\frac{6}{8}=\frac{3}{4}$$

(ii) P(E1’) = 1 – P(E1) = 1 – $$\frac{5}{8}=\frac{3}{4}$$

(iii) P(E2’) = 1 – P(E2) = 1 – $$\frac{2}{8}=\frac{6}{8}=\frac{3}{4}$$

(iv) P(E’1 ∪ E’2) = P(E1 ∩ E2)’ = 1 – P(E1 ∩ E2) = 1 – $$\frac{1}{8}=\frac{7}{4}$$

Question 13.
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଇଲେ ‘ଫଳ 5 କିମ୍ବା ଏକ ଅଯୁଗ୍ମ ସଂଖ୍ୟା’ ଆସିବାର ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
ସମାଧାନ :
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଇଲେ ସାମ୍ପଲ୍ ସ୍ପେସ୍
S = {1, 2, 3, 4, 5, 6} = |S| = 6
ମନେକର ଫଳ 5 ଏକ ଘଟଣା = E1 ଏବଂ ଫଳ ଏକ ଅଯୁଗ୍ମ ସଂଖ୍ୟା ଘଟଣା = E2
E1 = {5} |E1| = 1
E2 = {1, 3, 5} = |E2|=3
E1 ∩ E2 = {5} = |E1 ∩ E2| = 1
ଫଳ ‘5’ କିମ୍ବା ଏକ ଅଯୁଗ୍ମ ସଂଖ୍ୟା ଘଟଣା E1 ∪ E2
P(E1 ∪ E2) = P(E1) + P(E2) – P(E1 ∩ E2)
⇒ P(E1 ∪ E2)
= $$\frac{\left|E_1\right|}{|S|}+\frac{\left|E_2\right|}{|S|}-\frac{\left|E_1∩E_2\right|}{|S|}$$
= $$\frac{1}{6}+\frac{3}{6}-\frac{1}{6}-\frac{3}{6}=\frac{1}{2}$$

Question 14.
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଇବାରୁ ‘‘ଫଳ ଅଯୁଗ୍ମ କିମ୍ବା ଫଳ ≥ 3’ ଘଟଣାଟିର ସମ୍ଭାବ୍ୟତା ନିରୂପଣ କର ।
ସମାଧାନ :
ଗୋଟିଏ ଲୁଡୁଗୋଟିକୁ ଥରେ ଗଡ଼ାଇଲେ Sample space S = {1, 2, 3, 4, 5, 6} = | S | = 6
ମନେକର ‘‘ଫଳ ଅଯୁଗ୍ମ ଏବଂ ଫଳ ≥ 3’’ ଏକ ଘଟଣା = E2
∴ E = {1, 3, 5} = |E1| = 3 ଏବଂ E2 = {3, 4, 5, 6} = |E2| = 4
∴ (E1 ∩ E2) = {3, 5} = (E1 ∩ E2) = 2
ଫଳ ଅଯୁଗ୍ମ କିମ୍ବା ଫଳ ≥ 3 = E1 ∪ E2
∴ P(E1 ∪ E2) = P(E1) + P(E2) – P(E1 ∩ E2)
= $$\frac{\left|E_1\right|}{|S|}+\frac{\left|E_2\right|}{|S|}-\frac{\left|E_1∩E_2\right|}{|S|}$$
= $$\frac{3}{6}+\frac{4}{6}-\frac{2}{6}=\frac{5}{2}$$

## CHSE Odisha Class 12 Alternative English Grammar Prepositions

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

## CHSE Odisha Class 12 Alternative English Grammar Prepositions

A. Preposition
Definition:
A preposition is a world that is put before a noun or pronoun in order to show more relation between them.

Example:
1. The lamp is on the table
Here, the preposition ‘on’ shows the relation of the noun (the lamp) with another noun (the table)
2. She fell into the well.
Here, the preposition ‘into’ shows the relation of the pronoun (she) with the noun (well).
3. Akash gives respect to me.
Here, the preposition ‘to’ shows the relation of the noun (Akash) with the pronoun (me).
4. She is hot under me.
Here, the preposition ‘under’ shows the relation of the pronoun (she) with another pronoun (me).

Kinds Of Prepositions

Prepositions can be classified as the following five types: such as:

1.  Simple prepositions: as
At, after, by, to, in, up, with, over, etc.
2. Compound prepositions: as
about, across, against, before, beside, into, until, within, etc.
3. Participle prepositions: as
Accepting, during, etc.
4. Phrasal prepositions: as
Along with, by virtue of instead of on account of by way of on behalf of according to, etc.
5. Double prepositions: as
from among, from beneath, from under, out of, etc.

Usage Of Prepositions

Prepositions are generally used before a noun phrase. But sometimes, they are used at the end of a sentence.
Example:
A. Before the noun phrase:

• Don’t sit on that broken chair.
• That by writes with his left hand.

B. At the end of a sentence:

• Used in a wh-question sentence.
(i) What is that animal like?
(ii) What are you looking for?
• Used in a sentence with a relative clause.
(i) Is this the book you were looking for?
(ii) I know the may you are talking about

PREPOSITION OF POSITION

Use of ‘AT’
‘At’ is used in
(a) Public places:
Examples:
1. There are no buses at the bus stop now.
2. She will be at the cinema now.

Examples:
1. Mr. Patra is living at 24, Mahatma Gandhi Road, BBSR.
2. You can find him at plot no, 5, Basanti Colony, Rourkela.

(c) A point in space:
Examples:
1. The watchman is standing at the gate.
2. The child is sitting at his desk.
3. He is at home today.
4. The temple is at the end of the village.

(d) With events:
Examples:
1. I met the girl at my sister’s wedding.
2. They are busy at the meeting.
3. There was a big crowd at the match.
4. They were at the party at that time.

(e) A place on the journey:
Examples:
1. We got down at Puri
2. This train doesn’t stop at Ballikuda.
3. You have changed the bus at Bhubaneswar.

Use Of ‘In’

‘In’ is used with:
(a) Large areas:
Examples:
1. Jagannath temple is in Odisha.
2. He lives in a town.

(b) Enclosed places:
Examples:
1. They bathed in a pond.
2. My mother is in the kitchen.
3. Keep your money in the box.

Examples:
1. My uncle lives in Kedar Gouri Road.

(d) Places of work:
Examples:
1. My mother works in a hospital.
2. Mahesh works in a factory.
3. I work in an office.
4. He works in a hotel.
5. My brother works in a bank.

When it refers to an open place, ‘on’ is used instead of ‘in’.
Examples:
1. My father works on a firm.
2. Siba works on the railway.
3. Ajit works on a tea plantation.

In the case of indefinite office, ‘in’ is used, as
Examples:
1. Kunal works in a bank.
2. Puspa works in a library.

In the case of a definite office, ‘at’ is used.
Examples:
1. He works at the main branch of the Bank of India, Bhubaneswar.
2. Pinki works at the University library.

Use Of ‘On’

‘On’ is used with:
(a) Surface:
Examples:
1. The books is on the table.
2. The clock is on the walk.
3. The children are playing on the beach.
4. A man is standing on the roof of the house.
5. There is a beautiful picture of Nehru at page 5.

(b) A-line:
Examples:
1. We line on a small river that flows into Chilika.
2. This house stands on the main road.

But ‘in’ and ‘at’ with buildings:
1. I left my umbrella in school.
2. My youngest daughter is at school and the eldest one is at college.
3. I spent the whole evening at the public library.
4. There is a big hall in the library.

‘In’, ‘on’, ‘all’ with streets or roads:
Examples:
1. Kuni got a job in Netajee Road.
2. My house is in Netajee Road.
3. You will find meat 65, Netajee Road.

Position And Movement:

Some prepositions relate to both position and movements. Others relate to either position or movement.
1. Use of ‘to’ and ‘at’:
Examples:
1. The boy went to the blackboard.
2. He stood at the blackboard.
3. He threw the ball at me.
At is used for both position and movement, but ‘to’ is used for only movement.

2. Use of ‘above’ and ‘over’
Examples:
1. The clock is over/above the door.
2. There is a temple above my house.
3. A kite is flying over our house (Not above)
Hence, over is used for both position and movement, but above is used only for the position.

3. Use of ‘below’ and ‘under’:
Examples:
1. I stood at the door under/below the clock.
2. The children are playing under the tree. (Not below)

4. Use of ‘between’and‘morning’:
Examples:
1. Divide the cake between Kuni and Muni.
2. There is a beautiful garden between the house and the road.
3. The teacher is standing among the pupils.
4. Divide the sweets between the four children.

5. Use of ‘In front oF and behind:
Examples:
1. The car is in front of the bus.
2. The bus is behind the car.

6. Use of ‘into’:
Examples:
1. We jumped into the pond.
2. The teacher walked into the classroom.
3. Pour the milk into the cup.

7. Use of ‘out oF and ‘outside’:
Examples:
1. He is out of town.
2. When the class finished the children rushed out of the classroom
3. He asked me to sit on a bench outside the closed door.
4. Some people are shouting outside the office.

8. Use of ‘onto’:
Examples:
The cat jumped onto the table.

9. Use of ‘up’ and ‘down’:
Example:
Mantu is going up the stairs, but Puspa is coming down the stairs.

10. Use of ‘along’, ‘across’ and ‘through’:
Examples:
1. There are trees all along the road.
2. The car is running along the road.
3. There is a bridge across the stream.
4. He swam across the river.
5. A road goes through the village.
6. The train went through the tunnel.

11. Use of ‘round’:
Example:
The earth moves round the sun.

12. Use of ‘against’:
Example:
He placed the ladder against the wall.

13. Use of ‘near’:
Example:
The supermarket is very near the library.

14. Use of ‘try’:
Example:
Come and sit by me.

15. Use of ‘beside’:
Example:

Exercise For Practice:

1. Fill in the blanks choosing ‘in’, ‘on’, or ‘at’.
1. I will meet you __________ the bus stand.
2. We live __________ a bit building.
3. My village is __________ a river bank.
4. My family lives __________ 24, Gandhi Road.
5. You will find a very tall building __________ the main road.
6. Sunita lives __________ Park Street.
8. You will find him __________ the garden now.
9. He is not __________ home now.
10. Someone has drawn a picture the door.
11. He will be in two years more __________ school before the goes to any college.
12. The temple is __________ the top of the hill.
13. She is a teacher __________ a village school.
14. We walked for a long time __________ the beach.
15. Do not walk __________ the middle of the road.
16. Many laborers are working __________ my firm.
17. I last saw you __________ this theatre.
18. Sushanta left his bag __________ the college.

2. Fill in the blanks by choosing the appropriate prepositions:
1. He walked __________ my room to see me. (into/to)
2. A river flows __________ the hill. (under / below)
3. We will need about to get __________ the river. (across / through)
4. A fan is moving __________ my head (over / above)
5. He threw the book __________ the table. (to/into)
6. I felt __________ in my pocket and took out a coin (into / in)
7. The cat jumped __________ the mouse (on/into)
8. I thought someone was standing __________ me (in front of / behind)
9. We drove __________ the road for fifty miles (on/along)
10. Water flows __________the pipe (along / through)
11. The car turned __________ the comer of the road. (on / round)
12. He falls __________ the horse (down/ off)
13. All the five brothers were quarreling __________ themselves. (between/among)
14. The bird flew __________ the nest (from/out of)
15. The squirrel hid __________ a bush. (blow / under)
16. I came __________ the stairs to welcome my friend. (below / down)

Preposition of Time:

When the duration of time of a happening is expressed we use the preposition life for, during, from, to, till, until by, etc. These are called prepositions of time.
1. Use of ‘At’
‘At’ is used with:
(a) Clock time:
Examples:
1. The train left at 7.30.
2. He reached at half past nine.

(b) Exact moment of day or night:
Example:
1. It is hot at noon.
at sunrise, at sunset, at midnight, at dawn, at dusk, at midday, etc.

(c) Lunch time:
Example:
1. He will be here at lunchtime, at breakfast, at dinner, etc.

(d) Festival, short holidays:
Example:
1. He goes home at Dusserah every year, at Christmas, at Id, at Holi, etc.

(e) Age:
Example:
1. He was admitted to our school at the age of five.

(f) Others:
Example:
1. The villain was killed at the end. at the moment etc.

Use of ‘On’
‘On’ is used with:
(a) Day of the week:
Example:
1. Offices and schools are closed on Sundays.

(b) Dates:
Examples:
1. Mahatma Gandhi was bom on 2nd October 1869.
2. My friend usually visits us on my birthday.

(c) Day + part of day:
Example:
1. I will go home on Sunday morning in the night of the 25th, on a came after-noon, etc.

(d) Special days:
Example:
1. My father gave me gifts on my birthday on New Year’s Day on Republic Day etc.

Use of ‘In’
‘In’ us used with:
(a) Week/month/season/year/century:
Examples:
1. He was absent from college in the first week.
2. The First World War began in 1914.
3. We are now in the 21 st century.

(b) The time something takes to be completed:
Examples:
1. The chief guest finished his speech in five minutes.
2. The clerk typed the report in half an hour.

(c) Part of day/night:
Example:
1. I get up in the morning every day.
2. The doctor is free in the afternoon.

Use of ‘on time’ and ‘in time’:
Examples:
1. We reached the station in time to buy the tickets.
2. The train reached on time.

Use of at night and in the night:
Examples:
1. I cannot sleep at night.
2. I woke up on night.

Use of During:

(a) We use during with a period of time in which something took place for a comparatively short period of time.
Examples:
1. During the summer vacation, I was ill for ten days.
2. I had slept only for two hours during the last night.

(b) We use during with an event:
Examples:
1. He broke his leg during the match.
2. Somebody shouted during the meeting.
3. The woman wept during her speech.

Use of ‘For’:

We use for to say how long something continued.
Examples:
1. We lived there for thirty years.
2. My father works in the factory for eight hours.
3. She is staying only for a week.
4. Pintu was at college for three years.
5. I have been waiting for two hours.

Use of ‘From… To’:

When it implies ‘for’, we use from… to.
Example:
1. My father works in the factory from 10 am to 6 pm. every day:

Use of Since And From:

We use (a) since and (b) from with the starting point of a period of time.
(a) Examples:
1. I have not seen you since June last.
2. The baby has been crying since morning.
(b) Examples:
1. Books will be on sale from tomorrow.
2. I hope they will be friends from now.
3. She was deaf from birth.
4. I worked from 7 o’clock.

Use of Until, Till, Upto / By, Before:

We use until, till, upto, by, and before with terminal points of a period of time. Examples:
1. You can keep the book until /till Friday.
2. We didn’t get up until/till morning.
3. They worked up to 3 pm.
N.B: Both until and till are used in positive and negative sentences, but up to is not used in negative sentences.

We can’t write:
They didn’t work upto 3 pm

Use of By:

It is used for ‘on’ or ‘before’ but not for later than Examples:
1. You should return home by the evening.
(= not later than the evening)
2. Can you send the book by Friday?
(= on or before Friday)

Use of Before:

It is used for earlier than:
Example:
1. You must see me before Friday.

Prepositions are also used for:
1. Cause:
Examples:
1. The patient died of cholera.
2. Smita was lined for the offense.
3. The man died from a wound.
4. She helped me out of kindness.
5. The student is sick with a cold.

2. Means:
Examples:
1. My uncle came by bus.
2. Pinkicameinatrain.
3. He broke the lock with a hammer.
4. The village was destroyed by fire.

3. Accompaniment:
Examples:
1. The child is playing with his friends.
2. Mr. Patra came without his wife.

4. Support and Opposition:
Examples:
1. I stand for my brother.
2. She is against me.
3. We are with you all the time.

5. Having, possession:
Example:
1. She is a girl with long hair.

6. Concession:
Example:
1. We went out in spite of the heavy rain.

7. State/condition:
Change of State
Examples:
1. The road is under repair.
2. Translate the passage into English.

8. Aim / Goal / Purpose / Target Example:
1. I bought a pen for Raju.

9. Origin/Source:
Examples:
1. This road runs from Bhubaneswar to Puri.
2. I borrowed a hundred rupees from my friend.

10. Separation:
Examples:
1. The rich man was robbed of his money.
2. The cat jumped off the chair.

11. Relation/possession:
Examples:
1. He is a friend of mine.
2. What is the name of your village.

12. Material / Ingredient:
Examples:
1. The chair is made of wood.
2. Flour is made from wheat.

13. Part/portion:
Example:
1. This is the door of this house.

14. Manner:
Examples:
1. She is like her mother.
2. He treated me with kindness.

15. Subject matter:
Examples:
1. She is reading a book in English grammar.
2. I told him about my institution.

Exercise For Practice

1. Use in, on or at where necessary.

1. We go to the cinema usually __________ Sundays.
2. Rajat will return __________ the evening.
3. Where were you last Saturday?
4. The train arrived __________ 7 o ’ clock __________ the morning.
5. Tikun was born __________ the month of September.
6. The sun sets __________ 5 pm __________December.
7. My sister’s wedding will take place __________ 15th May.
8. He is busy __________ the moment.
9. Can you come __________ this evening?
10. My mother goes to the temple __________ Monday morning.
11. I will finish my breakfast __________ two minutes.
12. The bus left __________ time.
13. I heard a loud noise __________ night.
14. We reached station __________ time to catch the train.
15. Our college will close __________ two weeks time.
16. What are you doing __________ tomorrow morning.
17. I was born __________ 1985.
18. We visit Puri __________every summer.
19. Don’t move out __________ noon. It is very hot.
20. I will see you __________ next Friday.

2. Choose the correct prepositions and fill in the blanks.

1. I have known Smita __________ two years. (for/since).
2. They have been at college __________ 2005. (from/since)
3. You must finish the work __________ tomorrow. (by/until)
4. We visited many places __________ the holidays. (between/during)
5. They kept working __________ evening? (until / by)
6. My friend will be here any day __________ Monday and Wednesday. (by/between)
7. He will wait for me __________ 5 o’clock. (until/ by)
8. What have you been doing __________ morning? (from/since)

Preposition With Verbs, Adjectives & Nouns
Verb + Preposition

• accuse of – The police accused him of stealing a radio.
• add to – If you add 3 to 4, you get 7.
• admit to – Ajit is admitted to +2 Arts of our college.
• agree to – I don’t agree to his suggestion.
• agree on – I agree on a date to start our journey.
• agree with – I agree with my friends to go on a picnic.
• aim at – The hunter aimed at the bird.
• apologize to/for – I apologize to you for not replying to your letter.
• appeal to, for – I apologize to you for not replying to your letter
• apply to for – The poor man applied to the bank for a loan.
• approve of – A approve of your decision.
• argue for – The workers argued for a rise in their pay.
• arrive at – We arrived at the station on time.
• arrive in – We arrived in Bhubaneswar to attend to the meeting.
• attend on – The queen had a good doctor attending to her.
• attend to – Attend to what your teacher is saying.
• avail of – You should avail yourself of the opportunity.
• believe in – I believe in God.
• belong to – This cost belongs to my father.
• beware of – Beware of the dog.
• blame for – Ramesh blamed his teacher for his failure.
• blame on – He blamed his failure on his teacher.
• boast of – A wise man never boasts of his knowledge.

• care for – She cared for his father all through his long illness.
• charge with – The police charged the man with murder.
• compare to/with – The poet compares her face to/with the moon.
• complain about – The old man never complains about/of/ against his
• of/against – suffering.
• compete with /against – Ranjit is competing with Ajit for the first position.
• confide to – I confide my troubles to my friend.
• confide in – I confide in his honesty.
• congratulate on – We congratulate him on his success.
• consent on – The parents consented to their daughter’s marriage.
• consist of – The committee consists often of members.
• consist in – Happiness consists in contentment.
• cure of – The doctor can cure you of your illness.
• consult with – The head minister consulted with other teachers about the development of the school.
• deal in – The shop deals in dress materials only.
• deal with – Itisveryhardtodealwithhim.
• deprive of – A prisoner is deprived of his rights of freedom in jail.
• die from – The poor man died from hunger / a wound.
• die off – Many people died of Malaria.
• differs from – French differs from English.
• differ with on – I am sorry to differ with you on that question.
• escape from – The thief escaped from the jail.
• exempt from – Textbooks are exempted from tax.
• gaze at – We gazed at the stars.
• hope for – Let us hope for the best.
• impose on – We should not impose more opinions on others.
• insist on/upon – The children insisted on/upon visiting the park.
• Interfere in – It is not good to interfere in other ‘s private life.
• interfere with – Don’t interfere with this machine.
• knock at/on – The visitor knocked at/on the door.
• listen to – I often listen to good music.

• mock at – They mocked my shyness.
• object to – I object to your smoking in this room.
• part with – Little children don’t want to part with their toys.
• pick up – The mother picked up the baby.
• pick out – She was picked out from thousands of applicants for the job.
• point to – Both the hour hand and minute hand pointed to twelve.
• point at – The hunter pointed a gun at the bird.
• point at – Point out the spelling mistakes.
• protect out – An umbrella protects us from the sun and the rain.
• preside of – The Prime Minister presides at the meetings of the cabinet.
• preside over – The City Council is presided over by the Mayor.
• protest against – People protested against the government’s new food policy.
• remind of – The photo reminds me of many school days.
• recover from – I have recovered from my illness.
• repent of – The man repented of his misdeeds.
• restrain from – Restrain the child from mischief
• result from – The loss resulted from his negligence.
• result in – Fifty percent of road accidents result in heat injuries.
• retire from – My other retired from service last year.
• shout at – The traffic police shouted at the car driver.
• succeed in – He succeeded in his attempt.
• suffer from – The child is suffering from fever.
• supply with – The cyclone-affected people were supplied with relief materials.

• suspect of – The whole class suspended Biju for stealing the wall clock.
• sympathize with – We all sympathized with the unfortunate man.
• wait for – I am waiting for my friend.
• worry over – The mother is worried about the health of her sick child.
• wonder at – The village boy wondered at the tall buildings of the city.

• absent from – I was absent from college yesterday as I had a headache.
• accustomed to – I am not accustomed to such work.
• afraid of – Who is not afraid of snakes?
• angry with – My other was angry with me for my rude behavior.
• anxious about – I am always anxious about my father’s health.
• anxious for – I am anxious for my examination results.
• ashamed of – Kunal is ashamed of his bad conduct.
• astonished at – All were astonished at his unexpected behavior.
• aware of – I was quite aware of this before I joined this party.
• blind to – Mothers are usually blind to the faults of their children.
• busy with – Pravakar is now busy with his friends.
• capable of – This child is capable of handling computers.
• careful about/with – The rich man is very careful about I money.
• careful of – Be careful of the ice on the road.
• careful with – You should be careful with that man, he is dangerous.
• confident of – Our soldiers are confident of winning the war.
• conscious of – The boy is very conscious of his manners when he is among the elders.

• different from – Your habits are different from mine.
• delighted with – The teacher was delighted with my success.
• disappointed with- I am disappointed with your examination rescue.
• disappointed at – I was disappointed at not finding her at home.
• eligible for – Only graduates are eligible for this job.
• envious of – We should not be envious of others’ happiness.
• equal to – A pound is roughly equal to 500 grains.
• essential to – Vitamins are essential to good health.
• essential for – Land is essential for food.
• familiar with – I am familiar with this city.
• familiar to – This city is familiar to me.
• famous for – Alexandar was famous for his bravery.
• fit for – He is fit for the job.
• fond of – Children are fond of sweets I playing.
• good at /bad at hopeless at – The boy is very good at / bad at mathematics.
• free with – He is free with his money.
• free of – They are happy to give their services free of change.
• free from – He is now free from disease.
• grateful to for – The blind man was gratefûlto the child for his help.
• good / kind / nice of- I was kind of you to help me.
• guilty of – He was found guilty of murder.
• ignorant of – They were ignorant of any events outside.
• ill with – The baby is ill with influenza.
• innocent of – The young man pleaded that he was innocent of the charges made against him.
• interested in – Some students are not interested in mathematics.
• jealous of – My neighbors are jealous of me for my fortune.
• keen on – I am keen on buying a house.
• proud of – We are proud of our brave soldiers.
• responsible for – Who is responsible for this tembleness?
• responsible to – The cabinet is directly responsible to Parliament.
• satisfied with / pleased with – The boy is satisfied with his performance.

• sorry for / about – We are sorry for/about all our mistakes.
• successful in – The Indian soldiers were successful in defeating the Pakistanis.
• superior to – Man is superior to animals.
• sure of – Are you sure of his honesty?
• surprised/shocked at – We were surprised at the news.
• suspicious of – I am suspicious of the man’s intentions.
• tired of – He was tired to walk. He wanted to rest.
• useful for – Soap is useful for removing dirt.
• useful to – This book is very useful to me.
• weak in – I am weak in mathematics.
• worthy of – You are worthy of all my praise.
• worried about – The government is very worried about the defènce of the country.

Noun + Preposition

• admiration for – I have a lot of admiration for my teachers.
• affection for – He felt a great affection for the old man.
• agreement with – I have an agreement with my friend to exchange books between us.
• belief in – She has lost her belief in God.
• a consequence of – You have lost your job as a consequence of your laziness and rudeness.
• love for – A mother’s love for her children is genuine.
• love of – A soldier’s love of his country can’t be questioned.
• objection to – I have an objection to your going out at night.
• proof of – Gifts of nature are proof of God’s love for man.
• reaction to – My reaction to his proposal was favorable.
• demand for – There is a lot of demand for sugar during festival times.
• difficulty in – I face difficulty in teaching the child.
• difficulty with – IamhavingdifficultywithmyneigJbour.

• discussion on about of – We had a discussion on the discipline of the college.
• effect on – The medicine has no effect on him.
• influence on – Teachers have a great influence on their students.
• example of – Lal Bahadur was an example of patriotism and honesty.
• knowledge of – Our teacher has a good knowledge of English.
• reason for – You must give reasons for beating that boy.
• relief from – A heavy shower gave us great relief from the unbearable heat.
• remedy for – There is no remedy for cancer.
• restriction on – There is a restriction on people entering this building.
• solution to/for – Can you suggest a solution to/for this problem?
• sympathy for – I feel sympathy for this helpless child.
• need for – There is no need for you to come here again.
• congratulation – Congratulation on your excellent performance.
• a rise / an increase in – There has been a rise / an increase in the price of sugar recently.
• a fall/decrease in – There has been a fall/decrease in the supply of wheat to our state.
• cause of – What is the cause of Malaria?

Exercise For Practice.

1. Fill in the blanks with the appropriate prepositions.

1. Most people believe __________ God.
2. You can’t depend __________ a selfish person.
3. We laughed __________ his silly jokes.
4. Our class consists __________ forty students.
5. The baby is suffering __________ fever.
6. Every year many people die __________ hunger.
7. Children should care __________ their old parents.
8. Do not throw stones __________ others.
9. I agree __________ you on this proposal.
10. The people accused the old man __________ murder.
11. This shop deals __________ food items.
12. The two brothers diffèr __________ each other in many ways.
13. I succeeded __________ solving the problems.
14. Children should not be deprived __________ the joy of playing.
15. The students are going __________ a picnic.
16. Our house is built __________ bricks.
17. The property was divided __________ four parts.
18. I am looking __________ a furnished house.

2. Put in the correct prepositions.

1. I am not afraid __________ snakes.
2. You should be sorry __________ your misbehavior.
3. We are all angry __________ his carelessness.
4. Latamangeshkar is famous __________ her singing.
5. Ranjit is good __________ mathematics.
6. Smoking is bad __________ health.
7. I am aware __________ your difficulties.
8. The young dancer was satisfied __________ her performance.
9. The child was keen __________ having chocolate.
1o. The oldman is worried __________ his poor health.
11. It was very clever __________ you to avoid that boy.
12. My problems are similar __________ yours.
13. I am not used __________ bad language.
14. You are late __________ college.
15. I am weak __________ English.
16. He is blind __________ one eye.
17. He is busy __________ his office work.
18. I can’t rely __________ you in this matter.

3. Fill in the blanks with appropriate prepositions.

1. There has been an increase __________ the price of sugar.
2. He developed an interest __________ music in his childhood.
3. I have a lot of differences __________ you.
4. What is the difference __________ a car and a bus?
5. I have got an invitation __________ the dinner party
6. My method __________ working is different from yours.
7. She is an expert __________ dogs.
8. What is the cause __________ your worry?
9. We made a request __________ more money.
10. An accident caused damage __________ the car.
11. I don’t have sympathy __________ people like you.
12. My advice has no effect __________ him.

Fixed Prepositions

What are fixed prepositions?
The prepositions ofthe prepositional phrases which are fixed expressions in order to express a definite meaning in a sentence are called fixed prepositions. Their forms remain unchanged.
Examples:
1. I was attached by a dog.
2. The news was sent by telegram.
3. I go to school by bicycle.

Here, in the above sentences, the prepositional phrases; by a dog, by telegram, by bicycle remain unchanged. Because the sentence nos. can’t be written as:-
The news was sent by telegram.
Similarly, the third sentence can’t be written as:
1. I go to school by bicycle.
2. Rather we can write like this I go to school on my / bicycle.
3. We can only write, by bicycle.

Examples of Fixed Expressions

Use of ‘At’:

‘At’ as a fixed preposition is used when it denotes the following meanings.
1. at ease: comfortably
Ex:-1 am sitting at ease in this armchair.
2. at heart: deep inside
Ex:- At heart, he is very kind.
3. at length: in great detail or for a long time
Ex:- The matter was discussed at length in the meeting.
4. at war / at peace: in a state of war/peace
Ex:- The two countries are not at war now, they are at peace.
5. at rest: not doing anything active, not worrying about anything.
Ex:- you should see him when he is at rest.
6. at first: at the beginning
Ex:- I did not like him at first, but not I do.
7. at last: It has happened after he has been waiting for it for a long time.
Ex:- At last, he found a girl for his son to merry.

8. at play: playing
Ex:- There was a noise of children at play.
9. at short notice: to be done very soon without very much warning.
Ex:- He had to live his previous quarters at short notice.
10. at work : Ex:- He had to live his previous quarters at short notice, busy doing a particular activity.
Ex:- He had been at work on a book.
11. at sight: as soon as one saw.
Ex:- The police were ordered to shoot at sight.
12. at hand: near
Ex:- Is there a doctor at hand?
13. at least: the number or amount mentioned is a minimum, and the actual number or quantity is much greater.
Ex:- Lend me at least 50 rupees.
14. at once: immediately
Ex:- He received a telegram and left for home at once.
15. at present: at the time of speaking
Ex:- He is not doing anything at present.
16. at the end: Towards the last part of something.
Ex:- There was a test at the end of the lesson.
17. at times: occasionally
Ex:- My friend visits me at times.
18. at best: taking the most hopeful view.
Ex:- I can give you 500 rupees at best.
19. at all times: doing something always.
Ex: – He remains busy in his business at all times.
20. at full speed: the highest speed, but no increase of further speed is possible. Ex:- He drove his car at full speed.
21. at a profit: at least some income is possible.
Ex:- Today, he is at a profit in his business.
22. at a loss: no income is possible.
Ex:- I am at a loss in my business this year.

Use of ‘By’:

1. by oneself – alone, without help from anybody.
Ex:- One can’t play tennis by oneself
2. by name – using the name of somebody.
Ex:- The teacher knows all his students by name.
3. by accident/chance – meet someone or something accidentally.
Ex:- I only found it by accident/chance.
4. by heart: remember something entirely.
Ex:- Learn this poem by heart.
5. by mistake: as the result of carelessness and forgetfulness.
6. by sight: know by appearance only, not as an acquaintance.
Ex:- I only know her by sight.
7. by surprise: unexpectedly
Ex:- The Cuttack was made by surprise.
8. by letter/telephone: to intimate news.
Ex:- The news was sent to him by letter.
9. by means of: doing something by its help.
Ex:- He stood first by means of hard labor.
10. by no means: no way of doing something.
Ex:- It is not possible to do by no means.

11. by din’t of: doing something through it.
Ex:- He passed the examination by didn’t of hard labor.
12. by all means: certainly
Ex:- A graduate can complete all the tests by all means.
13. by hook or by crook: by fair means or foul
Ex:- The cunning people gain their ends by hook or by crook.
Ex:- This medicine will show its effect by and by.
15. by way of: in the shape of
Ex:- He passed a silly remark on his beloved by way of a joke.
16. by force of: by virtue of
Ex:- The Indian team wins the hockey match by force of their superior game.

Use of ‘In’:

1. in brief: in a few words
Ex:- Tell me the whole story in brief.
you can’t make it happens quicker, it will happen when the time is
2. in due course: right font.
Ex:- Every home will have a computer in due course.
3. in general: in most cases, usually
Ex:- Many think that in general men are more hard-working than women.
4. in pieces: broken
Ex:- The mirror is in pieces.
5. in tears: is crying
Ex:- She was in tears when she heard about her sister’s death.
6. in case: fit so happens
7. In fact: really, actually
Ex:- I do not like him, in fact, I hate him.
8. in public / in private: You say or do something when a group of people you do not know are present.
Ex:- He repeated in public what he said in private.

9. in order: the condition on which everything is carefully arranged.
Everything in the room is in order.
10. in common: having the same interests, characteristics features to make friends with.
Ex:- Both India and Pakistan have many things in common.
11. inflame: formally
Ex:- In the Koraput district some schools are rarely open, they exist in the name on.
12. in turn: In succession, one after the other.
Ex:- The patients saw the doctor in turn.

Other examples:
in-hospital/bed/prison/jail in all
in the sun/rain in the air
in trouble/danger/debt/difficulty in theory/practice in fun

Use of ‘Out of’:

1. out of print: not in print
Ex:- This book is out of print.
2. out of work: has no job
Ex:- He has been out of work for 10 years.
3. out of fashion: not fashionable
Ex:- Tight jeans are out of fashion now.
4. out of danger: not in danger
Ex:- The patient is now out of danger.
5. out of reach: Can’t be reached / not within reach.
Ex:- The picture on the wall is out of reach.
6. out of place: not suitable for a particular situation or occasion.
Ex:- I felt completely out of piece among all those smart rich people.
out of the question: not allowed, not possible
Ex:- You can’t go to the wedding in that old shirt, it is quite out of the question.
8. out of stock: no stock
Ex:- We haven’t any more wheat, we’re out of stock.
9. out of turn: before or after the permitted time.
Ex:- You must not speak out of turn.
10. out of doors: not inside a building but in the open air.
Ex:- Farmers spend most of their time out of doors.

More examples:
Ex:- I heard this news on the radio.
2. on television: news telecast on TV.
Ex:- You should have watched the 7.30 p.m. news on television today.
3. on fire: something is burning
Ex:- The house is on fire.
4. on sale: about to be sold
Ex:- These old books are on sale.
5. on duty: in the working hour
Ex:- The policeman is on duty.
6. on time: the reference of the perfect moment.
Ex:- The train arrived on time.

Exercise For Practice

1. Fill in the blanks with the correct prepositions.
1. The whole house was _________ fire (in/on/under)
2. I have got this book _________ loan from the library (by/on / under)
3. The patient is _________ treatment in hospital (under/ in, an)
4. I sold my old bike _________ a profit (at /on/with)
5. Mary is a Christian _________ birth. (in/with / by)
6. She has made this toy _________ hand (with/by/ through)
7. Minu’s father is a doctor _________ profession (in/by/ at)
8. We could not solve the problem _________ first, but we were able to do _________ it the end. (in/at/before)
9. The porter earns fifty rupees a day _________ an average. (on/by/with)
10. Father goes to his office _________ his car (with/in / by)
11. The policeman is _________ duty now (with/on / by)
12. The man sitting _________ my right is an engineer (to/on/ towards)

13. Please keep you things _________ order. (in/on/at).
14. Don’t disturb him, he is _________ work. (in/at/on)
15. I have a lot of work _________ on hand. (at/out/of)
16. Rice is _________ stock in the market. (with/out / of)
17. The thief is _________ prison now (in/inside/within)
18. My eldest daughter is _________ university now (in/at/inside)
19. Please send the letter_________ post (in/by/with)
20. You must keep your knowledge _________ date (out of! upto)
21. The crirninalis _________ arrest. (in/under at)
22. He is worthy _________ my praise. (to/of/for)
23. He is cure _________ his success. (fon in/of)
24. He is weak _________ mathematics (in/by/to)

## CHSE Odisha Class 12 Alternative English Grammar Phrasal Verbs

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

## CHSE Odisha Class 12 Alternative English Grammar Phrasal Verbs

Phrasal Verbs:
We have to remember the phrasal verbs with their meaning perfectly. Let us discuss some important phrasal verbs.

Intransitive:

1. break down: stop working.
Ex:-. My car broke down twice during our journey.
2. break out: start suddenly
Ex:- Cholera has broken out in our locality.
3. burst out: begin to do something suddenly
Ex:- The children burst out laughing.
Ex:- How did the accident come about?
5. come out: published, become known
Ex:- This magazine comes out once in a week. Our results came out yesterday.
6. come off: happen, take place
Ex:- My sister’s wedding came off in a grand way.
7. come on: say to encourage
Ex:- Come on, let’s try again.
8. come round: regain consciousness, cure, recover
Ex:- He is still unconscious. He has not come around.

9. die away: become weak, disappear gradually
Ex:- The noise gradually died away.
10. draw up: approach and stop
Ex:- A car drew up beside me.
11. drop in: call on somebody
Ex:- Why don’t you drop in and see me sometime?
12. drop out: withdraw
Ex:- Ajit has dropped out (of the team)
13. fall out: quarrel
Ex:- You should not fall out for such a trivial problem.
14. getaway: escape
Ex:- Two of the thieves got away.
15. get along: image
Ex:- It is very difficult to get along without money.
16. get on: make progress
Ex:- How are you getting on at college?
17. go off: explode
Ex:- Many people died when a bomb went off in the busy market area.
18. goon: continue
Ex:- The two friends, went on talking for hours.
19. go out: extinguish
Ex:- The lamp went out in the wind.
20. get up: the rise
Ex:- I get up early in the morning.
21. give in: surrender, accept defeat
Ex:- The tired soldiers finally gave in to their enemy.
22. give out: come to an end
Ex:- Our food supply gave out after a week.
23. give up: cease, stop, abandon
Ex:- I have given up smoking
24. hold on: maintain one’s position
Ex:- Out troops held on resolutely refusing to yield an inch.
25. look out: pay attention, be careful
Ex:- Look out: there is a heavy truck coming very fast behind us.
26. lookup: become better
Ex:- The weather is looking up now.

27. makeup: replace a loss
Ex: It will take a long time to make up the loss.
28. pull up: come to a stop
Ex:- The driver pulled up at the traffic lights.
29. put up: stay, live
Ex:- We are putting up in a small house.
30. run down: lose a store of energy
Ex:- The battery has run down.
31. run out: to come to an end
Ex:- All our food has run out.
32. set out / off: begin a journey
Ex:- We set out / off our journey early in the morning.
33. set in: begin
Ex:- The trains have set in early this year.
34. shut up: be quiet
Ex:- Shut up and leave me alone.
35. take off: leave the ground
Ex:- The airplane took off at 70’ clock.
36. turn up: come usually to a meeting
Ex:- The meeting was postponed, as only Haifa do-can people turned up.
37. wear out: became unfit for use
Ex:- This part of the machine has worn out. Cheap shoes wear out easily.

Transitive:

1. account for: give a reason for
Ex:- You must account for your absence at college yesterday.
2. agree with: be good for health.
Ex:- Egg does not agree with me.
3. break into: enter by force
Ex:- Thieves broke into our house last night.
4. (i) burst into: a sudden show of emotion.
Ex:- She burst into tears on getting the bad news.
(ii) to come in suddenly
Ex:- The angry men burst into my room and shouted at me.
5. call on: visit
Ex:- We called on our new neighbor yesterday.
6. come across: meet, find by chance.
Ex:- I come across some friends in the marketplace yesterday.

7. come over: influence
Ex:- He looks very sad, what has come over him?
8. count on: rely on
Ex:- Can I count on your help during my difficulties?
9. do without: manage without
Ex:- He can’t do without tea.
10. get on: progress, conduct
Ex:- How are you getting on with your study?
11. get over: to recover from an illness/loss/difficulty?
Ex:- Leena has not got over the shock of her husband’s death.
12. go into: investigate
Ex:- The auditors have gone into our accounts.
Ex:- I have gone through this novel.
14. jump at: accept eagerly
Ex: The children jumped at the proposal of visiting Nandakanan.
15. keep off: remain at a distance
Ex:- Keep off the grass
16. live on: have as food
Ex:- A baby lives on milk only.
17. look after: take care of
Ex:- The old man has nobody to look after him.
18. look after: to consider
Ex:- I look up to Rajesh as my own brother.
19. look into: examine carefully
Ex:- The police are looking into the theft at present.
20. look for: try to find
Ex:- I looked for my lost pen but found it nowhere
21. standby: help
Ex:- If they trouble you we’ll stand by you.

22. stand for: represent
Ex:- M.O. stands for a money order.
23. take after: resemble, look alike
Ex:- The baby takes after its mother.
24. take to: start a habit
Ex:- Rahul has taken to drinking after his wife’s death.

Verb + Object + Particle

2. count in: include
Ex:- If you are going to the circus, count me in.
3. order about: call to do something
Ex:- Don’t try to order me about, I am not your servant.
4. take for: mistake
Ex:- My aunt took me forAnil.
5. fell apart: consider separate
Ex:- I have never been able to fall the two brothers apart.
6. try on: put on a garment to see whether it fits
Ex:- You must try these shoes on before you buy them.

Verb + Particle + Object
(Or)
Verb + Object + Practical

1. blow up: break into pieces by an explosion
Ex:- The bridge was blown up by enemy soldiers.
2. bring about: cause to happen
Ex:- The new principal brought about several changes in the college.
3. bring out: Publish
Ex:- My father will bring out a new book next week.
4. bring up: rear, educate
Ex:- The mother worked hard to bring up her children.
5. call off: cancel
Ex:- We called off the strike after an agreement with the government.
6. carry on: Continue
Ex:- Smita carried on singing for a long time.

7. carry out: obey, do successfully
Ex:- You should carry out the order of your parents.
8. cut down: diminish, reduce
Ex:-You should cut down your expenses.
9. give up: stop
Ex:- You should give up smoking.
10. keep up: maintain, retain
Ex:- You should keep up the glory of your motherland.
11. keep away: remain at a distance Ex:- Keep away the children from fire.
12. lay by: keep for future use
Ex:- You should lay by something for old age.
13. leave out: omit
Ex:- You can leave out the questions you can not answer.
14. let down: opposite of back up.
Ex:- You have promised to stand by me. You won’t let me down, will you?
15. let off: not punish
Ex:- I’ll let you off this time, but I’ll punish you if you do it again.
16. lookup: search for a word in a dictionary.
Ex:- Look up a word in a dictionary, if you do not know its meaning.
17. make out: understand
Ex:- Can you make out the meaning of this sentence?
18. makeup: replace a loss
Ex:- It will take a long time to make up for the loss.
19. make over: to hand over charges
Ex:- The outgoing Principal made over the charge to the new Principal
20. pack up: stop working
Ex:- It is time to pack up and go home.
21. pick out: choose
Ex:- She picked out a frock that she liked most.
22. pulldown: Destroy
Ex:- The old building was pulled down.
23. put on: begin to wear, and dress oneself.
Ex:- Don’t forget to put your coat on.
24. put down: suppress by force
Ex:- The violent agitation was put down in no time.

25. put off: postpone, keep for a later time.
Ex:- Don’t put off today’s work for tomorrow.
26. put out: extinguish
Ex:- Put out the light before you sleep.
27. run over: knockdown – by traffic.
Ex:- Hundreds of pedestrians are run over in the streets every year.
28. setup: establish
Ex:- The government has set up a hospital in our village.
29. take in: cheat
Ex:- He was taken in by the shopkeeper.
30. takeoff: remove clothes, hat, etc.
Ex:- Take off your shoes before entering a temple.
31. take over: accept the duty
Ex:- Ramesh, took over the business from his father.
32. turndown: reject
Ex:- He turned down my request.
33. turn on: start the flow of
Ex:- I turned on the tap.
34. turnoff: stop the flow of
Ex:- Please turn off all lights before going to bed.
35. windup: bring to an end
Ex:- It is time for him to wind up his speech.
36. workout: calculate correctly
Ex:- An intelligent child can work out this sum.

Verb + Adverb Particle + Preposition + Object

1. catch up with: come from behind and reach someone in front by going faster.
Ex:- Drive fester, they are catching up with us.
2. do away with: abolish, get rid of
Ex:- You can’t do away with violence by using violence.
3. get up with: make progress in something you are doing.
4. go back on: fail to keep.
Ex:- I can’t go back on my word.

5. go in for: choose something as your job or interest.
Ex:- I thought of going in for teaching.
6. grow out of: became too big for
Ex:- He has grown out of that shirt.
7. look forward to: to be excited and pleased about something that is going to happen.
Ex:- We are looking forward to our uncle’s visit.
8. look down upon/on: hate, despise
Ex:- We should not look down upon the poor.
9. make up for: compensate for
Ex: – Hard work can often make up for lack of intelligence.
10. put up with: tolerate, bear
Ex:- I can’t put up with your rudeness any.
11. run out of: use all of something
Ex:-We have run out of sugar
12. watch out for: Keep looking and waiting for something/someone
13. keep up with: manage to go or learn as far as someone.
Ex:- The new boy can’t keep up with the class.

Exercise For Practice
1. Use appropriate phrasal verbs for the underlined verbs in the following sentences:

1. My brother has read this novel.
2. I can’t tolerate his insulting words.
3. His grandfather died yesterday.
4. He has postponed the meeting.
5. I can’t understand his speech.
6. We would not hate uncivilized people.
7. You should not try to cheat me.
8. Cholera has began in our locality.
9. The boy resembles his father.
10. That book has been published.
11. You should obey the words of the elders.
12. Pramila belongs to a royal family.
13. You should rise early in the morning.
14. Our Principal distributed the prizes.
15. You should maintain the prestige of your parents.
16. Stop the computer.
17. He has solved all the sums.
18. He has established a factory.
19. The police followed the thief.
20. The Pakistan army had to yield.

1. My brother has gone through this novel.
2. I can’t put up with his insulting words.
3. His grandfather passed away yesterday.
4. He has called off the meeting.
5. I can’t make out his speech.
6. We would not look down upon uncivilized people.
7. You should not try to take me in.
8. Cholera has broken out in our locality.
9. The boy takes after his father.
10. That book has been brought out.
11. You should carry out the words of the elders.
12. Pramila comes of a royal family.
13. You should get up early in the morning.
14. Our Principal gave away the prizes.
15. You should keep the prestige of your parents.
16. Turn off the computer.
17. He has worked out all the sums.
18. He has set up a factory.
19. The police ran after the thief.
20. The Pakistan army had to give in.

Exercise For Practice
2. Supply a phrasal verb of the same meaning as indicated in the brackets.

1. All the lights __________ when the power supply was cut off. (stopped giving light).
2. Please check if you have __________ any name. (omitted).
3. The workers __________ working for a long time. (continued)
4. You should __________ something for your children’s education. (save for future use).
5. Priya has __________ her illness only recently. (recovered from)
6. It is dishonest to __________ one’s words. (fail to keep a promise)
7. Ranjit was __________ by a car. (hit).
8. The tires of my cycle have __________. (become unfit for use)
9. Mr. Patra has __________ a school in his village. (established)
10. You should __________ wild animals in a jungle. (be careful of)
11. This clock has __________. (stopped working)
12. You must __________ your misbehavior. (give a reason for)
13. I __________ and an old friend at a shop. (met by chance)
14. The new boy __________ with almost everybody in the class. (quarrel)

Exercise For Practice
3. Choose the correct particles to make the sentences meaningful.

1. I called __________ my friend, (off / on)
2. We took a long time to work __________ the problem, (out / at)
3. The minister has promised to think the __________ matter. (about / over)
4. It is hard to make __________ their purpose, (out / upon)
5. We are looking __________ the problem carefully, (at/into/for)
6. Always keep __________ from danger, (out/away)
7. The thief got __________ with my car. (out / away)
8. An accident brought __________ a change in his life, (about / out)
10. The robbers broke __________ the bank at night, (down / into)
11. Take __________ the dirty clothes, (of / off / out)
12. The child is looking __________ the birds, (to/at/ for)
13. I rang __________ Aju in the morning, (to / up / for)
14. They are bringing __________ a new book, (about / out / to)
15. Would you care __________ a tea? (for/to / on)
16. They pulled __________ the old house, (off / top/down)
17. The reporters took __________ the speech, (down/off/to)
18. I am looking __________ your problem (to with/up with/in with)
19. I can’t put __________ rude people (to with / up with / in with)
20. The soldiers blew __________ the bridge, (down / off / up)
21. I can’t make __________ the meaning of this word, (to/of / out)
22. She carried __________ singing for a long time, (into/of / on)

## CHSE Odisha Class 12 Alternative English Grammar Tense and Aspect

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

## CHSE Odisha Class 12 Alternative English Grammar Tense and Aspect

We have already discussed about ”Tense” in the 1 st year. Let us do some exercises now.
Exercise For Practice (Solved):

1. Fill in the blanks with the correct very forms (Present Tense) from those in brackets.
1. My brother _________ (read) a play by Kalidas.
2. The students _________ (play) much attention to their studies.
3. Who _________ (say), I am the wrong.
4. _________ the birds not _________ (chirp) early in the morning?
5 _________ the students _________ (swim) in the river?
6 _________ your mother not _________ (keep) fit these days?
8. Puspa _________ not _________ I (iron) her clothes.
9. _________ your sister know how to swim?
10. Rakesh _________ not _________ (take) coffee without sugar.
11. _________ we not _________ (see) many ups and downs in the life?
12. I _________ drop a five rupee note somewhere.
13. _________ I not _________ (invite) him to dinner?
14. Pinki _________ not _________ (keep) awake till midnight these days.
15. Rajeswari _________ (travel) round the world.
16 _________ it _________ (drizzle) since room?
17. She _________ (withdraw) her name from the debate.
18. She _________ (wait) for you for an hour.
19 _________ the maidservant _________ (wash) the floor?
20. It _________ no _________ (rain) her for the last two days.

1. My brother is reading a play by Kalidas.
2. The students are playing much attention to their studies.
3. Who says. I am the wrong.
4. Do the birds not chirp early in the morning?
5. Are the students swimming in the river?
6. Does your mother not keep fit these days?
8. Puspa is not ironing her clothes.
9. Does your sister know how to swim?
10. Rakesh does not take coffee without sugar.
11. Have we not seen many ups and downs in the life?
12. I have dropped a five rupee note somewhere.
13. Am I not inviting him to dinner?
14. Pinki does not keep awake till midnight these days.
15. Rajeswari has traveled round the world.
16. Has it been drizzling since the room?
17. She has withdrawn her name from the debate.
18. She has been waiting for you for an hour.
19. Has the maidservant washed the floor?
20. It has not raining her for the last two days.

Exercise For Practice:
2. Fill in the blanks with correct verb forms (Present Tense) from those in brackets.

1. _________ God not _________ (protect) us all?
2. _________ you sister _________ (pass) the examination?
3. Hari _________ recently _________ (sell) his house.
4. I _________ (read) English for eight years.
5 _________ you _________ graze the cattle since morning?
6. Malaria _________ (rage) in the city for two years.
7. Vegetables and fruits _________ not _________ (harm) us in any way.
8 she not _________ (visit) her home every year?
9. She _________ not _________ (bathe) in hot water during summer.
10. Whom _________ you _________ (like) the most?
11. I _________ (learn) the verses from the Gita.
12. _________ they _________ (travel) by train?
13. Seema _________ not _________ (wash) her clothes.
14. _________ the police not _________ (chose) the thieves?
15. _________ those forests, not _________ (look) green?
16. How _________ you _________ (pull) on with your brother?
17. Who _________ (teach) you since morning?
18. _________ they been _________ (boil) since for ten minutes?
19. Tap _________ not _________ (run) for an hour.
20. Whose umbrella _________ you _________ (use) since last two days?

Exercise For Practice:
3. Fill in the blanks with correct verb forms (Past Tense) from those in brackets.

1. My father _________ (give) me this present on my birthday.
2. When I _________ (visit) her house, she _________(sleep).
3 _________ Suraj _________ (write) a romantic novel?
4. We _________ (reach) the station before the train _________ (leave).
5. She _________ (sleep) since 8 p.m.
6 _________ Gandhi always _________ (speak) the truth?
7. It _________ (drizzle) since 4 o’ clock.
8. It _________ (rain) heavily at 10 o’ clock.
9. Hari _________ (try) to grind his own axe.
10. The teacher _________ (not) _________ (punish) the naughty boys.
11. I _________ not _________ (talk) to Rahim the other day.
12. He _________ (go) to the post office after the rain _________ (stop).
13. I _________ (wait) for you when the bell _________(ring).
14. _________ the old man _________ (cross) the road very carefully.
15. Mother _________ (prepare) tea for five minutes.
16. Shakil _________ not _________(entertain) the guests with her titbits.
17. The train _________ (run) continuously for four hours.
18. Which God _________ you _________ (worship) in the temple?
19. Ranjana _________ not _________(call) on me last night.
20. In whose house _________ Sheela _________ (stay)?

1. My father gave me this present on my birthday.
2. When I visited her house, she was sleeping.
3. Was Suraj writing a romantic novel?
4. We had reached the station before the train left.
5. She had been sleeping since 8 p.m.
6. Did Gandhi always speak the truth?
7. It had been drizzling since 4 o’clock.
8. It had been raining heavily at 10 o’clock.
9. Hari was trying to grind his own axe.
10. The teacher did not punish the naughty boys.
11. I was not talking to Rahim the other day.
12. He want to go to the post office after the rain has stopped.
13. I was waiting for you when the bell rang.
14. Did the old man cross the road very carefully?
15. Mother had been preparing tea for five minutes.
16. Shakil was not entertaining the guests with her titbits.
17. The train had been running continuously for four hours.
18. Which God had you worshipping in the temple?
19. Ranjana did not call on me last night.
20. In whose house was Sheela staying?

Exercise For Practice:
4. Fill in the blanks with correct verb forms (Past Tense) from those in brackets.

1. You _________ (listen) to Radio for half an hour.
2. Whose clothes _________ you _________ (fold)?
3. Whom _________ you _________ (teach) Grammar?
4. When I _________ (teach), he _________ (doze).
5. It _________ not _________ (rain) when we _________ (leave) for.
6 _________ it _________ heavily at 10 o ’ clock yesterday (rain)?
7. I _________ (read) a novel the whole day long.
8. When you _________ (send) her a telegram?
9 _________ an accident not _________ (take) place here yesterday?
10. The police _________ not _________ (arrest) the thieves knowingly?
11. _________ I _________ (lend) her some money yesterday?
12. He _________ (solve) the difficult sum at once.
13. Mohan _________ not _________ (work) in the worship for several days.
14. _________ he not _________ (knock) at the door for five minutes?
15. Where _________ he _________ (hide) for two days?
16. Which book _________ you _________ (land) _________ to me ?
17. _________ the sun not _________ (set) when the farmers _________ (return) home ?
18. I _________ not _________ (receive) any letter from my uncle.
19. Whose like _________ (fly) high?
20. Who _________ (shatter) this glass into pieces?

## CHSE Odisha Class 12 Alternative English Solutions Unit 4 Text C: The Mushroom of Death

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.

## CHSE Odisha 12th Class Alternative English Solutions Unit 4 Text C: The Mushroom of Death

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)

(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

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.
(f) Someone of his poems was published in newspapers and magazines (so that / consequently) he thought of himself as an established poet.

(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.
(f) Someone of his poems was published in newspapers and magazines so he thought of himself as an established poet.

Activity – 15

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)

(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.

(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
(d) Work tranquility

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

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

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

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

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

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

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

Question 9.
Athorart:
(a) crosswise
(b) following
(c) flattened out
(a) crosswise

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

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

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

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

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

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

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

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

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

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

Question 20.
Lore:
(a) sentiment
(c) suspicion
(d) fabestories

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 5.
Bald:
(b) rash
(d) insulting

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

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

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

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

Question 10.
Wrought:
(b) broken
(c) complicated
(d) strengthened

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 33.
Solicit:
(a) to command
(b) to worry
(c) to sympathise with

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

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

TEST – III

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

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

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

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

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

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

Question 7.
(a) tofettenout
(b) to pacify
(c) to annoy
(d) to make secure
(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
(c) to place oneself in a favorable position

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

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

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

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

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

Question 14.
Cumulative:
(a) serious
(b) swollen
(c) rich

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

Question 16.
Disparate:
(b) discouraged
(c) reckless
(d) stingy

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

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

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

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

Question 21.
Circuitous:
(a) surrounded
(b) dizzy
(d) deceptive

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

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

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

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

Question 26.
Cursory:
(a) informal
(b) penetrating
(c) angry
(d) rapid and superficial
(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
(b) to put in action

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

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

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

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

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

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

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

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

TEST-IV

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 28.
(a) to stare
(b) to tease
(d) to criticize

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

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

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

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

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

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

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

TEST-V

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question 18.
Apotheosis:
(a) revelation
(b) pithy saying
(c) perfect example
(c) perfect example

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

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

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

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

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

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

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

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

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

Question 28.
Infraction:
(a) small portion
(b) collision
(c) oversight
(d) violation of law
(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.
plausible – evident, having proofs.
thermal – relating to heat or temperature
precipitation – rainfall.

## BSE Odisha 10th Class Maths Solutions Geometry Chapter 2 ବୃତ୍ତ Ex 2(a)

Odisha State Board BSE Odisha 10th Class Maths Solutions Geometry Chapter 2 ବୃତ୍ତ Ex 2(a) Textbook Exercise Questions and Answers.

## BSE Odisha Class 10 Maths Solutions Geometry Chapter 2 ବୃତ୍ତ Ex 2(a)

Question 1.
ଉକ୍ତିଟି ଠିକ୍ ଥିଲେ T ଏବଂ ଭୁଲ ଥିଲେ F ଲେଖ ।
(i) ଏକ ସମତଳରେ ଥିବା ଏକ ବକ୍ରରେଖାର ପ୍ରତ୍ୟେକ ବିନ୍ଦୁ ଉକ୍ତ ସମତଳ ଉପରିସ୍ଥ ଏକ ଦତ୍ତ ବିନ୍ଦୁଠାରୁ ଏକ ନିର୍ଦ୍ଦିଷ୍ଟ ଦୂରତାରେ ଥିଲେ ବକ୍ରରେଖାଟିକୁ ବୃତ୍ତ କୁହାଯାଏ ।
(ii) ବୃତ୍ତର ପ୍ରତ୍ୟେକ ବିନ୍ଦୁ କୌଣସି ଏକ ବ୍ୟାସାର୍କର ଏକ ପ୍ରାନ୍ତ ବିନ୍ଦୁ ଅଟେ ।
(iii) ଏକ ବୃତ୍ତର ଅସଂଖ୍ୟ ବ୍ୟାସ ରହିଛି ।
(iv) କେନ୍ଦ୍ର, ବୃତ୍ତର ଏକମାତ୍ର ବିନ୍ଦୁ ଯାହା ବୃତ୍ତର ପ୍ରତ୍ୟେକ ବ୍ୟାସ ଉପରେ ଅବସ୍ଥିତ ।
(v) ଏକ ଜ୍ୟା ବୃତ୍ତର ଅନ୍ତର୍ଦେଶକୁ ଯେଉଁ ଦୁଇ ଅଂଶରେ ବିଭକ୍ତ କରେ ସେମାନେ ପ୍ରତ୍ୟେକ ଉତ୍ତଳ ସେଟ୍ ଅଟନ୍ତି ।
(vi) ବୃତ୍ତର ଏକ ବ୍ୟାସ ଗୋଟିଏ ଜ୍ୟାକୁ ସମଦ୍ବିଖଣ୍ଡ କଲେ ସେମାନେ ପରସ୍ପର ପ୍ରତି ଲମ୍ବ ଅଟନ୍ତି ।
(vii) ପ୍ରତ୍ୟେକ ତ୍ରିଭୁଜର ପରିକେନ୍ଦ୍ର ଏହାର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।
(vii) ଏକ ବୃତ୍ତର କେନ୍ଦ୍ର, ଏହାର ଏକମାତ୍ର ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ଯାହାଠାରୁ ବୃତ୍ତର ପ୍ରତ୍ୟେକ ବିନ୍ଦୁର ଦୂରତା ସମାନ ।
(ix) ଏକ ରଶ୍ମି ବୃତ୍ତକୁ ଗୋଟିଏ ମାତ୍ର ବିନ୍ଦୁରେ ଛେଦ କରେ । ତେବେ ରଶ୍ମିର ଆଦ୍ୟ ବିନ୍ଦୁଟି ବୃତ୍ତର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ
(x) ଏକ ତ୍ରଭରେ $$\overline{\mathrm{AB}}$$ ଓ $$\overline{\mathrm{BC}}$$ ହୁକଟି ସବସମ କ୍ୟା 6ହ6ଲ B ଦିନ୍ଦନାମ କ୍ୟାମଣ ∠ABC କୁ ସମଦ୍ୱଖଣ୍ଡ ହେବ ।
(xi) ଗୋଟିଏ ବିନ୍ଦୁ ଦୁଇ ବା ତତୋଽଧ୍ଵକ ବୃତ୍ତର କେନ୍ଦ୍ର ହୋଇପାରିବ ନାହିଁ ।
(xii) ଗୋଟିଏ ସରଳରେଖା ଗୋଟିଏ ବୃତ୍ତରକୁ ସର୍ବଦା ଦୁଇଟି ବିନ୍ଦୁରେ ଛେଦ କରେ ।
Solution:
(i) F (ବକ୍ରରେଖାଟି ଏକ ବୃତ୍ତର ଚାପ ହୋଇପାରେ ।)
(ii) T (ଏକ ବ୍ୟାସାର୍କର ଦୁଇଟି ପ୍ରାନ୍ତ ବିନ୍ଦୁ ମଧ୍ୟରୁ ଗୋଟିଏ ବିନ୍ଦୁ କେନ୍ଦ୍ର ଓ ଅନ୍ୟଟି ବୃତ୍ତ ଉପରିସ୍ଥ ।)
(iii) T (ବୃତ୍ତ ଉପରିସ୍ଥ )
(iv) F
(v) T
(vi) T (ବ୍ୟାସରେ କେନ୍ଦ୍ର ଅବସ୍ଥିତ । କେନ୍ଦ୍ର ଓ ଜ୍ୟାର ମଧ୍ୟବିନ୍ଦୁକୁ ଯୋଗ କରୁଥିବା ରେଖାଖଣ୍ଡ ଜ୍ୟା ପ୍ରତି ଲମ୍ବ ।)
(vii) F (ସ୍ଥୂଳକୋଣୀ ତ୍ରିଭୁଜ ଓ ସମକୋଣୀ ତ୍ରିଭୁଜର ପରିକେନ୍ଦ୍ର ତ୍ରିଭୁଜର ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ନୁହେଁ ।)
(viii) T (ବୃତ୍ତର ସଂଜ୍ଞା ଅନୁସାରେ ।)
(ix) F (ରଶ୍ମିଟି ବୃତ୍ତପ୍ରତି ସ୍ପର୍ଶକ ହେବ ।)
(x) T (O ବୃତ୍ତର କେନ୍ଦ୍ର ହେଲେ A OAB = A OCB ହେବ ।
(xi) F (ଏକକୈନ୍ଦ୍ରିକ ବୃତ୍ତମାନଙ୍କର ଗୋଟିଏ କେନ୍ଦ୍ରବିନ୍ଦୁ ।)
(xii) F (ଆଦୌ ଛେଦ ନକରିପାରେ ବା ଗୋଟିଏ ବିନ୍ଦୁରେ ଛେଦ କରିପାରେ ।)

Question 2.
ପ୍ରଦତ୍ତ ସମ୍ଭାବ୍ୟ ଉତ୍ତରରୁ ଠିକ୍ ଉତ୍ତରଟି ବାଛି ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।

(i) ଦୁଇଟି ଅସମାନ୍ତର ଜ୍ୟାର ଛେଦବିନ୍ଦୁ …………………. ଅଟେ ।
(a) ବୃତ୍ତର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ
(b) ବୃତ୍ତର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ
(c) ବୃତ୍ତ ଉପରିସ୍ଥ ଏକ ବିନ୍ଦୁ
(d) ବୃତ୍ତ ଉପରିସ୍ଥ କିମ୍ବା ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ
Solution:
ହଉ ଭପରିମ୍କ କିମ୍ଵା ଆନ୍ତଃମ ଦିନ୍ଦୁ

(ii) P ବିନ୍ଦୁ ବୃତ୍ତର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ ହେଲେ ବୃତ୍ତ ଉପରେ P ଠାରୁ ସମଦୂରବର୍ତ୍ତୀ ……………………… ଯୋଡ଼ା ବିନ୍ଦୁ ଅଛି ।
(a) 1
(b) 2
(c) 8
(d) ଅସଂଖ୍ୟ
Solution:
ଅସଂଖ୍ୟ

(iii) 6ଗାଟିଏ ରେଖାଖଣ୍ଡ ସଦାଧକ ………….. ଟି ବୃତ୍ତର ଜ୍ୟା ହୋଇପାରିବ ।
(a) 1
(b) 2
(c) 4
(d) ଅସଂଖ୍ୟ
Solution:
2

(iv) 6ଗାଟିଏ ରେଖାଖଣ୍ଡ ସଦାଧକ ………….. ବ୍ୟାସାର୍ଦ୍ଧ ହୋଇ ପାରିବ ।
(a) 1
(b) 2
(c) 4
(d) ଅସଂଖ୍ୟ
Solution:
ଅଫଖ୍ୟ

(v) ଗୋଟିଏ ବୃତ୍ତରେ ଏକ ଜ୍ୟାର ଗୋଟିଏ ପ୍ରାନ୍ତବିନ୍ଦୁ କେନ୍ଦ୍ରଠାରୁ 5 ସେ.ମି. ଦୂରରେ ଏବଂ ଜ୍ୟାଟିର ମଧ୍ୟବିନ୍ଦୁ କେନ୍ଦ୍ରଠାରୁ 3 ସେ.ମି. ଦୂରରେ ଅଛି । ଜ୍ୟାଟିର ଦୈର୍ଘ୍ୟ …………….. 6ପ.ମି.
(a) 8
(b) 12
(c) 16
(d) 20
Solution:
8

Question 3.
ଏକଭର 16 6ସ.ମି. ଦେଣ୍ୟ ବିଶିଷ୍ଟ 6ଟାଟିଏ ବ୍ୟା ଏକ ବ୍ୟାସାଶୁ $$\overline{OP}$$ ଦାରା D ଜିନ୍ଦୁ 6ର ପମଦ୍ୱିଖଣ୍ଡତ ହୁଏ | ହୁଇର ଦ୍ୟାସାରୁ 10 6ସ.ମି. 6ଦ୍ର6କ DP ର 6ଦିଶ୍ୟ ନିଣ୍ଟଯ କର |

Solution:
ଦକ ହଉ6ର AB ର୍ଯ୍ୟର 6ଦିଣ୍ୟ = 16 ପେ.ମି.
⇒ AD = $$\frac { 16 }{ 2 }$$ ପେ.ମି. = 8 ପେ.ମି. (AD = $$\frac { 1 }{ 2 }$$ AB)
ହଉର ଦ୍ୟାସାର୍ଦ (OA) = 10 ପେ.ମି. = OP
∴ △ODA 6ର OD = $$\sqrt{\mathrm{OA}^2-\mathrm{AD}^2}$$ = $$\sqrt{\mathrm{10}^2-\mathrm{8}^2}$$ = $$\sqrt{100-64}$$ = $$\sqrt{36}$$ = 6 6ସ.ମି. |
DP = OP – OD = 10 6ସ.ମି. – 6 6ସ.ମି. = 4 6ସ.ମି. |
∴ $$\overline{DP}$$ ର 6ଦିଖ୍ୟ 4 6ସ.ମି. |

Question 4.
6ଟାଟିଏ ତ୍ରଭର 6କହ୍ O | ଏକ ଲ୍ୟା $$\overline{\mathbf{AB}}$$ ର ମଧ୍ୟଦିନୁ D 6ହ6ଲ ପ୍ରମାଣୀ କର ଯେ $$\overline{\mathbf{OD}}$$ , ∠AOB କୁ ସମଦ୍ୱିଖଣ୍ କ6ର |
Solution:

ଦଇ : ହଉର 6କହ O | ଖ୍ୟା AB ର ମଧ୍ୟଦିନ୍ଦୁ D |
ପ୍ରାମାଣ୍ୟ : $$\overline{\mathbf{OD}}$$, ∠AOB କୁ ସମଦିଖଣ୍ଡକ କ6ର |
ପ୍ରମାଣ : △AOD ଓ △BOD ମଧ୍ୟ6ର
AO = BO (ଏକା ଦୃତ୍ତର ବନ୍ଦ୍ୟାସାଦଁ)
$$\overline{\mathbf{OD}}$$ ସାଧାରଣ ଦାନ୍ଦୁ
△AOD ≅ △BOD (ଦା.ଦା.ଦା. ସଦପମତା)
⇒ ∠AOD ≅ ∠BOD (ର୍ଥ ନୁର୍ପ କୋଣ)
⇒ $$\overline{\mathbf{OD}}$$ , ∠AOB କୁ ସମଦ୍ୱିଖଣ୍ଡ କରେ |

Question 5.
6ଟାଟିଏ ତ୍ରଭର 6କହ୍ O | ଏକ ଲ୍ୟା $$\overline{\mathbf{AB}}$$ ଓ $$\overline{\mathbf{AC}}$$ ର ମଧ୍ୟଦିନୁ D 6ହ6ଲ ପ୍ରମାଣୀ କର ଯେ $$\overline{\mathbf{OA}}$$ , ∠BAC କୁ ସମଦ୍ୱିଖଣ୍ କ6ର |

Solution:
ଦଇ : O, ABC ହଉର 6କହ ଏକ AB = AC | (ତ୍ୟାଦୟ ସଦଂସମ)
ପ୍ରାମାଣ୍ୟ : $$\overline{\mathbf{OA}}$$, ∠BAC କୁ ସମଦିଖଣ୍ଡ କ6ର |
ଆଥାତ୍ m∠OAB = m∠OAC
ର୍ଥଙନ : $$\overline{\mathbf{OB}}$$ ଏବଂ $$\overline{\mathbf{OC}}$$ ଅଳନ କର |
ପ୍ରମାଣ : △AOB ଓ △AOC ମଧ୍ୟରେ
AB = AC (ଦଇ)
OB = OC (ଗୋଟିଏ ହଭର ବ୍ୟାପୀ ବଂ)
$$\overline{\mathbf{OA}}$$ ସାଧାରଣ ଦାନ୍ଦୁ
△AOB ≅ △AOD (ଦା.ଦା.ଦା. ସଦପମତା)
⇒ m∠OAB = m∠OAC (ର୍ଥ ନୁର୍ପ କୋଣ)

Question 6.
ଗୋଟିଏ ବୃତ୍ତର କେନ୍ଦ୍ର O ଏବଂ $$\overline{\mathbf{AB}}$$ ଓ CD ଏହାର ଦୁଇଟି ସମାନ୍ତର ଜ୍ୟା । P ଓ Q ଯଥାକ୍ରମେ AB ଓ CDର ମଧ୍ୟବିନ୍ଦୁ ହେଲେ ପ୍ରମାଣ କର ଯେ ( ବିନ୍ଦୁ, $$\stackrel{\leftrightarrow}{P}$$ ଉପରିସ୍ଥ ହେବ ।
Solution:
ଦତ୍ତ : AB ଓ CD ବୃତ୍ତର ଯେକୌଣସି ଦୁଇଟି ସମାନ୍ତର ଜ୍ୟା । P ଓ Q ଯଥାକ୍ରମେ AB ଏବଂ CD ଜ୍ୟାର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : ବୃତ୍ତର କେନ୍ଦ୍ର ‘O’, $$\overline{\mathbf{PQ}}$$ ଉପରିସ୍ଥ ହେବ ।
ପ୍ରମାଣ : ମନେକର ବୃତ୍ତର କେନ୍ଦ୍ର ‘O’, PQ ଉପରିସ୍ଥ ନୁହେଁ ।

OP ଓ $$\overline{\mathbf{OQ}}$$ ଅଙ୍କନ କର । O ବିନ୍ଦୁରେ AB ସହ ସମାନ୍ତର କରି $$\overrightarrow{\mathrm{OR}}$$ ଅଙ୍କନ କର।
m∠APO = 90° (∵ 0 କେନ୍ଦ୍ର ଏବଂ AB ଜ୍ୟାର ମଧ୍ୟବିନ୍ଦୁ P)
∴ m∠POR = 90° (∵ AB || OR ଏବଂ $$\overline{\mathbf{PO}}$$ ହେଲେ)
ସେହିପରି m∠CQO = 90° ଏବଂ m∠ROQ = 90° ($$\overline{\mathbf{CD}}$$ || $$\overline{\mathbf{OR}}$$ ଏବଂ QO ଛେଦକ)
∴m∠POR + m∠ROQ = 180°
⇒ P, O ଓ Q ଏକରେଖୀୟ ।
⇒ O, $$\overline{\mathbf{PQ}}$$ ଉପରିସ୍ଥ ଏକ ବିନ୍ଦୁ ହେବ ।

Question 7.
ଗୋଟିଏ ସମବାହୁ ତ୍ରିଭୁଜର ପରିକେନ୍ଦ୍ରଠାରୁ ତ୍ରିଭୁଜର ବାହୁମାନେ ସମଦୂରବର୍ତ୍ତୀ – ପ୍ରମାଣ କର ।
Solution:
ଆମେ ଜାଣୁ ସମବାହୁ ତ୍ରିଭୁଜର ବାହୁମାନଙ୍କର ଦୈର୍ଘ୍ୟ ସମାନ ।
ପୁନଶ୍ଚ ଏକ ବୃତ୍ତରେ ସମାନ ଦୈର୍ଘ୍ୟବିଶିଷ୍ଟ ଜ୍ୟାମାନ କେନ୍ଦ୍ରଠାରୁ ସମଦୂରବର୍ତ୍ତୀ ।
ତେଣୁ ସମବାହୁ ତ୍ରିଭୁଜର ପରିକେନ୍ଦ୍ରଠାରୁ ତ୍ରିଭୁଜର ବାହୁମାନ ସମଦୂରବର୍ତ୍ତୀ । (ପ୍ରମାଣିତ)

ବିକଳ୍ପ ପ୍ରମାଣ :
ଦତ୍ତ : A ABCର ବାହୁମାନ କେନ୍ଦ୍ରଠାରୁ ସମୟଦୂରବର୍ତ୍ତୀ ।
ପ୍ରମାଣ : $$\overline{\mathbf{AB}}$$, BC ଓ CA ର ସମଦ୍ଵିଖଣ୍ଡକ ଲମ୍ବମାନଙ୍କର ଛେଦବିନ୍ଦୁ O ।
ତେଣୁ O ପରିବୃତ୍ତର କେନ୍ଦ୍ର ।
AB, BC ଓ CA ହେତୁ ଜ୍ୟାମାନ କେନ୍ଦ୍ରଠାରୁ ସମଦୂରବର୍ତ୍ତୀ
କାରଣ AB = BC = CA (ଉପପାଦ୍ୟ – 8)

Question 8.
ପ୍ରମାଣ କର ଯେ ବୃତ୍ତରେ ଏକ ବ୍ୟାସ ଏହାର ବୃହତ୍ତମ ଜ୍ୟା । (ସୂଚନା : ଏକ କ୍ୟାର କେନ୍ଦ୍ରଠାରୁ ଦୂରତା d ≥ 0 ) ଏବଂ ବୃତ୍ତର ବ୍ୟାସାର୍ଦ୍ଧ r ହେଲେ, ଜ୍ୟାର ଦୈର୍ଘ୍ୟ $$2 \sqrt{r^2-d^2} \leq 2 r$$ = ବ୍ୟାସ) ।
Solution:

ଦତ୍ତ : ABC ବୃତ୍ତର $$\overline{\mathbf{AC}}$$ ବ୍ୟାସ । O ବୃତ୍ତର କେନ୍ଦ୍ର ।
ପ୍ରାମାଣ୍ୟ : AC ବୃହତ୍ତମ ଜ୍ୟା ।
ଅଙ୍କନ : AB ଜ୍ୟା ଅଙ୍କନ କର । OD ⊥ AB ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △AODରେ m∠ADO = 90°
⇒ 2AO > 2AD ⇒ AC > AB
ସେହିପରି ପ୍ରମାଣ କରାଯାଇ ପାରେ ଯେ, $$\overline{\mathbf{AC}}$$ ର ଦୈର୍ଘ୍ୟ
ଅନ୍ୟ ଯେକୌଣସି ଜ୍ୟାର ଦୈର୍ଘ୍ୟଠାରୁ ବୃହତ୍ତର ।
∴ AC ବ୍ୟାସ ବୃତ୍ତର ବୃହତ୍ତମ ଜ୍ୟା । (ପ୍ରମାଣିତ)

Question 9.
ଗୋଟିଏ ବୃତ୍ତରେ ଦୁଇଟି ସମାନ୍ତର ଜ୍ୟାର ଏକ ପାର୍ଶ୍ବରେ ବୃତ୍ତର କେନ୍ଦ୍ର ଅବସ୍ଥିତ । ପ୍ରମାଣ କର ଯେ ଜ୍ୟା ଦ୍ଵୟ ସର୍ବସମ ନୁହଁନ୍ତି ।
Solution:
S ବୃତ୍ତର O କେନ୍ଦ୍ର । କେନ୍ଦ୍ରର ଏକ ପାର୍ଶ୍ବରେ AB ଓ CD ଦୁଇଟି ଜ୍ୟା | AB || CD |
ପ୍ରାମାଣ୍ୟ : AB ≠ CD ଅର୍ଥାତ୍ AB ଓ CD ଜ୍ୟା ଦ୍ଵୟ ସର୍ବସମ ନୁହଁନ୍ତି ।
ଅଙ୍କନ : OM ⊥ CD ଅଙ୍କନ କର ଏବଂ OA ଓ OC ଅଙ୍କନ କର ।
ପ୍ରମାଣ : OM ⊥ AB ⇒ OM ⊥ CD

△OAM ରେ OA2 = OM2 + AM2 ….(i)
△ONC ରେ OC2 = ON2 + CN2 ….(ii)
OA = OC 6ହତ୍ର (i) ଓ (ii) ବ୍ ପାଲାଟା OM2 + AM2 = ON2 + CN2
⇒ AM2 – CN2 = ON2 – OM2 > 0 (∵ ON > OM)
⇒ CN < AM = $$\frac { 1 }{ 2 }$$ CD < $$\frac { 1 }{ 2 }$$ Ab
⇒ CD < AM ⇒ AB ≠ CD (ପ୍ରମାଣିତ) (∵ ON > OM ⇒ CD < AB)

Question 10.
AB ଓ CD ଗୋଟିଏ ବୃତ୍ତର ଦୁଇଟି ସମାନ୍ତର ଜ୍ୟା । AB = CD = 8 ସେ.ମି. । ବୃତ୍ତର ବ୍ୟାସାର୍ଦ୍ଧ 5 ସେ.ମି. ହେଲେ ଜ୍ୟାଦ୍ଵୟର ମଧ୍ୟବର୍ତ୍ତୀ ଦୂରତା ନିର୍ଣ୍ଣୟ କର ।
Solution:

AB = CD ଓ AB || CD
ହେତୁ ଜ୍ୟାଦ୍ଵୟ କେନ୍ଦ୍ରର ବିପରୀତ ପାର୍ଶ୍ଵରେ ରହିବେ ।
AB = 8 6ସ.ମି. ⇒ BP $$\frac { 1 }{ 2 }$$ × 8 ସେ.ମି. = 4 ସେ.ମି. |
OB = 5 ସେ.ମି., OP ⊥ AB ଓ OQ ⊥ CD ହେଉ ।
POB ସମ6କାଣା ତ୍ରିକୁଲ6ର OP = $$\sqrt{\mathrm{OB}^2-\mathrm{BP}^2}$$ = $$\sqrt{5^2-4^2}$$ = $$\sqrt{9}$$ = 3
∴ PQ = OP + OQ = 3 ସେ.ମି. + 3 ସେ.ମି. = 6 ସେ.ମି. |
∴ ଜ୍ୟାଦ୍ବୟର ମଧ୍ୟବର୍ତ୍ତୀ ଦୂରତା 6 ସେ.ମି. |

Question 11.
10 ସେ.ମି. ବ୍ୟାସାର୍ଦ୍ଧ ବିଶିଷ୍ଟ ଗୋଟିଏ ବୃତ୍ତର ଦୁଇଟି ସମାନ୍ତର ଜ୍ଯା $$\overline{\mathbf{AB}}$$ ଓ $$\overline{\mathbf{CD}}$$ ମଧ୍ୟରେ ଦୂରତା 10 ସେ.ମି. | $$\overline{\mathbf{AB}}$$ କ୍ୟା କେନ୍ଦ୍ରଠାରୁ 6 ସେ.ମି. ଦୂରରେ ଅବସ୍ଥିତ ହେଲେ $$\overline{\mathbf{AB}}$$ ଓ $$\overline{\mathbf{CD}}$$ ର ଦୈର୍ଘ୍ୟ ନିର୍ଣ୍ଣୟ କର ।
Solution:

ବୃତ୍ତର ବ୍ୟାସାର୍ଦ୍ଧ = 10 ସେ.ମି.
$$\overline{\mathbf{AB}}$$ ଓ $$\overline{\mathbf{CD}}$$ ମଧ୍ୟରେ ଦୂରତା = 10 ସେ.ମି. |
∴ $$\overline{\mathbf{AB}}$$ ଓ $$\overline{\mathbf{CD}}$$ କେନ୍ଦ୍ରର ବିପରୀତ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ ।
ମନେକର O ବୃତ୍ତର କେନ୍ଦ୍ର ।
OP ⊥ AB ଓ OQ ⊥ CD |
PQ = 10 ସେ.ମି. , OP = 6 ସେ.ମି.
∴ OQ = (10 – 6) 6 ସେ.ମି. . = 4 ସେ.ମି. |
A OBP ରେ BP = $$\sqrt{\mathrm{OB}^2-\mathrm{OP}^2}$$ = $$\sqrt{10^2-6^2}$$ ସେ.ମି. = $$\sqrt{64}$$ ସେ.ମି. = 8 ସେ.ମି.
⇒ AB = 2BP = 16 ସେ.ମି. |
△OQD ରେ QD = $$\sqrt{\mathrm{OD}^2-\mathrm{OQ}^2}$$ = $$\sqrt{10^2-4^2}$$ ସେ.ମି. = $$\sqrt{84}$$ ସେ.ମି.
= 2√21 ସେ.ମି. |
∴CD = 2QD = 4√21 ସେ.ମି. |

Question 12.
ଗୋଟିଏ ବୃତ୍ତରେ △ABC ଅନ୍ତର୍ଲିଖ୍ ହୋଇଛି । ଯଦି AB = ÀC ହୁଏ, ପ୍ରମାଣ କର ଯେ ∠BACର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମି ବୃତ୍ତର କେନ୍ଦ୍ର ବିନ୍ଦୁଗାମୀ ଅଟେ ।
Solution:

: O, ABC ଦ୍ରଦ୍ଭର କେନ୍ଦ୍ର ଏବଂ AB = AC |
ପ୍ରାମାଣ୍ୟ : AO, ZBACର ସମଦ୍ଵିଖଣ୍ଡକ ।
ଅଙ୍କନ : OB ଓ OC ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △ABO ଓ △ACO ମଧ୍ୟରେ

AB = AC (ଦତ୍ତ)
AO (ସାଧାରଣ ବାହୁ)
OB = OC (ଏକା ବୃତ୍ତର ବ୍ୟାସାର୍ଦ୍ଧ)
⇒ △ABO ≅ △ACO (ବା. ବା. ଦା . ସବଂସମତା)
m∠BAO = m∠CAO (ଅନୁରୁପ କୋଣ)
⇒ ∠BACର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମି $$\overrightarrow{\mathrm{AO}}$$ ବୃତ୍ତର କେନ୍ଦ୍ରବିନ୍ଦୁଗାମୀ ।

Question 13.
ଗୋଟିଏ ବୃତ୍ତର ଦୁଇଟି ଜ୍ୟା ଏକ ବ୍ୟାସ ଦ୍ବାରା ସମଦ୍ବିଖଣ୍ଡିତ ହେଲେ ପ୍ରମାଣ କର ଯେ ଜ୍ୟା ଦୁଇଟି ସମାନ୍ତର ।
Solution:

ଦତ୍ତ : ବୃତ୍ତର କେନ୍ଦ୍ର O । PQ ଓ RS ଜ୍ୟା ଦ୍ବୟ XY ବ୍ୟାସଦ୍ବାରା ଯଥାକ୍ରମେ M ଓ N ବିନ୍ଦୁରେ ସମଦ୍ବିଖଣ୍ଡିତ
ପ୍ରାମାଣ୍ୟ : PQ||$$\overline{\mathbf{RS}}$$
ପ୍ରମାଣ : PQ ର ମଧ୍ୟବିନ୍ଦୁ M ।
⇒ OM ⊥ PQ ⇒ m∠QMO = 90°
ସେହିପରି RS ର ମଧ୍ୟବିନ୍ଦୁ N |
⇒ ON ⊥ RS ⇒ m∠ONR = 90° |
∴ m∠QMO = m∠ONR = 90° |
ମାତ୍ର ଏହି କୋଣଦ୍ଵୟ ଏକାନ୍ତର, ତେଣୁ PQ || RS |

Question 14.
ପ୍ରମାଣ କର ଯେ ଗୋଟିଏ ବୃତ୍ତର ଦୁଇଟି ଜ୍ୟା କେନ୍ଦ୍ର ହେବ । (ସୂଚନା : ଅସମ୍ଭବାୟବ ପ୍ରଣାଳୀ ପରସ୍ପରକୁ ସମଦ୍ବିଖଣ୍ଡ କଲେ ସେମାନଙ୍କ ଛେଦବିନ୍ଦୁ ବୃତ୍ତର (Method of contradiction) ବ୍ୟବହାର କର)
Solution:

ଦତ୍ତ : PQ ଓ RS ଜ୍ୟା ଦ୍ବୟ ପରସ୍ପରକୁ O ବିନ୍ଦୁରେ ସମଦ୍ଵିଖଣ୍ଡ କରନ୍ତି ।
ପ୍ରାମାଣ୍ୟ : ‘O’ ବୃତ୍ତର କେନ୍ଦ୍ର ।
ପ୍ରମାଣ : ମନେକର ‘O’ ବୃତ୍ତର କେନ୍ଦ୍ର ନୁହେଁ । O’ ବୃତ୍ତର କେନ୍ଦ୍ର ହେଉ । O’ O ଅଙ୍କନ କର ।
O, RS ଜ୍ୟାର ମଧ୍ୟବିନ୍ଦୁ ⇒ m∠O′OS = 90°
ସେହିପରି O, PQ ଜ୍ୟାର ମଧ୍ୟବିନ୍ଦୁ ⇒ m∠O’OQ = 90°
∴ m∠O’OS = m∠O’OQ = 90°
କିନ୍ତୁ ଏହା ଅସମ୍ଭବ ।
କାରଣ Q ଓ S ବିନ୍ଦୁ O’O ର ଏକ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ ।
∴ O’ ଓ O ବିନ୍ଦୁଦ୍ଵୟ ଏକ ଓ ଅଭିନ୍ନ ।
⇒ PQ ଓ RS ଜ୍ୟା ଦ୍ଵୟର ଛେଦବିନ୍ଦୁ, ‘O’ ବୃତ୍ତର କେନ୍ଦ୍ର ହେବ ।

15. ଗୋଟିଏ ବୃତ୍ତର ଦୁଇଟି ଜ୍ଯା AB ଓ BC, B ଠାରେ ୨୦ କୋଣ ଉତ୍ପନ୍ନ କରନ୍ତି । ବୃତ୍ତର କେନ୍ଦ୍ର O ହେଲେ ପ୍ରମାଣ କର ଯେ A, O ଏବଂ C ଏକ ଏକରେଖୀୟ ।
Solution:
ଦତ୍ତ : ( କେନ୍ଦ୍ର ବିଶିଷ୍ଟ ଏକ ବୃତ୍ତର $$\overline{\mathrm{AB}}$$ ଓ $$\overline{\mathrm{BC}}$$ ଦୁଇଟି ଜ୍ୟା ।
m∠ABC = 90°
ପ୍ରାମାଣ୍ୟ : A – 0 – C ଅର୍ଥାତ୍ $$\overline{\mathrm{AC}}$$ ବ୍ୟାସ ।
ଅଙ୍କନ : $$\overline{\mathrm{BO}}$$ ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △ABO ରେ AO = BO ⇒ m∠OAB = m∠ABO = θ (ମ6ନକର)

△BOC ରେ BO = CO ⇒ m∠OBC = m∠OCB = α (ମ6ନକର)
∴ θ + α = 90° (·.· m∠ABC = 90°)
△ABO ରେ m∠AOB = 180° – 2θ ଏବଂ △BOC ରେ m∠BOC = 180° – 2α
∴ m∠AOB +m∠BOC = 360 – 2(θ + α) = 360° – 2(90°) (∵ θ + α = 90°) = 180°
⇒ A – O – C = $$\overline{\mathrm{AC}}$$ ଏକ ବ୍ୟାସ ।

Question 16.
ପ୍ରମାଣ କର ଯେ ଏକ ସମକୋଣୀ ତ୍ରିଭୁଜରେ କର୍ପୂର ମଧ୍ୟବିନ୍ଦୁ, ଏହାର ପରିବୃତ୍ତର କେନ୍ଦ୍ର ଅଟେ ।
Solution:

ଦତ୍ତ : △ABC ର m∠ABC = 90° । ‘O’ $$\overline{\mathrm{AC}}$$ କର୍ପୂର ମଧ୍ୟବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : A ABC ର ପରିବୃତ୍ତର କେନ୍ଦ୍ର O ।
ଅଙ୍କନ : $$\overrightarrow{\mathrm{BO}}$$ ଅଙ୍କନ କର । $$\overrightarrow{\mathrm{BO}}$$ ଉପରିସ୍ଥ ‘D’ ଏପରି ଏକ ବିନ୍ଦୁ ଯେପରିକି BO = OD |
DC ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △OBC ଓ △ODC ଦ୍ଵୟରେ AO = OC (∵ O, $$\overline{\mathrm{AC}}$$ ର ମଧ୍ୟବିନ୍ଦୁ)
BO = OD (ଅଙ୍କନ), m∠AOB = m∠COD (ପ୍ରତୀପ)
△ABO = △CDO ⇒ m∠BAO = m∠OCD 19° AB = CD
⇒ ମାତ୍ର ଏହି କୋଣଦ୍ଵୟ ଏକାନ୍ତର ⇒ $$\overline{\mathrm{AB}}$$ || $$\overline{\mathrm{CD}}$$
ପୁନଶ୍ଚ, m∠ABC = 90° ହେତୁ ABCD ଏକ ଆୟତଚିତ୍ର ।
∴ AC = BD = $$\frac { 1 }{ 2 }$$ AC = $$\frac { 1 }{ 2 }$$ BD ⇒ AO = BO
∴ AO = BO = CO
ଏଠାରେ ଠ ବିନ୍ଦୁଠାରୁ A, B ଓ C ବିନ୍ଦୁତ୍ରୟ ସମଦୂରବର୍ତ୍ତୀ ।
⇒ O ବିନ୍ଦୁ △ABC ର ପରିବୃତ୍ତର କେନ୍ଦ୍ର ।

Question 17.
PQ ବୃତ୍ତର ଜ୍ୟା । P ଓ Q ଠାରେ ଉକ୍ତ ଜ୍ୟା ରେ ଅଙ୍କିତ ଲମ୍ବଦ୍ଵୟ R ଓ S ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । ପ୍ରମାଣ କର ଯେ PQSR ଏକ ଆୟତ ଚିତ୍ର |
Solution:

ଦତ୍ତ : $$\overline{\mathrm{PQ}}$$ ଜ୍ୟାର P ଓ Q ଠାରେ ଅଙ୍କିତ ଲମ୍ବଦ୍ଵୟ ବୃତ୍ତକୁ R ଓ S ବିନ୍ଦୁରେ ଛେଦ କ6ର |
ପ୍ରାମାଣ୍ୟ : PQSR ଏକ ଆୟତଚିତ୍ର ।
ଅଙ୍କନ : $$\overline{\mathrm{RQ}}$$ ଓ $$\overline{\mathrm{PS}}$$ ଅଙ୍କନ କର ।
ପ୍ତମାଣ : △PRQ ରେ m∠RPQ = 90° ⇒ RQ ଏବ ଦ୍ୟାସ ….(i)
△SPQ ରେ m∠PQS = 90° ⇒ PQ ଏବ ଦ୍ୟାସ ….(ii)
ଆମେ ଜାଣିଛେ, ବ୍ୟାସଦ୍ଵୟ ପରସ୍ପରକୁ କେନ୍ଦ୍ର ‘O’ରେ ସମଦ୍ଵିଖଣ୍ଡ କରନ୍ତି ।
∴ PORS ଏକ ଆୟତଚିତ୍ର ।

ବିକଳ୍ପ ପ୍ରମାଣ:
PQRS ଦ୍ରଭନ୍ନକଖତ ଚତୁରୁଢର m∠P + m∠S = 180°
⇒ m∠S = 90° (∴ $$\overline{\mathrm{RQ}}$$ ବୃତ୍ତର ଏକ ବ୍ୟାସ)
⇒ m∠P = 90°, 6ସଦ୍ରପତି m∠R = 90° |
∴ PQSR ଏକ ଆୟତଚିତ୍ର ।

Question 18.
ଚିତ୍ରରେ A ଓ B ଦୁଇଟି ପରସ୍ପର ଛେଦୀ ବୃତ୍ତର କେନ୍ଦ୍ର ଏବଂ P ଓ Q ବୃତ୍ତ ଦ୍ଵୟର ଛେଦବିନ୍ଦୁ ଅଟନ୍ତି । ପ୍ରମାଣ କର ଯେ,
(i) $$\stackrel{\leftrightarrow}{\mathbf{AB}}$$, $$\overline{\mathrm{PQ}}$$ ସାଧାରଣ ଜ୍ୟାକୁ ସମର୍ଦ୍ଦିଖଣ୍ଡ କରେ ।
ଏବଂ (ii) $$\stackrel{\leftrightarrow}{\mathbf{A B}}$$ ⊥ $$\overline{\mathrm{PQ}}$$

Solution:
ଦତ୍ତ : A ଓ B ଦୁଇଟି ପରସ୍ପରଛେଦୀ ବୃତ୍ତର କେନ୍ଦ୍ର ।
P ଓ Q ବୃତ୍ତଦ୍ଵୟର ଛେଦବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : (i) $$\stackrel{\leftrightarrow}{\mathbf{AB}}$$, $$\overline{\mathrm{PQ}}$$ କୁ ସମର୍ଦ୍ଦିଖଣ୍ଡ କରେ । (ii) $$\stackrel{\leftrightarrow}{\mathbf{A B}}$$ ⊥ $$\overline{\mathrm{PQ}}$$
ଅଙ୍କନ : $$\overline{\mathrm{PA}}$$, $$\overline{\mathrm{AQ}}$$, $$\overline{\mathrm{BQ}}$$ ଏବଂ $$\overline{\mathrm{BP}}$$ ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △APB ଏବଂ △AQB ଦ୍ବୟରେ
PA = AQ (ଏକା ଦ୍ୱଭର ବ୍ୟାସାକିଂ), BP = BQ
ଏବଂ $$\overline{\mathrm{AB}}$$ ସାଧାରଣ ବାହୁ ।
∴△APB ≅ △AQB (ଦା .ଦା. ଦା)

⇒ m∠PAM = m∠QAM (ଅନୁରୂପ କୋଣ)
ବର୍ତ୍ତମାନ △APM ଏବଂ △AQM ଦ୍ବୟରେ
PA = AQ, m∠PAM = m∠QAM ଏବଂ $$\overline{\mathrm{AM}}$$ ସମତ୍ତିଖଣ୍ଡ କରିବ ।
∴ △APM ≅ △AQM (ଦା .ଦା. ଦା)
PM = MQ ⇒ $$\overleftrightarrow{\mathrm{AB}}$$, $$\overline{\mathrm{PQ}}$$ ….. (ii) (ପ୍ରମାଣିତ)
ଏବଂ m∠AMP = m∠AMQ
କିନ୍ତୁ ଏମାନେ ସନ୍ନିହିତ ପରିପୂରକ ହେତୁ m∠AMP = m∠AMQ = 90°
⇒ $$\overleftrightarrow{\mathrm{AB}}$$ ⊥ $$\overline{\mathrm{PQ}}$$ ….. (ii) (ପ୍ରମାଣିତ)

Question 19.
ଚିତ୍ରରେ ଦୁଇଟି ବୃତ୍ତ ପରସ୍ପରକୁ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । P ଠାରେ PQ ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବ ବୃତ୍ତ ଦ୍ଵୟକୁ A ଓ B ଠାରେ ଛେଦ କରେ ଓ ସେହିପରି () ଠାରେ PQ ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବ ବୃତ୍ତଦ୍ଵୟକୁ C ଓ D ଠାରେ ଛେଦ କରେ । ପ୍ରମାଣ କର ଯେ, AB = CD ।

Solution:
ଦତ୍ତ : ଦୁଇଟି ବୃତ୍ତ ପରସ୍ପରକୁ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । P ଠାରେ PQ ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବ ବୃତ୍ତଦ୍ଵୟକୁ A ଓ B ଠାରେ ଛେଦକରେ Q ଠାରେ PQ ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବ ବୃତ୍ତଦ୍ଵୟକୁ C ଓ D ଠାରେ ଛେଦ କରେ ।
ପ୍ରାମାଣ୍ୟ : AB = CD
ଅଙ୍କନ : ବୃତ୍ତଦ୍ଵୟର କେନ୍ଦ୍ର O1 ଓ O2 ନିଆ | O1X1 ⊥ AP ଏବଂ

O2 Y1 ⊥ PB ଅଙ୍କନ କର ।
X1 O1, ଓ Y1 O2, CDକୁ ଯଥାକ୍ରମେ X2 ଓ Y2 ରେ ଛେଦକରୁ ।
ପ୍ରମାଣ : AB || CD = X1 Y1 || X2 Y2
ପୁନଶ୍ଚ X1 X2 || Y1 Y2
∵ m∠X2X1Y1 +m∠X1Y1Y2 = 180°
⇒ X1Y1Y1X2 ଏକ ଆପ୍ତତତିତ୍ର |
X1Y1 = X2Y2
AB = AP + PB = 2PX1 + 2PY1
= 2(PX1 + PY1) = 2 X1Y1 = 2X2Y2
= 2(X2Q + QY2) = 2QX2 + 2QY2 = CQ + QD = CD (ପ୍ରମାଣିତ)

Question 20.
A ଓ B କେନ୍ଦ୍ର ବିଶିଷ୍ଟ ଦୁଇଟି ବୃତ୍ତ ପରସ୍ପରକୁ P ଓ ଠୁ ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । P ମଧ୍ୟ ଦେଇ AB ସହିତ ସମାନ୍ତର ସରଳରେଖା ବୃତ୍ତ ଦ୍ଵୟକୁ M ଓ N ବିନ୍ଦୁରେ ଛେଦ କଲେ ପ୍ରମାଣ କର ଯେ, MN = 2AB | (ସୂଚନା : $$\overline{\mathbf{AC}}$$ ଓ $$\overline{\mathbf{BD}}$$, $$\overline{\mathbf{MN}}$$ ପ୍ରତି ଲମ୍ବ ଅଙ୍କନ କରି ଦର୍ଶାଅ ଯେ, AB = CD)

ସମାଧାନ :
ଦତ୍ତ : A ଓ B କେନ୍ଦ୍ର ବିଶିଷ୍ଟ ଦୁଇଟି ବୃତ୍ତ ପରସ୍ପରକୁ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ।
P ବିନ୍ଦୁ ମଧ୍ୟଦେଇ ଅଙ୍କିତ $$\overline{\mathbf{MN}}$$, $$\overline{\mathbf{AB}}$$ ସହ ସମାନ୍ତର ।
ପ୍ରାମାଣ୍ୟ : MN = 2AB
ଅଙ୍କନ : AR ⊥ MP ଏବଂ BS ⊥ PN
ପ୍ରମାଣ : AR ⊥ MP ⇒ RP = $$\frac { 1 }{ 2 }$$ MP
ସେହିପରି BS ⊥ NP ⇒ PS = $$\frac { 1 }{ 2 }$$ PN
∴ RP + PS = $$\frac { 1 }{ 2 }$$ (MP + PN) ⇒ RS = $$\frac { 1 }{ 2 }$$ MN ……(i)
କିନ୍ତୁ RABS ଏକ ଆୟତଚିତ୍ର ⇒ RS = AB ….(ii)
(i) ଓ (ii) ରୁ AB = $$\frac { 1 }{ 2 }$$, MN (ପ୍ରମାଣିତ)

Question 21.
ଚିତ୍ରରେ ଗୋଟିଏ ସରଳରେଖା ଦୁଇଟି ଏକ କେନ୍ଦ୍ରିକ ବୃତ୍ତ S1 ଓ S2 କୁ ଯଥାକ୍ରମେ A, C, D ଓ B ବିନ୍ଦୁରେ ଛେଦ କରୁଛି । ପ୍ରମାଣ କର ଯେ, AC = DB |

Solution:
ଦତ୍ତ : S1 ଓ S2 ଦୁଇଟି ଏକ କେନ୍ଦ୍ରିକ ବୃତ୍ତ । ଏକ ସରଳରେଖା ବୃତ୍ତଦ୍ଵୟକୁ ଯଥାକ୍ରମେ A, C, D ଓ B ବିନ୍ଦୁରେ ଛେଦ କରୁଛି |
ପ୍ରାମାଣ୍ୟ : AC = DB
ଅଙ୍କନ : OM ⊥ AB ଅଙ୍କନ କର । $$\overline{\mathbf{OC}}$$ ଏବଂ $$\overline{\mathbf{OA}}$$ କୁ ଯୋଗକର ।
ପ୍ରମାଣ : $$\overline{\mathbf{OM}}$$ ⊥ $$\overline{\mathbf{AB}}$$ ଏବଂ $$\overline{\mathbf{OM}}$$ ⊥ $$\overline{\mathbf{CD}}$$

∴ AM = MB ଏବଂ CM = MD
⇒ AM – CM = MB – MD ⇒ AC = BD

Question 22.
ଗୋଟିଏ ବୃତ୍ତର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ P ମଧ୍ୟ ଦେଇ ଅଙ୍କିତ ଦୁଇଟି ଛେଦକ ବୃତ୍ତକୁ A, B ଏବଂ C, D ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ଯେପରି P – A – B ଏବଂ P – C – D। ଯଦି AB = CD ହୁଏ, ପ୍ରମାଣ କର ଯେ, PA = PC ଏବଂ AC || BD |

Solution:
ଦତ୍ତ : ବୃତ୍ତର ବହିଃସ୍ଥ ବିନ୍ଦୁ P ମଧ୍ୟଦେଇ ଏକ ଛେଦକ ବୃତ୍ତକୁ A, B ଏବଂ C, D ବିନ୍ଦୁରେ ଛେଦ କରୁଛନ୍ତି । AB = CD
ପ୍ରାମାଣ୍ୟ : (i) PA = PC (ii) $$\overline{\mathbf{AC}}$$||$$\overline{\mathbf{BD}}$$
ଅଙ୍କନ : କେନ୍ଦ୍ର O ଠାରୁ $$\overline{\mathbf{AB}}$$ ଓ DC ପ୍ରତି ଯଥାକ୍ରମେ OM ଏବଂ ON ଲମ୍ବ ଅଙ୍କନ କର । OP ଅଙ୍କନ କର ।
ପ୍ରମାଣ : AB = CD = $$\frac { 1 }{ 2 }$$ AB = $$\frac { 1 }{ 2 }$$ CD ⇒ AM = NC ଏବଂ MB = ND… (i)
△OMP ଏବଂ △ONP ଦ୍ୱୟରେ m∠OMP = m∠ONP (ପ୍ରତ୍ୟେକ ସମକୋଣ)
OM = ON ( ସମାନ ଦୈର୍ଘ୍ୟ ବିଶିଷ୍ଟ ଜ୍ୟା ମାନ କେନ୍ଦ୍ରଠାରୁ ସମଦୂରବର୍ତ୍ତୀ), $$\overline{\mathbf{OP}}$$ ସାଧାରଣ ବାହୁ ।
△OMP ≅ △ONP
⇒ MP= NP ⇒ MP – AM = NP – NC [(i)ରୁ] → PA = PC (ପ୍ରମାଣିତ) …(ii)
ପୁନଶ୍ଚ MP + MB = NP + ND [(i)ରୁ]
⇒ PB = PD ⇒ △PBD ସମଦିବାହୁ (ii) ରୁ PA = PC → A PAC ଏକ ସମଦ୍ବିବାହୁ ।
△PBD ରେ PB = PD ⇒ m∠PBD = m∠PDB = θ (ମନେକର)
ସେହିପରି △PAC m∠PAC = m∠ACP = α (ମନେକର)
∴ △PBD ରେ θ + θ + m∠BPD = 180°
⇒ 2θ + m∠BPD = 180° ….(iii)
ପେଦ୍ୱିପରି △PAC ରେ, 2α + m∠APC = 180° …(iv)
(iii) 2θ = 2α

Question 23.
ABC ବୃତ୍ତର କେନ୍ଦ୍ର O । ଏହାର ଦୁଇଟି ସର୍ବସମ ଜ୍ୟା ପରସ୍ପରକୁ ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ P ଠାରେ ଛେଦ କରନ୍ତି । B ଓ C, $$\overline{\mathbf{OP}}$$ ର ଏକ ପାର୍ଶ୍ୱସ୍ଥ ହେଲେ ପ୍ରମାଣ କର ଯେ, (i) PA = PC ଏବଂ (ii) $$\overline{\mathbf{AC}}$$ || $$\overline{\mathbf{BD}}$$ | (ସୂଚନା : $$\overline{\mathbf{OE}}$$ L $$\overline{\mathbf{AB}}$$ ଏବଂ $$\overline{\mathbf{OF}}$$ ⊥ $$\overline{\mathbf{CD}}$$ ଅଙ୍କନ କରି O, P ଯୋଗ କର)

ସମାଧାନ :
ଦତ୍ତ : ବୃତ୍ତର କେନ୍ଦ୍ର ( 1 AB = CD, AB ଓ CD ଜ୍ୟା ଦ୍ଵୟର ଛେଦବିନ୍ଦୁ P ।
B ଓ C, $$\overline{\mathbf{OP}}$$ ର ଏକ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ ।
ପ୍ତାମଣ୍ୟ : (i) PA = PC (ii) $$\overline{\mathbf{AC}}$$ || $$\overline{\mathbf{BD}}$$
ଅଙ୍କନ : $$\overline{\mathbf{OE}}$$ L $$\overline{\mathbf{AP}}$$ ଏବଂ $$\overline{\mathbf{OD}}$$ L $$\overline{\mathbf{CD}}$$ ଅଙ୍କନ କର ।
ପ୍ରମାଣ : AB = CD ⇒ $$\frac { 1 }{ 2 }$$ AB = $$\frac { 1 }{ 2 }$$ CD ⇒ AE = CF ଏବଂ BE = FD
△OPE ଏବଂ OPF ଦଯ6ର m≤OEP = m≤OFP = 90°, OE = OF (∵ ଲ୍ୟା ଦ୍ବାଦାସମ)
$$\overline{\mathbf{OP}}$$ ସାଧାରଣ ∴ △OPE ≅ △OPE ⇒ PE = PF
⇒ AE – PE = CF – PF ⇒ AP = CP ….(i)
⇒ △APC ସମଦିବାହୁ ।
ପୁନଶ୍ଚ BE + PE = DF + FP ⇒ PB = PD ⇒ △PBD ସମଦ୍ବିବାହୁ ।
△APC ସମଦିବାହୁ ।
m∠PAC = m∠PCA = θ (ମ6ନକର) ⇒ 2θ + m∠CPA = 180° ……(1)
△PBD ସମଙ୍ଗିବାହୁ = m∠PBD = m∠PDB = α (ମନେକର)
⇒ 2α + m∠BPD = 180° …..(2)
(1) ଓ (2)ରୁ 2θ + m∠CPA = 2α + m∠BPD (∵ m∠CPA = m∠BPD ପ୍ରତୀପ)
⇒ 2θ = 2α ⇒ θ = α ⇒ m∠PAC = m∠PBD
ମାତ୍ର ଏହି କୋଣଦ୍ଵୟ ଏକାନ୍ତର, ତେଣୁ $$\overline{\mathbf{AC}}$$ || $$\overline{\mathbf{BD}}$$ ……. (ii)

## CHSE Odisha Class 11 English Grammar Additional Questions

Odisha State Board CHSE Odisha Class 12 Invitation to English 4 Solutions Grammar Additional Questions Textbook Activity Questions and Answers.

## CHSE Odisha 11th Class English Grammar Additional Questions

A. Rewrite the passages after correcting all grammatical errors in it.

(1) There is great excitement in the planet of Venus this week. For the first time, Venusia scientists manage to land an unmanned spacecraft in the planet Venus, and it is sending back signals, as well as photographs, ever since. The craft directed into an area known as Gonebay.
There was great excitement on the planet of Venus this week. For the first time, Venusian scientists managed to land an unmanned spacecraft on the planet Venus, and it has been sending back signals, as well as photographs ever since. The craft was directed into an area known as Gonebay.
(2) How did birds know when to flew south for winter? How long do a bear sleep in winter? Do a porcupine really shoot its quills at an enemy? How do squirrels know where he buries nuts?
How do birds know when to fly south for the winter? How long does a bear sleep in winter? Does a porcupine really shoot his quills at an enemy? How does a squirrel know where he buried a nut?

B. Correct the errors.

(a) He is an European.
(b) I met the concerned clerk.
(c) It is high time you get up early.
(d) It has been five years since I last met you.
(e) I congratulate you for winning the prize.
(a) He is a European.
(b) I met the clerk concerned.
(c) It is high time you got up early.
(d) It is five years since I last met you.
(e) I congratulate you on winning the prize.

C. Supply the correct tense of the verbs given in brackets.

1. Water always (freeze) at 0 degrees centigrade.
2. Students frequently (make) mistakes of tense usage when they do this exercise.
3. I (have) my hair cut whenever it gets too long.
4. I (take) my dog for a walk every evening before it died.
5. He (come) to my office whenever he needed money.
6. Last year she (wear) the same dress at every party.
7. Whenever I climb a hill, my ear (boil).
8. She (sing) very beautifully before she was married, but nowadays she (not sing) anymore.
9. I seldom (see) him at concerts these days. He (go) to them regularly before the war.
10. She cooks very well but her sister (cook) much better when I knew her.
11. Every time he opens his mouth, he (say) something foolish.
12. He occasionally makes a big effort, but usually he (not bother).
13. Whenever I (go) to see him, he was out.
14. In the past men frequently (fight) duels. Nowadays they seldom (do).
15. How often you (go) to the theatre when you were in London?
16. You (play) with dolls when you were a little girl?
17. The ancient Egyptians (build) pyramids as tombs for their kings.
18. When I was young, my father always (give) me some money on Saturdays.
19. If he is wise, a pianist (practise) four hours a day.
20. His parents don’t know what to do with their child. He (lie) habitually.
21. My aunt Jane (hate) girls who made up.
22. We all (study) Latin when we were at school.
23. Wood always (float).

1. Water always freezes at 0 degrees centigrade.
2. Students frequently make mistakes of tense usage when they do this exercise.
3. I have my hair cut whenever it gets too long.
4. I took my dog for a walk every evening before it died.
5. He came to my office whenever he needed money.
6. Last year she wore the same dress at every party.
7. Whenever I climb a hill, my ear boils.
8. She sang very beautifully before she was married, but nowadays she does not sing anymore.
9. I seldom see him at concerts these days. He went to them regularly before the war.
10. She cooks very well but her sister cooked much better when I knew her.
11. Every time he opens his mouth, he says something foolish.
12. He occasionally makes a big effort, but usually, he does not bother.
13. Whenever I went to see him, he was out.
14. In the past men frequently fought duels. Nowadays they seldom do.
15. How often you go to. the theatre when you were in London?
16. Did you play with dolls when you were a little girl?
17. The ancient Egyptians built pyramids as tombs for their kings.
18. When I was young, my father always gave me some money on Saturdays.
19. If he is wise, a pianist practices four hours a day.
20. His parents don’t know what to do with their child. He lies habitually.
21. My aunt Jane hated girls who made up.
22. We all studied Latin when we were at school.
23. Wood always floats.

D. Put the verbs in brackets into the correct Present Tense, Continuous, or Simple.

1. Buses usually (run) along this street. but today they (not run) because it is under repair.
2. John (pass) the post office on his way to work every day.
3. She usually (sit) at the back of the class, but today she (sit) in the front row.
4. I rarely (carry) an umbrella, but I (carry) one now because it is raining.
5. What you generally (do) for a living?
6. You (enjoy) your English class today?
7. You (enjoy) washing dishes as a rule?
8. We nearly always (spend) our holidays at the seaside, but this year we are going to France.
9. Mr. Jones usually (sell) only newspapers, but this week he (sell) magazines as well.
10. You (wash) your hands before every meals.
11. Mary generally (begin) cooking at 11, but today she came home early and (cook) now, although il is only 10.30.
12. I’m sorry you can’t see her. She (sleep) still. She usually (wake) much earlier.
13. Why you (wear) a coat this morning? I never (wear) one till October.
14. Joan still (do) her homework. Her sister, who always (work) quicker, (play) already in the garden.
15. These builders generally (build) very rapidly. They (work) at present on two separate contracts.
17. John, who (study) medicine at present, hopes to go abroad after graduation.
18. He generally (come) to my office every clay, but today he (visit) his parents in the country.
19. You (watch) television often? The electrician (install) ours at this moment.
20. Mary usually (wear) a hat to go shopping. but today, as the sun (shine) she (not wear) one.

1. Buses usually run along this Street, but today they are not running because it is under repair.
2. John passes the post office on his way to work every day.
3. She usually sits at the back of the class, but today she is sitting in the front row.
4. I rarely carry an umbrella, but I am carrying one now because it is raining.
5. What do you generally do for a living?
6. Are you enjoying your English class today?
7. Do you enjoy washing dishes as a rule?
8. We nearly always spend our holidays at the seaside, but this year we are going to France.
9. Mr. Jones usually sells only newspapers, but this week he is selling magazines as well.
10. Do you wash your hands before every meal?
11. Mary generally begins cooking at 11, but today she came home early and is cooking now, although it is only 10.30.
12. I’m sorry you can’t see her. She is still sleeping. She usually wakes much earlier.
13. Why are you wearing a coat this morning? I never wear one till October.
14. Joan is still doing her homework. Her sister, who always works quicker, is already playing already in the garden.
15. These builders generally build very rapidly. They are working at present on two separate contracts.
16. What are you doing at this moment? If you are not doing anything, please help me.
17. John, who is studying medicine at present, hopes to go abroad after graduation.
18. He generally comes to my office every day, but today he (is visiting) his parents in the country.
19. Do you often watch television? The electrician is installing ours at this moment.
20. Mary usually wears a hat to go shopping, but today, as the sun is shining she is not wearing one.

E. Put the verbs in brackets into the correct tense, Continuous or Simple.

1. You (see) the house on the corner? That is where I was born.
2. You (listen) to what I am saying You (understand) me?
3. I (notice) Mary (wear) a new hat today.
4. She (not understand) what you (mean).
5. I (need) a new suit. They (offer) special prices at the tailor this week.
6. You (smell) gas? I (think) the new stove is leaking.
7. Look at Mary! She (drink) up her medicine, but I can say that she (hate) it.
8. John (seem) rather tired today.
9. it still (rain), but it (look) as if it will soon stop.
10. You (mind) helping me a moment? I (try) to mend this table.
11. Ask him what he (want).
12. You (remember) the name of that girl who (walk) on the other side of the street?
13. ‘Will you have some tea?’ ‘I (prefer) coffee, please.’
14. I (suppose) I must go now. My wife (wait) for me at home.
15. You (see) this box? It (contain) matches.
16. These twins, who (resemble) one another so strongly, (study) art at present.
17. After what has happened, you really (mean) to say that you still (believe) him?
18. You (suppose) the children still (sleep)?
19. ‘The train still (stand) in the station. You (think) we can just catch it?
20. I (notice) you (possess) a copy of Waugh’s latest book. Will you lend it me?

1. Do you see the house on the corner? That is where I was born.
2. Are you listening to what I am saying? Do you understand me?
3. I notice Mary is wearing a new hat today.
4. She does no: understand what you mean.
5. I need a new suit. They are offering special prices at the tailor this week.
6. Do you smell gás? I think the new stove is leaking.
7. Look at Mary! She is drinking up her medicine, but I can say that she hates it.
8. John seems rather tired today.
9. It is still raining, but it looks as if it will soon stop.
10. Do you mind helping me a moment? I urn Irving to mend this table.
11. Ask him what he wants.
12. Do you remember the name of that girl who is walking on the other side of the street?
13. ‘Will you have some tea?’ ‘I prefer coffee, please.’
14. 1 suppose I must go now. My wife is waiting for me at home.
15. Do you see this box? It contains matches.
16. These twins, who resemble one another so strongly, are studying art at present.
17. After what has happened, do you really mean to say that you still believe him?
18. Do you suppose the children still sleeping?
19. ‘The train is still standing in the station. Do you think we can just catch it
20. I notice you possess a copy of Waugh’s latest book. Will you lend it me?

F. Supply the correct form of the Present Perfect tense, Continuous or Simple in the place of the verbs in brackets.

1. They just (arrive) from New York.
2. They still (not succeed) in reaching the summit.
3. I this very minutes (receive) a telegram from my brother in India.
5. I now (study) your proposals and regret I cannot accept them.
6. They (live) here since January.
7. We (wait) on the platform since three o’ clock.
8. She already (ring) the bell twice.
9. I see you just (have) your hair cut.
10. She (write) letters all morning, but I (not start) to write any yet.
11. The children (sleep) all this afternoon.
12. How long you (stay) in that old hotel?
13. They (work) in the same factory for twenty years now.
14. Since when you (have) that new car?
15. I (knock) on the door for ten minutes now without an answer.
16. They (build) that bridge for over a year and still it isn’t finished.
17. I (try) three times and (be) successful only once.
18. How many times you (be) to the cinema this week?
19. He (go) to the dentist off and on for six months.
20. He (take) the exam. three times and (fail) every time.
21. William (marry) the eldest Jones girl at last.
22. I (try) to get in touch with you for several days now.
23. She just (spend) three weeks at her grandmother’s.
24. He (work) hard on his book for some time and (finish) it at last.
25. You ever (read) ‘War and Peace’?

1. They have just arrived from New York.
2. They have still not succeeded in reaching the summit.
3. I have this very minutes received a telegram from my brother in India.
5. I have now studied your proposals and regret I cannot accept them.
6. They have been living here since January.
7. We have been waiting on the platform since three o’ clock.
8. She has already rung the bell twice.
10. She has been writing letters all morning, but I have no: starred to write any yet.
11. The children have been sleeping all this afternoon.
12. How long have you been staying in that old hotel?
13. They have been working in the same factory for twenty years now.
14. Since when have you had that new car?
15. I have been knocking on the door for ten minutes now without an answer.
16. They have been building that bridge for over a year and still it isn’t finished.
17. I have tried three times and have been successful only once.
18. How many times have you been to the cinema this week?
19. He has been going to the dentist off and on for six months.
20. He has taken the exam. three times and has failed every time.
21. William has married the eldest Jones girl at last.
22. I have been trying to get in touch with you for several days now.
23. She has just spent three weeks at her grandmother’s.
24. He has been working hard on his book for some time and has just finished it at last.
25. Have you ever read ‘War and Peace’?

G. Supply the missing prepositions using only to or at.

1. He is quite blind _____ her faults.
to

2. He is extraordinarily clever _____ mimicking others.
at

3. She, on the other hand, is very efficient _____ her work.
at

4. He is an expert making himself understood in foreign languages.
at

5. Contrary my expectations, I quite enjoyed myself at the party.
to

6. She was standing too close _____ the fire and got burned.
to

7. That fellow’s no good games at all.
at

8. He carried out the project which had always been dear _____ his heart.
to

9. We are all very indignant _____ the injustice done to him.
at

10. I’m sorry! I’m very bad _____ explaining myself.
at

11. He’s the one who’s so very lucky _____ cards.
at

12. That device is entirely new _____ me.
to

13 He’s not equal _____ the job they’ve given him.
to

14. He remained faithful _____ his principles in spite of great pressure.
to

15. The delay proved fatal _____ our plans.
to

16. She was overjoyed _____ the prospect of meeting him again.
at

17. His activities are very harmful _____ my interests.
to

18. The government showed itself hostile _____ any progress.
to

19. She’s no terribly cruel _____ her dog.
to

20. I engaged him because he was so prompt in understanding my instructions.
at

21. He was heartbroken _____ her indifference to him.
at

22. He doesn’t like me although I’ve always been kind _____ him.
to

23. Yes, he’s the kind of person who is always quick figures.
at

24. This is much inferior _____ the one I bought last week.
to

25. You will be liable _____ a heavy fine if you do that.
to

26. This flower is not native _____ England.
to

27. Naturally she was sad _____ the death of her parrot.
at

28. He does his work carefully but he’s terribly slow ______ it.
at

29. Who wouldn’t be triumphant _____ their success in the examination?
at

30. A dutiful daughter and obedient _____ her parents.
to

31. You shouldn’t be surprised _____ a thing like that.
at

32. It should be obvious _____ the meanest intelligence.
to

33. The seeds are peculiar ______ this genus of plant.
to

34. Can’t you manage to be a little more polite your aunt?
to

35. He works in a factory. Previous _____ that he was in a laundry.
to

36. I have been truly astonished _____ the number of people who believe it.
at

37. This is quite irrelevant ______ the matter we are discussing.
to

38. They were shocked _____ his apparent lack of appreciation.
at

39. He was so rude _____ her that she never spoke to him again.
to

40. Sacred _____ the memory of Mary Jones.
to

41. They are very sensitive _____ people’s opinion of them.
to

42. I’ve got one similar _____ yours.
to

43. Subject _____ the exigencies of the service.
to

44. The people of this country are very skilful ______ making dolls.
at

45. It’s useful _____ me to have him about the house.
to

46. It’s vital _____ a proper understanding of the problem.
to

H. Supply the missing preposition using with, for or of.

1. Don’t be afraid _____ the dog! It won’t bite you.
of

2. I can’t be angry _____ him now that he’s apologized _____ what he has done.
with, for

3. He’s far ahead _____ the others in arithmetic.
of

4. You ought to be ashamed _____ yourself.
of

5. Are you aware _____ the fact that it is half past ten.
of

6. I’m sorry. They are simply not capable doing it.
of

7. Don’t disturb him! He’s busy _____ his accounts.
with

8. He’s ambitious and eager _____ honours.
for

9. For goodness sake! Do be careful _____ that vase. You could easily drop it.
with

10. Children must be taught to be careful _____ traffic.
of

11. I am not at all certain _____ the date of his arrival.
of

12. His explanation was not consistent _____ the facts.
with

13. Monsieur X was famous _____ his collection of pictures.
for

14. I was conscious _____ a feeling of uneasiness.
of

15. Kingston lies due west _____ London.
of

16. I’m sorry. I’m not content _____ your explanations.
with

17. The soldier was pronounced fit _____ service.
for

18. Mary was terribly envious _____ Joan’s new hat.
of

19. We are all very fond _____ going to the theatre.
of

20. John is very discontented _____ his salary.
with

21. I shall be grateful any advice you can give me.
for

22. This exercise is full ______ the most terrible mistakes.
of

23. Are you familiar _____ the works of Milton?
with

24. The manager is well qualified _____ his position.
for

25. That is something I am profoundly glad.
of

26. This chair is not identical _____ the one I bought last year.
with

27. He’s a sporting fellow, and always ready for anything.
for

28. That student is ignorant _____ the first rules of grammar.
of

29. Your explanation is incompatible _____ the story I heard.
with

30. John, you will be responsible for providing the drinks.
for

31. Judges must be independent of

political pressure.
of

32. I know he’s a difficult child, but you must be patient _____ him.
with

33. I am extremely sorry _____ the delay, but I was held up.
for

34. He was jealous _____ his brother’s good fortune.
of

35. Comedians are always popular holiday crowds.
with

36. I can’t bake a cake as we are short _____ eggs this week.
of

37. His income is sufficient _____ his needs.
for

38. He’s always very shy approaching his chief.
of

39. Let us be thankful _____ small mercies.
for

40. It is wise to be sure _____ your facts before you speak.
of

41. One is generally tolerant _____ small faults.
of

42. She is, unfortunately, devoid _____ a sense of humor.
of

43. They gave him a visa valid _____ all countries in Europe.
for

44. I’m tired _____ arguing with you.
of

45. They proved themselves unworthy _____ the trust which was placed in them.
of

K. Supply the missing prepositions, from, about, on or in.

1. Keep away _____ the machine while it is running.
from

2. He wsa singularly fortunate _____ his choice of wallpaper.
in

3. He is intent _____ attending the football match on Saturday.
on

4. I am very dubious _____ your chances of passing the examination.

5. They are proficient _____ the use of their fists.
in

6. This is quite different _____ what I expected.
from

7. The diet here is deficient _____ vitamins.
in

8. That young man is very keen _____ cycling.
on

9. We are all very enthusiastic ________ our next holiday.

10. It was far ______ my intention to suggest that he was unintelligent.
from

11. He was perfectly honest _____ his intentions to win the prize at all costs.

12. The secretary was not well qualified _____ shorthand.
in

13. Some people appear completely immune _____ this disease.
from

14. I am very reluctant _____ asking him to do this.

15. Our plans must remain dependent _____ the weather.
on

16. Everybody was very uneasy _____ the outcome of the negotiations.

17. Of course, you are quite right _____ that.

18. He was involved in an accident, resulting _____ the slippery condition of the road.
from

19. The enemy is weak _____ artillery.
in

20. Entomologists are still curious _____ the life-cycle of that month.

21. I am not very interested _____ the story of your life.
in

22. Put that cake back in the cupboard. where it will be safe _____ the cat.
from

23. I’m afraid he is quite wrong _____ the date of the invasion.

24. His father was very sad _____ his son’s failure in his final exams.

25. I am extremely doubtful _____ the wisdom of pursuing that course of action.

L. Supply the prepositions at or to as appropriate.

1. What time did you arrive _____ your home?
at

2. He finds it difficult to accustom himself _____ the climate.
to

3. AIl the visitors exclaimed _____ the beauty of the place.
at

4. Just glance _____ this for me, would you?
at

5. His debts amount _____ a considerable sum.
to

6 I can only guess _____ the extent of the damage.
at

7. It is useless to appeal _____ his better nature.
to

8. She hinted darkly _____ all sorts of wild actions in his youth.
at

9. If you want permission. you must apply _____ the caretaker.
to

10. He was attached _____ the French Army during the war.
to

11. He is too sick to attend _____ his duties.
to

12. _____ what do you attribute your success in life?
to

13. I’m sure this one doesn’t belong _____ me.
to

14. He challenged him _____ a game of chess.
to

15. It is very unkind to joke _____ the expense of the disabled.
at

16. Shall I compare thee _____ a summer’s day?
to

17. If you want them to hear, you’ll have to knock a good deal harder _____ the door.
at

18. The prisoner was condemned _____ penal servitude for life.
to

19. The fire was confined _____ the kitchen regions.
to

20. I will never consent _____ her marrying that man.
to

21. Just have a look _____ this for me, would you?
at

22. I have been entirely converted _____ the use of an electric razor.
to

23. Employees who have twenty-five years’ service become entitled _____ a pension.
to

24. You can safely entrust your little son _____ her care.
to

25. People who heard her voice marvelled _____ it.
at

26. Let’s invite them all _____ dinner.
to

27. Listen _____ me!
to

28. Do you object my smoking?
to

29. He peered _____ the exhibit on account of his short-sightedness.
at

30. Such an idea would never occur _____ me!
to

31. The patient is reacting very unsatisfactorily _____ the drug.
to

32. The children peeped _____ the guests as they were arriving.
at

33. I have been reduced ______ using oil for lack of fat.
to

34. The children are playing _____ Red Indians again.
at

35. She has been forced to resort _____ all sorts of devices to avoid him.
to

36. The patient has not responded _____ treatment.
to

37. It is very rude to point people in the street.
at

38. If you’ll bring the drinks, I’ll see _____ the food.
to

39. They have been subjected _____ all sorts of indignities.
to

40. I refuse to submit ______ that sort of treatment.
to

41. Can you wonder _____ it, if they are reduced _____ begging.
at, to

42. I’m sorry he finally succumbed _____ the temptation of stealing.
to

43. If you want to pass your examination, you il have to work very hard _____ your Latin.
at

44. We shall never surrender _____ that enemy.
to

45. I don’t know him, but he has been starting _____ me for ten minutes.
at

46. I have never subscribed _____ the general opinion of him.
to

47. Turn _____ page 22 and start reading!
to

48. The government has again yielded _____ the pressure from outside.
to

49. She always trusts _____ her neighbors to help her.
to

to

M. Turn the following into passive.

1. The government has called out troops.
Troops have been called out.

2. Fog held up the trains. (agent required)
Trains were held up by fog.

3. You are to leave this here. Someone will call for it later on.
This is to be left here. k will be called for.

4. We called in the police.
Police were called in.

5. They didn’t look after the children properly.
Children were not properly looked after.

6. They are flying in reinforcements.
Reinforcements are being flown in.

7. Then they called up men of 28.
Men of 28 were called up.

8. Everyone looked up to him. (agent required)
He was looked up to by everyone.

9. All the ministers will see him off at the airport. (agent required)
He will be seen off at the airport by all the ministers.

10. He hasn’t slept in his bed.
Bed hasn’t been slept in.

11. We can build on more rooms.
More rooms can be built on.

12. They threw him out.
He was thrown out.

13. They will have to adopt a different attitude.
Different attitude will have to be adopted.

14. He’s a dangerous maniac. They ought to lock him up.
He ought to be locked up.

15. Her story didn’t take them in. (agent required)
They weren’t taken in by her story.

16. Burglars broke into the house.
House was broken into.

17. The manufacturers are giving away small plastic toys with each packet of cereal.
Small plastic toys are being given away.

18. They took down the notice.
Notice was taken down.

19. They frown on smoking here.
Smoking is frown on.

20. After the government had spent a million pounds on the scheme they decided that it was impracticable and gave it up. (Make only the first and last verbs passive)
After a million pounds had been spent, the scheme was given up.

21. When I returned I found that they had towed my car away. I asked why they had done this and they told me that it was because I had parked it under a No Parking sign. (four passives)

22. People must hand in their weapons.
Weapons must be handed in.

23. The crowd shouted him down.
He was shouted down.

24. People often take him for his brother.
He is often taken for his brother.

25. No one has taken cut the cork.
The cork hasn’t been taken out.

26. The film company were to have used the pool for aquatic displays, but now they have changed their minds about it and are filling it in. (Make the first and last verbs passive)
Pool was to have been used it is being filled in.

27. This college is already full. We ne turning away students the whole time.
Students are being turned away.

28. You will have to pull down this skyscraper as you have not complied with the town planning regulations.
Skyscraper will have to be pulled down as the town planning regulations have not been complied with.

N. Put the following sentences into passive, using infinitive construction where possible.

1. We added up the money and found that it was correct.
Money was added up and found to be correct.

2. I’m employing a man to tile the bathroom.
I am having the bathroom tiled.

3. Someone seems to have made a terrible mistake.
A terrible mistake seem to have been made.

4. It is your duty to make tea at eleven o’ clock. (Use suppose.)
You are supposed to make tea.

5. People know that he is armed.
He is known to be armed.

6. Someone saw him pick up the gun.
He was seen to pick up?

7. We know that you were in town on the night of the crime.
You are known to have been.

8. We believe that he has special knowledge which may be useful to the police. (one passive)
He is believed to have special knowledge.

9. You needn’t have done this.
This needn’t have been done.

10. It’s a little too loose: you had better ask your tailor to take it in. (one passive)
You had better have it taken in.

11. He likes people to call him — ‘sir’.
He likes to be called sir’.

12. Don’t touch this switch.
This switch isn’t to be/mustn’t be touched.

13. You will have to get someone to see to it.
You will have to have/get it seen to.
(Or) It will have to be seen to.

14. It is impossible to do this. (Use can’t)
This can’t be done.

15. Someone is following us.
We are being followed.

## CHSE Odisha Class 11 Math Notes Chapter 3 Relations And Function

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 3 Relations And Function will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 3 Relations And Function

Order Pairs
An ordered pair consists of a pair of objects, or elements or numbers or functions in order.
We denote order pairs as (a, b)

• An order pair is not a set of two objects.
• (a, b) = (c, d) ⇒ a = c and b = d
• (a, b) × (b, a)

Cartesian Product Of Sets:
If A and B are non-empty sets, then their Cartesian product, denoted by A × B and defined by A × B = {(a, b): a ∈ A, b ∈ B} = Set of all ordered pairs (a,b) where a ∈ A and b ∈ B
Note:
1. For finite sets A and B |A × B| = |A| . |B|
2. A × B = Φ ⇔ A = Φ or B = Φ
3. A2 = A × A

Properties of Cartesian product:
1. A × B ≠ B × A (Cartesian product is non-commutative)
2. A × (B ∪ C) = (A × B) ∪ (A × C)
3. A × (B ∩ C) = (A × B) ∩ (A × C)
4. A × B = B × A ⇔ A = B
5. A × (B – C) = ( A × B) – (A × C)
6. A ⊂ B ⇒ A × A ⊂ (A × B) ∩ (B × A)
7. A ⊂ B ⇒ A × C ⊂ B × C
8. A ⊂ Band C ⊂ D ⇒ A × C ⊂ B × D
9. (A × B) ∩ (C × D) = ( A ∩ C) × (B ∩ D)

Relation
Let A and B be two arbitrary sets. A binary relation from A to B is a subset of A × B.
OR f is a relation from A to B if f ⊆ A × B
Note:

• If a of A is related to b of B by relation ‘f’ then we write (a,b) ∈ f or a f b
• As Φ ⊂ A × B we have Φ is a relation from A to B. This relation is known as a null of void or empty relation.
• As A × B ⊆ A × B, A × B is also a relation from A to B. This relation is known as universal relation.
• If |A| = m and |B| = n then number of relations from A to B is 2mn

Domain, co-domain, and Range of a relation:
Let f is a relation from A to B. Domain of f = Dom (f) or Df
={x ∈ A : (x, y) ∈ f for some y ∈ B) Co-domain of f = B
Range of ‘f’ = Rng (f) or Rf = {y ∈ B : (x, y) ∈ f for some x ∈ A}

Types Of Relation:
(a) One-many relation: A relation f from A to B is one many if (a, b) and (a, b’) ∈ f ⇒ b ≠ b’
(b) Many-one relations: A relation f from A to B is many-one if (a, b) and (a’, b) ∈ f ⇒ a ≠ a’
(c) One-one relation: A relation f from A to B is one-one if (a, b), (a, b’) ∈ f ⇒ b = b’ and (a, b), (a’, b) ∈ f ⇒ a = a’

Inverse of a relation: Let f is a relation from A to B. The inverse of f is denoted by f-1 is a relation from B to A defined as f-1 = {(b, a): (a, b) ∈ f}

Function:
A relation ‘f’ from X to Y is called a function if:
(a) Df = Dom (f) = X and
(b) (x, y) and (x, z) ∈ f ⇒ y = z or A relation from A to B is a function
if ⇒ Domain of f = X i.e All elements of X is engaged in the relation and
⇒ f is not one many.

Note:
(1) If a relation f from X to Y becomes a function then we write f: X → Y.
(2) If f is a function from A to B i.e. f: X → Y and (x, y) ∈ f then we write y = f(x)
(3) Mapping, map, transformation, transform, operator, and correspondence are different synonym terms of function.
(4) If f: X → Y is defined as y = f(x), then

• y is called the value of the function at x or the image of x under f or the dependent variable.
• x is called the independent variable or pre-image of y under f.

Domain, Co-domain or Range of a function:
Let f: X → Y defined as y = f(x)
(a) Domain of ‘f’ = Dom f = Df = {x ∈ X: y = f(x)}
(b) Range of f = Rng f = Rf = f(A) = {f(x) ∈ Y: x ∈ A } Clearly f(A) ⊆ y
(c) If |A| = m, |B| = n then number of functions from A to B = nm

Real valued function :
A function f: A → B is a real-valued function if B ⊆ R.
→ f is a real function if A ⊆ R and B ⊆ R

Techniques to find Domain and Range of a Real function:
(a) Techniques to find Domain: Let the function is defined as y = f(x).
Step -1: Check the values of x for which f(x) is well defined.
Step -2: The set of all values obtained from step -1 is the domain of ‘f.

(b) Techniques to find range: Let the function is y = f(x)

• Method-1 (By inspection):
→ Step -1: Get values of y for all values x ∈ Dom f.
→ Step -2: Set of all these values of y = Rng f
• Method-2:
→ Step -1: Write x in terms of y
→ Step -2: Get values of y for which x is well defined in Dom f.
→ Step -3: Rng (f) = The set of all y obtained from step 2.

Some Real Functions:
(a) Constant function: A function f: A → R defined as f(x) = k, for some k ∈ R is called a constant function.

(b) Identity function: Let A ⊆ R. The function f: A → A defined as f(x) = x, x ∈ A is called the identity function on A. We denote it by IA

(c) Polynomial function: A function f: A → R defined by f(x) = f(x) = a0 + a1x + a2x2 + anxn where a0, a1, a2, ….., an are real numbers and an ≠ 0 is called a polynomial function (polynomial) of degree n.

(d) Rational function: A function of form f(x) = $$\frac{\mathrm{P}(x)}{\mathrm{Q}(x)}$$ where P(x) and Q(x) are polynomial functions of x is known as a rational function.

(e) Absolute value function OR modulus function: The function f: R → R defined as  f(x) = |x| = $$\begin{cases}x, & x \geq 0 \\ -x, & x<0\end{cases}$$ is called as the modulus function.
→ Rng f = [ 0, ∞] = R+U {0}

Properties Of Modulus Function:
1. For any real number x, we have $$\sqrt{x^2}=|x|$$
2. If a and b are positive real numbers then

• x2 ≤ a2 ⇔ |x| ≤ a
• x2 ≥ a2 ⇔ |x| ≥ a
• a2 ≤ x2 ≤ b2 ⇔ a ≤ |x| ≤ b ⇔ x ∈ [-b, – a] ∪ [a, b]

(f) Signum function: The function f: R → R defined as f(x) = $$\begin{cases}\frac{x}{|x|}, & x \neq 0 \\ 0, & x=0\end{cases}$$ is called signum function.
→ We denote a signum function as f(x) = sgn(x)
→ Range of a signum function = {-1, 0, 1}

(g) Greatest integer function: The function f: R → R defined by f(x) = [x] is called the greatest integer function. [x] = The greatest among all integers ≤ x. OR [x] = n for n ≤ x < n + 1

Properties of the greatest integer function :
Let n is an integer and x is a real number between n and n + 1
(i) [-n]= -[n]
(ii) [x + k] = [x] + k (for an integer ‘k’)
(iii) [-x] = -[x] – 1
(iv) [x] + [-x] = $$\begin{cases}-1, & \mathrm{x} \notin \mathrm{Z} \\ 0, & \mathrm{x} \in \mathrm{Z}\end{cases}$$
(v) [x] – [-x] = $$\begin{cases}2[\mathrm{x}]+1, & \mathrm{x} \notin \mathrm{Z} \\ 2[\mathrm{x}], & \mathrm{x} \in \mathrm{Z}\end{cases}$$
(vi) [x] ≥ k ⇒ x ≥ k for k ∈ Z
(vii) [x] ≤ k ⇒ x < k +1 for k ∈ Z
(viii) [x] > k ⇒ x > k + 1 for k ∈ Z
(ix) [x] < k ⇒ x < k for k ∈ Z

(h) Exponential Function: A function f: R → R defined as f(x) = ax where a > 0 and a ≠ 1 is called the exponential function.

Properties Of Exponential Function:
1. ax+y =  ax . ay
2. (ax)y = axy
3. ax = 1 if x = 0
4. If a > 1, ax > ay ⇒ x > y
5. If a < 1, ax > ay ⇒ x < y

Logarithmic Function:
Let a ≠ 1 is a positive real number. The function f: (0, ∞) → R defined by f(x) = logax is called the logarithmic function, where y = loga ⇔ ay = x
→ Domain of a logarithmic function = (0, ∞) and Range = R

Properties of logarithmic function:
1. loga (xy) = logax + logay
2. loga (x/y) = logax – logay
3. logaa = 1
4. loga(x)y = y logax
5. loga x = 0 ⇔ x = 1
6. logax = $$\frac{1}{\log _a{ }_a}$$ , x ≠ 1
7. logab = $$\frac{\log _a b}{\log _c a}$$
8. $$\log _{a^n}\left(x^m\right)$$ = $$\frac{m}{n}$$ loga|x|

Different Categories of function:
(a) Algebraic Function: A function that can be generated by a variable by a finite number of algebraic operations such as addition, subtraction, multiplication, division, square root, etc. is called an algebraic function.

(b) Transcendental function: A non-algebraic function is a transcendental function.
⇒ Trigonometric, trigonometric, Exponential, and logarithmic functions are transcendental functions.

Even And Odd Functions:
A function ‘f’ is an even function  if f(-x) = x and is an odd function
if f(-x) = x and is an odd function: if f(-x) = -f(x)
Note:
1. If ‘f’ is any function f(x) + f(-x) is always an even function and f(x) – f(-x) is an odd function.
2. Every function f(x) can be expressed as the sum of an even and an odd function as f(x) = g(x) + h(x), where
g(x) = $$\frac{f(x)+f(-x)}{2}$$
h(x) = $$\frac{f(x)-f(-x)}{2}$$

Periodic Function:
A function is called a periodic function with period k if f(x + k) = f(x) for some constant k ≠ 0. The least positive value of k for which f(x + k) = f(x) holds is called the fundamental period of f.

Properties of periodic function :
(1) If k is the period of f then any non-zero integral multiples of k is also a period of f.
(2) If k is the period of f(x) then f(ax + b) is also periodic with period $$\frac{k}{a}$$
(3) If f1(x) + f2(x) and f3(x) are periodic functions with periods k1, k2, k3, respectively then the function a1f1(x) + a2f2(x) + a3f3(x) is also a periodic function with period, LCM (k1, k2, k3)

Algebra Of Real functions:
(a) Equality of two functions: Two functions f and g are equal iff ‘
(i) Dom f = Dom g
(ii) Co-Dom f = Co-Dom g
(iii) f(x) = g(x) for all x belonging to their common domain.

(b) Addition of two functions: Let f: D1 → R and g: D2 → R be two real functions.
The sum function f + g is defined by f + g: D1 ∩ D2 → R and (f + g)(x) = f(x) + g(x) ∀ x ∈ D1 n D2

(c) Subtraction of two functions: Let f: D1 → R and g: D2 → R. The difference function (f – g) is f – g: D1 ∩ D2) → R defined by (f – g) (x) = f(x) – g(x) ∀ x ∈ D1 ∩ D2

(d) Scalar multiplication: Let f: D → R and c is any scalar. The scalar multiple of f by the scalar c is cf: D → R defined as (cf)(x) = c. f(x) ∀ x ∈ D1.

(e) Multiplication of two functions: Let f: D1 → R and g: D2 → R are two real functions. The product function (fg) is (fg): D1 ∩ D2 → R defined as (fg)(x) = f(x)g(x) ∀ ∈ D1 ∩ D2

(f) The quotient of two functions: Let f: D1 → R and g: D2 → R are two real functions. the quotient function ($$\frac{f}{g}$$) i,e,. $$\frac{f}{g}$$: D1 ∩ D2 → R, defined by ($$\frac{f}{g}$$)(x) = $$\frac{f(x)}{g(x)}$$, ∀ x ∈ D1 ∩ D2

## CHSE Odisha Class 11 Math Notes Chapter 2 Sets

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 2 Sets will enable students to study smartly.

## CHSE Odisha 11th Class Math Notes Chapter 2 Sets

Set:
Set is an undefined term in mathematics. But we understand set as “a collection of well-defined objects”.

• Set is a collection.
• The objects (called elements) in a set must be well-defined.

Set Notation:
We denote set as capital alphabets like A, B, C, D…..and the elements by the small alphabets like x, y, z ….

• If x is an element of set A we say “x belongs to A” and write ‘x ∈ A’.
• If x is not an element of set A we say “x does not belong to A” and we write ‘x ∉ A’.

Set Representation:
(a) Extension or tabular or Roster Method: In this method, we describe a set listing the elements, separated by commas within curly brackets.
Note: While listing out the elements the repetition of an object has no effect. Thus, we don’t do this.

(b) Intention or set builder or set selector method: In this method: a set is described by a characterizing property p(x) of element x. In this case, the set is described as {x : p(x) holds}

Types Of Set:
(a) Empty of full or void set: It is a set with no element.

• We denote empty set by ‘Φ’
• There is only one empty set.

(b) Singleton set: It is a set with only one element.

(c) Finite set: A set is finite if it has a finite number of elements.

(d) Infinite Set: A set that is not finite is called an infinite set.

(e) Equal sets: Two sets A and B are equal if they have the same elements. Two sets A and B are equal if all elements of A are also elements of B and all elements of B are also elements of A.

(f) Equivalent set: Two finite sets A and B are equivalent if they have the same number of elements.

Subsets: Let A and B be two sets. If every element of A is an element of B then A is called a. subset of B (we write A ⊂ B) and B is called a superset of A (We write B ⊃ A)
Thus A ⊂ B is x ∈ A ⇒ x ∈ B
Note.
(i) A set is a subset of itself.
(ii) Empty set Φ is a subset of every set.
(iii) A is called a proper subset of B if B contains at least one element that is not in A
(iv) If A has n elements then total number of subsets of A = 2n.

Universal set:
A set ‘U’ that contains all sets in a given context is called the universal set.

Power set:
Let A is any set. The collection (or set) of all subsets of A is called the power set of A. We denote it as P(A)
P(A) = { S: S ⊂ A }

Set Operations:
(a) Union of sets :
The union of two sets A and B is the set of all elements of A or B or both.
∴ A ∪ B = {x ∈ A or x ∈ B}

(b) Intersection of sets:
Intersection of two sets A and B is the set of all those elements that belong to both A and B . (or all common elements of A and B)
∴ A ∩ B = {x: x ∈ A and x ∈ B }
Two sets A and B are disjoint if A ∩ B = Φ. Otherwise, A and B are intersecting or overlapping sets.

(c) Difference of sets: The difference of two sets ‘A and B’ is the set of all elements of A which do not belong to B.
∴ A- B = {x: x ∈ A and x ∈ B)

(d) Symmetric difference of two sets: Symmetric difference of two sets A and B is the set (A – B) ∪ (B – A)
∴ A Δ B = (A – B) ∪ (B – A) = (A ∪ B) – (A ∩ B)

(e) Complement of a set: Let the complement of a set A (denoted as A’ or Ac) be defined as U – A

• A’ = {x ∈ U) : x ∉ A)
• x ∈ A’ ⇔ x ∉ A

Laws Of Set Algebra:
(a) Idempotent law: For any set A we have
(i) A ∪ A = A
(ii) A ∩ A = A

(b) Identity laws: For any set A we have
(i) A ∪ Φ = A and
(ii) A ∩ U = A

(c) Commutative laws: For any three sets A, B, and C
(i) A ∪ B = B ∪ A
(ii) A ∩ B = B ∩ A

(d) Associative laws: For any three sets A, B, and C
(i) A ∪ (B ∪ C) = (A ∪ B) ∪ C
(ii) A ∩ (B ∩ C) = (A ∩ B) ∩ C

(e) Distributive laws: For any three sets A, B, and C
(i) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
(ii) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

(f) De-morgans laws: For any two sets A and B
(i) (A ∪ B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ ∪ B’

Some more properties of sets: For any three sets A, B, and C
(a) A ⊂ (A ∪ B) and (A ∩ B) ⊂ A
(b) A ∪ B = B ⇔ A ⊂ B
(c) A ∩ B = A ⇔ A ⊂ B
(d) B ⊂ A and C ⊂ A ⇒ (B ∪ C ) ⊂ A and A ⊂ B, A ⊂ C ⇒ A ⊂ (B ∩ C)
(e) B ⊂ C ⇒ A ∪ B ⊂ A ∪ C and A ∩ B ⊂ A ∩ C
(f) A – B = A ∩ B’
(g) A – B = A ⇔ A ∩ B = Φ
(h) (A – B) ∪ B = A ∪ B and (A – B ) ∩ B = Φ
(i) A ⊆ B ⇔  B’ ⊆ A’
(j) A Δ B = B Δ A

Cardinality or order of a finite set: The cardinality or order of a finite set A (denoted as |A| or O(A) or n (A)) is the number of elements in ‘A’.

Some important results on the cardinality of finite sets and applications of set theory:
If A, B, and C are finite sets and ‘U’ is the finite universal set then a number of elements belonging to at least one of A or B.
(a) |A ∪ B| = |A| + |B| – |A ∩ B|
(b) |A ∪ B| = |A| + |B| for A ∩ B = Φ i.e. for two disjoint sets A and B
(c) Number of elements belonging to at least one of A, B, or C
= |A ∪ B ∪ C|
= |A| + |B| + |C| – |A ∩ B| – |B ∩ C| – |C ∩ A| + |A ∩ B ∩ C|
(d) Number of elements belonging to exactly two of the three sets A, B, and C = |A ∩ B| + |B ∩ C| + |C ∩ A| – 3 |A ∩ B ∩ C|
(e) Number of elements belonging to exactly one of the three sets A, B, and C = |A| + |B| + |C| – 2 |A ∩ B| -2 |B ∩ C| – 2 |C ∩ A| + 3 |A ∩ B ∩ C|
(f) Number of elements belonging to A but not B = |A – B| = |A| – |A ∩ B|
∴ |A| = |A – B| + |A ∩ B|
(g) Number of elements belonging to exactly one of A or B
= |A Δ B| = |A – B| + |B – A|
= |A| + |B| – 2 |A ∩ B|
(h) |A’ ∪ B’| = |U| – |A ∩ B|
(i) |A’ ∩ B’| = |U| – |A ∪ B| = Number of elements belonging to neither A nor B.