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Odisha State Board BSE Odisha 8th Class English Solutions Lesson 2 The Thief and the Tiger Textbook Exercise Questions and Answers.

BSE Odisha Class 8 English Solutions Lesson 2 The Thief and the Tiger

Session – 1
Pre-Reading (ପ୍ରାକ୍-ପଠନ):

→ Man rides a horse or an elephant or a donkey. Does he ever ride a lion or a tiger or a bear? Why? See the picture below. What do you see? A man is riding a tiger. Is it possible? When? Where? Let’s read the story and see how.

ଲୋକଟି ଗୋଟିଏ ଘୋଡ଼ା, କିମ୍ବା ହାତୀ କିମ୍ବା ଗଧ ଉପରେ ଚଢ଼ିଛି । ସେ କେବେ ସିଂହ, କିମ୍ବା ବାଘ କିମ୍ବା ଭାଲୁ ଉପରେ ଚଢ଼ିଥାଏ କି ? କାହିଁକି ? ତଳେ ଥ‌ିବା ଚିତ୍ରଟିକୁ ଦେଖ । କ’ଣ ଦେଖୁଛ ? ଗୋଟିଏ ଲୋକ ବାଘ ଉପରେ ଚଢ଼ିଛି କେତେବେଳେ ? କେଉଁଠି ? ଆସ ଗପଟି ପଢ଼ି ସବୁକଥା ଜାଣିବା ।

II. While Reading

Text

• SGP – I
• Read paragraphs 1-3 silently and answer the questions that follow.
  (୧ମରୁ ୩ୟ ଅନୁଚ୍ଛେଦ ପର୍ଯ୍ୟନ୍ତ ପାଠକରି ନୀରବରେ ପ୍ରଦତ୍ତ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

1. Once …………………………………………… in the stable.
ଥରେ ଜଣେ ରାଜା ଥିଲେ । ସେ ତାଙ୍କର ବଳବାନ ଓ ଦ୍ରୁତଗାମୀ ଅଶୁମାନଙ୍କ ପାଇଁ ଖୁବ୍ ପ୍ରସିଦ୍ଧ ଥିଲେ । ସେ ସବୁଠାରୁ ଦକ୍ଷ ଶକ୍ତିଶାଳୀ ଘୋଡ଼ାମାନଙ୍କୁ ଆଣି ତାଙ୍କ ଘୋଡ଼ାଶାଳରେ ରଖୁଥିଲେ । ଦିନେ ଗୋଟିଏ ଚୋର ତାଙ୍କର ଗୋଟିଏ ଘୋଡ଼ାକୁ ଚୋରି କରିନେବାକୁ ଇଚ୍ଛା ପ୍ରକାଶ କଲା । ଗୋଟିଏ ବାଘ ଚୋରଟିର ଏ ଯୋଜନା ବିଷୟରେ ଜାଣିବାକୁ ପାଇଲା । ସେ ସେହି ଚୋରଟିର ମାଂସ ଖାଇବାକୁ ଚିନ୍ତାକଲା । ଏଣୁ ସେଇ ରାତିରେ ବାଘଟି ରାଜାଙ୍କ ଘୋଡ଼ାଶାଳରେ ଏକ ନିରାପଦ ସ୍ଥାନରେ ଛପିରହିଲା । ବାଘଟି ଘୋଡ଼ାମାନଙ୍କ ସହ ନୀରବରେ ଶାନ୍ତ ଅବସ୍ଥାରେ ଛିଡ଼ା ହୋଇ ରହିଲା । ସତେଯେମିତି ସେ ଘୋଡ଼ାଶାଳର ଅନ୍ୟ ଘୋଡ଼ାମାନଙ୍କ ପରି ଗୋଟିଏ ଘୋଡ଼ା ।

2. After sometime …………………………………………………… rode on it.
କିଛି ସମୟ ପରେ ଚୋରଟି ଘୋଡ଼ାଶାଳରେ ପ୍ରବେଶକଲା । ଘୋଡ଼ାଶାଳଟି ଅନ୍ଧାର ଥିଲା । ସେ ପ୍ରତ୍ୟେକ ଘୋଡ଼ାର ପିଠିରେ ହାତମାରି କେଉଁଟି ସବୁଠୁ ଶକ୍ତିଶାଳୀ ଘୋଡ଼ା ତାହା ପରୀକ୍ଷା କରିବାକୁ ଲାଗିଲା । ଶେଷରେ ସେ ବାଘର ପିଠିରେ ହାତ ପକାଇଲା । ଏଇଟି ସବୁଠୁ ଭଲ ଘୋଡ଼ା ବୋଲି ଚିନ୍ତାକଲା ଏବଂ ସେଇଟିକୁ ଘୋଡ଼ାଶାଳର ବାହାରକୁ ଆଣିଲା । ତା’ପରେ ସେ ବାଘର ପାଟିରେ ଲଗାମ ବାନ୍ଧିଲା ଏବଂ ତା’ଉପରେ ଚଢ଼ିଗଲା ।

3. The tiger ……………………………………… anything in the dark.
ବାଘର ଆଗରୁ ଏପରି କୌଣସି ଅଭିଜ୍ଞତା ନଥିଲା । ସେ ଚୋରକୁ ଅତି ଶକ୍ତିଶାଳୀ ବ୍ୟକ୍ତି ବୋଲି ଭାବିଲା । ସେ ଚୋରଟିକୁ ଭୟଙ୍କର ଭାବରେ ଡରିଯାଇଥିଲା । ଏଣୁ ଅତି ଦ୍ରୁତଗତିରେ ଦୌଡ଼ିବାକୁ ଲାଗିଲା । ଚୋରଟିର ମଧ୍ୟ ଏପରି ଏକ ପ୍ରାଣୀ ଉପରେ ଚଢ଼ିବାର ଅଭିଜ୍ଞତା ନଥିଲା । ସେ ଏହାକୁ ଅତି ଶକ୍ତିଶାଳୀ ଘୋଡ଼ା ବାଛି ଆଣିପାରିଛି ବୋଲି ଚିନ୍ତାକଲା । ସେ ଜାଣି ନଥିଲା ଯେ ସେ ଏକ ବାଘ ଉପରେ ଚଢ଼ିଛି କାରଣ ସ୍ଥାନଟି ସମ୍ପୂର୍ଣ୍ଣ ଅନ୍ଧକାର ଥିଲା ।

Word Meaning

famous : well-known to many people I renowned (ପ୍ରସିଦ୍ଧ, ବିଖ୍ୟାତ)
swift : fast moving (କ୍ଷିପ୍ର / ଶୀଘ୍ର ଗତି କରୁଥିବା)
stable : house for horses (ଘୋଡ଼ାଶାଳ)
steal : the act of taking something from someone unlawfully (ଚୋରିକରିବା)
plan : design / scheme (ଯୋଜନା)
flesh : meat (ମାଂସ)
hid : ଲୁଚାଇ

silently : without speaking / without making a sound (ନୀରବରେ )
touched: be in direct physical contact (ଛୁଇଁବା)
bridle: leather band put on the head of a horse to control its movement/reins
experience: the accumulation of knowledge or skills that results
from direct participation in events or activities (ଅନୁଭୂତି)
powerful : having great influence (କ୍ଷମତାଶାଳୀ / ବଳଶାଳୀ)
terribly : horribly / causing fear (ଭୟଙ୍କର ଭାବରେ)
imagine : fancy / think / suppose (କଳ୍ପନାକରିବା)

Comprehension Questions and Answers: (ସଂକ୍ଷିପ୍ତ ପ୍ରଶ୍ନୋତ୍ତର)

Question 1.
What was the king famous for?
(ରାଜା କେଉଁଥୂପାଇଁ ପ୍ରସିଦ୍ଧ ଥିଲେ ?)
Answer:
The king was famous for his strong and swift horses.

Question 2.
What did the thief plan to do?
(ଚୋରଟି କ’ଣ କରିବାପାଇଁ ଯୋଜନା କଲା ?)
Answer:
The thief wanted to steal a horse.

Question 3.
Why did the tiger hide in the stable?
(ବାଘଟି ଘୋଡ଼ାଶାଳରେ ଛପିରହିଥିଲା କାହିଁକି ?)
Answer:
The tiger came to know about the thief’s plan and thought of eating the thief s flesh.

Question 4.
Why did the thief touch the back of each horse?
(ଚୋରଟି କାହିଁକି ପ୍ରତି ଘୋଡ଼ା ପିଠିରେ ହାତ ମାରୁଥିଲା ?)
Answer:
The thief touched the back of each horse to steal the best one.

Question 5.
Why did he think the tiger to be the best horse ?
(ସେ ବାଘକୁ କାହିଁକି ସର୍ବଶ୍ରେଷ୍ଠ ଘୋଡ଼ା ବୋଲି ଭାବୁଥିଲା ?)
Answer:
He thought the tiger was the best horse because he felt the tiger’s back was different from the other horses.

Question 6.
How did he ride on it?
(ସେ କିପରି ଏହା ଉପରେ ଚଢ଼ିଲା ?)
Answer:
He rode on the tiger putting a bridle on its mouth thinking it was a horse.

Question 7.
He did not know that he was riding a tiger. Why?
(ସେ ଜାଣି ନଥୁଲା ଯେ ସେ ଏକ ବାଘ ଉପରେ ଚଢ଼ିଥିଲା, ଏହାର କାରଣ କ’ଣ ?)
Answer:
He didn’t know that he was riding a tiger, because there was darkness inside the stable.

Question 8.
Where did the tiger run into?
(ବାଘ କେଉଁଠାକୁ ଦୌଡ଼ି ପଳାଇଲା ।)
Answer:
The tiger ran into the forest.

Session – 2

• SGP – 2
• Read paragraph 4 silently and answer the questions that follow.
  (ଚତୁର୍ଥ ଅନୁଚ୍ଛେଦଟି ପାଠକରି (ନୀରବରେ) ପ୍ରଦତ୍ତ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

4. The day…………………………………… as it could.
ରାତି ପାହିଲା । ଅନ୍ଧକାର ଦୂରୀଭୂତ ହେଲା । ସେତେବେଳେ କେବଳ ସେ ଜାଣିପାରିଲା ଯେ ସେ ଭୁଲବଶତଃ ଗୋଟିଏ ବାଘ ଉପରେ ଚଢ଼ିଛି । ସେ ଭୟଭୀତ ହୋଇପଡ଼ିଲା । ବାଘଟି ମଧ୍ୟ ଭୟଭୀତ ହୋଇ ଅଧିକ ଦ୍ରୁତଗତିରେ ଦୌଡୁଥାଏ । ଚୋରଟି କ’ଣ କରିବ, ସ୍ଥିର କରିପାରିଲା ନାହିଁ । ସେ ପ୍ରାୟ ଅଚେତ ହେବାଭଳି ଅନୁଭବ କଲା । ଦୌଡୁଥିବାବେଳେ ବାଘ ଗୋଟିଏ ଗଛଦେଇ ଦୌଡ଼ୁଥିଲା । ଏହି ସମୟରେ ଚୋରଟି ଗଛର ଡାଳକୁ ଧରି ଗଛ ଉପରେ ଚଢ଼ିଗଲା । ବାଘ ଖୁସି ଅନୁଭବ କଲା । କାରଣ ସେ ଲୋକଟି ପାଇଁ ସେ ଅତ୍ୟଧ୍ୱ ଭୟଭୀତ ହୋଇପଡ଼ିଥିଲା । ପାରୁପର୍ଯ୍ୟନ୍ତ ଦ୍ରୁତ ବେଗରେ ବାଘଟି ବଣମଧ୍ୟକୁ ଦୌଡ଼ିଲା ।

Word Meaning

dawn : day break / beginning of the day (ପ୍ରଭାତ / ଉଷା)
disappear : vanish from sight (ଅନ୍ତର୍ଦ୍ଧାନ ହୋଇଯିବା / ଅଦୃଶ୍ୟ ହୋଇଯିବା)
frightened : scared / intense fear (ଭୟଭୀତ କରାଇବା)
fast : swift (ଦୃତ / ଦ୍ରୁତଗାମୀ)
faint : to feel weak and lose consciousness (ଅଚେତ ହୋଇଯିବା)
passed : to cross (ଅତିକ୍ରମ କରିବା)
branch : part of a tree (ଶାଖା)
climb : ascend (ଚଢ଼ିବା)

Comprehension Questions and Answers: (ସଂକ୍ଷିପ୍ତ ପ୍ରଶ୍ନୋତ୍ତର)

Question 1.
When did the thief come to know that he was riding a tiger?
(ଚୋରଟି କେତେବେଳେ ଜାଣିପାରିଲା ଯେ ସେ ଏକ ବାଘ ଉପରେ ଚଢ଼ିଛି ?)
Answer:
When the day dawned and darkness disappeared the thief came to know that he was riding a tiger.

Question 2.
How did he save himself?
(ସେ କିପରି ନିଜକୁ ରକ୍ଷାକଲା ?)
Answer:
When the tiger ran by a tree beside the road the thief caught hold of one of the branches of the tree and climbed up.

Question 3.
Why was the tiger happy?
(ବାଘ କାହିଁକି ଖୁସି ହୋଇଗଲା ?)
Answer:
The tiger was terribly afraid of the rider. So when the rider climbed up the tree he was happy.

• SGP – 3 (Text book page No. 34)
• Read paragraphs 5 – 8 silently and answer the questions that follow.
  (୫ମ ଠାରୁ ଅଷ୍ଟମ ଅନୁଚ୍ଛେଦ ପର୍ଯ୍ୟନ୍ତ ନୀରବରେ ପାଠକରି ପ୍ରଦତ୍ତ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

5. After…………………………………………. dead body.
କିଛି ସମୟ ପରେ ଚୋରଟି ଗଛ ଉପରୁ ଓହ୍ଲାଇ ଗଛତଳେ ବିଶ୍ରାମ ନେଲା । ଅତ୍ୟଧ‌ିକ ଭୟ ଓ କ୍ଳାନ୍ତ ଅନୁଭବ କରିଥିବାରୁ ସେ ସେଠାରେ ଶୋଇପଡିଲା । ଗଭୀର ନିଦ୍ରା ଯୋଗୁଁ ସେ ଗୋଟିଏ ମୃତ ଶବଭଳି ଜଣାପଡୁଥିଲା । ସେହି ବାଟଦେଇ ଗୋଟିଏ ଗଧ୍ଵ ଯାଉଥିଲା । ସେ ଲୋକଟିକୁ ଏକ ମୃତ ଶବ ଭାବି ତା’ର ମାଂସ ଖାଇବାକୁ ଇଚ୍ଛାକଲା । ସେ ମନକୁମନ କହିଲା, ମୁଁ କି ଭାଗ୍ୟବାନ, ଏଇ ଶବଟି ମୋର ସପ୍ତାହେରୁ ଅଧିକ କାଳ ଖାଦ୍ୟ ହୋଇପାରିବ । କିନ୍ତୁ କେହିଜଣେ ଏହାକୁ ବଣଭିତରକୁ ଟାଣିନେବାପାଇଁ ମୋତେ ସାହାଯ୍ୟ କରିବା ଦରକାର ।

6. Thinking so, ………………………………….. with a rope”.
ଏଇକଥା ଚିନ୍ତା କରି ଗଧ୍ଵ ଆଉ ଅନ୍ୟ ଗୋଟିଏ ପ୍ରାଣୀର ସାହାଯ୍ୟ ପାଇଁ ବଣ ଭିତରକୁ ଗଲା ଏବଂ ସେଇ ବାଘଟି ସହିତ ତା’ର ସାକ୍ଷାତ ହେଲା । ସେ ବାଘକୁ କହିଲା, ‘ବାଘ ମହାଶୟ’ ଗୋଟିଏ ମଣିଷର ଶବକୁ ଟାଣି ଆଣିବାରେ ମୋତେ ସାହାଯ୍ୟ କରିବ କି ? ମୁଁ ଏହାର ଅଧା ତୁମକୁ ଦେବି । ‘ଗତ ରାତିରେ ଘଟିଥିବା ଘଟଣାର ଅନୁଭୂତିରୁ ବାଘ କହିଲା, ‘ତୁମେ ମୋତେ ଠକିଦେବ ନାହିଁ ତ ? ମୋତେ ଏକା ଛାଡ଼ିଦେଇ ତୁମେ ଦୌଡ଼ି ପଳାଇଯିବନି ତ ? ‘ଗଧ୍ଵ କହିଲା, ‘ଆମେ ଦୁହେଁ ଦିହିଁଙ୍କୁ ଗୋଟିଏ ଦଉଡ଼ିରେ ବାନ୍ଧିଦେବା, ତା’ହେଲେ କେହି କାହାକୁ ଛାଡ଼ି ପଳାଇ ଯାଇପାରିବା ନାହିଁ ।’’

7. The wolf ……………………………………. wolf died.
ଗଧ୍ଵ ଓ ବାଘ ଦୁହେଁ ଦୁହିଁଙ୍କୁ ଗୋଟିଏ ଦଉଡ଼ିରେ ବାନ୍ଧି ସନ୍ତର୍ପଣରେ ଶବ ପାଖକୁ ଚାଲିଲେ । ଚୋରଟି ନିଦରୁ ଉଠିଯାଇଥିଲା । ସେ ଏମାନଙ୍କ ଆସିବା ଜାଣିପାରି ଭୟରେ ଚିତ୍କାର କରି କହିଲା, ‘ହଇରେ ବାଘ ତୁ ପୁଣି ଆସିଛୁ ?’’ ଏବେ ବାଘଟି ସେ ଚୋରକୁ ପୁଣି ଦେଖ୍ ଭୟଭୀତ ହୋଇପଡ଼ିଲା ଏବଂ ଗଧୂକୁ ଘୋଷାରିନେଇ ଯେତେ ପାରେ ସେତେ ଜୋର୍‌ରେ ଦଉଡ଼ିବାକୁ ଲାଗିଲା । ବିଚରା ଗଧୂଟି ବାଟରେ ମରିଗଲା ।

8. Since …………………………………………………… that day.
ସେବେଠୁ ବାଘ ଆଉ ମଣିଷ ମାଂସ ନ ଖାଇବା ପାଇଁ ପ୍ରତିଜ୍ଞାକଲା । ଚୋରଟି ମଧ୍ୟ ରକ୍ଷା ପାଇଯାଇଥିବାରୁ ଖୁସି ହେଲା ଏବଂ ସେ ମଧ୍ୟ ଆଉ ଚୋରି ନ କରିବାପାଇଁ ପ୍ରତିଜ୍ଞାକଲା ।

Word Meaning

tired: of strength or energy (କ୍ଳାନ୍ତ)
wolf: a wild carnivorous animal (ଗଧୂ)
luck : destiny/fate / fortune (ଭାଗ୍ୟ)
last : continue to stay (ଚାଲୁ ରଖୁବା | ତିଷ୍ଠିବା)
drag : to pull by force (ଘୋଷାରିଘୋଷାରି ଟାଣିନେବା)
suspicious: a doubtful condition, feeling that something is wrong (ସନ୍ଦେହପରାୟଣ )
cheat: betray/deceive (ଠକିବା)
leave: to give up/abandon (ଛାଡ଼ିବା)
alone : lonely (ଏକୁଟିଆ)
suggest : to give opinion (ମତାମତ ଦେବା)
tie : to tag/join (ବାନ୍ଧିବା / ସଂଯୁକ୍ତ କରିବା)
rope : twisted cord (ଦଉଡ଼ି / ରଜ୍ଜୁ)
awake: rise/get up (ଉଠିପଡ଼ିବା)
footsteps: found of feet/sound, of a person walking (ପାଦ ଶବ୍ଦ)
promised : to make a promise / assure (ପ୍ରତିଜ୍ଞାକରିବା)
desire: want/like to get (ଇଚ୍ଛା).
give up: to avoid/abandon (ଛାଡ଼ିଦେବା )
stealing: to take others without knowledge (ଚୋରି)

Comprehension Questions and Answers: (ସଂକ୍ଷିପ୍ତ ପ୍ରଶ୍ନୋତ୍ତର)

Question 1.
Why did the thief fall fast asleep?
(ଚୋରଟି ଗଭୀର ନିଦ୍ରାରେ ଶୋଇଁପଡ଼ିଲା କାହିଁକି ?)
Answer:
The thief fell fast asleep because he was afraid and tired.

Question 2.
Who saw him ? What was his plan ?
(କିଏ ତାକୁ ଦେଖୁଲା ? କ’ଣ ସେ ଚିନ୍ତାକଲା ?)
Answer:
A wolf saw him. He thought he as dead and planned to use him as his food for more than a week long.

Question 3.
What did he want the tiger to do?
(ସେ ବାଘକୁ କଣ କରିବାପାଇଁ ଚାହିଁଲା ?)
Answer:
He wanted the tiger to help him to drag the dead body into the forest.

Question 4.
What did the tiger say?
(ବାଘ କ’ଣ କହିଲା ?)
Answer:
The tiger asked him if he was going to cheat him or not.

Question 5.
“Won’t you run away leaving me alone ?” Who said this?
(ତୁ ମୋତେ ଏକୁଟିଆ ଛାଡ଼ି ପଳାଇଯିବୁନି ତ ? ଏକଥା କିଏ କହିଲା ?)
Answer:
“Won’t you run away leaving me alone ?” The tiger said this to the wolf.

Question 6.
What did the wolf say?
(ଗଧୂଟି କ’ଣ କହିଲା ?)
Answer:
The wolf suggested to the tiger that they would tie each other with a rope so that no one of them could run away leaving another.

Question 7.
What awoke the thief?
(ଚୋରଟି କାହିଁକି ଉଠିପଡ଼ିଲା ?)
Answer:
The footsteps of the tiger and the wolf awoke the thief.

Question 8.
Why did the tiger run away?
(ବାଘ କାହିଁକି ଦୌଡ଼ି ପଳାଇଲା ?)
Answer:
The tiger ran away frightened of the thief.

Question 9.
How did the wolf die?
(ଗଧୂ ମଲା କିପରି ?)
Answer:
The wolf died being dragged by the tiger as they tied each other with a rope.

Question 10.
What did the tiger promise?
(ବାଘ କ’ଣ ପ୍ରତିଜ୍ଞା କଲା ?)
Answer:
The tiger promised no to desire for human flesh any more.

Question 11.
What did the thief stop doing?
(ଚୋର ସେହିଦିନଠାରୁ କ’ଣ ନ କରିବାକୁ ସ୍ଥିରକଲା ?)
Answer:
The thief stopped stealing from that day.

Session – 3
III. Post-Reading (ପଢ଼ିବା ପରେ)

1. Visual Memory Development Technique (VMDT)
Whole Text : the tiger and the thief in the stable……
(ଘୋଡ଼ାଶାଳରେ ବାଘ ଏବଂ ଚୋର…
the tiger with the thief on its back ran for life
(ବାଘ ଚୋରକୁ ପିଠିରେ ବସାଇ ଜୀବନବିକଳରେ ପଳାଇଲା)
……….. the thief and the tiger save themselves
(ଚୋର ଏବଂ ବାଘ ଉଭୟେ ନିଜ ନିଜକୁ ରକ୍ଷାକଲେ ।
……………the wolf’s plan and he died.
(ଗଧୂର ଯୋଜନା ଏବଂ ତା’ର ମୃତ୍ୟୁ)

Part Text (Para-5): “What good luck !, This dead man will last me more than a week. But someone should help me drag the dead body”.
(କି ଭାଗ୍ୟ ! ଏହି ଶବଟି ମୋର ସପ୍ତାହେରୁ ଅଧିକକାଳ ଖାଦ୍ୟ ହୋଇ ପାରିବ । କିନ୍ତୁ ଏହାକୁ ଘୋଷାରିନେବାପାଇଁ ମୋତେ କେହିଜଣେ ସାହାଯ୍ୟ କରିବା ଦରକାର ।)

2. Comprehension Activity : (ବୋଧ ପରିମାପକ କାର୍ଯ୍ୟାବଳୀ)

(a) Choose the correct alternatives and complete each sentences.
(ଠିକ୍ ବିକଳ୍ପ ବାଛି ବାକ୍ୟଗୁଡ଼ିକ ପୂର୍ଣ କର ।)

Question 1.
The thief put a _______________ on the tiger’s mouth.
(A) saddle
(B) bridle
(C) chain
(D) rope
Answer:
(B) bridle

Question 2.
When the tiger and the thief saw each other, _______________.
(A) only the tiger was frightened
(B) only the thief was frightened
(C) both were frightened
(D) none was frightened
Answer:
(C) both were frightened

Question 3.
“What a good luck !“ said _______________.
(A) the king
(B) the tiger
(C) the thief
(D) the wolf
Answer:
(D) the wolf

Question 4.
The wolf’s final plan was to ________ the dead body.
(A) drag
(B) bury
(C) burn
(D) eat
Answer:
(D) eat

Question 5.
Seeing the thief, the tiger ran for life _______________ the wolf.
(A) dragging
(B) carrying
(C) leading
(D) following
Answer:
(A) dragging

(b) Given below are some sentences. They are about what happened in the story. But they are not in the right order. Fill in the boxes with correct serial numbers to rearrange the sentences. (ତଳେ କେତେକ ବାକ୍ୟ ଲେଖାହୋଇଛି । ସେଗୁଡ଼ିକ ଗଳ୍ପରେ ଘଟିଯାଇଥିବା ଘଟଣା ସମ୍ପର୍କିତ । କିନ୍ତୁ ସେଗୁଡ଼ିକ ଠିକ୍ କ୍ରମ ଅନୁସାରେ ନାହାନ୍ତି । ବାମପଟେ ଥ‌ିବା ଖାଲି ଘରଗୁଡ଼ିକରେ ଠିକ୍ କ୍ରମ ନମ୍ବର ଲେଖ୍ ବାକ୍ୟଗୁଡ଼ିକ ସଜାଅ ।)

[ ] The wolf requested the tiger to drag the man.
[ ] The thief got up.
[ ] The tiger among the horses stood silently.
[ ] Once, a thief came inside the stable to steal a horse.
[ ] It was dark everywhere.
[ ] The thief fell asleep like a dead man.
[ ] He climbed up a tree.
[ ] He thought the tiger to be the best horse.
[ ] The tiger ran for life dragging the wolf.
[ ] At night a tiger entered the stable.
[ ] The wolf and the tiger tied each other with a rope.

Answer:
[ 8 ] The wolf requested the tiger to drag the man.
[ 10] The thief got up.
[ 3 ] The tiger among the horses stood silently.
[ 1 ]Once, a thief came inside the stable to steal a horse.
[ 4 ] It was dark everywhere.
[ 7 ] The thief fell asleep like a dead man.
[ 6 ] He climbed up a tree.
[ 5 ] He thought the tiger to be the best horse.
[ 11] I The tiger ran for life dragging the wolf.
[ 2 ] At night a tiger entered the stable.
[ 9 ] The wolf and the tiger tied each other with a rope.

Session – 4

3. Listening : (ଶ୍ରବଣ)
On the chart below, the characters in the story are written in the boxes from left to right at the top. Some words related to the characters are given in the boxes from top to bottom at the left. Your teacher will read out the words one by one. Listen to him/her carefully and put a tick (✓) in the box on the word line below the character. One is done for you.
(ତଳ ଚାର୍ଟରେ ବିଷୟର ଚରିତ୍ରଗୁଡ଼ିକ ପୃଷ୍ଠାର ଅଗ୍ରଭାଗରେ ବାମରୁ ଡାହାଣକୁ ଲେଖାଯାଇଛି । ସେମାନଙ୍କ ଚରିତ୍ର ସହିତ ଖାପଖାଉଥ‌ିବା କେତେକ ଶବ୍ଦ ଉପରୁ ତଳକୁ ବାମପଟେ ଲେଖାଯାଇଛି । ତୁମ ଶିକ୍ଷକ ସେ ଶବ୍ଦଗୁଡ଼ିକୁ ଗୋଟିଏ ପରେ ଗୋଟିଏ ପାଠକରିବେ । ଯତ୍ନର ସହିତ ତାଙ୍କ ଶବ୍ଦଗୁଡ଼ିକ ଶୁଣ ଏବଂ ଶବ୍ଦ ଧାଡ଼ି ପାଖରେ ଥ‌ିବା ଖାଲି ଘରମାନଙ୍କରେ ସେମାନଙ୍କ ଚରିତ୍ରର ଗୁଣକୁ ଖାପଖାଉଥ‌ିବା ଶବ୍ଦ ପାଖରେ ଚିହ୍ନ ଦିଅ ।)

Answer:

Characters	thief	horse	tiger	wolf
Words
Strong			✓
flesh	✓
drag	✓
steal	✓
swift		✓	✓
speed			✓
Search	✓
ride	✓
stable	✓
run			✓
died				✓
branch
Forest			✓	✓
bridle		✓
slept	✓

4. Speaking : (କଥନ)
• Practise the following dialogues.
(ନିମ୍ନ ସଂଳାପଗୁଡ଼ିକ ଅଭ୍ୟାସ କର ।)
• Step : (କେଉଁ କେଉଁ ସ୍ତରଦେଇ ଏହି ସଂଳାପ ପାଠ କରାଯିବ ।)
• Rehearsal-teacher reads aloud, and students listen. The teacher reads aloud and students repeat after him/her dialogue by dialogue.
• Teacher vs Students.
• Students vs students (in two groups)
(ଶିକ୍ଷକ ପାଟି କରି ପଢ଼ିବେ – ଛାତ୍ରଛାତ୍ରୀମାନେ ଶୁଣିବେ)
ଶିକ୍ଷକ ପାଟି କରି ପାଠ କରିବାପରେ ଛାତ୍ରଛାତ୍ରୀ ସେଗୁଡ଼ିକୁ ଆଉଥରେ ଦୋହରାଇ କହିବେ ।
ଶିକ୍ଷକ –
କହିବେ –
ଛାତ୍ରଛାତ୍ରୀ କହିବେ – କହିବେ ।
ପରସ୍ପର ସହିତ – (ଦୁଇଟି ଦଳରେ ବିଭକ୍ତ ହୋଇ)
(They do this reading from the text.)

Wolf : Good Luck ! This dead man will last me more than a week.
But who will help me drag the dead body ?
Tiger: What are you looking for, Mr. Wolf?
Wolf: Mr. Tiger, will you help me drag this dead body?
Tiger: Why should I?
Wolf: I’ll give you half of it. ,
Tiger: Aren’t you going to cheat me?
Wolf: No, no, not at all. How can I?
Tiger: Won’t you run away leaving me alone?
Wolf: How can that be? We’ll tie each other with a rope.
Tiger: Good idea! That’ll do.

Session – 5

5. Vocabulary : (ଶବ୍ଦଜ୍ଞାନ)
Match who lives where. Write the serial numbers in brackets. One is done for you.

Answer:

Session  – 6

6.  Writing : (ଲିଖନ)

a. In comprehension Activity No. 2 b you have rearranged the sentences of the story. Use the sentences serially and write the story in the space given below. (ସଂପ୍ରତି କାର୍ଯ୍ୟ ପ୍ରକ୍ରିୟା No. 2(b)ରେ ଗଛର ବାକ୍ୟଗୁଡ଼ିକ ଠିକ୍ କ୍ରମରେ ସଜାଯାଇଛି ।ବାକ୍ୟଗୁଡ଼ିକ କ୍ରମ ଅନୁସାରେ ଲେଖୁ ଗଳ୍ପଟି ପ୍ରସ୍ତୁତ କର ।)

The Thief And The Tiger
______________________________________________
______________________________________________
______________________________________________
______________________________________________
Answer:

• Once a thief came inside the stable to steal a horse. At night a tiger entered the stable.
• The tiger among the horses stood silently.
• It was dark everywhere.
• He thought the tiger to be the best horse.
• He climbed up a tree.
• The thief fell asleep like a dead man.
• The wolf requested the tiger to drag the man.
• The wolf and the tiger tied each other with a rope.
• The thief got up.
• The tiger ran for life dragging the wolf.

(b). Write answers to the following questions.

Question (i).
Where did the thief and the tiger hide? Why?
(ଚୋର ଏବଂ ବାଘ କେଉଁଠି ଲୁଚିଥିଲେ ? କାହିଁକି ?)
Answer:
They hid in the king’s stable.
The thief wanted to steal a horse.
The tiger thought of eating man’s flesh.

Question (ii).
Why did the thief think the tiger to be the best horse?
(ଚୋର ବାଘକୁ କାହିଁକି ସର୍ବଶ୍ରେଷ୍ଠ ଘୋଡ଼ା ବୋଲି ଭାବିଲା ?)
Answer:
He thought so because the back of the tiger gave him a smooth silky touch.

Question (iii).
Why was the thief frightened when it was the day?
(ଦିନ ହୋଇଯିବା ପରେ ଚୋରଟି ଭୟଭୀତ ହୋଇପଡ଼ିଲା କାହିଁକି ?)
Answer:
The thief was frightened when it was the day because he could not see due to darkness inside the stable at night.

Question (iv).
How did he save himself?
(ସେ କିପରି ନିଜକୁ ରକ୍ଷାକଲା ?)
Answer:
When the tiger was running in fear with great speed passing under a tree the thief was caught hold of one branch of the tree and climbed up.

Question (v).
Where did the theif take rest? Why did he fall asleep?
(ଚୋର କେଉଁଠି ବିଶ୍ରାମ ନେଲା ? ସେ କାହିଁକି ଗଭୀର ନିଦ୍ରାରେ ଶୋଇପଡ଼ିଲା ?)
Answer:
The thief took rest under the tree.

Question (vi).
What was the wolf’s plan?
(ଗମ୍ଵାର ଯୋଜନା କ’ଣ ଥିଲା ?)
Answer:
The wolf planned to use the dead body as his food for more than a week long.

Question (vii).
What sort of help did the wolf want from the tiger? What was his offer to him?
(ଗଧୂଆ ବାଘ ନିକଟରୁ କିପରି ସାହାଯ୍ୟ ଆଶା କରିଥିଲା । ସେ ତାକୁ କେଉଁ ଉପହାର ଦେବାପାଇଁ ପ୍ରତିଶୃତି ଦେଲା ?)
Answer:
The wolf wanted the tiger to help him drag the body to the safest place in the forest.

Question (viii).
What was the tiger’s suspicion?
(ବାଘର ସନ୍ଦେହ କ’ଣ ଥିଲା ?)
Answer:
The tiger feared that the wolf would cheat him and run away leaving him alone.

Question (ix).
Why did the two animals tie each other with a rope?
(କାହିଁକି ଦୁଇଜଣ ପ୍ରାଣୀ ପରସ୍ପରକୁ ଏକ ଦଉଡ଼ିରେ ବାନ୍ଧିଲେ ?)
Answer:
The two animals tied each other with a rope so that neither of them could cheat or run away living another.

Question (x).
What did the tiger do when the thief shouted at him?
(ମଣିଷର ବଡ଼ପାଟି ଶୁଣି ବାଘ କ’ଣ କଲା ?)
Answer:
The tiger ran as fast as he could getting frightened and seeing the thief.

Question (xi).
What did the tiger promise?
(ବାଘ କ’ଣ ପ୍ରତିଜ୍ଞାକଲା ?)
Answer:
The tiger promised not to desire for human flesh anymore.

Question (xii).
What did the thief stop doing?
(ଚୋର କେଉଁ ଅଭ୍ୟାସ ବନ୍ଦ କରିଦେଲା ?)
Answer:
The thief stopped stealing anymore.

Session – 7

7. Mental Talk : (ମାନସିକ ଆଳାପ)

• The tiger was stronger than the thief, but not so clever. Fear made him weaker.
  (ବାଘ ମଣିଷଠାରୁ ବଳବାନ ଥିଲା; କିନ୍ତୁ ଚତୁର ନଥିଲା । ଭୟ ତାକୁ ଦୁର୍ବଳ କରିଦେଲା ।)
• Mind power is mightier than muscle power.
  (ମନର ବଳ ଶାରୀରିକ ବଳଠାରୁ ଅଧୁକ ଶକ୍ତିଶାଳୀ)

Tail-Piece

Did you like the story, “The Thief and The Tiger”?
(ଚୋର ଏବଂ ବାଘ ଗଳ୍ପଟି ଭଲ ଲାଗିଲା କି ?)
Read a story here, is more interesting than this.
(ଆଉ ଗୋଟିଏ ଏହିପରି ଗପ ପଢ଼ । ଏହାଠାରୁ ଅଧିକ ମନୋରଞ୍ଜନ କି)

The Liger On A Tiger (ବାଘ ଏବଂ ଫାଘ)

One day…………………………………………………. they could.

ଦିନେ ଗୋଟିଏ ଲୋକର ଘୋଡ଼ାଛୁଆଟି ହଜିଯାଇଥିଲା । ସେ ତାକୁ ଖୋଜି ଖୋଜି ବଣଭିତରକୁ ପଶିଯାଇଥିଲା । ସେ ବହୁତ କ୍ଳାନ୍ତ ହୋଇପଡ଼ିଥିଲା ଏବଂ ଆଗକୁ ଯିବାକୁ ଅକ୍ଷମ ହୋଇପଡ଼ିଥିଲା । ସେ ଘରକୁ ଫେରିଆସିବାକୁ ଇଚ୍ଛାକଲା । ମାତ୍ର ଘରକୁ ଫେରିବା ବାଟ ନପାଇ ବାଉଳା ହୋଇଗଲା । ସେତେବେଳକୁ ରାତି ହୋଇଯାଇଥିଲା । ଘରକୁ ଫେରିଯିବା ଆଉ ସମ୍ଭବ ନଥିଲା । ସେ ଗୋଟିଏ କୁଡ଼ିଆ ଦେଖିବାକୁ ପାଇଲା । ଏହା ଗୋଟିଏ ବୁଢ଼ୀଲୋକର କୁଡ଼ିଆ ଥିଲା । ଲୋକଟି ସେଠାରେ ଆଶ୍ରୟ ନେବାକୁ ବୁଢ଼ୀକୁ ଅନୁରୋଧ କଲା । ତା’ଘରେ ମାତ୍ର ଦୁଇଟି କୋଠରି ଥିଲା । ସେ ଗୋଟିଏ କୋଠରିରେ ତାରି ନାତୁଣୀ ସହିତ ରହୁଥିଲା ଏବଂ ଅନ୍ୟ କୋଠରିଟିରେ ଘରର ଜିନିଷପତ୍ର ସବୁ ରହିଥିଲା । ସେହି ଭଣ୍ଡାରଘରେ ତାକୁ ରହିବାକୁ ବୁଢ଼ୀ ଅନୁମତି ଦେଲା ।

ନାତୁଣୀଟି ଘୋଡ଼ାଛୁଆ ବିଷୟରେ କିଛିକିଛି ଶୁଣିଥିଲା । ଏଣୁ ସେ ଲୋକଟି ପାଖକୁ ଯାଇ ଘୋଡ଼ାଛୁଆ ବିଷୟରେ ସବୁକଥା ଜାଣିବାକୁ ଚାହିଁଲା । କିନ୍ତୁ ତା’ର ବୁଢ଼ୀ ମା’ ତାକୁ କହିଲା, ‘ଆଦୌ ନୁହେଁ । ତୁ ସେଠାକୁ ଆଦୌ ଯିବା ଉଚିତ ନୁହେଁ । ସେଠାରେ ବାଘ ଫାଘ ତୋତେ ଟେକିନେଇଯିବେ ।’’ ପ୍ରକୃତରେ ‘ଫାଘ’ ବୋଲି କୌଣସି ପ୍ରାଣୀ ନଥିଲା । କିନ୍ତୁ ବୁଢ଼ୀ ଘର ପଛପଟକୁ ପ୍ରତିଦିନ ଆସୁଥ‌ିବା ବାଘଟି ବୁଢ଼ୀ ମୁହଁରୁ ଫାଘ ସମ୍ପର୍କରେ ଶୁଣିବାକୁ ପାଇ ବିଚଳିତ ହୋଇପଡ଼ିଲା । ସେ ଭାବିଲା ବୋଧହୁଏ ଫାଘଟି ତା’ଅପେକ୍ଷା ଅଧିକ ଭୟଙ୍କର ଏକ ପ୍ରାଣୀ । ସେ ଗୋଟିଏ ରାକ୍ଷସ କିମ୍ବା ଭୂତ । ସେ ଖୁବ୍ ଭୟଭୀତ ହୋଇପଡ଼ିଲା । ସେ ସେଠାରୁ ପଳାଇଯିବାକୁ ଉପାୟ ଖୋଜିଲା । ଟିକିଏ ପରେ ଲୋକଟି ବାହାରକୁ ଆସି ଏହା ଦେଖିଲା ।

ସେତେବେଳକୁ ସକାଳ ହୋଇ ଆସୁଥାଏ । ସେଇ ଜାଲୁଜାଲୁଆ ଅନ୍ଧାର ଭିତରେ ଲୋକଟି ବାଘଟିକୁ ଦେଖି ତା’ର ଘୋଡ଼ାଛୁଆ ବୋଲି ଭାବିଲା । ସେ ସଙ୍ଗେ ସଙ୍ଗେ ସେଠାକୁ ଛୁଟିଯାଇ ବାଘର ମୁହଁକୁ ଗୋଟିଏ କନାରେ ବାନ୍ଧିପକାଇଲା । ଏହାଫଳରେ ବାଘର କାନ, ନାକ, ମୁଣ୍ଡ ଓ ବେକ ଜଣାପଡ଼ିଲା ନାହିଁ । ସେ ସଙ୍ଗେ ସଙ୍ଗେ ବାଘର ପିଠିରେ ବସିଗଲା । ବାଘର କି ଭୟ ! ସେ ଲୋକଟିକୁ ଫାଘ ବୋଲି ଭାବି ଜୀବନ ବିକଳରେ ଦୌଡ଼ିବାକୁ ଲାଗିଲା । ଯେତେବେଳେ ରାତି ପାହିଲା, ଲୋକଟି ଦେଖିଲା ଯେ ସେ ଗୋଟିଏ ବାଘ ପିଠିରେ ବସିଛି । କ’ଣ କରିବ ? କେମିତି ସେ ବାଘ ପିଠିରୁ ଖସି ଜୀବନ ବଞ୍ଚାଇବ ! ଯେତେବେଳେ ବାଘଟି ଗୋଟିଏ ବରଗଛ ତଳ ଦେଇ ଦୌଡୁଥିଲା, ଲୋକଟି ସେହି ସମୟରେ ବରଗଛର ଏକ ଡାଳକୁ ଧରି ଗଛ ଉପରେ ଚଢ଼ିଗଲା ଏବଂ କହିଲା, ‘ହେ ଭଗବାନ ! ତୁମକୁ ଧନ୍ୟବାଦ !

ଯାହାହେଉ ମୁଁ ବଞ୍ଚିଯାଇଛି ।’’ ବାଘଟି ମଧ୍ୟ ଫାଘ କବଳରୁ ମୁକ୍ତି ପାଇ ନିଜକୁ ଧନ୍ୟ ମନେକଲା । ବାଘଟି ଦୌଡ଼ି ନ ପଳାଇ ଗଛତଳେ ନିଶ୍ଵାସ ମାରିଲା ଏବଂ ଅନ୍ୟ ବାଘମାନଙ୍କୁ ଚିତ୍କାର କରି ଗଛ ଉପରେ ଥ‌ିବା ଭୟଙ୍କର ପ୍ରାଣୀ ଫାଘ ବିଷୟରେ ଜଣାଇଲା । ସବୁ ବାଘମାନେ ଆସି କହିଲେ, ‘ଖବର କ’ଣ ? କିଏ ତୁମର ମୁହଁ ବାନ୍ଧି ପକାଇଛି ?’’ ବାଘଟି ଦୀର୍ଘଶ୍ଵାସ ମାରି କହିଲା, ‘‘ଭାଇମାନେ ! ମୁଁ ମୃତ୍ୟୁମୁଖରୁ ବଞ୍ଚ୍ ଫେରିଆସିଛି । ମୋତେ ଗୋଟିଏ ‘ଫାଘ’ ଧରିନେଇଥିଲା । ମୁଁ ତାକୁ ପୂଜା ଅର୍ପଣ କରିବାକୁ ପ୍ରତିଜ୍ଞା କରିବାରୁ ସେ ମୋତେ ଛାଡ଼ିଦେଇଛି । ମୁଁ ଯଦି ତାକୁ ପୂଜା ଅର୍ପଣ ନଦିଏ; ତେବେ ସେ ମୋତେ ପୁଣି ଧରିନେଇଯିବ । ‘‘ଏକଥା ଶୁଣି ସବୁ ବାଘ ‘ଫାଘ’କୁ ପୂଜା କରି ଗଣ୍ଡା, ମଇଁଷି ପ୍ରଭୃତି ବିଭିନ୍ନ ଦ୍ରବ୍ୟ ଅର୍ପଣ କରିବାକୁ ଲାଗିଲେ । ଲୋକଟି ତା’ଜୀବନରେ କେବେ ହେଲେ ଏତେସଂଖ୍ୟକ ବାଘ ଏକାଠି ହେବାର ଦେଖି ନଥିଲା ।

ସେ ଖୁବ୍ ଭୟ ପାଇଗଲା । ସେ ଗଛ ଉପରେ ବସି ଥରିବାକୁ ଲାଗିଲା । ଉଭୟ ଲୋକ ଏବଂ ଗଛ ଦୋହଲୁଥିଲେ । ବାଘମାନେ ମଧ୍ୟ ଭୟ ପାଇ ଯାଇଥିଲେ । ସେମାନେ ଉପରକୁ ଚାହିଁଲେ; ମାତ୍ର ପତ୍ରଗହଳରେ ଲୋକଟିକୁ ଦେଖୁରିଲେ ନାହିଁ । ଲୋକର ପିନ୍ଧାଲୁଗାର ଶେଷ ଅଂଶଟି ଗୋଟିଏ ଡାଳରୁ ଓହଳିଥିଲା । ସେମାନେ ଏହା ପତ୍ରଗହଳ ମଧ୍ୟରେ ଜାଣିପାରିଲେ ନାହିଁ । ସେମାନେ ଏହାକୁ ଏକ ଲାଞ୍ଜ ବୋଲି ଭାବିଲେ । ଏହା ଦେଖ୍ ଗୋଟିଏ ବୁଢ଼ା ବାଘ କହିଲା, ‘ଏହା ଏକ ବିପଜ୍ଜନକ ଜୀବଭଳି ଲାଗୁଛି । ଏହା ନିଶ୍ଚୟ ଲାଇଗର ।’’ ଏହା ଶୁଣି ସମସ୍ତ ବାଘ ଚିତ୍କାର କରି ଉଠିଲେ, ‘‘ସେ ଆମକୁ ଧରିନେବ, ଜୀବନ ବଞ୍ଚାଇ ପଳାଇଯାଅ ।’’ ଏବଂ ସମସ୍ତେ ଯେତେ ଜୋର୍‌ରେ ପାରିଲେ ଦୌଡ଼ବାକୁ ଲାଗିଲେ ।

Word Meaning

dangle: to hang or swing loosely (ଉପରୁ ଓହଳିବା / ଝୁଲିରହିବା)
escape: to get free from something (ଖସି ପଳେଇବା)
flutter: to move by waving quickly and lightly
huge: very big (ବିରାଟ, ଖୁବ୍ ବଡ଼ ଆକାରର, ବିଶାଳ)
pant: to breathe quickly (ଅଣନିଶ୍ୱାସୀ ହୋଇପଡ଼ିବା)
rush out: to go or move suddenly with great speed (ହଠାତ୍ ଧାଇଁବା)
Search: to look for (ଖୋଜିବା)
shelter: a house or a place to stay (ଆଶ୍ରୟସ୍ଥଳୀ)

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

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

Question 1.
ନିମ୍ନ ଉକ୍ତିଗୁଡ଼ିକରେ ଠିକ୍ ଉକ୍ତି ପାଇଁ T ଓ ଭୁଲ ଉକ୍ତି ପାଇଁ F ଲେଖ ।
(i) ବୃତ୍ତର ଏକ ଉପସେଟ୍‌କୁ ଚାପ କହନ୍ତି ।
(ii) ଚାପର ଏକ ଅନ୍ତ୍ରମ୍ମ ଦିନ୍ଦୁ ସମ୍ରକ୍ର ଦୃଭର ଅନ୍ତ୍ରମ ବନ୍ଧୁ ନ୍ମ6ଦୃ |
(iii) ଗୋଟିଏ ବୃତ୍ତରେ P ଓ Q ଦୁଇଟି ଚାପର ସାଧାରଣ ପ୍ରାନ୍ତବିନ୍ଦୁ ହେଲେ ଚାପଦ୍ଵୟ ପରସ୍ପରର ପରିପୂରକ ଚାପ
(iv) ପ୍ରତ୍ୟେକ ଚାପର ପ୍ରାନ୍ତବିନ୍ଦୁକୁ କେନ୍ଦ୍ର ସହିତ ଯୋଗ କଲେ ଯେଉଁ କୋଣ ଉତ୍ପନ୍ନ ହୁଏ ତାହା ଉକ୍ତ ଚାପର କେନ୍ଦ୍ରସ୍ଥ କୋଣ ଅଟେ ।
(v) ଦୁଇଟି ଚାପର ଡିଗ୍ରୀ ପରିମାପର ସମଷ୍ଟି 360°ରୁ ଅଧ‌ିକ ହୋଇ ପାରିବ ନାହିଁ ।
(vi) ବୃତ୍ତ ଏକ ଉତ୍ତଳ ସେଟ୍ ନୁହେଁ ।
(vii) ଗୋଟିଏ ବୃତ୍ତରେ ଦୁଇଟି ଚାପର ଗୋଟିଏ ସାଧାରଣ ପ୍ରାନ୍ତ ବିନ୍ଦୁ ଥିଲେ ଚାପ ଦୁଇଟି ସନ୍ନିହିତ ଚାପ ହେବେ ।
(viii) ଦୁଇଟି ସର୍ବସମ ଜ୍ୟା ସହ ସମ୍ପୃକ୍ତ ଚାପଦ୍ଵୟ ସନ୍ନିହିତ ଚାପ ହେଲେ ଚାପଦ୍ଵୟର ସଂଯୋଗରେ ସର୍ବଦା ବୃହତ୍ ଚାପ ଗଠିତ ହେବ ।
(ix) ଦୁଇଟି ସର୍ବସମ ଜ୍ୟା ପରସ୍ପରକୁ ଲମ୍ବ ଭାବରେ ବୃତ୍ତର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ P ରେ ଛେଦ କରନ୍ତି । ବୃତ୍ତର କେନ୍ଦ୍ର O ଠାରୁ ସେମାନଙ୍କ ପ୍ରତି \(\overline{\mathrm{OQ}}\), \(\overline{\mathrm{OR}}\) ଲମ୍ବ ଗଠନ କରାଯାଇଛି । ତେବେ ଠ, Q, P ଓ R ଏକ ବର୍ଗଚିତ୍ରର ଶୀର୍ଷବିନ୍ଦୁ ହେବେ ।
(x) \(\overparen{B P C}\) ର ଡିଗ୍ରୀ ପରିମାପ 30° । A ବୃତ୍ତ ଉପରିସ୍ଥ ଏକ ବିନ୍ଦୁ ହେଲେ △ABC ରେ ∠Aର ପରିମାଣ ସର୍ବଦା 15° 6ଦ୍ଵଦା
(xi) ଗୋଟିଏ ଚାପ ଅସଂଖ୍ୟ ବିନ୍ଦୁର ସମାହାର ଅଟେ ।
(xii) ବୃତ୍ତାନ୍ତର୍ଲିଖ ରମ୍ବସ୍ ଏକ ବର୍ଗଚିତ୍ର ।
Solution:
(i) T
(ii) T
(iii) T
(iv) F
(v) F
(vi) T
(vii) T
(viii) F
(ix) F
(x) F
(xi) T
(xii) T

Question 2.
ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।
(i) ଏକ ବୃହତ୍ ଚାପର ଡିଗ୍ରୀ ପରିମାପ …………….. ରୁ ବେଶୀ ।
(ii) ଗୋଟିଏ ସୁଷମ୍ ଷଡ଼ଭୁଜର ପ୍ରତ୍ୟେକ ବାହୁ ଏହାର ପରିବୃତ୍ତର କେନ୍ଦ୍ରଠାରେ ଉତ୍ପନ୍ନ କରୁଥିବା କେନ୍ଦ୍ରସ୍ଥ କୋଣର ପରିମାଣ ……………….. |
(iii)
(iv) ଗୋଟିଏ ବୃତ୍ତରେ ଦୁଇଟି ସର୍ବସମ ଜ୍ଯା \(\overline{\mathrm{AB}}\) ଓ \(\overline{\mathrm{CD}}\) ପରସ୍ପରକୁ ବୃତ୍ତର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ Pରେ ଛେଦ କରନ୍ତି । O ବୃତ୍ତର କେନ୍ଦ୍ର ଏବଂ B ଓ C \(\overline{\mathrm{OP}}\)ର ଏକ ପାର୍ଶ୍ଵରେ ଥିଲେ \(\overparen{A D}\) ଓ ……………….. ଦୁହେଁ ସର୍ବସମ ।
(v) ଗୋଟିଏ ବୃତ୍ତରେ ଏକ ଜ୍ୟାର ଦୈର୍ଘ୍ୟ ବ୍ୟାସାର୍ଦ୍ଧ ସହ ସମାନ ହେଲେ ଉକ୍ତ ଜ୍ୟା ଦ୍ଵାରା ଛେଦିତ କ୍ଷୁଦ୍ର ଚାପର ତିଗ୍ରା ପରିମାପ ………….. |
(vi) \(\overline{\mathrm{AB}}\) ର ଏକ ପାଣ୍ଡରେ C ଓ D ଦୁଇଟି ଦିନ୍ଦୁ | m∠ACB = m∠ADB = 20° △ACD ର ପରିଦର୍ଭର 6କହ O 6ଦୃବେ m∠AOB ……………. |
(vii) m∠ABC = 90° ଚତୁର୍ଭୁକ △ABC ର ପରିଦଉଭେ AC ଏକ ………………… |
(viii) ABCD ଏକ ଦ୍ଵରାନ୍ତକଖର ଚତୁର୍ଭୁକ m∠BAD ………………. ଚାପର ଡିଗ୍ରୀ ପରିମାପର ଅର୍ଦ୍ଧେକ ।
(ix) ଏକ ଅର୍ଥବୃତ୍ତର ଡିଗ୍ରୀ ପରିମାପ ………….. |
(x) ଗୋଟିଏ ବୃତ୍ତରେ ଏକ ଚାପର ଡିଗ୍ରୀ ପରିମାପ 90° ହେଲେ, ସଂପୃକ୍ତ ଜ୍ୟା ଓ ବ୍ୟାସାର୍ଦ୍ଧର ଅନୁ ପାତ …………….. |
Solution:
(i) 180°
(ii) 60° \(\left(\frac{360^{\circ}}{6}=60^{\circ}\right)\)
(iii) 70° m∠D = 180° – 120° = 60°
m∠C = 180° – 50° = 130°
∴ m∠C – m∠D = 130° – 60° = 70°

(iv) \(\overparen{B C}\) \(\overparen{A C B}\) ≅ \(\overparen{C A D}\) ( ∵ AB ≅ CD (ଦତ୍ତ))
⇒ BC ≅ AD

(v) 60° ଜ୍ୟାର ପ୍ରାନ୍ତବିନ୍ଦୁଦ୍ଵୟ ଓ କେନ୍ଦ୍ରବିନ୍ଦୁ ଏକ ସମବାହୁ ତ୍ରିଭୁଜର ଶୀର୍ଷବିନ୍ଦୁ ହେବେ |
(vi) 40° O ବିଦୁଟି ବୃତ୍ତର କେନ୍ଦ୍ର ହେବ ।
(vii) ବ୍ୟାସ (ଅର୍ଦ୍ଧବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ ସମକୋଣ ।)
(viii) \(\overparen{B C D}\)
(ix) 90°
(x) √ 2 : 1 (ସଂପୃକ୍ତ ଜ୍ୟାଟି ବୃତ୍ତର ବ୍ୟାସ)

Question 3.
ଚିତ୍ରରେ △ABC ବୃତ୍ତାନ୍ତର୍ଲିଖ ଏବଂ ସୂକ୍ଷ୍ମକୋଣୀ । D, E, F ବୃତ୍ତ ଉପରିସ୍ଥ ତିନୋଟି ବିନ୍ଦୁ ହେଲେ ନିମ୍ନ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।

(i) ∠B କେଉଁ ଚାପର ଅନ୍ତର୍ଲିଖ୍ ?
(ii) ∠B ଦ୍ବାରା କେଉଁ ଚାପ ଛେଦିତ ?
(iii) \(\overline{\mathrm{AB}}\) ଜ୍ୟା ଦ୍ବାରା ଛେଦିତ କ୍ଷୁଦ୍ରଚାପ ଓ ବୃହତ୍ ଚାପ କିଏ ?
(iv) ∠A ର ପରିମାଣ କେଉଁ କେନ୍ଦ୍ରସ୍ଥ କୋଣ ପରିମାଣର ଅର୍ଦ୍ଧେକ ?
(v) △ABC ରେ ଯଦି AB = BC ହୁଏ ତେବେ କେଉଁ ଚାପ ଦ୍ଵୟ ସର୍ବସମ ହେବେ ?
(vi) ଦୁଇଟି ସନ୍ନିହିତ ଚାପର ନାମ ଲେଖ ଯେପରିକି ସେମାନଙ୍କ ସଂଯୋଗରେ \(\overparen{B A D}\) ଗଠିତ ହେବ ।
(vii) \(\overparen{B F C}\) ଉପରେ ଏପରି ଏକ ବିନ୍ଦୁ P ନିଅ ଯେପରିକି m∠BPA = m∠C । ଏପରି କେତୋଟି ବିନ୍ଦୁ ଅଛି ? ADC ଉପରେ ଏପରି କୌଣସି ବିନ୍ଦୁ ଅଛି କି ? \(\overparen{B E A}\) ଉପରେ ଏପରି କୌଣସି ବିନ୍ଦୁ ଅଛି କି ?
Solution::
(i) ∠B, \(\overparen{E B F}\) ର୍ଥିଥଚା \(\overparen{A B C}\) ର ଅନ୍ତ୍ର କିଣତ |
(ii) ∠Bଦ୍ଵାରା \(\overparen{A D C}\) ଛେଦିତ ।
(iii) \(\overline{\mathrm{BC}}\) ଖ୍ୟା ଦ୍ଵାରା ଛେଦିତ ସୁଦୃତାପ \(\overparen{B F C}\) ଦ୍ଦଦ୍ର ତ୍ତାପ \(\overparen{B A C}\) |
(iv) m∠A = \(\frac { 1 }{ 2 }\) m∠BOC
(v) △ABC ରେ \(\overparen{A E B}\) ≅ \(\overparen{B F C}\)
(vi) \(\overparen{B E A}\) ∪ \(\overparen{A D}\) = \(\overparen{B A D}\), \(\overparen{B E}\) ∪ \(\overparen{E A D}\) = \(\overparen{B A D}\)
(vii) ଏପରି ଅସଂଖ୍ୟ ବିନ୍ଦୁ ଅଛି । \(\overparen{A D C}\) ଉପରେ ମଧ୍ୟ ଅସଂଖ୍ୟ ବିନ୍ଦୁ ଅଛି । \(\overparen{B E A}\) ଚାପ ଉପରେ ଏପରି ବିନ୍ଦୁ ରହିବା ସମ୍ଭବ ନୁହେଁ ।

Question 4.
ଚିତ୍ରରେ ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜ ଯାହର କର୍ଣ୍ଣଦ୍ଵୟ ବୃତ୍ତର କେନ୍ଦ୍ରଠାରେ ଛେଦ କରନ୍ତି ।
m\(\overparen{A E B}\) = 100° 6 ଦ୍ଵ6କ |

(i) ଚତୁର୍ଭୁଜର ସମସ୍ତ କୋଣ ପରିମାଣ ନିର୍ଣ୍ଣୟ କର ।
(ii) \(\overparen{A H D}\) ଓ \(\overparen{B F C}\) ମଧ୍ୟରେ କି ସମ୍ପର୍କ ଦେଖୁଛ ?
(iii) ABCD କି ପ୍ରକାର ଚତୁର୍ଭୁଜ ?
Solution:

m \(\overparen{A E B}\) : 100°
⇒ m∠ADB = m∠ACB = 50°
m∠COD = 100° (ତ୍ପତାପ 6କାଶ)
⇒ m∠CBD = m∠CAD = 50°
m∠BOC = 180° – 100° = 80°
⇒ m∠BAC = m∠BDC = 40°
⇒ m∠AOD = 80°
⇒ m∠ACD = m∠ABC = 40°

(i) m∠A = m∠BAC + m∠CAD = 40° + 50° = 90°
6ସ ଦ୍ଵିପରି m∠С = 90°, m∠B = 90° ଓ m∠D = 90° |
(ii) \(\overparen{A H D}\) ≅ △\(\overparen{B F C}\)
(ii) ABCD ଏକ ଆୟତଚିତ୍ର ।

Question 5.
ଚିତ୍ରରେ \(\overline{\mathrm{AB}}\) ଓ \(\overline{\mathrm{CD}}\) ଜ୍ୟା ଦ୍ଵୟ ପରସ୍ପରକୁ ବୃତ୍ତର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ P ଠାରେ ଛେଦ କରନ୍ତି । m∠PBD = 80°, m∠CAP = 45° 6ଦ୍ଵଲେ :
(i) △BPDର କୋଣ ପରିମାଣଗୁଡ଼ିକ ନିର୍ଣ୍ଣୟ କର ।
(ii) △APCର କୋଣ ପରିମାଣଗୁଡ଼ିକ ନିର୍ଣ୍ଣୟ କର ।
(iii) △APC ଓ △BPD ମଧ୍ୟରେ କି ସମ୍ପର୍କ ଦେଖୁଛ ?

Solution:
m∠PBD = 80°, m∠CAP = 45°

(i) \(\overparen{B C}\) ଉପରିସ୍ଥ m∠CDB = m∠CAP = 45°
∴ m∠BPD = 180° – 80° – 45° = 55°
∴ △BPDର m∠BPD = 55°
m∠PDB = 45°, m∠PBD = 80°

(ii)
△APCର m∠CAP = 45° (ଦଉ)
m∠APC = m∠BPD = 55° (ପ୍ରତାପ 6କାଣ)
∴ m∠ACP = m∠PBD = 80° (\(\overparen{A D}\) ଉପରିସ୍ଥ ପରିଧ୍ଵସ୍ଥ ଏକ କୋଣ)

(iii) △APC ~ △BPD (କୋ . କୋ . କୋ . ପାଦଣ୍ୟ)

Question 6.
△ABCରେ ∠Aର ସମଦ୍ବିଖଣ୍ଡକ ତ୍ରିଭୁଜର ପରିବୃତ୍ତକୁ D ବିନ୍ଦୁରେ ଛେଦ କଲେ ପ୍ରମାଣ କର ଯେ, △BDC ସମଦ୍ବିବାହୁ ।
Solution:

ଦତ୍ତ : △ABCର ∠Aର ସମଦ୍ଵିଖଣ୍ଡକ ତ୍ରିଭୁଜର ପରିବୃତ୍ତକୁ D ବିନ୍ଦୁରେ ଛେଦକରେ ।
ପ୍ରାମାଣ୍ୟ : △BDC ସମଦ୍ବିବାହୁ ।
ପ୍ରମାଣ : ∠BAD ≅ ∠CAD (ଦଉ)
⇒ \(\overparen{B D}\) ≅ \(\overparen{D C}\) ⇒ \(\overline{\mathrm{BD}}\) ≅ \(\overline{\mathrm{DC}}\) (ଚାପ ସର୍ବସମ ହେତୁ ଜ୍ୟାଦ୍ଵୟ ସର୍ବସମ)
=> △BDC ସମଦିବାହୁ ।

Question 7.
ଚିତ୍ରରେ ଗୋଟିଏ ବୃତ୍ତର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ A ଠାରୁ \(\overrightarrow{\mathbf{AP}}\) ଓ \(\overrightarrow{\mathbf{AR}}\) ରଶ୍ମିଦ୍ଵୟ ବୃତ୍ତକୁ ଯଥାକ୍ରମେ P, Q ଏବଂ R, S ଠାରେ ଛେଦ କରନ୍ତି ଯେପରି A-P-Q ଏବଂ A-R-S |
(a) ପ୍ରମାଣ କର ଯେ △APR ~ △AQS
(b) ପ୍ରମାଣ କର ଯେ △APS ~ △ARQ

(C) ଯଦି \(\overline{\mathrm{PS}}\) ଓ \(\overline{\mathrm{QR}}\) ର ଛେନ୍ଦ୍ରବିନ୍ଦୁ T ହୁଏ, ତେଦେ
(i) ପ୍ରମାଣ କର ଯେ TP • TS = TR • TQ
(ii) ପ୍ତମାଣ କାର 6ପ m∠PTR = \(\frac { 1 }{ 2 }\) (m\(\overparen{Q S}\) + m\(\overparen{P R C}\))

(d) m∠PAR = 15° ଏବଂ m\(\overparen{Q X S}\) = 50° ହେଲେ m∠PTR ନିର୍ଣ୍ଣୟ କର ।
Solution:

(a) ପ୍ରମାଣ : PRSQ ଏକ ଦ୍ଵରାନୁଲଖତ ଚତୁରୁଜ |
⇒ m∠RSQ + m∠RPQ = 180°
କିନ୍ନ m∠RPQ + m∠APR = 180°
⇒ m∠RSQ + m∠RPQ = m∠RPQ + m∠APR
m∠RSQ = m∠APR
△APR ଓ △AQS ମଧ୍ୟ6ର m∠RAP = m∠QAS
ଓ m∠RSQ = m∠APR |
⇒ △APR ~ △AQS (6କା-6କା ସାଦ୍ୱଣ)

(b) △APS ଓ △ARQ ମଧ୍ୟ6ର
m∠PAS = m∠RAQ (ମଧ୍ୟ6ର 6କା)
m∠ASP = m∠AQR (ଏକ ଚାପ ଉପରିମ ପରିଧମ 6କାଣ)
⇒ △APS ~ △ARQ

(c) (i) △ TPQ ଓ △TRS ମଧ୍ୟ6ର
m∠TPQ = m∠TRS (ଏକ ଚାପ ଉପରିମ ପରିଧମ 6କାଣ)
m∠PTQ = m∠RTS (ପ୍ତତାପ 6କାଣ)
⇒ △TPQ ~ △TRS (6କା-6କା ସାଦ୍ୱଣ)
⇒ \(\frac { TP }{ TR }\) = \(\frac { TQ }{ TS }\) ⇒ TP • TS = TR • TQ
=m∠SPQ + m∠PQR = \(\frac { 1 }{ 2 }\)m\(\overparen{Q S}\) + \(\frac { 1 }{ 2 }\)m\(\overparen{P R}\) = \(\frac { 1 }{ 2 }\) (m\(\overparen{Q S}\) + m\(\overparen{P R}\)) (9flêe)

(d) m∠PAR = 15°
m\(\overparen{Q X S}\) = 50° ⇒ m∠QPS = \(\frac { 50° }{ 2 }\) = 25°
△APS 6ର 6କାଣ m∠QPS = m∠PAR + m∠PSR
⇒ 25° = 15° +m∠PSR ⇒ m∠PSR = 25° – 15° = 10°
m∠QRS = m∠QPS = 25°
∴ m∠PTR = m∠QRS + m∠TSR = 25° + 10° = 35°

Question 8.
ଚିତ୍ରରେ ABC ଦଉର \(\overparen{A X B}\) ଓ \(\overparen{B Y C}\) ହୁକରି ଚାପର ଗିଗାମାପ ଯଥାକୃମେ 80° ଓ 140° |

(i) m∠BAC କିଣ୍ଡଯ କର |
(ii) m\(\overparen{A B C}\) କିଣ୍ଡଯ କର |
(iii) m\(\overparen{A C B}\) କିଣ୍ଡଯ କର |
(iv) \(\overparen{A Z C}\) ଓ \(\overparen{B Y C}\) ମଧ୍ୟରେ କି ସମଳ ଅଛି ?
Solution:
m\(\overparen{A X B}\) + m\(\overparen{B Y C}\) + m\(\overparen{A Z C}\) = 360° ⇒ 80° + 140° + m\(\overparen{A Z C}\) = 360°
⇒ m\(\overparen{A Z C}\) = 360° – 80° – 140° = 140°
(i) M∠BAC = \(\frac { 1 }{ 2 }\) m\(\overparen{B Y C}\) = \(\frac { 1 }{ 2 }\) × 140° = 70°
(ii) \(\overparen{A B C}\) = 360° – 140° = 220°
(iii) \(\overparen{A C B}\) = M\(\overparen{A Z C}\) + m\(\overparen{B Y C}\) = 140° + 140° = 280°
(iv) \(\overparen{A Z C}\) ≅ \(\overparen{B Y C}\) (“: m∠AOC = m∠BOC)

Question 9.
ଏକ ବୃତ୍ତର କେନ୍ଦ୍ର O ଏବଂ \(\overline{\mathrm{AB}}\) ଏକ ବ୍ୟାସ । ବୃତ୍ତ ଉପରିସ୍ଥ P ଓ ଠୁ ବିଦୁ୍ୟଦ୍ୱୟ \(\overline{\mathrm{AB}}\) ର ଏକ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ । ଯଦି A ଓ P ପ୍ରାନ୍ତ ବିନ୍ଦୁ ବିଶିଷ୍ଟ ଚାପର ଡିଗ୍ରୀ ପରିମାପ 60° ଏବଂ B ଓ ( ପ୍ରାନ୍ତବିନ୍ଦୁ ବିଶିଷ୍ଟ ଚାପର ଡିଗ୍ରୀ ପରିମାପ 50° ହୁଏ ତେବେ–
(i) A ଓ Q ପ୍ରାନ୍ତବିନ୍ଦୁ ବିଶିଷ୍ଟ କ୍ଷୁଦ୍ରଚାପର ଡିଗ୍ରୀ
(ii) P ଓ B ପ୍ରାନ୍ତବିନ୍ଦୁ ବିଶିଷ୍ଟ ବୃହତ୍ ଚାପର ଡିଗ୍ରୀ
(iii) P ଓ Q ପ୍ରାନ୍ତବିନ୍ଦୁ ବିଶିଷ୍ଟ ବୃହତ୍ ଚାପର ଡିଗ୍ରୀ ପରିମାପ ନିର୍ଣ୍ଣୟ କର ।

Solution:
m∠AOP = 60°
m∠BOQ = 50°
⇒ m⇒POQ = 180° – (60° + 50°) = 70°
(i) m\(\overparen{A Q}\) = 60° + 70° = 130°
(ii) m\(\overparen{P B}\) = 70° + 50° = 120°
(iii) m\(\overparen{P Q}\) = 70°

Question 10.
\(\overline{\mathrm{AB}}\) ଓ \(\overline{\mathrm{CD}}\) ଦୁଇଟି ସମାନ୍ତର ଜ୍ୟା । ପ୍ରମାଣ କର ଯେ,
(i) M\(\overparen{A X C}\) = m\(\overparen{B Y D}\), (ii) AC = BD |

Solution:
ଦତ୍ତ : ABCD 96 \(\overline{\mathrm{CD}}\) || \(\overline{\mathrm{AB}}\) |
ପ୍ରାମାଣ୍ୟ :
(i) M\(\overparen{A X C}\) = m\(\overparen{B Y D}\)
(ii) AC = BD
ଅଙ୍କନ : \(\overline{\mathrm{BC}}\) ଅଙ୍କନ କର ।
ତ୍ପମାଣ : \(\overline{\mathrm{CD}}\) || \(\overline{\mathrm{AB}}\) (ଦଉ)

⇒ m∠ABC = m∠BCD
⇒ \(\frac { 1 }{ 2 }\) m\(\overparen{A X C}\) = \(\frac { 1 }{ 2 }\) m\(\overparen{B Y D}\)
⇒ m\(\overparen{A X C}\) = m\(\overparen{B Y D}\)

(ii) \(\overparen{A X C}\) = \(\overparen{B Y D}\) ⇒ AC = BD (ଚାପ ସବସମ ହେତ୍ ଲ୍ୟା ସଦସ୍ୟ)

Question 11.
ABCD ଏକ ଦ୍ଵରୀନ୍ତ୍ର କିଖବ ତତ୍କଲକ |
(i) AC = BD ଏବଂ \(\overline{\mathbf{AB}}\)||\(\overline{\mathbf{CD}}\) 6ଦ୍ର6କ ପ୍ରମାଣ କର ସେ, AD = BC |
(ii) AD = BC 6ଦ୍ର6କ ପ୍ରମାଣ କର ସେ, AC = BD ଏବଂ \(\overline{\mathbf{AB}}\)||\(\overline{\mathbf{CD}}\) |
Solution:
(i) ଦର : ABCD ଦୃଭାନ୍ତ୍ରରଖତ ଚତୁରୁକରେ
AC = BD ଏବଂ \(\overline{\mathbf{AB}}\)||\(\overline{\mathbf{CD}}\) |
ପ୍ରାମଣ୍ୟ: AD = BC
ପ୍ରମାଣ : AC = BD (ଦତ୍ତ)

⇒ \(\overparen{A D C}\) ≅ \(\overparen{B C D}\)
⇒ l \(\overparen{A X D}\) + l \(\overparen{D Y C}\) = l \(\overparen{D Y C}\) + l \(\overparen{B Z C}\)
⇒ l \(\overparen{A X D}\) = l \(\overparen{B Z C}\) ⇒ \(\overparen{A X D}\) ≅ \(\overparen{B Z C}\)
⇒ AD = BC

(ii) ଦଉ : ABCD ଏକ ତ୍ରଭାନ୍ତକଖତ ତଡୁରୁକ | AD = BC
ପ୍ତମାଣ୍ୟ: (i) AC = BD (ii) \(\overline{\mathbf{AB}}\)||\(\overline{\mathbf{CD}}\)
ପ୍ରମଣ : AD = BC ⇒ \(\overparen{A X D}\) ≅ \(\overparen{B Z C}\)

⇒ l \(\overparen{A X D}\) = l \(\overparen{B Z C}\)
⇒ l \(\overparen{A X D}\) + l \(\overparen{D Y C}\) = l \(\overparen{D Y C}\) + l \(\overparen{B Z C}\)
⇒ l \(\overparen{A D C}\) = l \(\overparen{B C D}\) ⇒ \(\overparen{A D C}\) ≅ \(\overparen{B C D}\) ⇒ AC = BD (i)
ପୁକଣ୍ଠ ∵ \(\overparen{A X D}\) ≅ \(\overparen{B Z C}\)
⇒ m \(\overparen{A X D}\) + m \(\overparen{B Z C}\) ⇒ \(\frac { 1 }{ 2 }\) m \(\overparen{A X D}\) = \(\frac { 1 }{ 2 }\) m \(\overparen{B Z C}\)
⇒ m∠ABD = m∠BDC (ଏକାନ୍ତ୍ରର) ⇒ \(\overline{\mathbf{AB}}\)||\(\overline{\mathbf{CD}}\) …(ii)

Question 12.
(i) ଗୋଟିଏ ବୃତ୍ତରେ \(\overparen{A X B}\) ଏକ ଚାପ । ପ୍ରମାଣ କର ଯେ \(\overparen{A X B}\)ର ଅନ୍ତଃସ୍ଥ ଗୋଟିଏ ଏବଂ କେବଳ ଗୋଟିଏ ବିନ୍ଦୁ C ଅଛି ଯେପରି \(\overparen{A C}\) ଓ \(\overparen{B C}\) ଚାପଦ୍ଵୟ ସର୍ବସମ ହେବେ । (C ବିନ୍ଦୁକୁ \(\overparen{A X B}\) ର ମଧ୍ୟବିନ୍ଦୁ କୁହାଯାଏ ।)
(ii) ଚାପର ମଧ୍ୟବିନ୍ଦୁ ଧାରଣାକୁ ବ୍ୟବହାର କରି ପ୍ରମାଣ କର ଯେ, AXBରେ ଅସଂଖ୍ୟ ବିନ୍ଦୁ ଅଛି ।

Solution:
(i) ଦତ୍ତ : O ବୃତ୍ତର କେନ୍ଦ୍ର । \(\overparen{A X B}\) ଏକ ଚାପ ।
ପ୍ରାମାଣ୍ୟ : (i) \(\overparen{A X B}\) ର ଅନ୍ତଃସ୍ଥ ଗୋଟିଏ
ଏବଂ କେବଳ ଗୋଟିଏ ବିନ୍ଦୁ C ଅଛି,
ଯେପରି \(\overparen{A C}\) = \(\overparen{C B}\)ହେବ ।
(ii) \(\overparen{A X B}\) ରେ ଅସଂଖ୍ୟ ବିନ୍ଦୁ ରହିଅଛି ।
∠AOB ର ସମଦ୍ଵିଖଣ୍ଡକ \(\overrightarrow{\mathrm{OC}}\) ଅଙ୍କନ କର ।
ଅଙ୍କନ :
ପ୍ରମାଣ :
(i) \(\overparen{A X B}\) ଚାପ ଉପରେ ଅବସ୍ଥିତ ଆବଶ୍ୟକ C ବିନ୍ଦୁଟି ଅନନ୍ୟ ଅର୍ଥାତ୍ ଗୋଟିଏ ଓ କେବଳ ଗୋଟିଏ ବିନ୍ଦୁ ହେବ ଯାହା m∠AOB ର ସମଦ୍ବିଖଣ୍ଡକ ରଶ୍ମି \(\overrightarrow{\mathrm{OC}}\) ଉପରେ ଅବସ୍ଥିତ ହେବ । ପୁନଶ୍ଚ ଚାପର ସର୍ବସମତା ଅନୁସାରେ ଦୁଇଟି ଚାପର ଡିଗ୍ରୀ ପରିମାପ ସମାନ ହେଲେ ଚାପଦ୍ଵୟ ସର୍ବସମ ହେବେ ।
ଦର ଚିତ୍ର6ର m∠AOC = m∠BOC 6ଦ୍ରଦ \(\overparen{A C}\) = \(\overparen{C B}\) 6ଦ୍ରଦ | (ପ୍ରମାଣିତ)

(ii) ଦତ୍ତ ଚିତ୍ରରେ A ଓ B ବିନ୍ଦୁ ସମେତ ‘A’ ଠାରୁ ‘B’ ପର୍ଯ୍ୟନ୍ତ ବୃତ୍ତ ଉପରିସ୍ଥ ସମସ୍ତ ବିନ୍ଦୁ ମାନଙ୍କର ସେଟ୍‌କୁ ଏକ ଚାପ କୁହାଯାଏ । A ଓ B ଏହି ଚାପର ଦୁଇଟି ପ୍ରାନ୍ତବିନ୍ଦୁ ଅଟନ୍ତି । ପ୍ରାନ୍ତବିନ୍ଦୁ ଭିନ୍ନ ଚାପ ଉପରିସ୍ଥ ଅନ୍ୟ ସମସ୍ତ ବିନ୍ଦୁମାନଙ୍କୁ ଚାପର ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ କୁହାଯାଏ; ଯାହା ଅସଂଖ୍ୟ ବିନ୍ଦୁମାନଙ୍କର ସେଟ୍ ।(ପ୍ରମାଣିତ)

Question 13.
ଚିତ୍ରରେ AB ବୃତ୍ତର ଏକ ବ୍ୟାସ ଏବଂ O କେନ୍ଦ୍ର । OD ଯେକୌଣସି ଏକ ବ୍ୟାସାର୍ଦ୍ଧ | \(\overline{\mathbf{AC}}\)||\(\overline{\mathbf{OD}}\) ହେଲେ, ପ୍ରମାଣ କର ଯେ, \(\overparen{B X D}\) ଓ \(\overparen{D Y C}\) ସର୍ବସମ ଅର୍ଥାତ୍ D, \(\overparen{B D C}\)ର ମଧ୍ୟବିନ୍ଦୁ । (ସୂଚନା : \(\overline{\mathbf{OC}}\) ଅଙ୍କନ କରି ଦର୍ଶାଅ ଯେ, m∠BOD = m∠DOC)

Solution:
ଦତ୍ତ : ବୃତ୍ତର କେନ୍ଦ୍ର O | \(\overline{\mathbf{AB}}\) ବୃତ୍ତର ବ୍ୟାସ । \(\overline{\mathbf{AC}}\)||\(\overline{\mathbf{OD}}\) |
ପ୍ରାମାଣ୍ୟ : \(\overparen{B X D}\) ≅ \(\overparen{D Y C}\) ଅର୍ଥାତ୍ D, \(\overparen{B D C}\) ର ମଧ୍ୟବିନ୍ଦୁ ।
ଅଙ୍କନ : \(\overline{\mathbf{CO}}\) ଅଙ୍କନ କର ।

ପ୍ରମାଣ : △AOC ରେ OA = OC ⇒ m∠OAC = m∠OCA
କିନ୍ତୁ m∠OAC – m∠BOD (ଅନୁରୁପ)
∴ m∠OAC = m∠BOD (i)
ପୁନଶ୍ଚ m∠OCA=m∠COD …(ii)
(i) ଓ (ii) ରୁ m∠BOD = m∠COD ⇒ \(\overparen{B X D}\) ≅ \(\overparen{D Y C}\)
⇒ D, \(\overparen{B D C}\) ର ମଧ୍ୟବିନ୍ଦୁ ।

Question 14.
ଚିତ୍ରରେ \(\overline{\mathbf{CD}}\) ଜ୍ୟା \(\overline{\mathbf{AB}}\) ବ୍ୟାସ ସହ ସମାନ୍ତର ଏବଂ CD = OB |
ପ୍ରମାଣ୍ୟ କର ଯେ m∠BDC = 2m∠OBD |

Solution:
ଦତ୍ତ : ବୃତ୍ତର \(\overline{\mathbf{AB}}\) ବ୍ୟାସ । \(\overline{\mathbf{CD}}\) ଜ୍ୟା । CD = OB \(\overline{\mathbf{CD}}\) ର ଦୈର୍ଘ୍ୟ, ବୃତ୍ତର ବ୍ୟାସାର୍ଦ୍ଧ ସହ ସମାନ ।
ପ୍ରାମାଣ୍ୟ : m∠BDC = 2m∠OBD
ଅଙ୍କନ : \(\overline{\mathbf{OC}}\) ଏବଂ \(\overline{\mathbf{OD}}\) ଅଙ୍କନ କର ।
ସ୍ତମାଣ : CD = OB = OC = OD ∴ △OCD ସମବାହୁ ।

⇒ m∠OCD = 60°
ପୁନଶ୍ଚ, COBD ଏକ ରମଣ (∵ CD = OB, \(\overline{\mathbf{CD}}\)||\(\overline{\mathbf{OB}}\) ଏବଂ OB = OC = CD)
∴ m∠OBD =m∠OCD = 60° ଏବଂ m∠BDC = 120°
∴ m∠BDC = 2m∠OBD (ପ୍ରମାଣିତ)

Question 15.
ABCD ବୃତ୍ତାନ୍ତର୍ଲିଖୂତ ଚତୁର୍ଭୁଜର \(\overline{\mathbf{AC}}\) ଓ \(\overline{\mathbf{BD}}\) କର୍ଣ୍ଣଦ୍ୱୟ ପରସ୍ପରକୁ P ଠାରେ ଛେଦ କରନ୍ତି । O ବୃତ୍ତର କେନ୍ଦ୍ର ଏବଂ B ଓ C, \(\overleftrightarrow{O P}\) ର ବିପରୀତ ପାର୍ଶ୍ଵରେ ଅବସ୍ଥିତ । ଯଦି AC = BD ହୁଏ, ତେବେ ପ୍ରମାଣ କର ଯେ,
(i) AB = CD, (ii) PA = PD 1° (iii) \(\overline{\mathbf{BC}}\) || \(\overline{\mathbf{AD}}\) |
Solution:
ଦତ୍ତ : ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜ । \(\overline{\mathbf{AC}}\) ଓ \(\overline{\mathbf{BD}}\) କର୍ଣ୍ଣଦ୍ଵୟ ପରସ୍ପରକୁ P ଠାରେ ଛେଦ କରନ୍ତି ।
AC = BD |

ପ୍ରାମାଣ୍ୟ : (i) AB = CD (ii) PA = PD (iii) \(\overline{\mathbf{BC}}\) || \(\overline{\mathbf{AD}}\)
ପ୍ତମାଣ: (i) AC = BD ⇒ \(\overparen{A D C}\) ≅ \(\overparen{B A D}\)
⇒ \(\overparen{A Y B}\) ≅ \(\overparen{C Z D}\)
(∵ \(\overparen{A X D}\) ଭରଯ \(\overparen{A D C}\) ଓ \(\overparen{B A D}\) ର ପାଧାରଣ ଚାପ )
⇒ AB = CD ….(i)

(ii) △ABD ଏବଂ △ADC ଦଯ6ର \(\overline{\mathbf{AD}}\) ପାଧାରଣ, AB = CD [(i) ରେ ପ୍ତମାଣିତ]
ଏବଂ AC = BD (ଦର)
∴ m∠ADB = m∠CAD ⇒ m∠ADP = m∠PAD
⇒ PA = PD …(ii)

(iii) m∠DAC = m∠DBC (ଏକ ଦ୍ଵରଖଣ୍ଡମ 6କାଣ)
କିନ୍ତୁ m∠DAC = m∠ADB
∴ m∠ADB = m∠PBC
କିନ୍ତୁ ଏମା6ନ ଏକାନ୍ତ୍ରର |
∴ \(\overline{\mathbf{BC}}\) || \(\overline{\mathbf{AD}}\)

Question 16.
(i) ପ୍ରମାଣ କର ଯେ ପ୍ରମାଣ କରଯେ ର ଥନ୍ତ୍ର କିଣିତ 6କାଶ ଏକ ପୁର6କାଣ |
(ii) ପ୍ରମାଣ କର ଯେ ଏକ ବୃହତ୍ ଚାପର ଅନ୍ତର୍ଲିଖ କୋଣ ଏକ ସୂକ୍ଷ୍ମକୋଣ ।
(ସୂଚନା : \(\overparen{A P B}\) ଏକ କ୍ଷୁଦ୍ର ଚାପ ଓ \(\overparen{A Q B}\) ଏକ ବୃହତ୍ ଚାପ ହେଉ ।
\(\overline{\mathbf{AD}}\) ଦ୍ୟାସ ଅକନ କର | m∠APD = 90° m∠APB)

Solution:
ଏବଂ m∠AQB < 90°
ଅଙ୍କନ : \(\overline{\mathbf{AR}}\) ବ୍ୟାସ ଅଙ୍କନ କରି । \(\overline{\mathbf{PR}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ : (i) \(\overline{\mathbf{AR}}\) ବ୍ୟାସ । m∠APR ଅର୍ଥବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ । ∴ m∠APR = 90°
କିନ୍ତ୍ର R, ∠APB ର ଅନ୍ତ୍ ମ ହୋଇଥିବାରୁ m∠APR + m∠RPB = m∠APB
m∠APR = 90° ହେତୁ m∠APB > 90° ….. (i)
ଅର୍ଥାତ୍ ∠APB ଏକ ସ୍ଥୂଳକୋଣ ।

(iii) \(\overparen{A P B}\) କ୍ଷୁଦ୍ର ଚାପର ବିପରୀତ ଚାପ \(\overparen{A Q B}\) ଏକ ବୃହତ୍ ଚାପ ।
m∠APB+m∠AQB = 180° (·.· AQBP ଏକ ବୃହତ୍ ଚାପ ହେଉ )
(i) ରେ ପ୍ରମାଣିତ m∠APB > 90°
∴ m∠AQB < 90° ଅର୍ଥାତ୍ m∠AQB ଏକ ସୂକ୍ଷ୍ମକୋଣ । (ପ୍ରମାଣିତ)

Question 17.
(i) △ABCର ପରିବୃତ୍ତର କେନ୍ଦ୍ର ( ତ୍ରିଭୁଜଟିର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ହେଲେ ପ୍ରମାଣ କର ଯେ,
m∠BAC + m∠OBC = 90° |
(ii) △ABCର ପରିବୃତ୍ତର କେନ୍ଦ୍ର O ତ୍ରିଭୁଜଟିର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ । O ଏବଂ A, \(\overline{\mathbf{BC}}\) ର ବିପରୀତ ପାଣମ 6ଦୃକେ, ଦେଲେ ପ୍ରମାଣ କର ଯେ, m∠BAC – m∠OBC = 90° |
Solution:
(i) ଦତ୍ତ : △ABCର ପରିବୃତ୍ତର କେନ୍ଦ୍ର ଠ, ତ୍ରିଭୁଜର ଏକ ଅନ୍ତଃସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ତାମାଣ୍ୟ: m∠BAC + m∠OBC = 90°
ଅଙ୍କନ : \(\overrightarrow{\mathrm{BO}}\) ବୃତ୍ତର ପରିଧ‌ିକୁ P ବିନ୍ଦୁରେ ଛେଦ କରୁ । \(\overline{\mathbf{PA}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ : ABC ବୃତ୍ତର ବ୍ୟାସ \(\overline{\mathbf{BP}}\) |

⇒ m∠BAP = 90° (ଅଦି ଦ୍ଵରଖଣ୍ଡମ କୋଣ)
⇒m∠BAC+m∠CAP = 90°
ମାତ୍ର m∠CAP = m∠PBC (ଏକ ବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ)
⇒ m∠PBC=m∠OBC
⇒ m∠BAC+m∠OBC = 90°

(ii) ଦତ୍ତ : △ABCର ପରିବୃତ୍ତର କେନ୍ଦ୍ର ଠ ତ୍ରିଭୁଜଟିର ଏକ ବହିଃସ୍ଥ ବିନ୍ଦୁ ।
ପ୍ତାମାଣ୍ୟ: m∠BAC – m∠OBC = 90°
ଅଙ୍କନ : \(\overrightarrow{\mathrm{BO}}\) ବୃତ୍ତର P ବିନ୍ଦୁରେ ଛେଦକରୁ । \(\overline{\mathbf{PA}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ: ABC ଦ୍ଦଭର ବ୍ୟାସ \(\overline{\mathbf{BP}}\) |

⇒ m∠BAP = 90° (ଅଦି ଦ୍ଵରଖଣ୍ଡମ କୋଣ)
⇒ m∠BAC – m∠CAP = 90°
⇒ m∠BAC – m∠CBP = 90°
(∵ m∠CAP – m∠CBP ଏକ ବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ)
⇒ m∠BAC – m∠OBC = 90°

Question 18.
ପ୍ରମାଣ କର ଯେ ଏକ ଟ୍ରାପିଜିୟମ୍‌ର ଅସମାନ୍ତର ବାହୁଦ୍ୱୟ ସର୍ବସମ ହେଲେ ଟ୍ରାପିଜିୟମ୍ ବୃତ୍ତାନ୍ତର୍ଲିଖୁତ ହେବ ।
Solution:
ଦତ୍ତ : ABCD ଏକ ଟ୍ରାପିଜିୟମ୍ । \(\overline{\mathrm{AD}}\) || \(\overline{\mathrm{BC}}\) ଏବଂ \(\overline{\mathrm{AB}}\) ≅ \(\overline{\mathrm{CD}}\) |
ପ୍ରାମାଣ୍ୟ : ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜ ।

ଅଙ୍କନ : \(\overline{\mathrm{DM}}\) || \(\overline{\mathrm{AB}}\) ଅଙ୍କନ କର ।.
\(\overline{\mathrm{DM}}\) , \(\overline{\mathrm{BC}}\) କୁ M ବିନ୍ଦୁରେ ଛେଦ କରୁ ।
ପ୍ରମାଣ : ADBM ଏକ ସାମାନ୍ତରିକ ଚିତ୍ର ।
(: \(\overline{\mathrm{AD}}\) || \(\overline{\mathrm{BM}}\) ଏବଂ \(\overline{\mathrm{DM}}\) || \(\overline{\mathrm{AB}}\) )
⇒ \(\overline{\mathrm{AB}}\) ≅ \(\overline{\mathrm{DM}}\)
କିନ୍ତୁ ଦକ \(\overline{\mathrm{AB}}\) ≅ \(\overline{\mathrm{DC}}\)
∴ \(\overline{\mathrm{DM}}\) ≅ \(\overline{\mathrm{DC}}\) ⇒ m∠DMC = m∠DCM
କିନ୍ତୁ \(\overline{\mathrm{AB}}\) || \(\overline{\mathrm{DM}}\), \(\overline{\mathrm{BC}}\) ଛେଦକ । ⇒ m∠ABM = m∠DMC (ଅନୁରୂପ)
m∠ABM = m∠DCM ⇒ m∠ABC = m∠DCB
\(\overline{\mathrm{AD}}\) || \(\overline{\mathrm{BC}}\) ହେତ୍ର m∠DAB + m∠ABC = 180°
⇒ m∠DAB + m∠DCB = 180° (∵ m∠ABC = m∠DCB)
⇒ ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜ । (ପ୍ରମାଣିତ)

Question 19.
ଦୁଇଟି ବୃତ୍ତ ପରସ୍ପରକୁ P ଓ () ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । P ବିନ୍ଦୁ ମଧ୍ୟଦେଇ ଏକ ସରଳରେଖା ବୃତ୍ତଦ୍ଵୟକୁ K ଓ L ବିନ୍ଦୁରେ ଛେଦ କରେ । ସେହିପରି Q ମଧ୍ୟଦେଇ ଏକ ସରଳରେଖା ବୃତ୍ତଦ୍ଵୟକୁ M ଓ N ବିନ୍ଦୁରେ ଛେଦ କରେ । K ଓ M \(\overline{\mathrm{PQ}}\) ର ଏକ ପାର୍ଶ୍ବରେ ଥିଲେ ପ୍ରମାଣ କର ଯେ, \(\overline{\mathrm{KM}}\) || \(\overline{\mathrm{LN}}\) |

Solution:
ଦତ୍ତ : S₁ ଓ S₂ ବୃତ୍ତଦ୍ଵୟ ପରସ୍ପରକୁ P ଓ Q ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି ।
P ଓ Q ବିନ୍ଦୁ ମଧ୍ୟଦେଇ ଅଙ୍କିତ ସରଳରେଖା ବୃତ୍ତଦ୍ଵୟକୁ ଯଥାକ୍ରମେ
K, L ଏବଂ M, N ବିନ୍ଦୁରେ ଛେଦ କରୁଛନ୍ତି ।
ପ୍ରାମାଣ୍ୟ : \(\overline{\mathrm{KM}}\) || \(\overline{\mathrm{LN}}\)
ଅଙ୍କନ : \(\overline{\mathrm{PQ}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ : m∠KMQ = m∠QPL (∵ ବୃତ୍ତାନ୍ତର୍ଲିଖୂତ ଚତୁର୍ଭୁଜର ବହିଃସ୍ଥ କୋଣର ପରିମାଣ, ଏହାର ବିପରୀତ ଅନ୍ତଃସ୍ଥ କୋଣର ପରିମାଣ ସହ ସମାନ ।)
କିନ୍ତୁ m∠QPL + m∠QNL = 180° (ଦୃଭାନ୍ତ୍ରକଖତ ଚତୁରୁକର ବିପର।ତ କୋଣ)
∴ m∠KMQ + m∠QNL = 180°; ମାତ୍ର ଏହି କୋଣଦ୍ଵୟ ଏକ ପାର୍ଶ୍ୱସ୍ଥ ଅନ୍ତଃସ୍ଥ କୋଣ ।
⇒ \(\overline{\mathrm{KM}}\) || \(\overline{\mathrm{LN}}\) (ପ୍ରମାଣିତ)

Question 20.
ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜରେ ∠B ଓ ∠Dର ସମତ୍ତିଖଣ୍ଡକ ଦ୍ଵୟ ପରସ୍ପରକୁ E ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । \(\stackrel{\longleftrightarrow}{\mathbf{D} E}\) ବୃତ୍ତକୁ F ବିନ୍ଦୁରେ ଛେଦ କଲେ ପ୍ରମାଣ କର ଯେ, \(\overline{\mathrm{BE}}\) ⊥ \(\overline{\mathrm{BF}}\) |

Solution:
ଦତ୍ତ : ABCD ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖତ ଚତୁର୍ଭୁଜ । ∠B ଓ ∠D ର
ସମଦ୍ୱିଖଣ୍ଡକଦ୍ୱୟ ପରସ୍ପରକୁ E ବିନ୍ଦୁରେ ଛେଦ କରୁଛନ୍ତି ।
\(\overrightarrow{\mathrm{DE}}\) ବୃତ୍ତକୁ F ବିନ୍ଦୁରେ ଛେଦ କରୁଛି ।
ପ୍ରାମାଣ୍ୟ : \(\overline{\mathrm{BE}}\) ⊥ \(\overline{\mathrm{BF}}\)
ପ୍ରମାଣ : m∠ADC + m∠ABC = 180° (ABCD ବୃତ୍ତାନ୍ତର୍ଲିଖ ଚତୁର୍ଭୁଜ)
\(\frac { 1 }{ 2 }\) m∠ADC + \(\frac { 1 }{ 2 }\) m∠ABC = 90°
⇒ m∠CDF + m∠EBC = 90°
କିନ୍ତୁ m∠CDF = m∠CBF (ଏକ ବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ)
m∠CBF + m∠EBC = 90° ⇒ m∠ERF = 90°
⇒ \(\overline{\mathrm{BE}}\) ⊥ \(\overline{\mathrm{BF}}\) (ପ୍ରମାଣିତ)

Question 21.
△ABCର କୋଣମାନଙ୍କର ସମଦ୍ବିଖଣ୍ଡକମାନେ ତ୍ରିଭୁଜର ପରିବୃତ୍ତକୁ X, Y ଓ Z ବିନ୍ଦୁରେ ଛେଦ କରନ୍ତି । ପ୍ରମାଣ କର ଯେ, △XYZର କୋଣମାନଙ୍କର ପରିମାଣ ଯଥାକ୍ରମେ 90° – \(\frac { 1 }{ 2 }\) m∠A, 90° – \(\frac { 1 }{ 2 }\) m∠B ଓ 90° – \(\frac { 1 }{ 2 }\) m∠C |
Solution:
ଦତ୍ତ : △ABC ଦୁଲାନ୍ତ୍ରଖତ ∠A, ∠B ଓ ∠C ର ସମଦ୍ଵିଖଣ୍ଡକ ବୃତ୍ତକୁ ଯଥାକ୍ରମେ X, Y ଏବଂ Z ବିନ୍ଦୁରେ ଛେଦ କରେ ।
ପ୍ରାମାଣ୍ୟ : m∠X = 90° \(\frac { 1 }{ 2 }\)m∠A, \(\frac { 1 }{ 2 }\) m∠B ଏଦ m∠Z = 90° – \(\frac { 1 }{ 2 }\) m∠C

ପ୍ରମାଣ : \(\overparen{A Z}\) ର ଦିପରାତ ଗାପାନୁଇଖତ m∠AXZ = m∠ACZ
ଏବଂ \(\overparen{A Y}\) ଚାପର ବିପରୀତ ଚାପାନ୍ତର୍ଲିଖ କୋଣ m∠AXY = m∠ABY
∴ m∠AXZ + m∠AXY = m∠ACZ = m∠ABY
⇒ m∠X = \(\frac { m∠C }{ 2 }\) + \(\frac { m∠B }{ 2 }\)
⇒ m∠X = 90° – \(\frac { m∠A }{ 2 }\) (∵ \(\frac { m∠A }{ 2 }\) + \(\frac { m∠B }{ 2 }\) + \(\frac { m∠C }{ 2 }\) = 90°)
ସେହିପରି ପ୍ରମାଣ କରାଯାଇ ପାରେ ଯେ,
m∠Y = 90° – \(\frac { m∠B }{ 2 }\) ଏବଂ m∠Z = 90° – \(\frac { m∠C }{ 2 }\) ହେବ ।

Question 22.
△ABC ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ ସମବାହୁ ତ୍ରିଭୁଜ । \(\overline{\mathbf{BC}}\) ଜ୍ୟା ସହ ସମ୍ପୃକ୍ତ କ୍ଷୁଦ୍ର ଚାପ ଉପରେ P ଏକ ବିନ୍ଦୁ । ପ୍ରମାଣ କର ଯେ PA = PB + PC । (ସୂଚନା : \(\overrightarrow{\mathbf{B P}}\) ଉପରେ D ନିଅ ଯେପରି PC = PD ହେବ । △BCD ଓ △ACP ର ତୁଳନା କର ।)
Solution:
ଦତ୍ତ : △ABC ଏକ ବୃତ୍ତାନ୍ତର୍ଲିଖ ସମବାହୁ ତ୍ରିଭୁଜ । \(\overline{\mathbf{BC}}\) ଜ୍ୟା ସହ ସଂପୃକ୍ତ କ୍ଷୁଦ୍ରଚାପ ଉପରେ P ଏକ ବିନ୍ଦୁ ।
ପ୍ରାମାଣ୍ୟ : PA = PB + PC
ଅଙ୍କନ : \(\overrightarrow{\mathbf{B P}}\) ଉପରେ D ଏକ ବିନ୍ଦୁ ନିଅ ଯେପରିକି PC = PD ହେବ । \(\overline{\mathbf{CD}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ : △ABC ଏକ ସମବାହୁ ତ୍ରିଭୁଜ ।
m∠BAC = m∠CPD (ବୃତ୍ତାନ୍ତର୍ଲିଖ୍ ଚତୁର୍ଭୁଜର ବହିଃସ୍ଥ କୋଣ)
∴ m∠CPD = 60°
ସୁନଶ୍ଚ, △PCD ରେ PC = PD |

∴ △PCD ଏକ ସମବାହୁ ତ୍ରିଭୁଜ । ⇒ PC = CD = PD
ବର୍ତ୍ତମାନ m ∠ACB = m∠PCD = 60°
⇒ m∠ACB+m∠BCP=m∠PCD+m∠BCP
⇒ m∠ACP=m∠BCD
△APC ଓ △BCD ଦୟରେ AC = BC, PC = CD
ଏକ m∠ACP=m∠BCD
∴ △ACP ≅ △BCD
⇒ AP = BD ⇒ AP = BP + PD ⇒ AP = BP + PC

Question 23.
△ABCରେ ∠Aର ସମଦ୍ବିଖଣ୍ଡକ △ABCର ପରିବୃତ୍ତକୁ P ବିନ୍ଦୁରେ ଛେଦ କରେ । P ବିନ୍ଦୁରୁ \(\overrightarrow{\mathbf{AB}}\) ଓ \(\overline{\mathbf{AC}}\) ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବ ଦ୍ବୟର ପାଦବିନ୍ଦୁ ଯଥାକ୍ରମେ Q ଏବଂ R । ପ୍ରମାଣ କର ଯେ, AQ = AR = \(\frac { AB+AC }{ 2 }\) | (ସ୍ମତନା : ଦଶାଥ ଯେ △PBQ ≅ △PCR ⇒ BQ = CR )
Solution:
ଦତ୍ତ : △ABC ର ∠A ର ସମଦ୍ବିଖଣ୍ଡକ ତ୍ରିଭୁଜର ପରିବୃତ୍ତକୁ P ବିନ୍ଦୁରେ ଛେଦ କରୁଛି । P ବିନ୍ଦୁରୁ \(\overrightarrow{\mathbf{AB}}\) ଓ \(\overrightarrow{\mathbf{AC}}\) ପ୍ରତି ଅଙ୍କିତ ଲମ୍ବର ପାଦବିନ୍ଦୁ ଯଥାକ୍ରମେ Q ଏବଂ R । (ଏଠାରେ △ABCର AC > AB)
ପ୍ରାମାଣ୍ୟ : AQ = \(\frac { AB+AC }{ 2 }\) = AR
ଅଙ୍କନ : \(\overline{\mathbf{PB}}\) ଓ \(\overline{\mathbf{PC}}\) ଅଙ୍କନ କର ।
ପ୍ରମାଣ : ∠A ର ସମଦ୍ବିଖଣ୍ଡକ ପରିବୃତ୍ତକୁ P ବିନ୍ଦୁରେ ଛେଦକରେ ।
⇒ \(\overparen{B P}\) = \(\overparen{P C}\) ⇒ BP = PC
△BPQ ଏକ △CPR ଦଯରେ
BP = PC, m∠BQP = m∠CRP (= 90°)
ଏବଂ PQ = PR

(∵ କୋଣର ବାହୁମାନଙ୍କଠାରୁ ସମଦୂରବର୍ତ୍ତୀ ବିନ୍ଦୁମାନ, କୋଣ ସମଦ୍ବିଖଣ୍ଡକ ଉପରେ ଅବସ୍ଥାନ କରିବେ ।)
∴ △BPQ ≅ △CPR ⇒ BQ = CR
ପୁନଶ୍ଚ, △AQP ଓ △APR ଦ୍ବୟରେ
PQ = PR, \(\overline{\mathbf{AP}}\) ସାଧାରଣ ଏବଂ M∠AQP = m∠ARP
∴ △AQP ⇒ △APR ⇒ AQ = AR

ଚଇଂଲାନ 2AQ = AQ + AQ = AQ + AR = AB + BQ + AC – CR
= AB + AC (∵ BQ = CR)
∴ AQ = \(\frac { AB + AC }{ 2 }\) ⇒ AR = \(\frac { AB + AC }{ 2 }\)
⇒ AQ = \(\frac { AB + AC }{ 2 }\) = AR

Question 24.
△ABCରେ ∠Aର ସମଦ୍ବିଖଣ୍ଡକ △ABCର ପରିବୃତ୍ତକୁ P ବିନ୍ଦୁରେ ଛେଦ କରେ । \(\overline{\mathbf{AP}}\) ଓ \(\overline{\mathbf{BC}}\)ର ଛେଦ ବିନ୍ଦୁ D ହେଲେ ପ୍ରମାଣ କର ଯେ △ABD ଓ △APC ସଦୃଶ ଅଟନ୍ତି । ସୁତରାଂ ଦର୍ଶାଅ ଯେ, AB • AC = BD • DC + AD² |
(ପୁଚନା : △ABD ଓ △APC ପଦଣ ⇒ AB.AC = AD.AP, AD² = AD (AP – PD))
Solution:
ଦତ୍ତ : △ABC ର ∠A ର ସମଦ୍ବିଖଣ୍ଡକ, ଏହାର ପରିବୃତ୍ତକୁ P ବିନ୍ଦୁରେ ଛେଦ କରେ । \(\overline{\mathbf{BC}}\) ଓ \(\overline{\mathbf{AP}}\) ର ଛେଦବିନ୍ଦୁ D |
ପ୍ରାମାଣ୍ୟ : (i) △ABD ~ △APC
(ii) AB AC = BD · DC + AD²
ପ୍ରମାଣ : △ABD ଓ △APC ଦ୍ବୟରେ
m∠ABD = m∠APC (ଏକ ବୃତ୍ତଖଣ୍ଡସ୍ଥ କୋଣ)
m∠BAD=m∠PAC ଅବଶିଷ m∠ADB = m∠ACP

∴ △ABD ~ △APC
⇒ \(\frac { AB }{ AP }\) = \(\frac { AD }{ AC }\) ⇒ AB . AC = AD . AP
⇒ AB . AC = AD (AD + DP)
= AD² + AD . DP …..(i)
ପୁନଣ୍ଡ △ABD ~ △PDC
(∵m∠BAD = m∠DCP, m∠ADB = m∠PDC)
⇒ \(\frac { BD }{ DP }\) = \(\frac { AD }{ DC }\) ⇒ BD . DC = AD . DP
(i) ରେ ପ୍ତ6ଯାଗ କଲେ AB . AC = AD² + BD . DC

Question 25.
(ଟଲେମୀଙ୍କ ଉପପାଦ୍ୟ) ABCD ଏକ ବୃତ୍ତାନ୍ତଲିଖତ ଚତୁର୍ଭୁଜ ହେଲେ ପ୍ରମାଣ କର ଯେ,AC · BD = AB · CD + BC · AD | ଗୁଣଫଳ, ଚତୁର୍ଭୁଜର ସମ୍ମୁଖୀନ ବାହୁମାନଙ୍କର ଦୈର୍ଘ୍ୟର ଗୁଣଫଳର ସମଷ୍ଟି ସଙ୍ଗେ ସମାନ ।)
(ସୂଚନା : ମନେକର m∠ADB > m∠BDC | E, AC ଉପରେ ଏପରି ଏକ ବିନ୍ଦୁ ହେଉ ଯେପରି m∠BDC = m∠ADE | ବର୍ତ୍ତମାନ △ADE ଏବଂ △BDC ସଦୃଶ ⇒ \(\frac { AE }{ BC }\) = \(\frac { AD }{ BD }\) ପୁନଶ୍ଚ △ADB ଏବଂ △EDC ସଦୃଶ ⇒ \(\frac { CD }{ BD }\) = \(\frac { EC }{ AB }\) | )
Solution:
ଦଭ : ABCD ଏକ ବୃତ୍ତାନ୍ତଲିଖତ ଚତୁର୍ଭୁଜ ହେଲେ |
ପ୍ରାମାଣ୍ୟ : AC . BD = AB . CD + BC . AD
ଅକନ : ମନେକର m∠ADB > m∠BDC |
\(\overline{\mathbf{AC}}\) ଉପରିସ୍ଥ E ଏପରି ଏକ ବିନ୍ଦୁ ନିଅ ।
ଯେପରିକି m∠ADE = m∠BDC ହେବ ।
ପ୍ରମାଣ : ବର୍ତ୍ତମାନ △ADE ଏବଂ △BDC ଦ୍ଵୟରେ
m∠ADE = m∠BDC ଏବଂ m∠DAE = m∠DBC
ଥଗଣିପୁ m∠AED = m∠BCD

∴ △ADE ~ △BDC
⇒\(\frac { AE }{ BC }\) = \(\frac { AD }{ BD }\) ⇒ AE . BD = AD . BC
ପୁନ୍ଦଣ, △ADB ଏବଂ △EDC ଦ୍ଵପ୍ରେଭେ
m∠ABD + m∠ECD = m∠ADB + m∠EDC)
(∵m∠ADE = m∠BDC ⇒ m∠ADE + m∠EDB = m∠BDC + m∠EDB)
∴ △ADB ~ △EDC
⇒\(\frac { BD }{ CD }\) = \(\frac { AB }{ EC }\) ⇒ EC . BD = AB . CD
(i) ଓ (ii) ରୁ AE . BD + EC. BD = AD. BC + AB. CD
⇒ BD (AE + EC) = AB. CD + BC. AD
⇒ BD. AC = AB. CD + BC. AD

Odisha State Board BSE Odisha 8th Class English Solutions Test-1 Textbook Exercise Questions and Answers.

BSE Odisha Class 8 English Solutions Test-1

Test -1

1. Your teacher gives you a dictation of five 3/4 lettered words. Write them.    [05]
(ତୁମ ଶିକ୍ଷକ ତୁମକୁ ୫ଟି ତିନି କିମ୍ବା ଚାରି ଅକ୍ଷରିଆ ଇଂରାଜୀ ଶବ୍ଦ ଡାକିବେ । ତୁମେ ଶୁଣି ଲେଖିବ ।)
huge, pant, drag, old, do.

2. Your teacher will read aloud the following lines. Listen to him/her and fill in the gaps.   [07]
(ତୁମ ଶିକ୍ଷକ ଏହି ଧାଡ଼ିଗୁଡ଼ିକ ପାଟି କରି ପଢ଼ିବେ । ତୁମେ ତାହା ସାବଧାନତା ସହ ଶୁଣି ଦ୍ବିତୀୟ ଥର ପଢ଼ିବା ପରେ ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କରିବ ।) (Questions with Answers)
“They _____________ into the king’s ___________. The ___________ man said to the king’s officer, Sir, I was ___________ to the town in my __________. This __________ wanted to __________ to the town market. He is ____________. So, I brought him to the ____________ on my horse. _________ he says that the horse is __________. Sir, ___________ help me to ___________ my horse from ____________.

Answer:
“They entered into the king’s court. The first man said to the king’s officer, Sir, I was rushing to the town in my horse. This man wanted to take my horse to the town market. He is a thief. So, I brought him to the court on my horse. Now he says that the horse is not mine. Sir, please help me to get my horse from the thief.

3. There is some relationship between spelling and pronunciation. Generally, there is, ie, ea, oo, ee, or ou, in the spelling of a word; this signals a long sound. And Odia speakers of English have problems with long sounds. They have a tendency to pronounce long sounds as short sounds. Given below are some words, underline which of them have long sounds.    [07]
(ସାଧାରଣତଃ ବନାନ ଏବଂ ଉଚ୍ଚାରଣ ମଧ୍ୟରେ କିଛିଟା ସମ୍ବନ୍ଧ ରହିଥାଏ । ସାଧାରଣତଃ ଗୋଟିଏ ଇଂରାଜୀ ଶବ୍ଦ ବନାନରେ je, ea, o୦, ce କିମ୍ବା ou ଦୀର୍ଘ ଭାବରେ ଉଚ୍ଚାରିତ ହୋଇଥା’ନ୍ତି ଏବଂ ଇଂରାଜୀ କହୁଥ‌ିବା ଓଡ଼ିଆ ବକ୍ତାମାନଙ୍କର ସମସ୍ୟା ହୋଇଥାଏ । ସେମାନେ ସାଧାରଣତଃ ଦୀର୍ଘ ଉଚ୍ଚାରଣକୁ ସୂକ୍ଷ୍ମ ଭାବରେ ଉଚ୍ଚାରଣ କରିଥା’ନ୍ତି । ତଳେ କେତେକ ଶବ୍ଦ ଦିଆଯାଇଛି । ସେଗୁଡ଼ିକରେ ଦୀର୍ଘ ଉଚ୍ଚାରିତ ଶବ୍ଦଗୁଡ଼ିକ ତଳେ ଗାର ଦିଅ ।
(Questions with Answers)

agree, market, reach, please, cover, thought, punish, village, need. speed, under, thief, steal, peace, deep, honey, seed, fields, spring, bean.

4. Write the following Odia names in English. (Teacher will give four names of persons in Odia.) (Questions with Answers)        [08]
____________________  _______________________
____________________  _______________________
____________________  _______________________
____________________  _______________________
Answer:
ମଧୁ — Madhu
ରମେଶ — Ramesh
ସଦାନନ୍ଦ — Sadananda
ବୀରେନ୍ଦ୍ର — Birendra

5. Write the following names of places in English. (Teacher will give four names of places in Odia.)     [08]
____________________  _______________________
____________________  _______________________
____________________  _______________________
____________________  _______________________
Answer:
କଟକ — Cuttack
ଭୁବନେଶ୍ୱର — Bhubaneswar
ରାଉରକେଲା — Rourkela
ସୁନ୍ଦରଗଡ଼ — Sundargarha

6. Match the words under ‘A’ with the words under ‘B’ —(who lives where).    [06]

Answer:

7. Read the following text and answer the questions. (ନିମ୍ନଲିଖ ଅନୁଚ୍ଛେଦଗୁଡ଼ିକ ପାଠ କରି ପ୍ରଦତ୍ତ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

1. Akbar was very fond of jewellery. He had hundreds of rings, rings in which diamonds and many other gems had been set. But all of the rings he had, he liked one of them the most. It was a large ring with a number of pearls and diamonds set in it. The ring was a present to Akbar from the Queen.
2. At Akbar’s palace, there were eight servants who looked after the Emperor’s clothes and jewellery. Every day one of these eight servants used to help Akbar get ready to go to the court. None other than these eight servants could enter the Emperor’s room.
3. One day the Emperor was getting ready to go to the court. He wanted to wear his favourite ring that day. He asked one of his servants to bring it. But the servants came back saying that he could not find the ring. Akbar ordered to search for the ring, but it could not be found.
4. Akbar was very angry. He felt that one of his servants had stolen the ring. He sent for Birbal. When Birbal came, he told him what had happened and asked him to find out the thief.

(a). Answer the following questions each in one complete sentence. [05 ]
(ଏହି ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଗୋଟିଏ ଗୋଟିଏ ପୂର୍ଣ୍ଣ ବାକ୍ୟରେ ପ୍ରକାଶ କର ।)

Question 1.
What was Akbar fond of?
Answer:
Akbar was fond of jewellery.

Question 2.
Who presented the ring to Akbar?
Answer:
The Queen presented the ring to Akbar.

Question 3.
Why was Akbar angry?
Answer:
Akbar was angry when he felt that one of his servants had stolen his most favourite ring.

Question 4.
Who did Akbar tell what had happened?
Answer:
Akbar told Birbal what had happened.

Question 5.
How many servants looked after Akbar’s clothes and jewellery?
Answer:
Eight servants looked after Akbar’s clothes and jewellery.

(b). From the text, write five sentences about Akbar.   [10]
(ଏହି ପାଠରୁ ପାଞ୍ଚୋଟି ବାକ୍ୟ ଆକବରଙ୍କ ସମ୍ପର୍କରେ ଲେଖ ।)
_______________________________________________
_______________________________________________
_______________________________________________
_______________________________________________
Answer:
Akbar was a great Emperor of India in the Mughal period. He was a good administrator. He was liked by all classes of people in India. He married a Hindu girl and made her Queen. He was fond of jewellery.

(c). Given below are some sentences. As per the text, the sentences are not in order. Order them by putting serial numbers in brackets provided against each sentence. (Questions with Answers)    [09]
(ନିମ୍ନରେ କେତେକ ବାକ୍ୟ ରହିଛି । ବହି ପାଠ ଅନୁଯାୟୀ ଏହା ଠିକ୍ କ୍ରମରେ ନାହିଁ । ସେମାନଙ୍କୁ ଠିକ୍ କ୍ରମରେ ଲେଖିବା ପାଇଁ ଡାହାଣ ପାଖରେ ଥ‌ିବା ଖାଲି ଘରେ କ୍ରମିକ ନମ୍ବରଗୁଡ଼ିକୁ ଲେଖ ।)

Akbar was very angry. [  ]
Akbar was fond of jewellery. [  ]
The queen presented the ring to Akbar. [  ]
The ring was not to be found. [  ]
Eight servants were in charge of the jewellery. [  ]
Akbar sent for Birbal. [  ]
Akbar ordered to search for the ring. [  ]

Answer:
Akbar was very angry. [ 5 ]
Akbar was fond of jewellery. [ 1 ]
The queen presented the ring to Akbar. [ 3 ]
The ring was not to be found. [ 4 ]
Eight servants were in charge of the jewellery. [ 2 ]
Akbar sent for Birbal. [ 6 ]
Akbar ordered to search for the ring. [ 7 ]

(d). See the use of the following four phrases in the text. The paragraph number is given against each phrase. Try to understand the meaning and use of the phrase from the context. Next, read the paragraph given and fill in the gaps with the right phrases. [8]
(ନିମ୍ନଲିଖ୍ ୪ଟି ଖଣ୍ଡବାକ୍ୟର ବ୍ୟବହାର ଲକ୍ଷ୍ୟ କର । ପ୍ରତ୍ୟେକ ଖଣ୍ଡବାକ୍ୟର ଡାହାଣରେ ଅନୁଚ୍ଛେଦର ସୂଚନା ଦିଆଯାଇଛି । ସେଗୁଡ଼ିକର ବ୍ୟବହାର ଏବଂ ଅର୍ଥ ତୁମ ବିଷୟ ମାଧ୍ୟମରେ ଜାଣିବାକୁ ଚେଷ୍ଟା କର । ଏହାପରେ ପ୍ରଦତ୍ତ ଅନୁଚ୍ଛେଦଟିକୁ ପଢ଼ି ଠିକ୍ ଭାବରେ ଖଣ୍ଡବାକ୍ୟଗୁଡ଼ିକୁ ଯୋଗ କର ।)
(Question with Answer)

Find out (5), looked after (2), sent for (5), fond of (1)

Abdul had a pet baby donkey. He was very __________ the baby donkey. He _____________ the donkey very well. One day the baby donkey went somewhere. He ___________ his faithful servant Ali. He asked Ali to _____________ the baby donkey.
Answer:
Abdul had a pet baby donkey. He was very fond of the baby donkey. He looked after the donkey very well. One day the baby donkey went somewhere. He sent for his faithful servant Ali. He asked Ali to find out the baby donkey.

8. Read the following text and do the tasks that.
(ନିମ୍ନ ବିଷୟଟିକୁ ପାଠ କର ଏବଂ ପ୍ରଦତ୍ତ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

1. Long long ago, on the bank of the river Nagabali there was a small village named Hatibadi, and at the one end of this village was the chatasali, or village school, run by Ghana Rath, where many children from villages nearby came to study Ghana Ratha taught all the subjects himself, including Mathematics, Literature and Social Sciences.
(ବହୁଦିନ ତଳେ ନାଗାବଳୀ ନଦୀ କୂଳରେ ହାତୀବାଡ଼ି ବୋଲି ଏକ କ୍ଷୁଦ୍ର ଗ୍ରାମ ଥିଲା ଏବଂ ଏହି ଗ୍ରାମର ଶେଷମୁଣ୍ଡରେ ଗୋଟିଏ ଚାଟଶାଳୀ ବା ଗାଁ ସ୍କୁଲ ଘନରଥ ନାମକ ଏକ ବ୍ୟକ୍ତିଙ୍କଦ୍ୱାରା ଚାଲୁଥୁଲା, ଯେଉଁଠି ଆଖାପାଖ ଗାଁର ବହୁ ପିଲା ପଢ଼ିବାକୁ ଆସୁଥିଲେ । ଘନରଥ ନିଜେ ଗଣିତ, ସାହିତ୍ୟ ଓ ସାମାଜିକ ଶିକ୍ଷାସହ ସବୁ ବିଷୟ ଶିକ୍ଷାଦାନ କରୁଥିଲେ ।)

2. By the side of the Chatasali ran a narrow road that led to the river and on this road, early every morning, you could see a boy named Hatia riding a donkey and leading another by a rope. He was the son of a washerman. But as his parents were dead, he supported himself by washing the dirty clothes in the village. Every day he took a donkey load of clothes to the river, where he washed and dried them. When his work was finished, he returned home by the same road, together with his two donkeys. One was named Bhadra and the other Madri.
(ଚାଟଶାଳୀର ପାଖଦେଇ ନଈ ଆଡ଼କୁ ଗୋଟିଏ ଅଣଓସାରିଆ ରାସ୍ତା ଯାଉଥିଲା ଏବଂ ଏହି ରାସ୍ତା ଉପରେ ତୁମେ ଦେଖିବ ପ୍ରତିଦିନ ବଢ଼ିଭୋରରୁ ହଟିଆ ନାମରେ ଜଣେ ଯୁବକ ଗୋଟିଏ ଗଧ ଉପରେ ଚଢ଼ି ଏବଂ ଆଉ ଗୋଟାକୁ ଦଉଡ଼ିରେ ବାନ୍ଧି ନେଇ ଯାଉଥବ । ସେ ଗୋଟିଏ ଧୋବାର ପୁଅ । ତା’ର ବାପ ମା’ ମରିଯାଇଥିବାରୁ ସେ ନିଜେ ସବୁ ମଇଳା ଲୁଗାପଟା ଗାଁ ଲୋକଙ୍କର ନେଇ ନଈକୁ ଯାଏ । ପ୍ରତ୍ୟେକ ଦିନ ଏକ ଗଧ ବୋଝେଇ ଗାଡ଼ିରେ ମଇଳା ଲୁଗା ନଦୀକୁ ନେଉଥିଲା । ଯେଉଁଠାରେ ସେ ତାକୁ ଧୋଇସାରି ଶୁଖାଇ ଦେଉଥିଲା ଏବଂ ସେଇ ବାଟଦେଇ ପୁଣି ଘରକୁ ଫେରୁଥିଲା ଦି ଗଧଙ୍କୁ ଧରି । ଗୋଟିକର ନାମ ଥିଲା ଭଦ୍ର ଏବଂ ଅନ୍ୟଟିର ନାମ ମାଦ୍ରି ।)

(a). Answer the following questions. [07]
(ନିମ୍ନଲିଖ ପ୍ରଶ୍ନଗୁଡ଼ିକର ଉତ୍ତର ଦିଅ ।)

Question (i)
What was the name of the river?
Answer:
The name of the river was Nagabali.

Question (ii)
What was the name of the village?
Answer:
The name of the village was Hatibadi.

Question (iii)
What was the name of the teacher?
Answer:
The name of the teacher was Ghana Ratha.

Question (iv)
What subjects did Ghana Ratha teach?
Answer:
Ghana Ratha taught all the subjects including Mathematics. Literature and Social Science.

Question (v)
What was the name of the boy?
Answer:
The name of the boy was Hatia.

Question (vi)
How many donkeys did Hatia have?
Answer:
Hatia had two donkeys.

Question (vii)
What were their names?
Answer:
The names of his two donkeys were Bhadra and Madri.

(b). Write four sentences about Hatia. [10]
(ହଟିଆ ବିଷୟରେ ୪ଟି ବାକ୍ୟ ଲେଖ ।)
(Question with Answer)
__________________________________________________________
__________________________________________________________
__________________________________________________________
__________________________________________________________
Answer:
1. Hatia was the son of a washerman.
2. Every day he was going to the rider riding a donkey and leading another by a rope for washing clothes.
3. His parents were dead.
4. Every day he took a donkey load of clothes to the river and there he washed and dried them.

(c). Rewrite paragraph 1 of the text replacing the following words/ phrases at the right places. [10]
(ପ୍ରଥମ ଅନୁଚ୍ଛେଦକୁ ଆଉ ଥରେ ଲେଖ ଯେପରିକି ନିମ୍ନରେ ପ୍ରଦତ୍ତ ଇଂରାଜୀ ଶବ୍ଦଗୁଡ଼ିକ ତାଙ୍କ ମଧ୍ୟରେ ରହିବେ ।)

Mahanadi, town, Cuttack, college, principal, Satpathy, Odia, Sanskrit and English.

Answer:
Long long ago on the bank of the river Mahanadi. There was a small town named Cuttack and at the one end of this small town was a college or Mahavidyalaya run by Mr Satpathy. Where many children from the town nearby came to study. Mr Satapathy taught all the subjects himself including Odia, Sanskrit and English.

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

CHSE Odisha 11th Class Math Notes Chapter 11 Straight Lines

Distance formula:
Distance between two points A (x₁, y₁) and A (x₂, y₂) = \(\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}\)

Section Formula:
If C(x, y) divides the join of A (x₁, y₁) and A (x₂, y₂) in the ratio m: n internally then, x = \(\frac{m x_2+n x_1}{m+n}\), y = \(\frac{m y_2+n y_1}{m+n}\)

Note:

• If the division is external then, x = \(\frac{m x_2-n x_1}{m-n}\), y = \(\frac{m y_2-n y_1}{m-n}\)
• If C(x, y) is the midpoint then x = \(\frac{x_1+x_2}{2}\), y = \(\frac{y_1+y_2}{2}\)

Area of triangle formula:
The area of triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by  = \(\frac{1}{2}\)[x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)]

Different points related to a triangle:
(a) Centroid of the triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is = G \(\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\)

(b) In centre of a triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is = I \(\left(\frac{a x_1+b x_2+c x_3}{3}, \frac{a y_1+b y_2+c y_3}{3}\right)\)

Slope Of A Line:
(a) Angle of inclination: the angle θ made by a line with positive x-axis is the angle of inclination.
(b) Slope of a line: Slope of a line is the tangent of angle of inclination. i,.e m = tan θ.
(c) Slope of a line joining A(x₁, y₁), and B(x₂, y₂) = \(\frac{y_2-y_1}{x_2-x_1}\)

Note:

(i) Slope of x-axis = 0
Slope of any line parallel to x-axis = 0

(ii) Slope of y-axis  = ∞
Slope of any line parallel to y-axis = ∞

Angle Between two Lines:
Angle Φ between two lines with slope m₁ and m₂ is given by tan Φ = \(\pm \frac{\left(m_1-m_2\right)}{1+m_1 m_2}\)

Note:

• To find the acute angle between two lines use the formula. tan Φ = \(\left|\frac{m_1-m_2}{1+m_1 m_2}\right|\)
• Two lines are parallel if m₁ = m₂
• Two lines are perpendicular if m₁m₂ = (-1).

Collinearity Of Three Points:
Three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are collinear if
(i) Sum of distances between two pairs of points = Distance between the 3rd pair.
Or, (ii) Area of Δ ABC = 0
Or, (iii) Let B(x₂, y₂) divides the join of AC in ratio k: 1
∴ \(x_2=\frac{k x_3+x_1}{k+1}, y_2=\frac{k y_3+y_1}{k+1}\)
The value of k obtained from two cases are equal.
Or, (iv) Slope of AB = Slope of AC.

Equation of a straight line:
Lines parallel to co-ordinate axes:
(i) Equation of any line parallel to x-axis is, y = k
⇒ Equation of x-axis is, y = 0

(ii) Equation of any line parallel to y-axis is, x = k
⇒ Equation of y-axis is, x = 0

Lines Not Parallel To Any Axes:
(i) Slope intercept form:
Equation of a line with slope ‘m’ and y-intercepts ‘c’ is: y = mx + c

(ii) Point slope form:
Equation of a line with slope ‘m’ and passing through a point A(x₁, y₁) is: y – y₁ = m(x – x₁)

(iii) Two point form:
Equation of the line passing through A(x₁, y₁) and B(x₂, y₂) is : \(\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}\)

(iv) Intercept form:
Equation of a line with x-intercept ‘a’ and y-intercept ‘b’ is \(\frac{x}{a}+\frac{y}{b}=1\)

(v) Normal form:
Equation of a line whose distance form origin is P and the perpendicular drawn form origin to the line makes an angle α with positive direction of x-axis is: x cos α + y sin α = P

(vi) Parameteric form or symmetric form:
Equation of the line passing through A(x₁, y₁) and making an angle θ with positive direction of x-axis is: \(\frac{x-x_1}{\cos \theta}=\frac{y-y_1}{\sin \theta}\) = r
Or, x = x₁ + r cos θ, y = y₁ + r sin θ
where r = The directed distance between points P(x, y) and A(x₁, y₁)

(vii) General form:
General equation of a straight line is Ax + By + C = 0

Note:

• Slope of this line = –\(\frac{\mathrm{A}}{\mathrm{B}}\)
• x-intercept = –\(\frac{\mathrm{C}}{\mathrm{A}}\)
• y-intercept = – \(\frac{\mathrm{C}}{\mathrm{B}}\)
• Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are parallel if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}\) perpendicular if a₁a₂ + b₁b₂ = 0 and coincident if \(\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\)

Condition of concurrency of three lines:
Three lines a₁x + b₁y + c₁ = 0 a₂x + b₂y + c₂ = 0 and a₃x + b₃y + c₃ = 0 are concurrent if \(\left|\begin{array}{lll}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{array}\right|\) = 0

Family Of Lines:

(i) Equation of lines parallel to the line ax + by + c = 0 is given by: ax + by + λ = 0
(ii) Equation of lines perpendicular to the line ax + by + c = 0 is given by bx – ay + λ = 0
(iii) Equation of lines passing through the point of intersection of two lines.
a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is given by: (a₁x + b₁y + c₁) + λ(a₂x + b₂y + c₂)

Distance of a point from a line:
The perpendicular distance of A(x₁, y₁) from the line ax + by + c  = 0 is: d = \(\left|\frac{a x_1+b y_1+c}{\sqrt{a^2+b^2}}\right|\)

Distance between two parallel lines:
ax + by + c₁ = 0 and  ax + by + c₂ = 0 is d = \(\left|\frac{c_1-c_2}{\sqrt{a^2+b^2}}\right|\)

Position of a point with respect to a line:
A point A(x₁, y₁) lies
(i) above the line ax + by + c = 0 if \(\frac{a x_1+b y_1+c}{b}\) > 0
(ii) below the line ax + by + c = 0 if \(\frac{a x_1+b y_1+c}{b}\) < 0

Equation of bisectors of angle between two intersecting lines:
(i) Equation of angle bisector of two lines. a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 is given by \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=\pm \frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)

Note:

Out of two bisector take one and find the angle between that bisector and one line. If the angle is less than 45° then that bisector is the bisector of acute angle, otherwise, the other bisector is the bisector of acute angle.

(ii) Bisector of angle containing a given point (h, k):

Step – 1: Check the sign of a₁h + b₁k + c₁  and a₂h + b₂k + c₂

• If they have same sign then the bisector of angle containing (h, k) is: \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=\frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)
• If they have opposite sign then the bisector of angle containing (h, k) is: \(\frac{a_1 x+b_1 y+c_1}{\sqrt{a_1^2+b_1^2}}=-\frac{a_2 x+b_2 y+c_2}{\sqrt{a_2^2+b_2^2}}\)

Change Of Axes (Shifting Of Origin):

(i) Translation of coordinate axes.
Let O'(h, k) is the origin of system S’ with respect to origin O(0, 0) of the system S. S’ is the translation of S. If (x, y) and (x’, y’) are the coordinate of a point P in the system S and S’ respectively then
x’ = x – h and y’ = y – k Or, x = x’ + h, y = y’ + k

(ii) Rotation of axes:
Let S’ is a rotation of S, α is the measure of rotation
If (x, y) and (x’, y’) are the coordinate of a point P with respect to S and S’ then x = x’ cos α – y’ sin α and y = x’ sin α + y’ cos α

(iii) Translation as well as a rotation:
If S’ is a combination of translation followed by a rotation then x = h + x’ cos α – y’ sin α, y = k + x’ sin α + y’ cos α

Odisha State Board BSE Odisha 10th Class Maths Solutions Algebra Chapter 5 ପରିସଂଖ୍ୟାନ Ex 5(c) Textbook Exercise Questions and Answers.

BSE Odisha Class 10 Maths Solutions Algebra Chapter 5 ପରିସଂଖ୍ୟାନ Ex 5(c)

Question 1.
ଦତ୍ତ ଉକ୍ତିମାନଙ୍କ ମଧ୍ୟରୁ ଯେଉଁଟି ଠିକ୍ ତା’ ପାଖରେ ‘ନ’ ଓ ଯେଉଁଟି ଭୁଲ୍ ତା’ ପାଖରେ F ଲେଖ ।
(i) ଏକ ତଥ୍ୟାବଳୀର ସମସ୍ତ ଲବ୍‌ଧାଙ୍କ ସମାନ ସମାନ ଥର ରହିଲେ ଏହି ତଥ୍ୟାବଳୀର ଗରିଷ୍ଠକ ନାହିଁ ।
(ii) ବାରମ୍ବାରତା ସାରଣୀ ଅନ୍ତର୍ଭୁକ୍ତ ତଥ୍ୟାବଳୀର ସର୍ବାଧ‌ିକ ବାରମ୍ବାରତା ହିଁ ଉକ୍ତ ତଥ୍ୟାବଳୀ ଗରିଷ୍ଠକ ।
(iii) ଏକ ତଥ୍ୟାବଳୀର ଯଦି ଗରିଷ୍ଠକ ଥାଏ, ତେବେ ଏହାର ସର୍ବଦା ଗୋଟିଏ ମାତ୍ର ଗରିଷ୍ଠକ ଥବ ।
ଉ :
(i) T (ଦ୍ରଷ୍ଟବ୍ୟରେ ଏହା ଲିଖ୍)
(ii) F (ଗରିଷ୍ଠକର ସଂଜ୍ଞା ଅନୁଯାୟୀ )
(iii) F ( କାରଣ 5, 7, 7, 7, 8, 8, 9, 9, 9ରେ ଗରିଷ୍ଠକ 7 ଏବଂ 9 ଅଟେ ।)

Question 2.
ଦତ୍ତ ତଥ୍ୟାବଳୀର ଗରିଷ୍ଠକ ନିର୍ଣ୍ଣୟ କର ।
(i) 5, 6, 7, 7,8, 9, 9, 9, 10, 10, 11, 12, 12
(ii) 12, 8, 15, 9, 11, 8, 10, 11, 13, 9, 12, 10, 14, 11, 13, 10
ସମାଧାନ :
(i) 5, 6, 7, 7,8, 9, 9, 9, 10, 10, 11, 12, 12 ଲବ୍‌ଧାଙ୍କଗୁଡ଼ିକ ସାନରୁ ବଡ଼ କ୍ରମେ ସଜ୍ଜିତ ।
ଏଠାରେ ଗରିଷ୍ଠକ M = 9 (∵ 9ର ବାରମ୍ବାରତା ସର୍ବାଧିକ ।)

(ii) 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 15
ଲବ୍‌ଧାଙ୍କଗୁଡ଼ିକ ସାନରୁ ବଡ଼ କ୍ରମେ ସଜ୍ଜିତ ।
ଏଠାରେ ଗରିଷ୍ଠକ M_o = 10 ଓ 11 (∵ 10 ଓ 11ର ବାରମ୍ବାରତା ସର୍ବାଧିକ ।)

Question 3.
ନିମ୍ନ ସାରଣୀ ଅନ୍ତର୍ଭୁକ୍ତ ତଥ୍ୟାବଳୀର ଗରିଷ୍ଠକ ନିର୍ଣ୍ଣୟ କର ।

ଉଚ୍ଚତା (ସେ.ମି.) ଲବ୍‌ଧାଙ୍କ :	120	121	122	123	124
ବାରମ୍ବାରତା :	5	8	18	10	9

ସମାଧାନ :
ସାରଣୀରୁ ସ୍ପଷ୍ଟ ଯେ ଲବ୍‌ଧାଙ୍କ 122 ର ବାରମ୍ବାରତା ସର୍ବାଧ‌ିକ 18 ।
∴ ଦତ୍ତ ତଥ୍ୟାବଳୀର ଗରିଷ୍ଠକ 122 ।

Question 4.
ଦୁଇଟି ଲୁଡୁଗୋଟିକୁ ଏକା ସାଙ୍ଗରେ 15 ଥର ଗଡ଼ାଇବାରେ ମିଳିଥିବା ଲବ୍‌ଧାଙ୍କଗୁଡ଼ିକ 7, 8, 10, 10, 11, 7, 12, 9, 7, 9, 8, 12, 11, 10, 7 । ଉକ୍ତ ବଣ୍ଟନର ଗରିଷ୍ଠକ ନିର୍ଣ୍ଣୟ କର ।
ସମାଧାନ :
ଦୁଇଟି ଲୁଡୁଗୋଟିକୁ ଏକା ସାଙ୍ଗରେ 15 ଥର ଗଡ଼ାଇବାରେ ମିଳିଥିବା ଲବ୍‌ଧାଙ୍କଗୁଡ଼ିକ –
7, 8, 10, 10, 11, 7, 12, 9, 7, 9, 8, 12, 11, 10, 71
ଲବ୍‌ଧାଙ୍କଗୁଡ଼ିକ ସାନରୁ ବଡ଼ କ୍ରମେରେ ସଜାକ ରଖିଲେ
7, 7, 7, 7, 8, 8, 9, 9, 10, 10, 10, 11, 11, 12, 12
ଏଠାରେ ଗରିଷ୍ଠକ M_o = 7 (∵ 7ର ବାରମ୍ବାରତା ସର୍ବାଧ‌ିକ 4)

Question 5.
ଗୋଟିଏ ଜୋତା ଦୋକାନରେ ବିଭିନ୍ନ ମାପ ବିଶିଷ୍ଟ ଜୋତା ବିକ୍ରୟର ବାରମ୍ବାରତା ବଣ୍ଟନ ନିମ୍ନରେ ଦିଆଯାଇଛି ।

କୋତାମପ	5	6	7	8	9	10
ବିକ୍ରି ସଂଖ୍ୟା	20	33	40	85	15	8

(i) ଉପରିସ୍ଥ ବଣ୍ଟନକୁ ଲକ୍ଷ୍ୟ କରି କେଉଁ ମାପର ଜୋତାକୁ ମହଜୁଦ ରଖୁବା ଲାଗି ଦୋକାନୀ ଅଧ୍ବକ ଧ୍ୟାନ ଦେବ, ସ୍ଥିର କର ।
(ii) ଦତ୍ତ ତଥ୍ୟାବଳୀର କେଉଁ ପ୍ରକାର କେନ୍ଦ୍ରୀୟ ପ୍ରବଣତା ତୁମେ ନିଶ୍ଚୟ କଲ ?
ସମାଧାନ :
(i) ଦତ୍ତ ତଥ୍ୟାବଳୀରେ ସ୍ପଷ୍ଟ ଯେ ୫ ନମ୍ବର ଜୋତାର ବିକ୍ରିସଂଖ୍ୟା (ବାରମ୍ବାରତା) ସର୍ବାଧ‌ିକ ‘85” ଯୋଡ଼ା । ତେଣୁ ଦୋକାନୀ ୫ ନମ୍ବର ଜୋତା ମହଜୁଦ୍ ରଖ୍ ପ୍ରତି ଅଧ‌ିକ ଧ୍ୟାନ ଦେବେ ।
(ii) ଦତ୍ତ ତଥ୍ୟାବଳୀର କେଉଁ ପ୍ରକାର କେନ୍ଦ୍ରୀୟ ପ୍ରବଣତା ତୁମେ ନିର୍ଣ୍ଣୟ କଲ ?

Odisha State Board CHSE Odisha Class 11 Math Notes Chapter 10 Sequences and Series will enable students to study smartly.

CHSE Odisha 11th Class Math Notes Chapter 10 Sequences and Series

Sequence:
A sequence is a function whose domain is N (The set of natural numbers).
Note: We can use the set of whole numbers as a domain.

Real Sequence:
If the range of a sequence is a subset of ‘R’, then it is a real sequence.
⇒ If: N → R is a sequence then f(n) for n = 1, 2, 3, ….. are the terms of the sequence.

Finite and infinite sequence:
A sequence with a finite number of terms is a finite sequence otherwise it is infinite.
Note: We denote a sequence by (t_n) or {t_n} where f(n) = t_n

Series:
An expression of the type t₁ + t₂ + t₃ + ….. (or ∑t_n) where t_n is the nth term of a sequence is a series.

Partial sums:
If \(\sum_{n=1}^{\infty} t_n\) is a series then a sum \(\mathrm{S}_n=\sum_{k=1}^n t_k\) is called the nth partial sum of the series for n = 1, 2, 3 …..
∴ s₁ = t₁, s₂ = t₁ + t₂, s₃ = t₁ + t₂ + t₃ and so on.

Progression:
Progression is a sequence whose terms follow as pattern.
Arithmetic progression (A.P):
A sequence  (t_n) is an A.P. If t_n+1 – t_n = d (constant) for n = 1, 2, 3, …..

(a) General form: a, a + d, a + 2d, a + 3d …..
(b) n^th term: t_n = a + (n – 1)d, where t₁ = a, and the common difference = d
(c) Sum of first n terms (Sn):
S_n = \(\frac{n}{2}\)[2a + (n – 1)d]
= \(\frac{n}{2}\)[a + l]
where a = first term
d = common difference
l = last term (or nth term)

Note:

1. If a, b, c are in A.P. then 2b = a + c.
2. If 3 numbers are in A.P. then we take them as a – d, a, a + d.
3. If 4 numbers are in A.P. then we take rhem as a – 3d, a – d, a + d, a + 3d.

(d) Insertion of arithmetic means between two given number:
Let m₁, m₂, m₃ …. m_n are ‘n’ arithmetic means between ‘a’ and ‘b’ then m_k = a + \(\frac{k(b-a)}{n+1}\) for k = 1, 2, ….. n.

Geometric progression (G.P):
If \(\frac{t_{n+1}}{t_n}\) = r (constant), for n = 1, 2, 3, ….. then the sequence (t_n) is a geometrical progression.

(a) General form: a, ar, ar², ar³ …..
(b) nth term of GP: nth term of G.P. = t_n = ar^n-1.
(c) sum of first n terms of a G.P.: S_n = \(\frac{a\left(1-r^n\right)}{1-r}(\text { for } r \neq 1)\)
(d) sum of an infinite G.P.: If |r| < 1 then the sum of the infinite G.P. a, ar, ar² ….. is S_∞ = \(\frac{a}{1-r}\)

Note:

1. If a, b, c are in G.P. then b² = ac
2. If 3 numbers are in G.P. we take them as \(\frac{a}{r}\), a, ar.
3. If 4 numbers are in G.P. then we take them as \(\frac{a}{r^3}, \frac{a}{r}, a r, a r^3\)

(e) Insertion of  geometric means between  two numbers:
If g₁, g₂ ……g_n are n geometric means between a and b then g_k = a \(\left(\frac{b}{a}\right)^{\frac{k}{n+1}}\), k = 1, 2, 3, …. n

Harmonic Progression (H.P):
A sequence a₁, a₂, a₃ ….. of non zero numbers is called a Harmonic progression if the sequence \(\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}\) ….. is an A.P.

(a) Harmonic mean:
Harmonic mean(H) between two numbers a and b is \(\frac{1}{\mathrm{H}}=\frac{\left(\frac{1}{a}+\frac{1}{b}\right)}{2}\)
= \(\frac{a+b}{2 a b}\)
⇒ H = \(\frac{2 a b}{a+b}\)

(b) Insertion of n harmonic means between two numbers:
Let H₁, H₂ ….. H_n are n harmonic means between a and b then \(\frac{1}{\mathrm{H}_K}=\frac{1}{a}\) + k_D, where D = \(\frac{a-b}{(n+1) a b}\).

Relation among A.M., G.M. and H.M.
AM ≥ GM ≥ HM

Arithmetic co-geometric sequence(AGP):
If (a_n) is an A.P. and (b_n) is an G.P. then the series (a_nb_n) is called an arithmetic co-geometric sequence.

(a) General form: a, (a + d) r, (a + 2d) r², (a + 3d)r³,…..
(b) nth term of A.G.P.: t_n = {a + (n – 1)d} r^n-1.
(c) sum of first terms of A.G.P.:
The sum of first n terms of A.G.P. a, (a + d) r, (a + 2d) r², …..  is
S_n = \(\frac{a}{1-r}+d r\left(\frac{1-r^{n-1}}{(1-r)^2}\right)-\frac{[a+(n-1) d] r^n}{1-r}\) for r ≠ 1
(d) sum of infinite A.G.P.: If |r| < 1 then we have \(S_{\infty}=\frac{a}{1-r}+\frac{d r}{(1-r)^2}\)

Sum of special sequences.:

Binomial Series:
(a) Binomial theorem for any real index:

• (1 + x) ⁿ = 1 + nx + \(\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{3 !} x^3+\ldots\) for |x| < 1
• (1 – x)^-1 = 1 + x + x² …..
• (1 + x)^-1 = 1 – x + x² – x³ …..
• (1 + x)^-2 = 1 – 2x + 3x² – 4x³ …..
• (1 – x)^-2 = 1 + 2x + 3x² + …..

Exponential series:

Logarithmic Series:

Odisha State Board CHSE Odisha Class 11 Invitation to English 1 Solutions Chapter 1 Standing Up for Yourself Textbook Exercise Questions and Answers.

CHSE Odisha 11th Class English Solutions Chapter 1 Standing Up for Yourself

CHSE Odisha Class 11 English Standing Up for Yourself Text Book Questions and Answers

UNIT – I
Gist with Glossary:

Gist:
The author was alone and living in Moscow. His parents were separated. His father seldom sent letters to him. His mother was a geologist, but later she gave it up to become a singer. She was entertaining the troops. These circumstances drove the author to live in the street. The street was his world which taught him how to overcome his fear of the stronger.

Glossary:
divorced : (husband and wife) separated by a court of law
front: war front, a place where two armies are engaged in a battle (ଯୁଦ୍ଧକ୍ଷେତ୍ର)
geologist: a person who studies rocks and crust of the earth to know its history (ଭୂତତ୍ତ୍ବବିତ୍)
concert: musical entertainment
elegantly: neatly, showing a good sense of style (ସୁନ୍ଦର ଭାବରେ)
overcome : conquer (ଜୟ କରିବା)

Think it out:
Question 1.
Is the narrator a child or an adult narrating his childhood experiences?
Answer:
The narrator is an adult narrating his childhood experiences.

Question 2.
Does the narrator have happy experiences in his childhood? Why/Why not?
Answer:
The narrator has both happy and unhappy experiences in his childhood. His parents were divorced. His mother left him. These circumstances made him lonely. The street became his teacher. He cultivated all bad habits. Another good habit he developed was the spirit of fearlessness which he has kept intact till today.

Question 3.
What was his relationship with his father?
Answer:
The narrator’s father lived somewhere in Kazakhstan with his new wife and hardly he got letters from his father. In short, there was no genuine relationship between father and son.

Question 4.
How did his mother spend his time?
Answer:
His mother spent her time in singing and giving entertainment to the troops.

Question 5.
What does ‘my education was left to the street’ mean here?
Answer:
After the divorce, his father lived with his new wife and his mother spent time in singing and giving entertainment to the troops. As a result, he became lonely and the street became his master. It taught him both good and bad habits.

Question 6.
What were two habits that remained with him all his life?
Answer:
The two habits that remained with him all his life were his preparedness to face the battle of life at any moment and his spirit of fearlessness.

Question 7.
What in your opinion was the best lesson that the street taught to the writer?
Answer:
In my opinion, the best lesson that the street taught to the writer was the habit of fearlessness. It taught him not to fear anyone.

UNIT – II
Gist with Glossary:

Gist:
A boy named Red was the monarch of the street. He possessed a peculiar appearance. He with two or three of his lieutenants was a terror in the street. The whole street was in the throes of fear at the sight of Red.

Glossary:
masterfully : carelessly (ବେଖାତର ଭାବରେ )
gait : manner of walking (ଚାଲିବା ଢଙ୍ଗ)
peak: a lock of hair growing just above the forehead (ମୁଣ୍ଡ ଉପରେ ଥ‌ିବା ଜଟ)
tumble : fall (ପଡ଼ିଯିବା)
cascade : waterfall (ଜଳପ୍ରପାତ)
pock-marked face: face with marks left after the smallpox
lieutenants : (here) supporters (ସମର୍ଥକ)
impressively : (here) without the slightest hesitation (ଦ୍ବିଧାହୀନଭାବେ )
tripped : followed by walking or running (ଜୋର୍ ରେ ପାଦ ପକାଇ ଚାଲିବା)
knuckle-duster: metal covering for the knuckles, for attack or defense

Think it out :
Question 1.
What made Red look older than he really was?
Answer:
His big and broad shoulders made Red look older than he really was.

Question 2.
How did he roam in the street?
Answer:
He roamed carelessly in the street with his legs wide. He walked like a seaman on the floor of a ship.

Question 3.
How did he dress himself?
Answer:
He dressed himself in a peculiar manner. He put on a cap. From under his cap, the writer noticed its peak at the back of his head.

Question 4.
Did he intentionally dress and walk in the manner described?
Answer:
A villain as he was, Red intentionally dressed and walked in the manner described.

Question 5.
Why did his lieutenants also wear their caps back to front?
Answer:
His lieutenants also wore their caps back to front, because they, like Red, wanted to evoke fear in everyone’s mind.

Question 6.
What pet animal comes to your mind when you read the expression ‘tripped at his heels’?
Answer:
When we read the expression ‘tripped at his heels’, the pet animal that comes to my mind is a dog.

Question 7.
What was his way of forcing money out of other boys?
Answer:
His way of forcing money out of other boys was to stop them and say simply but firmly the one-word ‘money’.

Question 8.
How did he rule the street?
Answer:
He ruled the street by stopping any boy and saying simply but firmly nothing but the one word ‘money’. His hangers-on emptied his pockets, and they beat him ruthlessly in case he resisted.

Question 9.
Was the narrator afraid of Red? Quote the sentence from the text in support of your answer.
Answer:
The narrator was certainly afraid of Red. The line ‘So was /’ is a case in point.

UNIT – III
Gist with Glossary:

Gist:
The author wrote a poem about him which by the next day became very popular. The people got the poem by heart and were filled with joy in expressing their hatred for Red. The author became a victim of Red. The ruffian gave him a violent strike with his knuckle duster. The author suffered fierce wounds. He lay unconscious and was bedridden for several days. After some days he went out with his bandaged head. As soon as he saw Red, he sprang to his feet. The author’s reaction was one of disgrace. He made up his mind to face Red without any fear.

Glossary:
exulted : showed great joy and pride (ଗର୍ବ ଓ ଆନନ୍ଦ ପ୍ରଦର୍ଶନ କଲେ)
triumphant: showing great joy or satisfaction (ଅତ୍ୟଧ୍ଵ ଆନନ୍ଦ ଓ ସନ୍ତୋଷ ପ୍ରଦର୍ଶନ)
hatred : dislike exhibited with joy (ଘୃଣା)
bore : make a bore, (here) looked pointedly (ତୀକ୍ଷ୍ଣ ଦୃଷ୍ଟିରେ ଚାହିଁଲେ )
drawled : spoke slowly so that the sounds of the vowels are longer than as usual (ଧୀର କଣ୍ଠରେ ସ୍ଵର ଲୟେଇ କହିବା)
crookedly : dishonestly, shrewdly (ଚତୁରତାପୂର୍ଣ୍ଣ ଢଙ୍ଗରେ)
remuneration: reward; (here the word has been used satirically) (ପାଉଣା )
pounding : beating heavily and repeatedly (ନିର୍ଘାତ ମାଡ଼ଦେବା)
impotent : helpless or powerless (ଶକ୍ତିହୀନ)
vanquishe : defeat completely (ସମ୍ପୂର୍ଣ୍ଣ ପରାସ୍ତ କରିବା)

Think it out :
Question 1.
What was the first thing the narrator did to overcome his fear of Red?
Answer:
The first thing that the narrator did to overcome his fear of Red was to write a poem about him.

Question 2.
How did the people in the street respond to the poem?
Answer:
The people in the street learnt the poem by heart. They were filled with great joy and excitement. Their hatred for Red ruled the most.

Question 3.
Explain the expression ‘triumphant hatred’.
Answer:
The narrator’s poem about Red filled the people in the street with great joy and excitement. They gave up their fear of Red and expressed their hatred for him with great satisfaction.

Question 4.
How did Red sneer at the narrator?
Answer:
Red addressed the narrator as a poet slowly with a mischievous smile on his face. He commented sarcastically that at last he wrote verses and asked if they rhymed.

Question 5.
What was the result of his first encounter with Red?
Answer:
Red struck the narrator’s head with a metal covering for the knuckles. As a result, he fell down with blood gushing out of his head and lost consciousness. He was confined to bed for several days. This was the result of his first encounter with Red.

Question 6.
‘This was my first remuneration as a poet’ – was the narrator happy with his reward as a poet?
Answer:
The narrator was not happy at all with his reward as a poet.

Question 7.
What was a more difficult situation for the narrator: to be injured by Red or to overcome his fear of Red when he saw Red after his injury?
Answer:
A more difficult situation for the narrator was overcoming his fear when he saw Red after his injury.

Question 8.
What was the result of his second encounter with Red?
Answer:
The result of his second encounter with Read was his determination to defeat the fear of Red despite suffering shame and experiencing futile anger at his cowardice.

UNIT – IV
Gist and Glossary:

Gist:
The author made up his mind to face Red without any fear. He trained himself with parallel bars and weights. He bought one textbook on ju-jitsu. After practicing the Japanese form of self-defense at home, he went out again. He encountered Red and his associates when they were absorbed in playing vingt-et-un. He went to them, kicked, and scattered the cards with utter contempt. The author’s rudeness made Red violently furious. Fierce fighting broke out.

Red was bewildered at his amazing fearlessness. He was disgracefully defeated. He was sobbing and wiping out his tears with his dirty hands. Red no longer became the monarch of the street. He learnt a great lesson from his encounter with Red. There is no need to be afraid of the strong. The strong can be challenged fearlessly, but one should be trained in the Japanese art of self-defense to overcome them. Besides, to be a poet, one should write not only poetry but abide by its essence.

Glossary:
parallel bars : pair of parallel bars on posts for physical exercise (ଶାରୀରିକ ବ୍ୟାୟମ ନିମନ୍ତେ ବ୍ୟବହୃତ ଏକଯୋଡ଼ ସମାନ୍ତର ଦଣ୍ଡ )
session : single meeting
miraculous : most remarkable ( ଉଲ୍ଲେଖନୀୟ )
ju-jitsu : Japanese art of self- defence (ଆତ୍ମରକ୍ଷା ନିମନ୍ତେ ଏକ ଜାପାନୀ କୌଶଳ)
vingt-et-un : a kind of card game (ଏକପ୍ରକାର ତାଳ)
impudence : utter disrespect
menacingly : in a threatening manner (ଧମକ ଦେବା ଭଙ୍ଗୀରେ)
divied into pocket : rushing quickly into pocket (ହଠାତ୍ ପକେଟ୍‌ରେ ପୂରାଇଲେ)
jabbing : aiming a sudden blow (ଲକ୍ଷ୍ୟକରି ଦୃତ ଆଘାତ ଦେବା)
bewildered : puzzled, confused
grubby : dirty (ମଇଳା)
stand up for: back up (ସହାୟତା ଦେବା)

Think it out :
Question 1.
How did the narrator train himself to grow stronger?
Answer:
To grow stronger, the narrator trained himself with a pair of parallel bars meant for gymnastic exercises. Besides, he resorted to weights.

Question 2.
How did he get a textbook on ju-jitsu?
Answer:
He got a textbook on ju-jitsu in exchange of a week’s ration card.

Question 3.
How long did he train himself before the final encounter with Red?
Answer:
He trained himself for three weeks before the final encounter with Red.

Question 4.
Where did the final encounter take place? What was Red doing then?
Answer:
The final encounter took place on the lawn in their yard, when Red was lost in playing a card game called vingt-et-un with his hangers-on.

Question 5.
How did the narrator attack Red?
Answer:
The narrator kicked and scattered cards played by Red and his lieutenants.

Question 6.
How did Red react to the narrator’s attack?
Answer:
Red reacted to the narrator’s attack in a state of surprise and asked him mockingly if he was looking for more.

Question 7.
How did the narrator tackle Red?
Answer:
The narrator tackled Red by making a fast sudden blow to him. Confused, he came towards the former furiously. The narrator cut him to size by catching his wrist and squeezing slowly.

Question 8.
How did Red suffer at the hands of the narrator?
Answer:
Crying loudly in pain, Red rolled on the ground. His fingers suffered injuries. The narrator made him sob and rub the tears over his small-pox-marked face with his dirty fist.

Question 9.
What lesson did the narrator learn during his encounter with a bully like Red?
Answer:
During his encounter with a bully like Red, the narrator learns that one needn’t be afraid of the strong. Besides, it is imperative for all to know the technique of vanquishing them.

Question 10.
What career did the narrator prepare himself for?
Answer:
The narrator prepared himself for becoming a poet. Besides writing poems, he should defend their themes at any cost.

Question 11.
Which of these do you think is true: courage means not having fear at all or courage means conquering fear? Justify your choice.
Answer:
I think courage means conquering fear. The narrator’s strong determination to defeat Red, the ruler of the street, bore a fruitful result and Red was defeated by him. This is a glittering example on point.

Post-Reading Activities:

Doing the Words
(A) In order to understand what you are reading from an English text, you need to guess the meaning of unfamiliar words/expressions from the context. Guess at the meaning of words from the way it is used in the sentences/text below. This will help you read faster and easier.

Question (i).
Alexander was so good with horses that he could ride any horse masterfully.
(clue: Did Alexander behave as a ‘master’ ?)
Answer:
skilfully

Question (ii).
His courage used to bewilder many including his father. One example was when he tried to master Bucephalus, an unruly horse, everyone was bewildered at his behavior.
(clue: Bigger words come from smaller ones whose meaning you would know: be + wild + er. ‘wild’ is something that we don’t know.)
Answer:
confuse

Question (iii).
He never worked as a lieutenant in the Greek army but was always its leader.
(clue: Is a lieutenant senior or junior to the leader of an army ?)
Answer:
sub-ordinate

Question (iv).
Alexander wanted to conquer the whole world. He actually conquered most of it.
(clue: Is it to rule or defeat ?)
Answer:
rule

Question (v).
Although he died very young, his military achievement was impressive.
(clue: Bigger words come from smaller ones that you may know, impress + ive. Was Alexander’s achievement remarkable or ordinary ?)
Answer:
remarkable

(B) Some words are in italics in each of the following sentences, and three different meanings are given below the sentence. Identify the meaning that best fits the words in italics.

Question 1.
What matters in the struggle for life is overcoming fear?
(i) present situation
(ii) have an important effect
(iii) problems
Answer:
(ii) have an important effect

Question 2.
Most of the play is written in verse.
(i) prose
(ii) dialogues
(iii) poetry
Answer:
(iii) poetry

Question 3.
We have decided to complete the project at whatever cost.
(i) no matter what the risk or loss may be
(ii) without considering how much money is needed
(iii) in the least expensive way
Answer:
(i) no matter what the risk or loss may be

Question 4.
I know for certain that daffodils bloom in Spring.
(i) want to make sure
(ii) know without a doubt
(iii) declare with confidence
Answer:
(ii) know without a doubt

Question 5.
She has made up her mind to buy a car this month.
(i) hoped
(ii) decided
(iii) thought
Answer:
(ii) decided

Question6.
We must stand up for what is right even if we are standing alone.
(i) defend
(ii) represent
(iii) face boldly
Answer:
(i) defend

(C) Mark the use of ‘heel’ in “Two or three lieutenants, in peaked caps back to front like Red’s, tripped at his heels.” Look up the word ‘heel’ in the dictionary. Study the meanings of ‘heel’ in the following idiomatic expressions. Use each of them in a sentence of your own.
(a) take to one’s heels – To run away
(b) on the heels of – Following closely behind somebody
(c) cool one’s heels – Completely controlled by somebody
(d) kick up one’s heels – To be relaxed and enjoy yourself
(e) turn on one’s heels – To turn around suddenly
(f) head over heels – Loving somebody very much
(g) come to heel – To agree to obey somebody and accept his order
(h) show a clean pair of heels
Answer:
(a) take to one’s heels – The two suspicious-looking men standing outside the jeweler’s shop took to their heels when the police car drew up.
(b) on the heels of – TV camera team arrived on the heels of the police.
(c) cool one’s heels – The interviewer hadn’t arrived, so the Secretary kept the four of us cooling our heels in the corridor for hours.
(d) kick up one’s heels – They were forced to keep up their heels for nearly an hour.
(e) turn on one’s heels – She turned on her heel and went back to her room.
(f) head over heels – They met in 2000 and felt head over heels in love.
(g) come to heel – A few government rebels refused to come to heel and had to be expelled from the party.
(h) show a clean pair of heels – Butler showed them all a clean pair of heels as he raced for the finishing line.

CHSE Odisha Class 11 English Standing Up for Yourself Important Questions and Answers

I. Short Answer Type Questions with Answers

1. Read through the extracts and answer the questions that follow.
In 41, I was living alone in an empty flat on a quiet Moscow street. My parents were divorced and my father was somewhere in Kazakhstan with his new wife and their two children. I seldom received letters from him. My mother was at the front. She had given up her work as a geologist to become a singer and was giving concerts for the troops. My education was left to the street. The street taught me to swear, smoke, spit elegantly through my teeth, and to keep my fists at the ready – a habit that I have to this day. The street taught me not to be afraid of anything or anyone – this is another habit I have kept. I realized that what mattered in the struggle for life was to overcome my fear of those who were stronger.

The ruler of our street was a boy of about sixteen who was nicknamed Red. Red was big and broad-shouldered beyond his years. Red walked masterfully up and down our street, legs wide and with a slightly rolling gait, like a seaman on his deck. From under his cap, its peak always at the back of his head, his forelock tumbled down in a fiery cascade, and, out of his round pock-marked face, green eyes, like a cat’s, sparkled with scorn for everything and everyone. Two or three lieutenants in peaked caps back to front like Red’s tripped at his heels. Red could stop any boy and say impressively the one-word ‘money’. His lieutenants would turn out the boy’s pockets, and if he resisted they beat him up hard. Everyone was afraid of Red. So was I. I knew he carried a heavy metal knuckle duster in his pocket.

Questions :
(i) Throw light on the writer’s father.
(ii) What bad habits did the narrator learn from the street?
(iii) Describe the physical appearance of Red.
(iv) How did his green eyes look like? What did they convey?
(v) What was the narrator aware of?

Answers :
(i) The writer’s parents were divorced. Then his father married again. He was living somewhere in Kazakhstan with his new wife and their two children.
(ii) The bad habits the narrator learnt from the street were smoking and spitting through his teeth in a nice manner.
(iii) Red had a round pock-marked face with green eyes.
(iv) Red’s green eyes looked like those of a cat. They conveyed his contempt for everything and everyone.
(v) The narrator was aware of the fact that Red carried a heavy metal covering for the knuckles in his pocket.

2. Read through the extract and answer the questions that follow.
I wanted to conquer my fear of Red. So I wrote a poem about him. This was my first piece of journalism in verse. By the next day, the whole street knew it by heart and exulted with triumphant hatred. One morning on my way to school, I suddenly came upon Red and his lieutenants. His eyes seemed to bore through me. “Ah, the poet,” he drawled, smiling crookedly. “So you write verses. Do they rhyme ?” Red’s hand darted into his pocket and came out armed with its knuckle duster; it flashed like lightning and struck my head. I fell down streaming with blood and lost consciousness. This was my first remuneration as a poet. I spent several days in bed. When I went out, with my head still bandaged, I again saw Red. I struggled with myself but lost and took to my heels. At home, I rolled on my bed, biting my pillow and pounding it in shame and impotent fury at my cowardice. I made up my mind to vanquish it at whatever cost.

Questions :
(i) When did the narrator suddenly meet Red?
(ii) Why did he faint?
(iii) ‘It flashed like lightning.’ What does ‘It’ refer to?
(iv) What happened to the narrator after Red had attacked him ruthlessly?
(v) Explain the expression ‘impotent fury’.

Answers :
(i) One morning when the narrator was going to school, he suddenly met Red.
(ii) He fainted because Red struck him on the head with his knuckle duster.
(iii) ‘It’ refers to Red’s knuckle-duster.
(iv) The narrator lay unconscious and was confined to bed after Red had attacked him ruthlessly.
(v) The expression ‘impotent fury’ signifies the narrator’s futile anger at his cowardice. He was incapable of encountering cruel and mischievous Red.

3. Read through the extract and answer the questions that follow.
I went into training with parallel bars and weights. After every session, I would feel my muscles they were getting bigger, but slowly. Then I remembered something I had read in a book about a miraculous Japanese method of wrestling which gave an advantage to the weak over the strong. I exchanged a week’s ration card for a textbook on ju-jitsu. For three weeks, I stayed at home, practicing with two other boys. Then I went out. Red was sitting on the lawn in our yard, playing vingt-et-un with his lieutenants. He was absorbed in the game.

Fear was still deep in me, urging me to go back. But I went up to the players and kicked and scattered the cards. Red looked up, surprised at my impudence after my recent flight. He got up slowly. “You looking for more ?” he asked menacingly. As before, his hand dived into his pocket for the knuckle duster. But I made a quick jabbing movement and Red, howling with pain, rolled on the ground. Bewildered, he got up and came at me swinging his head furiously from side to side like a maddened bull.

I caught his wrist and squeezed slowly, as I had read in the book until the knuckle-duster dropped from his limp fingers. Nursing his hand, Red fell down again. He was sobbing and smearing the tears over his pock-marked face with his grubby fist. That day Red ceased to be the monarch of our street. And from that day on, I knew for certain that one need not fear the strong. All one needs is to know the way to beat them. For every strong man, there is a special ju-jitsu. What I also learned on this occasion was that to be a poet, I had not only to write poems but know how to stand up for them.

Questions :
(i) What is the importance of a textbook on ju-jitsu?
(ii) In what state did the narrator attack Red?
(iii) Where did Red always keep the knuckle duster?
(iv) What quality in the narrator prevented Red from becoming the monarch of their street again?
(v) Suggest a suitable title to the extract.

Answers :
(i) The textbook on ju-jitsu deals with a miraculous Japanese technique of wrestling that gives benefits to the weak over the strong.
(ii) The narrator attacked Red in a state of great determination, though fear was still haunting him.
(iii) Red always kept the knuckle duster in his pocket.
(iv) The narrator’s firm determination to overcome the fear of the strong prevented Red from becoming the monarch of the street again.
(v) The value of fearlessness.

II. Multiple Choice Questions (MCQs) with Answers
Choose the correct option.

UNIT-I
WARM-UP
Have you ever…………..up in the streets.

Question 1.
The author belongs to which of the following country?
(a) England
(b) America
(c) Russia
(d) Germany
Answer:
(c) Russia

Question 2.
The author is:
(a) a poet
(b) a novelist
(c) a dramatist
(d) all of the above
Answer:
(d) all of the above

Question 3.
In the Warm-up section there is a reference about a child, who is :
(a) given all comfort by parents
(b) deserted by parents
(e) looked after by a destitute center
(d) none of the above
Answer:
(b) deserted by parents

Question 4.
In the Warm-up para, the name of a movie is mentioned. The movie is:
(a) The Wizard Millionaire
(b) The Millionaire of London
(c) Slumdog Millionaire
(d) The Mumbai Crorepati
Answer:
(c) Slumdog Millionaire

Question 5.
Who is Danny Boyle?
(a) film producer
(b) film director
(c) film actor
(d) none of these
Answer:
(b) film director

Question 6.
Who is Loveleen Tandan ?
(a) a co-director of the film
(b) director of the film
(c) film producer
(d) film actor
Answer:
(a) a co-director of the film

Question 7.
The film in the Warm-up para is an adaptation of a novel. What is that?
(a) The Novel A and Q
(b) The Novel of the Young Generation
(c) The Fall of Paradise
(d) The Novel Q and A
Answer:
(d) The Novel Q and A

Question 8.
The adaptation of the novel is done by an Indian author and diplomat. Who is he?
(a) Vikash Swarup
(b) Akas Swarup
(c) Nancy Swarup
(d) Loveleen Tandan
Answer:
(a) Vikash Swaru

Question 9.
The story in the text is about :
(a) a French child
(b) an Indian child
(c) a Russian child
(d) a Chinese child
Answer:
(c) a Russian child

Question 10.
The child in the story is :
(a) a young film actor
(b) the author himself
(c) an English writer
(d) a storyteller
Answer:
(b) the author himself

The Text:
In 41, I was living alone ……….. were stronger.

Question 11.
Where did the boy live?
(a) in a bungalow in Moscow
(b) in a slum in London
(c) in an empty flat in Moscow
(d) in a modest house in Paris
Answer:
(c) in an empty flat in Moscow

Question 12.
What had happened to his parents?
(a) They were divorced.
(b) They were living together.
(c) They were always quarreling.
(d) They were dead.
Answer:
(a) They were divorced.

Question 13.
Where did his father live with his new wife?
(a) in Mumbai
(b) in Baluchistan
(c) in Kazakhstan
(d) in Moscow
Answer:
(c) in Kazakhstan

Question 14.
He received letters from his father :
(a) often
(b) seldom
(c) every month
(d) none of these
Answer:
(b) seldom

Question 15.
What was his mother?
(a) an anthropologist
(b) a stenographer
(c) a historian
(d) a geologist
Answer:
(d) a geologist

Question 16.
Why did she (his mother) give up her work as a geologist?
(a) to become an actress
(b) to become a politician
(c) to become a singer
(d) to become a dancer
Answer:
(c) to become a singer

Question 17.
What was she doing after leaving her work as a geologist?
(a) giving concerts for the troops as a singer
(b) giving training to the dancers
(c) giving guidance to the youth for social work
(d) working as a film director
Answer:
(a) giving concerts for the troops as a singer

Question 18.
What do you mean by the word ‘front’ here?
(a) the border of a state
(b) in front of a place or street
(c) place where two armies are fighting in a war
(d) all of these
Answer:
(c) place where two armies are fighting in a war

Question 19.
What do you mean by the word ‘concert’?
(a) cooperation
(b) confrontation
(c) musical entertainment
(d) mutual effort
Answer:
(c) musical entertainment

Question 20.
The word ‘troop’ refers to :
(a) dance party
(b) army
(c) group
(d) herd
Answer:
(b) army

Question 21.
The author’s parents were divorced when he was :
(a) a young man
(b) a boy
(c) a middle-aged man
(d) quite grown up
Answer:
(b) a boy

Question 22.
His fate pushed him to life :
(a) in the street
(b) in Luxury
(c) in danger
(d) none of these
Answer:
(a) in the street

Question 23.
To swear, smoke and spit were his :
(a) vice
(b) virtue
(c) good habits
(d) none of these
Answer:
(a) vice

Question 24.
What do you mean by the word “elegantly”?
(a) showing a bad sense of style
(b) showing a good sense of style
(c) attack somebody with style
(d) it is a style of dance
Answer:
(b) showing a good sense of style

Question 25.
Which habit he has kept to this day?
(a) smoking
(b) spit elegantly through his teeth
(c) keeping his fists at the ready
(d) all of these
Answer:
(c) keeping his fists at the ready

Question 26.
Another habit he had kept was :
(a) fearful
(b) fearlessness
(c) cowardice
(d) none of these
Answer:
(b) fearlessness

Question 27.
What kind of fear the boy had to overcome?
(a) fear of those who were stronger
(b) fear of losing friendship
(c) fear of being robbed
(d) none of these
Answer:
(a) fear of those who were stronger

Question 28.
What was the experience of the narrator in his childhood?
(a) unhappy
(b) happy
(c) feeling of danger
(d) none of these
Answer:
(a) unhappy

Question 29.
What in your opinion was the best lesson that the street taught to the narrator?
(a) spit elegantly
(b) to keep his fists at the ready
(c) not to be afraid of anything or anyone
(d) none of these
Answer:
(c) not to be afraid of anything or anyone

Unit – II

Warm-up
The Text
Do you think a duster……….. in his pocket.

Question 30.
The narrator was scared of someone in his childhood. Who was he?
(a) a boy nicknamed Robbin
(b) a man who was a murderer
(c) a boy nicknamed Red
(d) a friend of his who was a terror
Answer:
(c) a boy nicknamed Red

Question 31.
How old was Red?
(a) twenty years
(b) sixteen years
(c) eighteen years
(d) nineteen years
Answer:
(b) sixteen years

Question 32.
Who was called the ruler of our street?
(a) the child
(b) the author
(c) the boy Red
(d) none of these
Answer:
(c) the boy Red

Question 33.
How was Red look like?
(a) big and broad-shouldered
(b) short and fatty
(c) tall and slim
(d) all of these
Answer:
(a) big and broad-shouldered

Question 34.
What do you mean by the word ‘masterfully’?
(a) to behave like a master
(b) to do the mastermind work
(c) not caring for others
(d) none of these
Answer:
(c) not caring for others

Question 35.
What do you mean by the word ‘gait’?
(a) manner of standing
(b) manner of sitting
(c) manner of talking
(d) manner of walking
Answer:
(d) manner of walking

Question 36.
‘Like a seaman on his deck’ refers to :
(a) the author
(b) a seaman of the past
(c) the boy Red
(d) none of these
Answer:
(c) the boy Red

Question 37.
What do you mean by the word ‘peak’ here?
(a) the highest point
(b) lock of hair growing just above the forehead
(c) the top of a mountain
(d) all of these
Answer:
(b) lock of hair growing just above the forehead

Question 38.
What is the meaning of the word ‘cascade’?
(a) vast grassland
(b) waterfall
(c) a green belt
(d) the surface of the body of water
Answer:
(b) waterfall

Question 39.
How was the face of Red looked like?
(a) red flat face
(b) round pock-marked face
(c) black round pock-marked face
(d) none of these
Answer:
(b) round pock-marked face

Question 40.
His green eyes were compared with like that of a :
(a) monkey
(b) tiger
(c) cat
(d) deer
Answer:
(c) cat

Question 41.
Which word from the following is the synonym of the word ‘scorn’?
(a) angry
(b) contempt
(c) remark
(d) conserve
Answer:
(b) contempt

Question 42.
Lieutenants here refers to :
(a) assistants of Red
(b) friends of Red
(c) supporters of Red
(d) none of these
Answer:
(c) supporters of Red

Question 43.
What was Red demanding from the boys stopping them on the road?
(a) to obey him
(b) money
(c) to steal for him
(d) to rob people on the way
Answer:
(b) money

Question 44.
What did his lieutenants do if somebody resisted?
(a) beat him up hard
(b) keep him confined in a place
(c) leave him to go silently
(d) none of these
Answer:
(a) beat him up hard

Question 45.
What did Red always carry in his pocket?
(a) a knife
(b) a rifle
(c) a knuckle-duster
(d) a blade
Answer:
(c) a knuckle-duster

Question 46.
Why did he carry a heavy metal duster?
(a) to attack
(b) for defense
(c) both for attack and defense
(d)none of these
Answer:
(c) both for attack and defense

Unit – III

Warm-up
The text
Does the narrator ……………….. whatever cost.

Question 47.
What did the narrator want to conquer?
(a) fear of God
(b) fear of ghost
(c) fear of himself
(d) fear of Red
Answer:
(d) fear of Red

Question 48.
What did he do to win over his fear of Red?
(a) wrote a poem about him
(b) prayed to God to give the strength
(c) decided to avoid him
(d) none of these
Answer:
(a) wrote a poem about him

Question 49.
This poem of the narrator was in the form of :
(a) poem
(b) verse
(c) dialogue
(d) dilect
Answer:
(b) verse

Question 50.
Which was the first piece of journalism of the narrator in verse?
(a) the first poem of the author
(b) the first article of the author
(c) the first short story of the author
(d) none of these
Answer:
(a) the first poem of the author

Question 51.
What do you mean by the statement “the whole street knew it by heart”?
(a) It was distributed to people of the street.
(b) The poem was popular among the people of the street.
(c) The street people sang the poem standing together.
(d) all of these
Answer:
(b) The poem was popular among the people of the street.

Question 52.
What do you mean by the word ‘exulted’?
(a) showed displeasure and unhappiness
(b) showed physical and mental strength
(c) showed great joy and excitement
(d) showed bad temperament and falsehood
Answer:
(c) showed great joy and excitement

Question 53.
What do you mean by the word “triumphant”?
(a) the feeling of victory with satisfaction
(b) the feeling of victory with anger
(c) showing bad manners by winning something
(d) all of these
Answer:
(a) the feeling of victory with satisfaction

Question 54.
What do you mean by the word phrase ‘triumphant hatred’?
(a) the feeling of victory over something or somebody you dislike
(b) feeling very bad after the victory
(c) showing bad manners after winning a match
(d) all of these
Answer:
(a) the feeling of victory over something or somebody you dislike

Question 55.
At what time the narrator one day met Red and his lieutenants?
(a) in the evening
(b) at night
(c) in the afternoon
(d) in the morning
Answer:
(d) in the morning

Question 56.
What is the meaning of the phrase ‘bore through’?
(a) to look at somebody aggressively
(b) to threaten someone to kill
(c) to stare in a way that makes somebody feel uncomfortable
(d) all of the above
Answer:
(c) to stare in a way that makes somebody feel uncomfortable

Question 57.
What do you mean by the word ‘drawl’?
(a) to say something speedily and angrily
(b) to say something slowly with longer vowel sounds
(c) to say something slowly with longer consonant sounds
(d) none of the above
Answer:
(b) to say something slowly with longer vowel sounds

Question 58.
What do you mean by the phrasal words ‘impotent fury’?
(a) fruitful anger
(b) futile anger
(c) fruitful with no anger
(d) none of these
Answer:
(b) futile anger

Question 59.
What do you mean by the word ‘vanquish’?
(a) defeat completely
(b) win completely
(c) break somebody’s power
(d) all of these
Answer:
(a) defeat completely

Question 60.
Who was smiling crookedly at the narrator?
(a) people in the street
(b) Red
(c) Red’s lieutenants
(d) none of the above
Answer:
(b) Red

Question 61.
Who told this to whom? “So you write verses. Do they rhyme ?”
(a) Red to his lieutenants
(b) Red to the people in the street
(c) Red to the narrator
(d) none of the above
Answer:
(c) Red to the narrator

Question 62.
What do you mean by the word ‘darted’?
(a) moved slowly
(b) moved suddenly
(c) moved secretly
(d) moved upward
Answer:
(b) moved suddenly

Question 63.
What is a knuckle-duster?
(a) It is an iron knife.
(b) It is an iron hammer.
(c) A metal covering for the knuckles for attack or defense.
(d) An axe-type weapon.
Answer:
(c) A metal covering for the knuckles for attack or defense.

Question 64.
What was the author’s first remuneration as a poet?
(a) a word of admiration by the people
(b) struck on his head with a knuckle-duster
(c) streaming with blood and lost consciousness
(d) none of the above
Answer:
(b) struck on his head with a knuckle-duster

Question 65.
Avoiding meeting Red in the second time, the author considers this act on his part as
(a) bravery
(b) cowardice
(c) cleverness
(d) wise
Answer:
(b) cowardice

Question 66.
What did he want to do at whatever cost?
(a) to conquer
(b) to win
(c) to vanquish
(d) to get rid of
Answer:
(c) to vanquish

Unit – IV

Warm-up
The text
Fear of Red ………….. stand up for them.

Question 67.
Which method of wrestling is an advantage to the weak over the strong?
(a) Chinese method of wrestling
(b) Japanese method of wrestling
(c) German method of wrestling
(d) French method of wrestling
Answer:
(b) Japanese method of wrestling

Question 68.
Which game Red was playing sitting in the lawn when the narrator went out to meet him?
(a) chess
(b) vingt-et-un
(c) length-ut-sn
(d) oxat-ut-bun
Answer:
(b) vingt-et-un

Question 69.
Which textbook on martial art did the narrator buy?
(a) on fu-fitsu
(b) on ju-jitsu
(c) on lu-litsu
(d) gu-bitsu
Answer:
(b) on ju-jitsu

Introducing the Author:
Yevgeny Yevtushenko is a promising Russian poet. His poetic career started with the publication of his first volume of poems in 1952. He represented the young generation of poets. He became the voice of the young poets who were bent upon seeing the old truths about socialist realism in a newer light. Flowers and Bullets, Freedom to Kill, and Stolen Apples testify to his poetic genius. Besides, he is a novelist, essayist, dramatist, screenwriter, actor, editor and director of films.

About the Topic:
The author’s parents were divorced when he was still a boy. The divorce plunged him into a state of loneliness. He turned this loneliness into a challenge. Thus his life began on a note of challenge. This topic is built around an incident in his boyhood while he was living in Moscow. It demonstrates that fear can be met effectively and successfully.

Summary:
The present essay gives a poetic description of the author’s bold encounter with a hoodlum while as a boy he was living in Moscow. The author injects realism in his narration of the incident. The essay begins with the author’s description of his divorced parents. Their divorce drove him into the street. His life started in the street. It became his home. He found his world in it. He describes his life in the street in a lighter vein. It was in the street that he swore and smoked. Here he learned the spirit of fearlessness.

The author gives a vivid account of ‘the ruler’ of the street. A boy about sixteen nicknamed Red was the hero of the street. He was a picture of panic to all. He was walking up and down the street careless of the people around him. His wide legs, fiery forelock, and green eyes dominated the street. He was always moving with two or three lieutenants. Red was a The author gives a vivid account of ‘the ruler’ of the street. A boy about sixteen nicknamed Red was the hero of the street. He was a picture of panic to all.

He was walking up and down the street careless of the people around him. His wide legs, fiery forelock, and green eyes dominated the street. He was always moving with two or three lieutenants. Red was a ruffian. His lieutenants could stop any boy unhesitatingly and empty out his pocket. The whole ‘street reacted in blatant fear. The author was also not free. But still, he was seized with a desire to conquer his fear of Red. He wrote a poem about the ruffian who always carried a heavy metal knuckle duster in his pocket.

The poem was on everybody’s lips. The whole street exhibited their dislike for Red with great joy. One morning on the way to school, the author accidentally met Red and his lieutenants. His eyes were fixed on the author with hatred and vengeance. He struck his head violently with his knuckle duster. He was unconscious and was confined to bed for several days. He again saw Red and tom in panic, he quickly escaped his notice. A sense of shame and cowardice overwhelmed him. He mustered the courage.

Fierce determination stared him in the face. He would defeat Red come what may. It was a challenge to him. He took to training with parallel bars and weights. He also remembered the Japanese method of wrestling which he had read in a book. He had to barter his ration card for a textbook on ju-jitsu which deals with the Japanese art of self-defense. He was engrossed in practicing this art with his friends for three weeks. He was filled with renewed confidence. He went out to meet the challenge of his life.

Sitting on the lawn, Red was lost in playing vingt-et-un with his lieutenants. In spite of lurking fear, he faced them with utter rudeness scattering their cards. Red flew into anger and before he used the knuckle-duster, the author crushed him. Writhing with pain, Red lay on the ground. He came to him like a maddened bull. What that writer had read in the book prompted him to deal with him without a shred of fear. He caught his wrist and squeezed him.

His knuckle duster could not help him. He fell down and was left to sob and wipe out his tears of defeat. The strong ruffian’s vanity and rule of the street were toms to pieces. The incident still clings to the author’s memory. His encounter with the monarch of the street taught him a lesson. One can easily overcome the fear of the strong, and the secret to beating them is to master the Japanese art of self-defense. Another lesson that he learned was that to be a poet one need not only write poems but to support their essence.

ସାରାଂଶ:
ଲେଖକ ଯେତେବେଳେ ବାଳକ ଥିଲେ ସେତେବେଳେ ସେ ପିତାମାତାଙ୍କଦ୍ୱାରା ଉପେକ୍ଷିତ ହୋଇଥିଲେ । ଫଳରେ ସେ ଏକାନ୍ତ ଭାବରେ ନିଃସଙ୍ଗ ଅବସ୍ଥାରେ ଜୀବନ କଟାଇଥିଲେ । ମସ୍କୋର ରାସ୍ତା ହିଁ ଥିଲା ତାଙ୍କର ସଂସାର । ସେ କ୍ଵଚିତ୍ ବାପାଙ୍କଠାରୁ ପତ୍ର ପାଉଥିଲେ । ରାସ୍ତାର ଶିକ୍ଷା ତାଙ୍କର ଜୀବନକୁ ପ୍ରଭାବିତ କରିଥିଲା । ରାସ୍ତାରେ ଯେଉଁମାନେ ଜୀବନ କଟାନ୍ତି ସେମାନଙ୍କ ପରି ଶପଥ କରିବା, ଧୂମପାନ କରିବା ଏବଂ ହାତମୁଠା ଟାଣ କରିବା ସେ ଶିଖୁଥିଲେ । କୌଣସି କଥାକୁ ଭୟ ନ କରିବା ଏବଂ କାହାକୁ ଭୟ ନ କରିବା ମନୋଭାବ, ରାସ୍ତା ହିଁ ତାଙ୍କୁ ଶିକ୍ଷା ଦେଇଥିଲା । ସେ ଅନୁଭବ କରିଥିଲେ ଯେ ଯେଉଁମାନେ ତାଙ୍କଠାରୁ ବଳବାନ୍ ତାଙ୍କୁ ବଳରେ ଟପିଯିବା ହିଁ ଜୀବନର ଆବଶ୍ୟକତା । ସେ ସମୟରେ ରାସ୍ତାର ଦାଦା ଥିଲା ଜଣେ ଷୋହଳ ବର୍ଷ ବୟସର ତୁରଣ, ଯା’ର ଡାକ ନାମ ଥିଲା ‘ରେଡ୍’ ।

ବିଲେଇର ଆଖ୍ ପରି ତା’ର ଆଖ୍ ଦୁଇଟି ସବୁବେଳେ ଜଳୁଥିଲା । ତା’ ପାଖରେ ଦୁଇ କିମ୍ବା ତିନିଜଣ ସମପୋଷାକ ପରିଧାନ କରିଥିବା ପାଖଲୋକ ଥାଆନ୍ତି । ରାସ୍ତାରେ ଯାଉଥ‌ିବା ଯେକୌଣସି ପିଲାକୁ ସେ ଅଟକାଇ ପାରୁଥିଲା ଏବଂ ସେତେବେଳେ ତା’ର ଦୁଷ୍କର୍ମରେ ସହାୟକ ପାଖଲୋକ ଦୁଇଟି ତା’ର ପକେଟ୍‌ରେ ଥ‌ିବା ପଇସା କାଢ଼ି ନେଉଥିଲେ । ଯଦି ପିଲାଟି ବାଧା ଦେଉଥୁଲା, ତେବେ ସେମାନେ ତାକୁ ନିସ୍ତୁକ ମାଡ଼ ଦେଉଥିଲେ । ଅନ୍ୟମାନଙ୍କ ପରି ଲେଖକ ବି ତାକୁ ଭୟ କରୁଥିଲେ । ସେ ଜାଣିଥିଲେ ରେଡ୍‌ର ପକେଟ୍‌ରେ ଗୋଟିଏ ଓଜନଦାର ଧାତୁନିର୍ମିତ ଅସ୍ତ୍ର ଅଛି । ରେଡ୍ ପ୍ରତି ଭୟ କିପରି ଦୂର ହୋଇପାରିବ ଲେଖକ ସେହି କଥା ଚିନ୍ତା କରୁଥିଲେ । ସେ ତେଣୁ ତା’ ବିଷୟରେ ଗୋଟିଏ କବିତା ଲେଖୁଲେ ।

ତା’ପରଦିନ ରାସ୍ତାର ସମସ୍ତେ ତାହା ମନେରଖ‌ିଲେ । ଅନ୍ୟମାନଙ୍କ ପରି ଲେଖକ ବି ତାକୁ ଭୟ କରୁଥିଲେ । ସେ ଜାଣିଥିଲେ ରେଡ୍‌ର ପକେଟ୍‌ରେ ଗୋଟିଏ ଓଜନଦାର ଧାତୁନିର୍ମିତ ଅସ୍ତ୍ର ଅଛି । ରେଡ୍ ପ୍ରତି ଭୟ କିପରି ଦୂର ହୋଇପାରିବ ଲେଖକ ସେହି କଥା ଚିନ୍ତା କରୁଥିଲେ । ସେ ତେଣୁ ତା’ ବିଷୟରେ ଗୋଟିଏ କବିତା ଲେଖୁଲେ । ତା’ପରଦିନ ରାସ୍ତାର ସମସ୍ତେ ତାହା ମନେରଖ‌ିଲେ । ଦିନେ ସକାଳେ ସ୍କୁଲ ଯିବା ରାସ୍ତାରେ ସେ ରେଡ୍ ଓ ତା’ର ସହକର୍ମୀମାନଙ୍କର ସମ୍ମୁଖୀନ ହେଲେ । ରେଡ୍ ତା’ ପକେଟ୍‌ରୁ ସେହି ଓଜନଦାର ଜିନିଷଟି କାଢ଼ିଲା । ତାହା ଚିକ୍ ଚିକ୍ କରିଉଠିଲା । ରେଡ୍‌ର ଆଘାତରେ ଲେଖକ ଆହତ ହୋଇ ଶଯ୍ୟାଶାୟୀ ହେଲେ । ତାଙ୍କ ଭାଷାରେ କବିରୂପେ ଏହା ତାଙ୍କର ପ୍ରଥମ ପାଉଣା ।

ମୁଣ୍ଡରେ ବେଣ୍ଡେଜ୍ ଥ‌ିବା ଅବସ୍ଥାରେ ରେଡ୍ ସହିତ ପୁଣି ତାଙ୍କର ସାକ୍ଷାତ ହୋଇଥିଲା, କିନ୍ତୁ ସେ ଦୂରେଇ ଯାଇଥିଲେ । ଭୀରୁତାଜନିତ ବ୍ୟଥୀରେ ସେ ଅଧୀର ହୋଇ ପଡ଼ିଥିଲେ । ରେଡ୍‌ର ସମ୍ମୁଖୀନ ହେବାପାଇଁ ଶକ୍ତି ସଞ୍ଚୟ କରିବାକୁ ସେ ମନେ ମନେ ସ୍ଥିର କଲେ । ତାଙ୍କର ମନେପଡ଼ିଲା ଗୋଟିଏ ବହିରେ ପଢ଼ିଥିବା କଥା । ଜାପାନୀ ମୁଷ୍ଟିଯୁଦ୍ଧର କୌଶଳଦ୍ଵାରା କିପରି ବଳବାନ୍‌ଠାରୁ ଆତ୍ମରକ୍ଷା କରିହୁଏ ତା’ ଜାଣିବାପାଇଁ ନିଜୟ ପଡ଼ିକାର୍ଡ ବଦଳରେ ସେ ବହିଟି କିଣିଲେ । ତିନି ସପ୍ତାହ ଦୁଇଟି ପିଲାଙ୍କ ସହ କୌଶଳ ଅଭ୍ୟାସ କରିବା ପରେ ସେ ବାହାରକୁ ବାହାରିଲେ । ତାଙ୍କ ଭିତରେ ତଥାପି ଭୟ ଥିଲା । ସେ କିନ୍ତୁ ମନରେ ଦୃଢ଼ଭାବ ପୋଷଣ କରି ରେଡ୍‌ର ସମ୍ମୁଖୀନ ହେଲେ ।

ରେଡ୍ ପକେଟ୍‌ରେ ଥ‌ିବା ଓଜନଦାର ଅସ୍ତ୍ରଟି ତାକୁ ସାହାଯ୍ୟ କରିପାରି ନଥିଲା । ଜାପାନୀ କୌଶଳରେ ସେ ତାକୁ ଆଘାତ କଲେ । ଯନ୍ତ୍ରଣାରେ ରେଡ୍ ଭୂଇଁରେ ଗଡ଼ିଗଲା । ଗୋଟିଏ ପାଗଳ ଷଣ୍ଢ ପରି ରେଡ୍ ତା’ପରେ ତାଙ୍କୁ ଆଘାତ କରିବାକୁ ଆସିଲା; କିନ୍ତୁ ବହିରେ ପଢ଼ିଥିବା କୌଶଳ ଅନୁଯାୟୀ ସେ ତା’ର ମଣିବନ୍ଧକୁ ଏପରିଭାବେ ଚାପିଦେଲେ ଯେ, ରେଡ୍‌ର ହାତର ଅସ୍ତ୍ର ତଳେ ପଡ଼ିଗଲା । ସେଇଦିନଠାରୁ ରାସ୍ତାର ଦାଦାର ଗରିମା ରେଡ୍‌ର ରହିଲା ନାହିଁ ଏବଂ ସେହିଦିନଠାରୁ ଲେଖକ ଜାଣିଲେ ଯେ, ଜଣେ ବଳବାନ୍‌କୁ ଭୟ କରିବା ଉଚିତ ନୁହେଁ । କେବଳ ବଳବାନ୍‌କୁ କିପରି ପରାସ୍ତ କରିବାକୁ ପଡ଼ିବ ତା’ର ଉପାୟ ଜାଣିବା ଉଚିତ । ଲେଖକ ପୁଣି ଶିକ୍ଷା କରିଥିଲେ ଯେ, କେବଳ କବିତା ଲେଖୁବା ସବୁକିଛି ନୁହେଁ, କବିତାର ମୂଲ୍ୟବୋଧକୁ ବଞ୍ଚାଇ ରଖୁ ବଡ଼ କଥା ।

Odisha State Board Elements of Mathematics Class 11 Solutions CHSE Odisha Chapter 14 Limit and Differentiation Ex 14(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 11 Math Solutions Chapter 14 Limit and Differentiation Exercise 14(b)

Question 1.

Using the ε – δ definition prove that
(i) \(\lim _{x \rightarrow 0}\) (2x + 3) = 3
Solution:
Let f(x) = 2x + 3
Here a = 0 and ℓ = 3
Let ε be any positive real number however small it may be.
Now |f(x) – ℓ| =| 2x + 3 – 3| =|2x|
Thus |f(x) – ℓ| < ε whenever
|2x| < ε i.e |x| < \(\frac{\varepsilon}{2}\)
Then |f(x) – ℓ| < ε
whenever |x – 0| < δ
under the condition δ = \(\frac{\varepsilon}{2}\)
∴ \(\lim _{x \rightarrow 0}\) (2x + 3) = 3

(ii) \(\lim _{x \rightarrow 1}\) (2x – 1) = 1
Solution:
Here f(x) = 2x – 1, ℓ = l and a = 1
Now |f(x)| = | 2x – 1 – 1|
= |2x – 2| = 2|x – 1|
Thus |f(x) –  ℓ| < ε
whenever 2|x – 1| < ε
i,e. |x – 1| < \(\frac{\varepsilon}{2}\) put δ = \(\frac{\varepsilon}{2}\)
Then |f(x) – ℓ| < ε
whenever|x – 1| < δ
Hence \(\lim _{x \rightarrow 1}\) (2x – 1) = 1

(iii) \(\lim _{x \rightarrow -2}\) (3x + 8) = 2
Solution:
|(3x + 8) – 2|
= |3x + 6| = 3|x + 2|
So |3x + 8 – 2| < ε
whenever 3|x + 2| < ε
i.e. |x + 2| < \(\frac{\varepsilon}{3}\)
Hence |(3x + 8) – 2| < ε
whenever | x + 2 | < δ
∴ \(\lim _{x \rightarrow -2}\) (3x + 8) = 2

(iv) \(\lim _{x \rightarrow 3}\) (x² + 2x – 8) = 7
Solution:
|(x² + 2x – 8) – 7|
= |(x² + 2x – 15|
= |(x + 5) (x – 3)|
=| x + 5| | x – 3|
If |x – 3| < 1 then| x + 5| =| x – 3 + 8| < |x – 3| + 8 < 9
Thus |(x² + 2x – 8) – 7| < 9 |x – 3|
So |(x² + 2x – 8) – 7| < ε
whenever 9|x – 3| < ε
i.e.| x – 3| < \(\frac{\varepsilon}{9}\)
Choose δ = minimum of 1 and \(\frac{\varepsilon}{9}\)
Then |(x² + 2x – 8) – 7| < ε
whenever |x – 3| < δ
∴ \(\lim _{x \rightarrow 3}\) (x² + 2x – 8) = 7   (proved)

(v) \(\lim _{x \rightarrow 9}\) √x = 3
Solution:
|√x – 3| = |\(\frac{(\sqrt{x}-3)(\sqrt{x}+3)}{\sqrt{x}+3}\)|
= \(\frac{|x-9|}{|\sqrt{x}+3|}\)

(v) \(\lim _{x \rightarrow a}\) √x = √a, a > 0
Solution:

(vii) \(\lim _{x \rightarrow 1}\) |3x + 2| = 5
Solution:
When x → 1, 3x + 2 is always positive.
So |3x + 2| = 3x + 2
Thus ||3x + 2| -5| = |3x + 2 – 5|
= 3|x – 1|
∴ ||3x + 2| – 5 | < ε
whenever 3|x – 1| < ε
i.e. |x – 1| < \(\frac{\varepsilon}{3}\)
put δ = \(\frac{\varepsilon}{3}\)
Hence ||3x + 2| – 5| < ε
whenever |x – 1| < δ
∴ \(\lim _{x \rightarrow 1}\) |3x + 2| = 5

(viii) \(\lim _{x \rightarrow 2}\) |5x – 7| = 3
Solution:
Let any arbitrary ε > 0
then |5x – 7 – 3| < ε
If |5(x – 2)| < ε
i.e. if lx – 2| < \(\frac{\varepsilon}{5}\)
Choosing δ = \(\frac{\varepsilon}{5}\) we have
for any arbitrary ε > 0 there exists a δ > 0 depending on ε
Such that
|x – 2| < δ ⇒ |(5x – 7) – 3| < ε
∴ \(\lim _{x \rightarrow 2}\) |5x – 7| = 3

Question 2.
If \(\lim _{x \rightarrow a}\) f(x) = ℓ then prove that \(\lim _{x \rightarrow a}\) |f(x)| = | ℓ | Is the converse true ? Justify your answer with reasons.
Solution:
Let \(\lim _{x \rightarrow a}\) f(x) = ℓ
Then |f(x) – ℓ| < ε whenever |x – a| < δ
Now |f(x)| – ℓ| < |f(x) – ℓ| < ε
whenever |x – a| < δ
So \(\lim _{x \rightarrow a}\) |f(x)| = | ℓ |
The converse is not always true because | ℓ | = | -ℓ |
So \(\lim _{x \rightarrow a}\) f(x) = ℓ or -ℓ

Question 3.
(i) Prove that \(\lim _{x \rightarrow a}\) x = a
Solution:
Let ε is any positive number
Let f(x) = x
Now |f(x) – a| < ε
if |x – a| < ε
Choosing δ = ε we see that for each ε > 0 we find a δ > 0 depending on ε such that
|x – al < d ⇒ |f(x) – a| < ε
⇒ \(\lim _{x \rightarrow a}\) f(x) = a i,e. \(\lim _{x \rightarrow a}\) x = a

(ii) Using (i) and the laws of limits prove that \(\lim _{x \rightarrow a} x^n=a^n\), when n is an integer.
Solution:
Case-1: Let n > 0 and n ε z
Now \(\lim _{x \rightarrow a} x^n=\lim _{x \rightarrow a}\) (x. x. x…….. n factors)
= a. a …… n factors = aⁿ
Case-2: Let n = 0

(iii) Using (ii) and the laws of limits prove that \(\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}\) where n is an integer.
Solution:

Case-3: n = 0  Hence the case is obvious

(iv) Using (iii), the laws of limits and assuming that \(\lim _{x \rightarrow a} \frac{1}{x^m}=a^{\frac{1}{m}}\) where m is a non-zero integer prove that for any rational number n, \(\lim _{x \rightarrow a} \frac{x^n-a^n}{x-a}=n a^{n-1}\).
Solution:

Question 4.
Evaluate the following :
(i) \(\lim _{x \rightarrow 1}\) (1 + 2x – 3x² + 4x³ – 5x⁴)
Solution:
\(\lim _{x \rightarrow 1}\) (1 + 2x – 3x² + 4x³ – 5x⁴)
= 1 + 2 – 3 + 4 – 5 = 7 – 8 = -1

(ii) \(\lim _{x \rightarrow 0}\) (3x² + 4x – 1)(x⁴ + 2x³ – 3x² + 5x + 2)
Solution:
\(\lim _{x \rightarrow 0}\) (3x² + 4x – 1)(x⁴ + 2x³ – 3x² + 5x + 2)
=(-1). 2 = -2

(iii) \(\lim _{x \rightarrow 2}\) \(\frac{x^2+3 x-9}{x+1}\)
Solution:
\(\lim _{x \rightarrow 2}\) \(\frac{x^2+3 x-9}{x+1}\)
\(\frac{2^2+3 \cdot 2-9}{2+1}=\frac{1}{3}\)

(iv) \(\lim _{x \rightarrow 3}\) \(\frac{x^2-9}{x-3}\)
Solution:
\(\lim _{x \rightarrow 3}\) \(\frac{x^2-9}{x-3}\)
= \(\lim _{x \rightarrow 3} \frac{(x-3)(x+3)}{x-3}\)
= \(\lim _{x \rightarrow 3}\) (x + 3) = 3 + 3 = 6

(v) \(\lim _{x \rightarrow 1} \frac{x^3-1}{x-1}\)
Solution:
\(\lim _{x \rightarrow 1} \frac{x^3-1}{x-1}\)
= \(\lim _{x \rightarrow 1} \frac{(x-1)\left(x^2+x+1\right)}{x-1}\)
= \(\lim _{x \rightarrow 3}\) (x² + x + 1)
= 1 + 1 +1 = 3

(vi) \(\lim _{x \rightarrow 2} \frac{x-2}{x^4-16}\)
Solution:

(vii) \(\lim _{x \rightarrow 2} \frac{x^3-8}{x^5-32}\)
Solution:

(viii) \(\lim _{x \rightarrow 3} \frac{x^2+2 x-15}{x^2-x-6}\)
Solution:

(ix) \(\lim _{x \rightarrow 0} \frac{(3+x)^3-27}{x}\)
Solution:

(x) \(\lim _{x \rightarrow 2} \frac{\frac{1}{x^2}-\frac{1}{4}}{x-2}\)
Solution:

(xi) \(\lim _{x \rightarrow 1} \frac{1}{(x-1)}\left\{\frac{1}{x+3}-\frac{2}{3 x+5}\right\}\)
Solution:

(xii) \(\lim _{h \rightarrow 0} \frac{(x+h)^3-x^3}{h}\)
Solution:

(xiii) \(\lim _{h \rightarrow 0} \frac{(x+h)^4-x^4}{h}\)
Solution:

(xiv) \(\lim _{x \rightarrow 1} \frac{x^m-1}{x^n-1}\), where m, n are integers.
Solution:

(xv) \(\lim _{x \rightarrow 1} \frac{x^2-2 x+1}{x^2-x}\)
Solution:

(xvi) \(\lim _{x \rightarrow 1} \frac{x^2+x-2}{x^3-x^2-x+1}\)
Solution:

Question 5.
Evaluate the following :
(i) \(\lim _{x \rightarrow \infty} \frac{2 x+1}{3 x-2}\)
Solution:
\(\lim _{x \rightarrow \infty} \frac{2 x+1}{3 x-2}\)
= \(\lim _{x \rightarrow \infty} \frac{2+\frac{1}{x}}{3-\frac{2}{x}}=\frac{2}{3}\)
[ ∵ As x → ∞, \(\frac{1}{x}\) → 0]

(ii) \(\lim _{x \rightarrow \infty} \frac{3 x^2+x-1}{2 x^2-7 x+5}\)
Solution:
\(\lim _{x \rightarrow \infty} \frac{3 x^2+x-1}{2 x^2-7 x+5}\)
\(=\lim _{x \rightarrow \infty} \frac{3+\frac{1}{x}-\frac{1}{x^2}}{2-\frac{7}{x}+\frac{5}{x^2}}=\frac{3}{2}\)

(iii) \(\lim _{x \rightarrow \infty} \frac{x^3+2 x^2+3}{x^4-3 x^2+1}\)
Solution:
\(\lim _{x \rightarrow \infty} \frac{x^3+2 x^2+3}{x^4-3 x^2+1}\)
\(\lim _{x \rightarrow\infty}\frac{\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^4}}{1-\frac{3}{x^2}+\frac{1}{x^4}}=\frac{0}{1}\) =0
[ ∵ As x → ∞, \(\frac{1}{x}\) → 0]

(iv) \(\lim _{x \rightarrow \infty} \frac{x^4-5 x+2}{x^3-3 x+1}\)
Solution:
\(\lim _{x \rightarrow \infty} \frac{x^4-5 x+2}{x^3-3 x+1}\)
\(\lim _{x \rightarrow \infty} \frac{x-\frac{5}{x^2}+\frac{2}{x^3}}{1-\frac{3}{x^2}+\frac{1}{x^3}}\) = ∞

(v) \(\lim _{x \rightarrow \infty}\left(\frac{x^3}{2 x^2-1}-\frac{x^2}{2 x+1}\right)\)
Solution:

(vi) \(\lim _{n \rightarrow \infty} \frac{n}{n+1}\)
Solution:
\(\lim _{n \rightarrow \infty} \frac{n}{n+1}\)
= \(\lim _{n \rightarrow \infty} \frac{n}{1+\frac{1}{n}}\) = 1

(vii) \(\lim _{n \rightarrow \infty}\left(\frac{n^2+n+1}{5 n^2+2 n+1}\right)\)
Solution:

(viii) \(\lim _{n \rightarrow \infty}\left(\frac{\sqrt{n}-1}{\sqrt{n}+1}\right)\)
Solution:
\(\lim _{n \rightarrow \infty}\left(\frac{\sqrt{n}-1}{\sqrt{n}+1}\right)\)
= \(\lim _{n \rightarrow \infty} \frac{1-\frac{1}{\sqrt{n}}}{1+\frac{1}{\sqrt{n}}}\) = 1

(ix) \(\lim _{n \rightarrow \infty}\left(\frac{6 n^5+2 n+1}{n^5+n^4+3 n^3+2 n^2+n+1}\right)\)
Solution:

(x) \(\lim _{n \rightarrow \infty} \frac{1+2+3+\cdots+n}{n^2}\)
Solution:

(xi) \(\lim _{n \rightarrow \infty} \frac{1^2+2^2+3^2+\ldots+n^2}{n^3}\)
Solution:

(xii) \(\lim _{n \rightarrow \infty} \frac{1^3+2^3+3^3+\ldots+n^3}{n^4}\)
Solution:

(xiii)  \(\lim _{n \rightarrow \infty} \frac{1+\frac{1}{2}+\frac{1}{2^2}+\ldots+\frac{1}{2^n}}{1+\frac{1}{3}+\frac{1}{3^2}+\ldots \frac{1}{3^n}}\)
Solution:

(xiv) \(\lim _{n \rightarrow \infty} \frac{\lfloor n}{\mid n+1-\lfloor n}\)
Solution:

Question 6.
Examine the existence of the following limits :
(i) \(\lim _{x \rightarrow \sqrt{3}}\) [x]
Solution:
L.H.L. = \(\lim _{x \rightarrow \sqrt{3}-}\) [x] = \(\lim _{h \rightarrow 0}\) [√3 – h] = 1
R.H.L. = \(\lim _{x \rightarrow \sqrt{3}+}\) [x] = \(\lim _{h \rightarrow 0}\) [√3 + h] = 1
Thus L.H.L., R.H.L both
exist and L.H.L. = R.H.L.
So the limit exists and its value is 1.

(ii) \(\lim _{x \rightarrow 0}[x]\)
Solution:

(iii) \(\lim _{x \rightarrow-2}[x]\)
Solution:
L.H.L. = \(\lim _{x \rightarrow-2-} \frac{x-2}{|x-2|}\)
= \(\lim _{h \rightarrow 0}\)[-2 – h] = -3
R.H.L. \(\lim _{x \rightarrow-2+}\) [x] = \(\lim _{h \rightarrow 0}\)[-2 + h] = -2
Thus L.H.L. ≠ R.H.L.
So the limit does not exist.

(iv) \(\lim _{x \rightarrow 0} \frac{|x|}{x}\)
Solution:

(v) \(\lim _{x \rightarrow 2} \frac{x-2}{|x-2|}\)
Solution:

(vi) \(\lim _{x \rightarrow \frac{1}{2}} \frac{|2 x-1|}{2 x-1}\)
Solution:

(vii) \(\lim _{x \rightarrow 1}[2 x+3]\)
Solution:

(viii) \(\lim _{x \rightarrow \infty} \frac{x}{[x]}\)
Solution:

(ix) \(\lim _{x \rightarrow \infty} \frac{x^2-x}{\left[x^2-x\right]}\)
Solution:

(x) \(\lim _{x \rightarrow 1} \frac{\left|x^2-3 x+2\right|}{x^2-3 x+2}\)
Solution:

(xi) \(\lim _{x \rightarrow \infty}(-1)^{[x]}\)
Solution:
\(\lim _{x \rightarrow \infty}(-1)^{[x]}\)
[Put n ≤ n + 1,As n→ ∞, x → ∞
= \(\lim _{x \rightarrow \infty}(-1)^n\) [ ∵ [x] = n
= ± 1 [If n is odd, (-1)ⁿ = – 1 and if n is even (-1)ⁿ = 1 ]
We know that whenever the limit exists it must be unique.
So \(\lim _{x \rightarrow \infty}(-1)^{[x]}\) does not exist.

(xii) \(\lim _{x \rightarrow \infty} \sin x\)
Solution:

(xiii) \(\lim _{x \rightarrow \infty} \cos x\)
Solution:

(xiv) \(\lim _{x \rightarrow 0} \cos \frac{1}{x}\)
Solution:

(xv) \(\lim _{x \rightarrow \infty} \sin \frac{1}{x}\)
Solution:

(xvi) \(\lim _{x \rightarrow 1} f(x) \text { if } f(x)= \begin{cases}2 x-1, & x \leq 1 \\ 2 x+1, & x>1\end{cases}\)
Solution:

(xvii) \(\lim _{x \rightarrow 0} f(x) \text { and } \lim _{x \rightarrow 1} f(x)\)
if \(f(x)=\left\{\begin{array}{l}
0 . x \leq 0 \\
1-2 x, 0<x \leq 1 \\
3-4 x, x>1
\end{array}\right.\)
Solution:

Question 7.
Let f(x) = {1 if x is rational, 0 if x is irrational then show that \(\lim _{x \rightarrow a}\) f(x) does not exist for any a ∈ R.
Solution:
Let x → a through rational numbers.
Then \(\lim _{x \rightarrow a}\) f(x) = 1
If x → a through rational numbers.
Then \(\lim _{x \rightarrow a}\) f(x) = 0
Thus \(\lim _{x \rightarrow a}\) f(x) does not exist.