Class 12

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

CHSE Odisha Class 12 Alternative English Grammar Tense and Aspect

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

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

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 She has withdrawn her name from the debate.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

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

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

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

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

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

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

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

Odisha State Board CHSE Odisha Class 12 Economics Solutions Chapter 15 ବ୍ୟାଙ୍କ Objective Questions.

CHSE Odisha 12th Class Economics Chapter 15 Objective Questions in Odia Medium

ବସ୍ତୁନିଷ୍ଠ ଓ ଅତିସଂକ୍ଷିପ୍ତ ଉତ୍ତରମୂଳକ ପ୍ରଶ୍ନୋତ୍ତର
A. ସମ୍ଭାବ୍ୟ ଚାରୋଟି ଉତ୍ତର ମଧ୍ୟରୁ ଠିକ୍ ଉତ୍ତରଟି ବାଛି ଲେଖ ।

1. ଚଳନ୍ତି ଜମାର ଅନ୍ୟ ନାମ ହେଉଛି :
(A) ଚାହିଦା ଜମା
(B) ମିଆଦି ଜମା
(C) ସ୍ଵୟ ଜମା
(D) ପୌନଃପୁନିକ ଜମା
Answer:
(A) ଚାହିଦା ଜମା

2. କେଉଁ ଜମା ସର୍ବାଧ‌ିକ ସୁଧ ପ୍ରଦାନ କରେ ?
(A) ଚଳନ୍ତି ଜମା
(B) ସଞ୍ଚୟ ଜମା
(C) ସ୍ଥାୟୀ ଜମା
(D) ପୌନଃପୁନିକ ଜମା
Answer:
(C) ସ୍ଥାୟୀ ଜମା

3. ମିଆଦି ଜମା କେଉଁ ଜମାର ଅନ୍ୟ ନାମ ?
(A) ଚଳନ୍ତି ଜମା
(B) ସଞ୍ଚୟ ଜମା
(C) ଚାହିଦା ଜମା
(D) ସ୍ଥାୟୀ ଜମା
Answer:
(A) ଚଳନ୍ତି ଜମା

4. କେଉଁ ଜମା ସୁଧ ଅର୍ଜନ କରେ ନାହିଁ ?
(A) ଚାହିଦା ଜମା
(B) ସଞ୍ଚୟ ଜମା
(C) ମିଆଦି ଜମା
(D) ଉପରୋକ୍ତ କୌଣସିଟି ନୁହେଁ
Answer:
(A) ଚାହିଦା ଜମା

5. କେଉଁ ବ୍ୟାଙ୍କ୍ ମୁଦ୍ରା ସୃଜନ କରିପାରେ ?
(A) ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍
(B) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(C) ବିନିମୟ ବ୍ୟାଙ୍କ
(D) ସଞ୍ଚୟ ବ୍ୟାଙ୍କ
Answer:
(B) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍

6. ନିମ୍ନଲିଖତ ବ୍ୟାଙ୍କମାନଙ୍କ ମଧ୍ୟରୁ କେଉଁ ବ୍ୟାଙ୍କ ସ୍ଵଳ୍ପକାଳୀନ ଋଣ ପ୍ରଦାନ କରେ ?
(A) ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍
(B) କୃଷି ବ୍ୟାଙ୍କ୍
(C) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(D) ସଞ୍ଚୟ ବ୍ୟାଙ୍କ୍
Answer:
(C) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍

7. ନିମ୍ନୋକ୍ତ କେଉଁ ଋଣ ପଦ୍ଧତି ବ୍ୟବସାୟୀଙ୍କଦ୍ୱାରା ସବୁଠାରୁ ଅଧିକ ଆଦୃତ ?
(A) ନଗଦୀ ଋଣ
(B) ପ୍ରତ୍ୟକ୍ଷ ଋଣ
(C) ଅତିରିକ୍ତ ଉଠାଣ
(D) ଉପରୋକ୍ତ କୌଣସିଟି ନୁହେଁ
Answer:
(A) ନଗଦୀ ଋଣ

8. ନିମ୍ନଲିଖ ମଧ୍ୟରୁ କେଉଁଟି ବାଣିଜ୍ୟ ବ୍ୟାଙ୍କର କାର୍ଯ୍ୟ ନୁହେଁ ?
(A) ଋଣ ପ୍ରଦାନ
(B) ଜମା ଗ୍ରହଣ
(C) ଋଣ ସୃଷ୍ଟି
(D) ନୋଟ୍ ପ୍ରଚଳନରେ ଏକାଧିକାର
Answer:
(D) ନୋଟ୍ ପ୍ରଚଳନରେ ଏକାଧିକାର

9. ଋଣ ଦେବାବେଳେ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ସୃଷ୍ଟି କରୁଥିବା ଜମାକୁ କ’ଣ କୁହାଯାଏ ?
(A) ପ୍ରାଥମିକ ଜମା
(B) ବ୍ୟୁତ୍ପନ୍ନ ଜମା
(C) ମିଆଦି ଜମା
(D) ଚଳନ୍ତି ଜମା
Answer:
(C) ମିଆଦି ଜମା

10. ଋଣ ପ୍ରଦାନ ଓ ଜମା ଗ୍ରହଣ ପ୍ରକ୍ରିୟାରେ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ସୃଷ୍ଟି କରୁଥିବା ମୁଦ୍ରାକୁ କ’ଣ କୁହାଯାଏ ?
(A) ଋଣ ମୁଦ୍ରା
(B) ବ୍ୟାଙ୍କ୍ ମୁଦ୍ରା
(C) ଜମା ମୁଦ୍ରା
(D) ଉପରୋକ୍ତ କୌଣସିଟି ନୁହେଁ
Answer:
(B) ବ୍ୟାଙ୍କ୍ ମୁଦ୍ରା

11. ଯଦି ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 10% ତେବେ 1000 ଟଙ୍କାର ମୂଳ ଜମା ସର୍ବାଧ‌ିକ କେତେ ଟଙ୍କା ହେବ ?
(A) 8000 ଟଙ୍କା
(B) 9000 ଟଙ୍କା
(C) 10,000 ଟଙ୍କା
(D) 11,000 ଟଙ୍କା
Answer:
(C) 10,000 ଟଙ୍କା

12. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 20% ହେଲେ ଋଣ ଗୁଣାଙ୍କ କେତେ ହେବ ?
(A) 20
(B) 10
(C) 15
(D) 5
Answer:
(D) 5

13. ଗ୍ରାହକମାନଙ୍କ ଅବଗତି ନିମନ୍ତେ ପ୍ରତ୍ୟେକ ଆର୍ଥିକ ବର୍ଷର ଶେଷଭାଗରେ ବ୍ୟାଙ୍କ୍ ନିଜ ଆୟ ବ୍ୟୟ ସଂକ୍ରାନ୍ତୀୟ
ବିବରଣୀ ଯେଉଁ ପତ୍ର ମାଧ୍ୟମରେ ପ୍ରକାଶ କରିଥାଏ ତାହାକୁ କ’ଣ କୁହାଯାଏ ?
(A) ଦେୟତା
(B) ପରିସମ୍ପରି
(C) ସନ୍ତୁଳନ ପତ୍ର
(D) ପୁଞ୍ଜି
Answer:
(C) ସନ୍ତୁଳନ ପତ୍ର

14. କେଉଁ ବ୍ୟାଙ୍କ୍‌ ସମଗ୍ର ବ୍ୟାକ୍ ବ୍ୟବସ୍ଥାର ଶୀର୍ଷ ଅନୁଷ୍ଠାନ ?
(A) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(B) ବିନିମୟ ବ୍ୟାଙ୍କ୍
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(D) କେନ୍ଦ୍ରୀୟ ସମବାୟ ବ୍ୟାଙ୍କ୍
Answer:
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

15. ପୃଥ‌ିବୀର ପ୍ରାଚୀନତମ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍‌ର ନାମ କ’ଣ ?
(A) ରିକ୍ ବ୍ୟାଙ୍କ୍
(B) ବ୍ୟାଙ୍କ୍ ଅଫ୍ ଇଂଲଣ୍ଡ
(C) ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍
(D) ଫେଡ଼େରାଲ ରିଜର୍ଭ ସିଷ୍ଟମ
Answer:
(A) ରିକ୍ ବ୍ୟାଙ୍କ୍

16. କେଉଁ ମସିହାରେ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍ ସ୍ଥାପିତ ହେଲା ?
(A) 1950
(B) 1951
(C) 1947
(D) 1935
Answer:
(D) 1935

17. କେଉଁ ଅନୁଷ୍ଠାନ ରାଷ୍ଟ୍ରର ମୁଦ୍ରାନୀତି ପ୍ରଣୟନ କରେ ?
(A) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(B) ବିତ୍ତ ମନ୍ତ୍ରଣାଳୟ
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(D) ଉପରୋକ୍ତ କୌଣସିଟି ନୁହେଁ
Answer:
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

18. ଭାରତରେ ଏକଟଙ୍କିଆ ନୋଟ କିଏ ପ୍ରଚଳନ କରେ ?
(A) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(B) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(C) ବିତ୍ତ ମନ୍ତ୍ରଣାଳୟ
(D) ଉପରୋକ୍ତ କୌଣସିଟି ନୁହେଁ
Answer:
(C) ବିତ୍ତ ମନ୍ତ୍ରଣାଳୟ

19. କେଉଁ ଅନୁଷ୍ଠାନ ଋଣ ନିୟନ୍ତ୍ରଣ କରେ ?
(A) ସମବାୟ ବ୍ୟାଙ୍କ୍
(B) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(C) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(D) ବିନିମୟ ବ୍ୟାଙ୍କ୍
Answer:
(B) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

20. କେଉଁ ବ୍ୟାକୁ ପୁନଃ ଅବମୂଲ୍ୟାୟନ ବ୍ୟାଙ୍କ୍ ମଧ୍ଯ କୁହାଯାଏ ?
(A) ବିନିମୟ ବ୍ୟାଙ୍କ୍
(B) ଔଦ୍ୟୋଗିକ ବ୍ୟାଙ୍କ୍
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(D) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
Answer:
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

21. ନିମ୍ନୋକ୍ତ କେଉଁଟି ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତିରେ ଏକ ଆୟୁଧ ?
(A) ବ୍ୟାଙ୍କ୍ ହାର
(B) ଖୋଲା ବଜାର କାରବାର
(C) ପରିବର୍ତ୍ତନଶୀଳ ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ
(D) ପ୍ରଚାର
Answer:
(D) ପ୍ରଚାର

22. କେଉଁ ବ୍ୟାଟ୍ ସରକାରଙ୍କୁ ଋଣ ଦିଏ ?
(A) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(B) ଉନ୍ନୟନ ବ୍ୟାଙ୍କ୍
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(D) କେନ୍ଦ୍ରୀୟ ସମବାୟ ବ୍ୟାଙ୍କ୍
Answer:
(C) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

23. ନିମ୍ନୋକ୍ତ କେଉଁଟି ପରିମାଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତିର ଏକ ଆୟୁଧ ?
(A) ବ୍ୟାଙ୍କ୍ ସୁଧ ହାର
(B) ଖୋଲା ବଜାର କାରବାର
(C) ନଗଦ ସଂରକ୍ଷିତ ଅନୁପାତରେ ପରିବର୍ଭନ
(D) ଉପରୋକ୍ତ ସମସ୍ତ
Answer:
(D) ଉପରୋକ୍ତ ସମସ୍ତ

24. କେଉଁ ବ୍ୟାଙ୍କ୍‌କୁ ଅନ୍ତିମ ଋଣଦାତା ବ୍ୟାଙ୍କ୍ କୁହାଯାଏ
(A) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍
(B) ଉନ୍ନୟନ ବ୍ୟାଙ୍କ୍
(C) ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍
(D) ଭାରତୀୟ ଷ୍ଟେଟ୍ ବ୍ୟାଙ୍କ୍
Answer:
(A) କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍

(B) ଶୂନ୍ୟସ୍ଥାନ ପୂରଣ କର ।

1. ଯେଉଁ ଅନୁଷ୍ଠାନରେ ଋଣସମୂହ ଲୋକମାନଙ୍କର ପରସ୍ପର ଋଣ ପରିଶୋଧ ନିମିତ୍ତ ବ୍ୟାପକ ଭାବରେ ଗୃହୀତ ହୁଏ, ସେହି ଅନୁଷ୍ଠାନକୁ ___________ କୁହାଯାଏ ।
Answer:
ବ୍ୟାଙ୍କ୍

2. ବ୍ୟାଙ୍କ୍ ଶବ୍ଦ ଇଟାଲାର ___________ ଶବ୍ଦରୁ ଉଦ୍ଧୃତ ।
Answer:
ବ୍ୟାଙ୍କା

3. ଇଟାଲୀ ଭାଷାରେ ‘ବ୍ୟାଙ୍କୋ’ ଶବ୍ଦର ଅର୍ଥ ହେଲା ___________ ।
Answer:
ବେଞ୍ଚ

4. ଯେଉଁ ଜମାକୁ ଜମାକାରୀ ଯେକୌଣସି କାର୍ଯ୍ୟକାରୀ ଦିନରେ ପୂର୍ବରୁ କୌଣସି ନୋଟିସ୍ ନ ଦେଇ ନିଜର ଆବଶ୍ୟକ ମୁତାବକ ମୁଦ୍ରା ଚେକ୍ ସାହାଯ୍ୟରେ ବ୍ୟାଙ୍କୁରୁ ଉଠାଇ ପାରନ୍ତି, ତାହାକୁ ___________ ଜମା କୁହାଯାଏ ।
Answer:
ଚଳନ୍ତି

5. ଯେଉଁ ଜମାକୁ ଜମାକାରୀ ନିର୍ଦ୍ଧାରିତ ସମୟକାଳ ପୂର୍ବରୁ ବିନା ନୋଟିସ୍‌ରେ ବ୍ୟାଙ୍କୁରୁ ଉଠାଇପାରେ ନାହିଁ, ତାହାକୁ ___________ ଜମା କୁହାଯାଏ ।
Answer:
ମିଆଦି

6. ସ୍ଵଳ୍ପ ସଞ୍ଚୟକାରୀ ଓ ନିମ୍ନ ଆୟକାରୀ ଲୋକମାନଙ୍କ ପାଇଁ ବ୍ୟାଙ୍କ୍ ___________ ଜମା ସୁବିଧା ପ୍ରଦାନ କରିଥାଏ ।
Answer:
ସଞ୍ଚୟ

7. ଯେଉଁ ଜମା ହିସାବରେ ଜମାକାରୀ ପ୍ରତି ମାସରେ ଏକ ନିର୍ଦ୍ଦିଷ୍ଟ ପରିମାଣର ମୁଦ୍ରା କିଛି ବର୍ଷ ପାଇଁ ଜମାଖାତାରେ ପୈଠ କରିଥା’ନ୍ତି, ତାହାକୁ ___________ ଜମା କୁହାଯାଏ ।
Answer:
ପୌନଃପୁନିକ

8. ଯେଉଁ ଋଣଗୁଡ଼ିକ ଗ୍ରାହକମାନଙ୍କର ପ୍ରତିଶ୍ରୁତି ପତ୍ର ବଦଳରେ ପ୍ରଦାନ କରାଯାଏ, ତାହାକୁ ___________ ଋଣ କୁହାଯାଏ ।
Answer:
ନଗଦୀ

9. ଯେଉଁ ବ୍ୟବସ୍ଥାଦ୍ୱାରା ଜମାକାରୀ ନିଜର ଚାହିଦା ଜମାଖାତାରେ ଥିବା ପରିମାଣଠାରୁ ଅଧିକ ମୁଦ୍ରା ବ୍ୟାଙ୍କରୁ ଉଠାଇବାକୁ ବ୍ୟବସ୍ଥା ସକ୍ଷମ ହୋଇଥାଏ, ତାହାକୁ ___________ କୁହାଯାଏ ।
Answer:
ଓଭରଡ୍ରାଫ୍‌ଟ

10. ବ୍ୟାଙ୍କ୍ ଯେଉଁ ଅର୍ଥ ନଗଦ ଜମା ଆକାରରେ ପ୍ରାପ୍ତ ହୋଇଥା’ନ୍ତି, ତାହାକୁ ___________ କୁହାଯାଏ ।
Answer:
ପ୍ରାଥମିକ ଜମା

11. ଋଣ ଦାନ ସମୟରେ ବ୍ୟାକ୍ ଜଣେ ଋଣଗ୍ରହୀତାଙ୍କୁ ଜମାକାରୀରେ ପରିଣତ କରେ, ବ୍ୟାକ୍ ଏପରି ଯେଉଁ ଜମା ସୃଷ୍ଟି କରେ ତାହାକୁ ___________ କୁହାଯାଏ ।
Answer:
ବ୍ୟୁତ୍ପନ୍ନ

12. ବ୍ୟାଙ୍କ୍ ସୃଷ୍ଟି କରୁଥିବା ମୁଦ୍ରା ନଗଦ ମୁଦ୍ରା ନୁହେଁ, ଏହା ଦୃଶ୍ୟ ହୋଇନଥାଏ, ଏହା କାଗଜ କଲମରେ ସୃଷ୍ଟ, ଏହାକୁ ___________ ମୁଦ୍ରା କୁହାଯାଏ ।
Answer:
ବ୍ୟାଙ୍କ୍

13. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ ଓ ରଣାଙ୍କ ମଧ୍ୟରେ ___________ ସମ୍ପର୍କ ରହିଛି ।
Answer:
ପରୋକ୍ଷ ଆନୁପାତିକ

14. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 10% ହେଲେ ଋଣ ଗୁଣାଙ୍କ ___________ ହେବ ।
Answer:
10

15. ଖୁବ୍ କମ୍ ସମୟ ପାଇଁ ପ୍ରଦାନ କରାଯାଉଥ‌ିବା ଏକ ସ୍ଵତନ୍ତ୍ର ଋଣ ହେଉଛି ଚାହିଁବା ମାତ୍ରେ ଫେରସ୍ତଯୋଗ୍ୟ ମୁଦ୍ରା ବା ଋଣ ଯାହାକୁ ନୋଟିସ୍ ଦେବାର 24 ଘଣ୍ଟାରୁ ___________ ଦିନ ମଧ୍ୟରେ ବ୍ୟାଙ୍କ୍ ଋଣକାରୀଠାରୁ ଏହି ଋଣ ଫେରସ୍ତ ଆଣିପାରେ ।
Answer:
14

16. ମିଆଦୀ ଜମା କ୍ଷେତ୍ରରେ, ମିଆଦ କାଳ ଯେତେ ଦୀର୍ଘତର ହୋଇଥାଏ, ସୁଧ ହାର ସେତିକି ___________ ହୋଇଥାଏ ।
Answer:
ଅଧ୍ୟା

17. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କଗୁଡ଼ିକ ଗ୍ରାହକମାନଙ୍କର ___________ ହିସାବରେ ସେମାନଙ୍କ ତରଫରୁ ଅନେକ କାର୍ଯ୍ୟ କରିଥାଏ ।
Answer:
ପ୍ରତିନିଧ୍ଵ

18. ସନ୍ତୁଳନ ପତ୍ରରେ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କର ଆର୍ଥିକ ଅବସ୍ଥା ଅର୍ଥାତ୍ ଦେୟତା ଓ ___________ ପ୍ରତିଫଳିତ ହୋଇଥାଏ ।
Answer:
ପରି ସମ୍ପତ୍ତି

19. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଅନ୍ୟମାନଙ୍କୁ ଯେଉଁ ପାଉଣା ଆଇନତଃ ଦେବାକୁ ବାଧ୍ୟ, ତାହା ହେଉଛି ବ୍ୟାଙ୍କ୍‌ର ___________ ।
Answer:
ଦେୟତା

20. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଅନ୍ୟମାନଙ୍କଠାରୁ ଆଇନତଃ ଯେଉଁ ପାଉଣା ପାଇବାକୁ ହକ୍‌ର, ତାହା ହେଉଛି ବ୍ୟାଙ୍କ୍‌ର ___________ ।
Answer:
ପରି ସମ୍ପତ୍ତି

21. ଅଂଶୀଦାରମାନେ ଯେଉଁ ପରିମାଣର ବ୍ୟାଙ୍କର ଅଂଶ କ୍ରୟ କରିବାକୁ ସ୍ବୀକୃତି ଦେଇଥା’ନ୍ତି, ___________ ପୁଞ୍ଜି କୁହାଯାଏ ।
Answer:
ଅଭିଦର

22. ଅଭିଦତ୍ତ ପୁଞ୍ଜିର ଯେଉଁ ଅଂଶ ନଗଦ ମୁଦ୍ରା ଦେଇ ଅଂଶୀଦାରମାନେ କ୍ରୟ କରିଥା’ନ୍ତି, ତାକୁ ___________ ପୁଞ୍ଜି କହନ୍ତି ।
Answer:
ପ୍ରଦତ୍ତ

23. ଲାଭର ଯେଉଁ ଅଂଶ ଅଂଶୀଦାରମାନଙ୍କ ମଧ୍ୟରେ ବଣ୍ଟନ କରା ନ ଯାଇ ଭବିଷ୍ୟତ ନିରାପତ୍ତା ପାଇଁ ସଂରକ୍ଷିତ ହୋଇଥାଏ, ତାହାକୁ ___________ ପାଣ୍ଠି କୁହାଯାଏ ।
Answer:
ସଂରକ୍ଷିତ

24. ଜମା ବ୍ୟାଙ୍କର ସର୍ବବୃହତ୍ ___________ ଅଟେ ।
Answer:
ଦେୟ

25. ପ୍ରତ୍ୟେକ ବ୍ୟାଙ୍କ୍ ଗ୍ରାହକମାନଙ୍କ ଚାହିଦା ମାତ୍ରକେ ସେମାନଙ୍କ ଆବଶ୍ୟକତା ପୂରଣ କରିବାପାଇଁ ନିଜ ପାଖେ କିଛି ନଗଦ ମୁଦ୍ରା ରଖନ୍ତି, ଏହାକୁ ___________ ମୁଦ୍ରା କୁହାଯାଏ ।
Answer:
ହସ୍ତସ୍ଥ

26. ପରମ୍ପରାଗତ ଭାବେ ହେଉ ବା ଆଇନ ଯୋଗୁଁ ହେଉ, ବାଣିଜ୍ୟିକ ବ୍ୟାଟ୍ସମାନେ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ମାର୍ସତରେ ମୋଟ ଜମାର କିଛି ଶତାଂଶ ନଗଦ ଆକାରରେ ଗଚ୍ଛିତ ରଖୁଥା’ନ୍ତି, ଯାହାକୁ ___________ କୁହାଯାଏ ।
Answer:
ଗଚ୍ଛିତ ଅନୁପାତ

27. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କର ସନ୍ତୁଳନ ପତ୍ରର ___________ ପାର୍ଶ୍ଵରେ ଦେୟତା ରହିତାଏ ।
Answer:
ବାମ

28. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କର ସନ୍ତୁଳନ ପତ୍ରର ___________ ପାର୍ଶ୍ବରେ ପରିସମ୍ପଭି ରହିଥାଏ ।
Answer:
ଡାହାଣ

29. ସରକାରଙ୍କର ପ୍ରଧାନ ଆର୍ଥିକ କାରବାରମାନ ସମ୍ପାଦନ କରୁଥିବା, ଏହାର କାର୍ଯ୍ୟ ପରିଚାଳନା ଓ ଅନ୍ୟ ଉପାୟରେ ସରକାରଙ୍କ ଆର୍ଥିକ ନୀତିକୁ ସମର୍ଥନ କରୁଥିବା ଓ ଆର୍ଥିକ ଅନୁଷ୍ଠାନମାନଙ୍କର କାର୍ଯ୍ୟାବଳୀକୁ ପ୍ରଭାବିତ କରୁଥିବା ବ୍ୟାଙ୍କ୍ ହିଁ ___________ ବ୍ୟାକ୍ ।
Answer:
କେନ୍ଦ୍ରୀୟ

30. ___________ ବ୍ୟାଙ୍କର ନୋଟ୍ ପ୍ରଚଳନ କ୍ଷେତ୍ରରେ ଏକାଧିକାର ରହିଛି ।
Answer:
କେନ୍ଦ୍ରୀୟ

31. ପ୍ରତ୍ୟେକ ଦେଶରେ କେବଳ ___________ ମାତ୍ର କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ ଥାଏ ।
Answer:
ଗୋଟିଏ

32. ___________ ବ୍ୟାଙ୍କ୍ ଅର୍ଥନୈତିକ ବିକାଶର ବାହକ ।
Answer:
କେନ୍ଦ୍ରୀୟ

33. ___________ ବ୍ୟାଙ୍କ୍ ସର୍ବଦା ଲାଭ ଅର୍ଜନ ପାଇଁ କାର୍ଯ୍ୟ କରିଥା’ନ୍ତି ।
Answer:
ବାଣିଜ୍ୟିକ

34. ନିର୍ଦ୍ଦିଷ୍ଟ କ୍ଷେତ୍ରରେ ଋଣ ନିୟନ୍ତ୍ରଣ ପାଇଁ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍ଗ୍‌ମାନଙ୍କ ପ୍ରତି ___________ ଜାରି କରେ ।
Answer:
ନିର୍ଦ୍ଦେଶନାମା

35. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ କେତେକ ନିର୍ଦ୍ଦିଷ୍ଟ କ୍ଷେତ୍ରରେ ଋଣ ନିୟନ୍ତ୍ରଣ ପାଇଁ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କମାନଙ୍କ ଉପରେ ___________ ସୃଷ୍ଟି କରେ ।
Answer:
ନୈତିକ ଚାପ

36. ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତିକୁ ___________ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତି ମଧ୍ୟ କୁହାଯାଏ ।
Answer:
ଚୟନାତ୍ମକ

37. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ବାଣିଜ୍ୟିକ ବ୍ୟାଟ୍ସମାନଙ୍କୁ ଯେଉଁ ହାରରେ ଋଣ ଦେଇଥାଏ, ତାହାକୁ ___________ କୁହାଯାଏ ।
Answer:
ବ୍ୟାଙ୍କ ହାର

38. ଜମାର ଯେଉଁ ଅନୁପାତ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଋଣ ଦେଇପାରେ ନାହିଁ, ତାହାକୁ ___________ ଅନୁପାତ କୁହାଯାଏ ।
Answer:
ନଗଦୀ ସଂରକ୍ଷଣ

39. ଅସ୍ଥାୟୀ ଆର୍ଥିକ ଅସୁବିଧା ଦୂର କରିବାପାଇଁ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କୁ ସରକାରଙ୍କୁ ସ୍ୱଳ୍ପକାଳୀନ ଋଣ ଓ ଅଗ୍ରିମ ଋଣ ପ୍ରଦାନ ଋଣ କରେ, ଏହି ଋଣକୁ ___________ କୁହାଯାଏ ।
Answer:
କାମଚଳା

40. ସରମାନଙ୍କ ___________ ସ୍ଵରୂପ କାର୍ଯ୍ୟକରି କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ସରକାରଙ୍କୁ ଦେଶର ମୁଦ୍ରାନୀତି ଓ ଆର୍ଥିକ ନୀତି ଉପରେ ମଧ୍ୟ ପରାମର୍ଶ ଦେଇଥାଏ ।
Answer:
ଉପଦେଷ୍ଟା

41. ଦେୟତା ପରିଶୋଧ କରିବାପାଇଁ ବ୍ୟାଙ୍କ୍-ବ୍ୟାଙ୍କ୍ ମଧ୍ୟରେ ନଗଦ ମୁଦ୍ରା ଚଳପ୍ରଚଳ ନ କରି କେବଳ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ନିକଟରେ ଥିବା ହିସାବରେ ପରିବର୍ତ୍ତନ କରି ଏହା କରାଯାଏ, ତେଣୁ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ସସମୂହ ପାଇଁ ଏକ ___________ ବ୍ୟାଙ୍କ୍ ।
Answer:
ଶୋଧନ

42. ___________ ନିୟନ୍ତ୍ରଣ କରି ଦେଶରେ ଦରଦାମ୍ କ୍ଷେତ୍ରରେ ସ୍ଥିରତା ରକ୍ଷା କରିବା କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ହିଁ ମୁଖ୍ୟ ଦାୟିତ୍ୱ ।
Answer:
ଋଣ

C. ନିମ୍ନଲିଖ ଉକ୍ତିଗୁଡ଼ିକ ଭୁଲ୍ କି ଠିକ୍ ଲେଖ । ରେଖାଙ୍କିତ ଅଂଶର ପରିବର୍ତ୍ତନ ନ କରି ଆବଶ୍ୟକ ସ୍ଥଳେ ସଂଶୋଧନ କର ।

1. ସମବାୟ ବ୍ୟାଙ୍କୁ କୃଷି ପାଇଁ ଦୀର୍ଘକାଳୀନ ଋଣ ଦେଇଥାଏ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ସମବାୟ ବ୍ୟାଙ୍କ୍ କୃଷି ପାଇଁ ସ୍ବଳ୍ପକାଳୀନ ଋଣ ଦେଇଥାଏ !

2. ଭାରତରେ ଏକକ ବ୍ୟାଙ୍କ ବ୍ୟବସ୍ଥା ପ୍ରଚଳିତ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଆମେରିକାରେ ଏକକ ବ୍ୟାଙ୍କ୍ ବ୍ୟବସ୍ଥା ପ୍ରଚଳିତ ।

3. ସ୍ଥାୟୀ ଜମାର ଅନ୍ୟନାମ ଚାହିଦା ଜମା ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ସ୍ଥାୟୀ ଜମାର ଅନ୍ୟନାମ ମିଆଦି ଜମା ।

4. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ ଓ ଋଣ ପରିମାଣ ମଧ୍ଯରେ ପ୍ରତ୍ୟକ୍ଷ ସମ୍ପର୍କ ରହିଛି ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ ଓ ଋଣ ପରିମାଣ ମଧ୍ଯରେ ପରୋକ୍ଷ ସମ୍ପର୍କ ରହିଛି ।

5. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ଋଣ ସୃଷ୍ଟି କରେ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଋଣ ସୃଷ୍ଟି କରେ ।

6. ମିଆଦି ଜମା ନଗଦ ମୁଦ୍ରା ସହିତ ସମାନ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଚାହିଦା ଜମା ନଗଦ ମୁଦ୍ରା ସହିତ ସମାନ ।

7. ବ୍ୟାଙ୍କ୍ ଅଫ ଇଂଲଣ୍ଡ ବିଶ୍ବର ପ୍ରଥମ କେନ୍ଦ୍ରୀୟ ବ୍ୟାକ୍ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ରିକ୍ ବ୍ୟାଙ୍କ୍ ଅଫ୍ ସ୍ବିଡ଼େନ୍ ବିଶ୍ବର ପ୍ରଥମ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ।

8. 1947 ମସିହାରେ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ ସ୍ଥାପିତ ହୋଇଥିଲା ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – 1935 ମସିହାରେ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କୁ ସ୍ଥାପିତ ହୋଇଥିଲା ।

9. ଭାରତରେ ଏକ ଟଙ୍କିଆ ନୋଟ୍ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍ ପ୍ରଚଳନ କରେ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଭାରତରେ ଏକ ଟଙ୍କିଆ ନୋଟ୍, ଭାରତ ସରକାରଙ୍କ ବିତ୍ତ ମନ୍ତ୍ରଣାଳୟ ପ୍ରଚଳନ କରେ ।

10. ଭାରତରେ ପାଞ୍ଚଶହ ଟଙ୍କିଆ ନୋଟ୍ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଦ୍ୱାରା ପ୍ରଚଳିତ ହୁଏ ।
Answer:
ଠିକ୍ ।

11. ବାଣିଜ୍ୟିକ ବ୍ୟାକ୍ ସରକାରଙ୍କ ବ୍ୟାଙ୍କ୍ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ସରକାରଙ୍କ ବ୍ୟାଙ୍କ୍ ।

12. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଋଣ ନିୟନ୍ତ୍ରଣ କରିଥାଏ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ଋଣ ନିୟନ୍ତ୍ରଣ କରିଥାଏ ।

13. ମିଆଦି ଜମାର ଉଠାଣ ଚାହିଁବା ମାତ୍ରେ ସମ୍ଭବ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଚାହିଦା ଜମାର ଉଠାଣ ଚାହିଁବା ମାତ୍ରେ ସମ୍ଭବ ।

14. ଭାରତୀୟ ଷ୍ଟେଟ୍ ବ୍ୟାକ୍ ବ୍ୟାକ୍‌ମାନଙ୍କର ବ୍ୟାକ୍ ଅଟେ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍ ବ୍ୟାଟ୍ସମାନଙ୍କର ବ୍ୟାଙ୍କ୍ ଅଟେ ।

15. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ଅନ୍ତିମ ଋଣଦାତା ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ଅନ୍ତିମ ଋଣଦାତା ।

16. ବ୍ୟାଙ୍କ୍ ରେଟ ବୃଦ୍ଧି ଘଟିଲେ ଋଣ ପରିମାଣ ବୃଦ୍ଧି ପାଏ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ବ୍ୟାଙ୍କ୍ ରେଟ ବୃଦ୍ଧି ଘଟିଲେ ଋଣ ପରିମାଣ ହ୍ରାସ ପାଏ ।

17. ଭାରତୀୟ ଷ୍ଟେଟ୍ ବ୍ୟାଟ୍ ଭାରତର କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କୁ ଭାରତର କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ।

18. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କୁ ଜନସାଧାରଣଙ୍କ ସହ ଋଣ କାରବାର କରିଥାଏ ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କୁ ସରକାରଙ୍କ ସହ ଋଣ କାରବାର କରିଥାଏ ।

19. ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କମାନେ ଦୀର୍ଘକାଳୀନ ଋଣ ପ୍ରଦାନ କରନ୍ତି ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କମାନେ ସ୍ଵଳ୍ପକାଳୀନ ଋଣ ପ୍ରଦାନ କରନ୍ତି ।

20. ଲାଭ ଅର୍ଜନ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ସର୍ବପ୍ରଧାନ ନୀତି ।
Answer:
ଭୁଲ୍ ।
ଠିକ୍ – ସାଧାରଣ କଲ୍ୟାଣ ସାଧନ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ସର୍ବପ୍ରଧାନ ନୀତି ।

D. ଗୋଟିଏ ବାକ୍ୟରେ ଉତ୍ତର ଦିଅ ।

1. ବାଣିଜ୍ୟ ବ୍ୟାଙ୍କ୍ କାହାକୁ କହନ୍ତି ?
Answer:
ଯେଉଁ ବିତ୍ତୀୟ ଅନୁଷ୍ଠାନ ଜନସାଧାରଣଙ୍କଠାରୁ ଜମା ଗ୍ରହଣ, ଋଣ ପ୍ରଦାନ, ମୁଦ୍ରା ଯୋଗାଣ, ମୁଦ୍ରା ସୃଜନ, ଅନ୍ତର୍ଦେଶୀୟ ଓ ବୈଦେଶିକ ବାଣିଜ୍ୟିକ ସୁବନ୍ଦୋବସ୍ତ ସହିତ ବିନିମୟ ପତ୍ରର ପୂର୍ବପ୍ରାପଣ କରନ୍ତି, ସେହି ଅନୁଷ୍ଠାନକୁ ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ କୁହାଯାଏ ।

2. ବିନିମୟ ବ୍ୟାଟ୍‌ର ମୁଖ୍ୟ କାର୍ଯ୍ୟ କ’ଣ ?
Answer:
ବିନିମୟ ବ୍ୟାଙ୍କ୍ ବୈଦେଶିକ ବାଣିଜ୍ୟ ପାଇଁ ଆବଶ୍ୟକ ହେଉଥ‌ିବା ଅର୍ଥ ଋଣ ଆକାରରେ ପ୍ରଦାନ କରେ ।

3. କେଉଁ ବ୍ୟାଙ୍କ୍ କୃଷି ପାଇଁ ଦୀର୍ଘକାଳୀନ ଋଣ ଦିଏ ?
Answer:
ଭୂବନ୍ଧକ ବ୍ୟାଙ୍କ କୃଷି ପାଇଁ ଦୀର୍ଘକାଳୀନ ଋଣ ଦିଏ ?

4. ଆଞ୍ଚଳିକ ଗ୍ରାମ୍ୟ ବ୍ୟାଙ୍କ୍ ଭାରତରେ କେବେ ସ୍ଥାପିତ ହୋଇଥିଲା ?
Answer:
1975 ମସିହାରେ ଭାରତରେ ଆଞ୍ଚଳିକ ଗ୍ରାମ୍ୟ ବ୍ୟାଙ୍କ୍ ସ୍ଥାପିତ ହୋଇଥିଲା ।

5. ଚଳନ୍ତି ଜମାକୁ କାହିଁକି ଚାହିଦା ଜମା ମଧ୍ୟ କୁହାଯାଏ ?
Answer:
ଚାହିଁବା ମାତ୍ରେ ଚଳନ୍ତି ଜମାର ଉଠାଣ ସମ୍ଭବ ହେଉଥ‌ିବାରୁ ଏହାକୁ ଚାହିଦା ଜମା ମଧ୍ୟ କୁହାଯାଏ ।

6. ସ୍ଥାୟୀ ଜମାର ଅନ୍ୟ ନାମ କ’ଣ ?
Answer:
ସ୍ଥାୟୀ ଜମାର ଅନ୍ୟ ନାମ ହେଲା ମିଆଦି ଜମା ।

7. ଅତିରିକ୍ତ ଉଠାଣ କାହାକୁ କହନ୍ତି ?
Answer:
ଯେତେବେଳେ ଜଣେ ଜମାକାରୀଙ୍କୁ ତାଙ୍କ ଜମାରାଶିଠାରୁ ଅଧିକ ଉଠାଣ ପାଇଁ ବ୍ୟାଙ୍କ୍ ଯେଉଁ ଅନୁମତି ପ୍ରଦାନ କରିଥାଏ, ତାହାକୁ ଅତିରିକ୍ତ ଉଠାଣ କୁହାଯାଏ ।

8. ବ୍ୟାକ୍ ମୁଦ୍ରା କ’ଣ ?
Answer:
ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ ସୃଷ୍ଟି କରୁଥିବା ଋଣମୁଦ୍ରାକୁ ବ୍ୟାଙ୍କ ମୁଦ୍ରା କୁହାଯାଏ ।

9. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ ଓ ଋଣ ଗୁଣାଙ୍କ ମଧ୍ୟରେ କି ସମ୍ପର୍କ ରହିଛି ?
Answer:
ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ ଓ ଋଣ ଗୁଣାଙ୍କ ମଧ୍ୟରେ ପରୋକ୍ଷ-ଆନୁପାତିକ ସମ୍ପର୍କ ରହିଛି ।

10. ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 20% ହେଲେ ଋଣ ଗୁଣାଙ୍କ କେତେ ହେବ ?
Answer:
ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 20% ହେଲେ ଋଣ ଗୁଣାଙ୍କ 5 ହେବ ।

11. ଯଦି ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 5% ତେବେ ପାଞ୍ଚ ହଜାର ଟଙ୍କାର ଜମା କେତେ ଟଙ୍କାର ସର୍ବାଧ‌ିକ ଋଣ ସୃଷ୍ଟି କରିପାରିବ ?
Answer:
ଯଦି ନଗଦୀ ସଂରକ୍ଷଣ ଅନୁପାତ 5% ତେବେ ପାଞ୍ଚ ହଜାର ଟଙ୍କାର ଜମା 95,000 ଟଙ୍କାର ସର୍ବାଧ‌ିକ ଋଣ ସୃଷ୍ଟି କରି ପାରିବ ।

12. ବିଶ୍ବର ପ୍ରଥମ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ନାମ କ’ଣ ?
Answer:
ସ୍ବିଡ଼େନ୍‌ର ରିକ୍ ବ୍ୟାଙ୍କ୍ ବିଶ୍ବର ପ୍ରଥମ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ !

13. ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍ କେବେ ପ୍ରତିଷ୍ଠିତ ହୋଇଥିଲା ?
Answer:
1935 ମସିହାରେ ଭାରତୀୟ ରିଜର୍ଭ ବ୍ୟାଙ୍କ୍ ପ୍ରତିଷ୍ଠିତ ହୋଇଥିଲା ।

14. ଭାରତରେ ଏକଟଙ୍କିଆ ନୋଟ୍ କିଏ ପ୍ରଚଳନ କରିଥାଏ ?
Answer:
ଭାରତ ସରକାରଙ୍କ ବିତ୍ତ ମନ୍ତ୍ରଣାଳୟ, ଭାରତରେ ଏକ ଟଙ୍କିଆ ନୋଟ୍ ପ୍ରଚଳନ କରିଥାଏ ।

15. ଇଂଲଣ୍ଡର କେନ୍ଦ୍ରୀୟ ବ୍ୟାକ୍ଚର ନାମ କ’ଣ ?
Answer:
ଇଂଲଣ୍ଡର କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ନାମ ହେଲା ବ୍ୟାଙ୍କ୍ ଅଫ୍ ଇଂଲଣ୍ଡ ।

16. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍‌କୁ ଅନ୍ତିମ ଋଣଦାତା ବୋଲି କାହିଁକି କୁହାଯାଏ ?
Answer:
କୌଣସି ଆର୍ଥିକ ସଙ୍କଟ ସମୟରେ ଯଦି ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ ଅନ୍ୟ କୌଣସି ସୂତ୍ରରୁ ଅର୍ଥ ଯୋଗାଡ଼ କରିବାପାଇଁ ଅସମର୍ଥ ହୁଏ, ତେବେ ଏହା ପରିଶେଷରେ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ଦ୍ବାରସ୍ଥ ହୋଇଥାଏ ଏବଂ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ଆର୍ଥିକ ସହାୟତା ପ୍ରଦାନ କରେ । ତେଣୁ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍‌କୁ ଅନ୍ତିମ ଋଣଦାତା କୁହାଯାଏ ।

17. କେଉଁ ସଂସ୍ଥା ଋଣ ନିୟନ୍ତ୍ରଣ କରିଥାଏ ?
Answer:
କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ଋଣ ନିୟନ୍ତ୍ରଣ କରିଥାଏ ।

18. ବ୍ୟାଙ୍କ୍ ହାର କାହାକୁ କହନ୍ତି ?
Answer:
କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ବାଣିଜ୍ୟିକ ବ୍ୟାଟ୍ସମାନଙ୍କୁ ଯେଉଁ ହାରରେ ଋଣ ଦେଇଥାଏ, ତାହାକୁ ବ୍ୟାଙ୍କ ହାର କୁହାଯାଏ ।

19. କେଉଁ ବ୍ୟାଙ୍କ୍ ପରିଶୋଧନର ମାଧ୍ୟମ ?
Answer:
କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ୍ ପରିଶୋଧନର ମାଧ୍ୟମ ।

20. ରାଷ୍ଟ୍ରରେ ବୈଦେଶିକ ବିନିମୟ ମୁଦ୍ରାର ତତ୍ତ୍ବାବଧାନ କିଏ କରିଥାଏ ?
Answer:
ରାଷ୍ଟ୍ରରେ କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କ ବୈଦେଶିକ ବିନିମୟ ମୁଦ୍ରାର ତତ୍ତ୍ୱାବଧାନ କରିଥାଏ ।

21. ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତିର ଅନ୍ୟ ନାମ କ’ଣ ?
Answer:
ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପଦ୍ଧତିର ଅନ୍ୟ ନାମ ହେଉଛି ଚୟନାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ।

22. ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପାଇଁ ଉଦ୍ଦିଷ୍ଟ ଦୁଇଟି ଆୟୁଧର ନାମ ଲେଖ ।
Answer:
ଗୁଣାତ୍ମକ ଋଣ ନିୟନ୍ତ୍ରଣ ପାଇଁ ଉଦ୍ଦିଷ୍ଟ ଦୁଇଟି ଆୟୁଧର ନାମ ହେଲା – (i) ପ୍ରତ୍ୟକ୍ଷ କାର୍ଯ୍ୟାନୁଷ୍ଠାନ, (ii) ନୈତିକ ପ୍ରବର୍ତ୍ତନ ।

23. ଖୋଲାବଜାର କାରବାର କହିଲେ କ’ଣ ବୁଝ ?
Answer:
କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କଦ୍ୱାରା ଖୋଲା ବଜାରରେ ସରକାରୀ ପ୍ରତିଭୂତିର କ୍ରୟ ବିକ୍ରୟ ପ୍ରକ୍ରିୟାକୁ ଖୋଲା ବଜାର କାରବାର କୁହାଯାଏ ।

24. ବ୍ୟାକ୍ ହାର ଓ ଋଣ ପରିମାଣ ମଧ୍ଯରେ କି ସମ୍ପର୍କ ରହିଛି ?
Answer:
ବ୍ୟାଙ୍କ୍ ହାର ଓ ଋଣ ପରିମାଣ ମଧ୍ଯରେ ପରୋକ୍ଷ ସମ୍ପର୍କ ରହିଛି ।

25. କେଉଁ ଜମା ଚାହିଁବାମାତ୍ରେ ପରିଶୋଧନୀୟ ନୁହେଁ ?
Answer:
ମିଆଦୀ ଜମା ଚାହିଁବାମାତ୍ରେ ପରିଶୋଧନୀୟ ନୁହେଁ ।

26. ବ୍ୟାଟ୍ସମାନଙ୍କର ସର୍ବବୃହତ୍ ଦେୟ କ’ଣ ?
Answer:
‘ଜମା’ ବ୍ୟାଙ୍କମାନଙ୍କର ସର୍ବବୃହତ୍ ଦେୟ ।

27. ପ୍ରଦତ୍ତ ପୁଞ୍ଜି ବା ପରିଶୋଧ ପୁଞ୍ଜି କ’ଣ ?
Answer:
ଅଭିଦତ୍ତ ପୁଞ୍ଜିର ଯେଉଁ ଅଂଶ ନଗଦ ମୁଦ୍ରାଦେଇ ଅଂଶୀଦାରମାନେ କ୍ରୟ କରିଥା’ନ୍ତି, ତାହାକୁ ପ୍ରଦତ୍ତ ପୁଞ୍ଜି ବା ପରିଶୋଧ ପୁଞ୍ଜି
କୁହାଯାଏ ।

28. ଅଭିଦତ୍ତ ପୁଞ୍ଜି କ’ଣ ?
Answer:
ଅଂଶୀଦାରମାନେ ଯେଉଁ ପରିମାଣର ଅଂଶ କ୍ରୟ କରିବାକୁ ସ୍ବୀକୃତି ଦେଇଥା’ନ୍ତି, ତାହାକୁ ଅଭିଦତ୍ତ ପୁଞ୍ଜି କୁହାଯାଏ ।

29. ସଂରକ୍ଷିତ ପାଣ୍ଠି କ’ଣ ?
Answer:
ଲାଭର ଯେଉଁ ଅଂଶ ଅଂଶୀଦାରମାନଙ୍କ ମଧ୍ୟରେ ବଣ୍ଟନ କରା ନ ଯାଇ ଭବିଷ୍ୟତ ନିରାପତ୍ତା ପାଇଁ ସଂରକ୍ଷିତ ହୋଇଥାଏ, ତାହାକୁ ସଂରକ୍ଷିତ ପାଣ୍ଠି କୁହାଯାଏ ।

30. ହସ୍ତସ୍ଥ ମୁଦ୍ରା କ’ଣ ?
Answer:
ପ୍ରତ୍ୟେକ ବ୍ୟାଙ୍କ୍ ଗ୍ରାହକମାନଙ୍କ ଚାହିଦାମାତ୍ରକେ ସେମାନଙ୍କ ଆବଶ୍ୟକତା ପୂରଣ କରିବା ପାଇଁ ନିଜ ପାଖେ କିଛି ନଗଦ ମୁଦ୍ରା ରଖନ୍ତି, ଏହାକୁ ହସ୍ତସ୍ଥ ମୁଦ୍ରା କୁହାଯାଏ ।

31. ବ୍ୟାଙ୍କ୍‌ ସର୍ବାଧ୍ଵକ ତରଳ ପରିସମ୍ପତ୍ତି କିଏ ?
Answer:
ନଗଦ ମୁଦ୍ରା ବ୍ୟାଙ୍କର ସର୍ବାଧ‌ିକ ତରଳ ପରିସମ୍ପତ୍ତି ।

32. କେଉଁ ଜମା ନିର୍ଦ୍ଦିଷ୍ଟ ସମୟ ଅତିବାହିତ ହେବା ପୂର୍ବରୁ ପ୍ରତ୍ୟାହାର ହୋଇପାରେ ନାହିଁ ?
Answer:
ମିଆଦୀ ଜମା ନିର୍ଦ୍ଦିଷ୍ଟ ସମୟ ଅତିବାହିତ ହେବା ପୂର୍ବରୁ ପ୍ରତ୍ୟାହାର ହୋଇପାରେ ନାହିଁ ।

33. ସନ୍ତୁଳନ ପତ୍ର କାହାକୁ କୁହାଯାଏ ?
Answer:
ବାଣିଜ୍ୟିକ ବ୍ୟାଙ୍କ୍ସର ପରିସମ୍ପତ୍ତି ଓ ଦେୟତାର ଏକ ବାର୍ଷିକ ବିବରଣୀକୁ ତା’ର ସନ୍ତୁଳନ ପତ୍ର କୁହାଯାଏ ।

34. ଦେୟତା କାହାକୁ କୁହାଯାଏ ?
Answer:
ଅଂଶୀଦାରମାନଙ୍କୁ ଓ ଜମାକାରୀମାନଙ୍କୁ ଯାହା ବ୍ୟାଙ୍କ୍ ଦେବାକୁ ବାଧ୍ୟ ହୋଇଥାଏ ତାହା ହେଉଛି ଦେୟତା ।

35. ପରିସମ୍ପତ୍ତି କ’ଣ ?
Answer:
ବ୍ୟାଙ୍କୁ ବିଭିନ୍ନ ସୂତ୍ରରୁ ମିଳୁଥିବା ପୁଞ୍ଜି ଓ ଅନ୍ୟାନ୍ୟ ସାମଗ୍ରୀକୁ ପରିସମ୍ପତ୍ତି କୁହାଯାଏ ।

36. କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ପୁନଃ ଅବମୂଲ୍ୟାୟନ ହାରର ଅନ୍ୟନାମ କ’ଣ ?
Answer:
କେନ୍ଦ୍ରୀୟ ବ୍ୟାଙ୍କର ପୁନଃ ଅବମୂଲ୍ୟାୟନ ହାରର ଅନ୍ୟ ନାମ ହେଉଛି ବ୍ୟାଙ୍କ୍ ହାର ।

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(b)

Question 1.
Each question given below has four possible answers, out of which only one is correct. Choose the correct one.
(i) (2î – 4ĵ) . (î + ĵ + k̂) = _______.
(a) -3
(b) +2
(c) -1
(d) -2
Solution:
(d) -2

(ii) If a = î + 2ĵ – k̂, b = î + ĵ + 2k̂, c = 2î – ĵ; then
(a) \(\vec{a} \perp \vec{b}\)
(b) \(\vec{b} \perp \vec{c}\)
(c) \(\vec{a} \perp \vec{c}\)
(d) no pair of vectors are perpendicular.
Solution:
(c) \(\vec{a} \perp \vec{c}\)

(iii) (-3, λ, 1) ⊥ (1, 0, -3) ⇒ λ = _______.
(a) 0
(b) 1
(c) impossible to find
(d) any real number
Solution:
(c) impossible to find

(iv) If \(\vec{a} \cdot \vec{b}=\vec{c} \cdot \vec{a}\) for all vectors \(\vec{a}\), then
(a) \(\vec{a} \perp(\vec{b}-\vec{c})\)
(b) \(\vec{b}-\vec{c}\) = 0
(c) \(\vec{b} \neq \vec{c}\)
(d) \(\vec{b}+\vec{c}=\overrightarrow{0}\)
Solution:
(b) \(\vec{b}-\vec{c}\) = 0

Question 2.
Find the scalar product of the following pairs of vectors and the angle between them.
(i) 3î – 4ĵ and -2î + ĵ
Solution:

(ii) 2î – 3ĵ + 6k̂ and 2î – 3ĵ – 5k̂
Solution:

(iii) î – ĵ and ĵ + k̂
Solution:

(iv) \(\vec{a}\) = (2, -2, 1) and \(\vec{b}\) = (0, 2, 4)
Solution:

Question 3.
If A, B, C are the points (1, 0, 2), (0, 3, 1) and (5, 2, 0) respectively, find m∠ABC.
Solution:

Question 4.
Find the value of λ so that the vectors \(\vec{a}\) and \(\vec{b}\) are perpendicular to each other.
(i) \(\vec{a}\) = 3î + 4ĵ, \(\vec{b}\) = -5î + λĵ
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (3î + 4ĵ) . (-5î + λĵ) = 0
⇒ -15 + 4λ = 0
⇒ λ = \(\frac{15}{4}\)

(ii) \(\vec{a}\) = î + ĵ + λk̂, \(\vec{b}\) = 4î – 3k̂
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ ( î + ĵ + λk̂) . (4î – 3k̂) = 0
⇒ 4 + 0 – 3λ = 0
⇒ λ = \(\frac{4}{3}\)

(iii) \(\vec{a}\) = 2î – ĵ – k̂, \(\vec{b}\) = λî + ĵ + 5k̂
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (2î – ĵ – k̂) . (λî + ĵ + 5k̂) = 0
⇒ 2λ – 1 – 5 = 0
⇒ 2λ = 6
⇒ λ = 3

(iv) \(\vec{a}\) = (6, 2, -3), \(\vec{b}\) = (1, -4, λ)
Solution:
If \(\vec{a}\) and \(\vec{b}\) are perpendicular \(\vec{a} \cdot \vec{b}\) = 0
⇒ (6, 2, -3) . (1, -4, λ) = 0
⇒ 6 – 8 – 3λ = 0
⇒ -2 – 3λ = 0
⇒ λ = –\(\frac{2}{3}\)

Question 5.
Find the scalar and vector projections of \(\vec{a}\) on \(\vec{b}\).
(i) \(\vec{a}\) = î, \(\vec{b}\) = ĵ
Solution:

(ii) \(\vec{a}\) = î + ĵ, \(\vec{b}\) = ĵ + k̂
Solution:

(iii) \(\vec{a}\) = î – ĵ – k̂, \(\vec{b}\) = 3î + ĵ + 3k̂
Solution:

Question 6.
In each of the problems given below, find the work done by a force \(\overrightarrow{F}\) acting on a particle, such that the particle is displaced from a point A to a point B.
(i) \(\overrightarrow{F}\) = 4î + 2ĵ + 3k̂
A (1, 2, 0), B (2, -1, 3)
Solution:
Displacement of the particle \(\overrightarrow{S}=\overrightarrow{AB}\)
= (2 – 1)î + (-1 – 2)ĵ + (3 – 0)k̂
=î – 3ĵ + 3k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (4î + 2ĵ + 3k̂) . (î – 3ĵ + 3k̂)
= 4 – 6 + 9
= 7 units.

(ii) \(\overrightarrow{F}\) = 2î + ĵ – k̂
A (0, 1, 2), B (-2, 3, 0)
Solution:
Displacement
\(\vec{S}\) = (-2 – 0)î + (3 – 1)ĵ + (0 – 2)k̂
= -2î + 2ĵ – 2k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (2î + ĵ – k̂) . (-2î + 2ĵ – 2k̂)
= -4 + 2 + 2
= 0 units.

(iii) \(\overrightarrow{F}\) = 4î – 3k̂
A (1, 2, 0), B (0, 2, 3)
Solution:
Displacement \(\vec{S}\) = -î + 3k̂
Work done = \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (4î – 3k̂) . (-î + 3k̂)
= -4 – 9
= -13 units.

(iv) \(\overrightarrow{F}\) = 3î – ĵ – 2k̂
A (-3, -4, 1), B (-1, -1, -2)
Solution:
Displacement \(\vec{S}\) = 2î + 3ĵ – 3k̂
Work done \(\overrightarrow{F} \cdot \overrightarrow{S}\)
= (3î – ĵ – 2k̂) . (2î + 3ĵ – 3k̂)
= 6 – 3 + 6
= 9 units.

Question 7.
If \((\vec{a}+\vec{b}) \cdot(\vec{a}-\vec{b})\) = 0 show that \(|\vec{a}|=|\vec{b}|\).
Solution:

Question 8.
(i) If a and b are perpendicular vectors show that

Solution:

(ii) Prove that two vectors are perpendicular iff \(|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2\)
Solution:

Question 9.
If \(\vec{a}, \vec{b}, \vec{c}\) are mutually perpendicular vectors of equal magnitude, show that \(\vec{a}+\vec{b}+\vec{c}\) is equally inclined to \(\vec{a} \cdot \vec{b} \cdot \vec{c}\)
Solution:

Question 10.
Prove the following by vector method.
(i) Altitudes of a triangle are concurrent;
Solution:
Let ABC be a triangle.

⇒ CF is perpendicular to AB.
Hence the altitudes of a triangle are concurrent.

(ii) Median to the base of an isosceles triangle is perpendicular to the base;
Solution:

⇒ OD is perpendicular to the base AB.
Hence the median to the base of an isosceles triangle is perpendicular to the base. (Proved)

(iii) The parallelogram whose diagonals are equal is a rectangle;
Solution:

⇒ m∠COA = 90°
Hence OABC is a rectangle. (Proved)

(iv) The diagonals ofa rhombus are at right angles;
Solution:

Hence the diagonals of a rhombus are at right angles. (Proved)

(v) An angle inscribed in a semi-circle is a right angle;
Solution:

∴ m∠ABC = 90°
Hence the angle inscribed in a semi-circle is a right-angle. (Proved)

(vi) In any triangle ABC; a = b cos C + c cos B;
Solution:

(vii) In a triangle AOB, m∠AOB = 90°. If P and Q are the points of trisection of AB, prove that OP2 + OQ2 = \(\frac{5}{9}\) AB2;
Solution:

(viii) Measure of the angle between two diagonals of a cube is cos-1\(\frac{1}{3}\).
Solution:

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(c) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(c)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) (î + k̂) × (î + ĵ + k̂) = ______.
(a) î – k̂
(b) k̂ – î
(c) k̂ – 2î – ĵ
(d) 2
Solution:
(î + k̂) × (î + ĵ + k̂) = \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array}\right|\)
= î (0 – 1) – ĵ (1 – 1) + k̂ (1 – 0)
= -î + k̂ = k̂ – î

(ii) A vector perpendicular to the vectors î + ĵ and î + k̂ is ______.
(a) î – ĵ – k̂
(b) ĵ – k̂ + î
(c) k̂ – ĵ – î
(d) ĵ + k̂ + î
Solution:
A vector perpendicular to the vectors î + ĵ and î + k̂ is
(î + ĵ) × (î + k̂) = \(\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
1 & 1 & 0 \\
1 & 0 & 1
\end{array}\right|\)
= î (1 – 0) – ĵ (1 – 0) + k̂ (0 – 1)
= î – ĵ – k̂

(iii) The area of the triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) is ______.
(a) \(\frac{1}{2}\)
(b) 1
(c) \(\frac{\sqrt{3}}{2}\)
(d) 2
Solution:

(iv) If â and b̂ are unit vectors such that â × b̂ is a unit vector, then the angle between â and b̂ is ______.
(a) of any measure
(b) \(\frac{\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) π
Solution:
|a × b| = ab sin θ = sin θ
⇒ sin θ = 1
⇒ θ = \(\frac{\pi}{2}\)

(v) If \(\vec{a}, \vec{b} \text { and } \vec{c}\) are non-zero vectors, then \(\vec{a} \times \vec{b}=\vec{a} \times \vec{c}\) ______.
(a) \(\vec{b}=\vec{c}\)
(b) \(\vec{a} \|(\vec{b}-\vec{c})\)
(c) \(\vec{b} \| \vec{c}\)
(d) \(\vec{b} \perp \vec{c}\)
Solution:

Question 2.
Let \(\vec{a}\) = 2î + ĵ, \(\vec{b}\) = -î + 3ĵ + k̂ and \(\vec{c}\) = î + 2ĵ + 5k̂ be three vectors. Find
(i) \(\vec{c} \times \vec{a}\)
Solution:

(ii) \(\vec{a} \times(-\vec{b})\)
Solution:

(iii) \((\vec{a}-2 \vec{b}) \times \vec{c}\)
Solution:

(iv) \((\vec{a}-\vec{c}) \times \vec{c}\)
Solution:

(v) \((\vec{a}-\vec{b}) \times(\vec{c}-\vec{a})\)
Solution:

Question 3.
Find the unit vectors perpendicular to the vectors
(i) î, k̂
Solution:

(ii) î + ĵ, î – k̂
Solution:

(iii) 2î + 3k̂, î – 2ĵ
Solution:

(iv) 2î – 3ĵ + k̂, -î + 2ĵ – k̂
Solution:

Question 4.
Determine the area of parallelogram whose adjacent sides are the vectors
(i) 2î, ĵ
Solution:

(ii) î + ĵ, -î + 2ĵ
Solution:

(iii) 2î + ĵ + 3k̂, î – ĵ
Solution:

(iv) (1, – 3, 1), (1, 1, 1).
Solution:

Question 5.
Calculate the area of the traingle ABC (by vector method) where
(i) A (1, 2, 4), B (3, 1, -2), C (4, 3, 1)
Solution:

(ii) A (1, 1, 2), B (2, 2, 3), C (3, -1, -1).
Solution:

Question 6.
Determine the sine of the angle between the vectors
(i) 5î – 3ĵ, 3î – 2k̂
Solution:

(ii) î – 3ĵ + k̂, î + ĵ + k̂
Solution:

Question 7.
Show that \((\vec{a} \times \vec{b})^2\) = a²b² – \((\vec{a}, \vec{b})^2\)
Solution:

Question 8.
If \(\vec{a} \times \vec{b}=\vec{b} \times \vec{c} \neq \overrightarrow{0}\), prove that \(\vec{a}+\vec{c}=m \vec{b}\), where m is a scalar.
Solution:

Question 9.
If \(\vec{a}\) = 2î + ĵ – k̂, \(\vec{b}\) = -î + 2ĵ – 4k̂, \(\vec{c}\) = î + ĵ + k̂, find \((\vec{a} \times \vec{b}) \cdot(\vec{a} \times \vec{c})\).
Solution:

Question 10.
If \(\vec{a}\) = 3î + ĵ – 2k̂, \(\vec{b}\) = 2î – 3ĵ + 4k̂ then verify that \(\vec{a} \times \vec{b}\) is perpendicular to both \(\vec{a}\) and \(\vec{b}\).
Solution:

Question 11.
Find the area of the parallelogram whose diagonals are vectors 3î + ĵ – 2k̂ and î – 3ĵ + 4k̂.
Solution:

Question 12.
Show that \((\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})=2(\vec{a} \times \vec{b})\). Interpret this result geometrically.
Solution:

= Vector area of the parallelogram ABCD.
Hence twice the vector area of a parallelogram ABCD is equal to the vector area of the parallelogram whose adjacent sides are the diagonals of the parallelogram ABCD.

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Additional Exercise Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Additional Exercise

(A) Multiple Choice Questions (Mcqs) With Answers

Question 1.
If f is an odd function, then write the value of \(\int_{-a}^a \frac{f(\sin x)}{f(\cos x)+f\left(\sin ^2 x\right)}\) dx
(a) 1
(b) 0
(c) -1
(d) 2
Solution:
(b) 0

Question 2.
If p and q are respectively degree and order of the differential equation y = e^dy/dx then write the relation between p and q.
(a) p ≠ q
(c) p ≡ q
(b) p = q
(d) None of these
Solution:
(b) p = q

Question 3.
Write the value of \(\int_0^1\){x} dx where {x} stands for fractional part of x.
(a) \(\frac{1}{2}\)
(b) \(\frac{3}{2}\)
(c) \(\frac{1}{4}\)
(d) \(\frac{2}{3}\)
Solution:
(a) \(\frac{1}{2}\)

Question 4.
Write the value of:
\(\int_0^{\pi / 2} \frac{\sin x}{\sin x+\cos x}\) dx – \(\int_0^{\pi / 2} \frac{\cos x}{\sin x+\cos x}\) dx
(a) 1
(b) 2
(c) 0
(d) π
Solution:
(c) 0

Question 5.
Write the value of \(\int_{\frac{\pi}{4}}^{\frac{\pi}{4}}\)sin⁵ x cos x dx
(a) 0
(b) 1
(c) cos x
(d) sin x
Solution:
(a) 0

Question 6.
Write the particular solution of the equation \(\frac{d y}{d x}\) = sin x given that y(π) = 2
(a) y = cos x + 1
(b) y = -cos x + 1
(c) y = -cos x – 1
(d) y = -sin x + 1
Solution:
(b) y = -cos x + 1

Question 7.
Write the degree of the following differential equation:
\(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(d) 3

Question 8.
Write the order ofthe following differential equation:
\(\frac{d^2 y}{d x^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(c) 2

Question 9.
What is F(x) if F(x) = \(\int_0^x\)e^2t sin 3t dt?
(a) e^2x sin 3x
(b) e^2x cos 3x
(c) e^x sin 3x
(d) e^2x sin x
Solution:
(a) e^2x sin 3x

Question 10.
\(\int \frac{d x}{\cos ^2 x \sin ^2 x}\) = ?
(a) -2 cos 2x + C
(b) -2 cot 2x + C
(c) -2 sin 2x + C
(d) 2 cot 2x + C
Solution:
(b) -2 cot 2x + C

Question 11.
If \(\int_1^2\)f(x) dx= λ, then what is the value of \(\)f(3 – x) dx?
(a) λ
(b) λ²
(c) 1λ
(d) 2λ
Solution:
(a) λ

Question 12.
What is the value of \(\int_{-1}^1 \frac{d x}{1+x^2}\)?
(a) \(\frac{2 \pi}{2}\)
(b) 2π
(c) π
(d) \(\frac{\pi}{2}\)
Solution:
(d) \(\frac{\pi}{2}\)

Question 13.
Write the order of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^4\) + y
(a) 1
(b) 3
(c) 2
(d) 0
Solution:
(b) 3

Question 14.
Write the degree of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^4\) + y
(a) 1
(b) 2
(c) 3
(d) 0
Solution:
(a) 1

Question 15.
Write the particular solution of \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = (1 + x)⁴, y = 0 when x = -1.
(a) y = \(\frac{(1+x)^2}{5}\)
(b) y = \(\frac{(2+x)^5}{5}\)
(c) y = \(\frac{(1-x)^5}{5}\)
(d) y = \(\frac{(1+x)^5}{5}\)
Solution:
(d) y = \(\frac{(1+x)^5}{5}\)

Question 16.
Evaluate the integral ∫2x cosec² x² dx?
(a) cot x² + C
(b) -cot x² + C
(c) -cot 2x² + C
(d) cot 2x² + C
Solution:
(b) -cot x² + C

Question 17.
What is the value of \(\frac{d}{d x} \int_{250}^{300}\left(x^4+5 x^3\right)^2\) dx
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
(a) 0

Question 18.
Write down the integral of ∫\(e^{x^2}\) 2x dx.
(a) \(e^{2 x^2}\)
(b) 2\(e^{2 x^2}\)
(c) \(e^{x^2}\)
(d) None of the above
Solution:
(c) \(e^{x^2}\)

Question 19.
What is the integral of ∫log e^x dx?
(a) \(\frac{2 x^2}{2}\) + C
(b) \(\frac{2 x^2}{3}\) + C
(c) \(\frac{x^2}{2}\) + C
(d) None of the above
Solution:
(c) \(\frac{x^2}{2}\) + C

Question 20.
What is the value of \(\int_{-2}^2\)|x| dx?
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(a) 0

Question 21.
\(\int_{-1}^1\)|1 – x| dx = ______.
(a) 0
(b) 1
(c) 2
(d) -1
Solution:
(c) 2

Question 22.
If ∫x³\(e^{c x^4}\)dx = \(\frac{1}{20} \mathrm{e}^{\mathrm{cx}}\) then C = ______.
(a) 0
(b) 2
(c) 4
(d) 5
Solution:
(d) 5

Question 23.
\(\int_a^b\)f(x) dx = 1 ⇒ \(\int_a^b\)k f(t)dt ______.
(a) k
(b) -k
(c) 2k
(d) None of the above
Solution:
(b) -k

Question 24.
\(\int_{-1}^1\)f(x) dx = k and f is an even function then \(\int_{-1}^1\)f(x) = ______.
(a) k
(b) -k
(c) 2k
(d) None of the above
Solution:
(c) 2k

Question 25.
If ∫\(\int_0^1\)f(x) dx = 4, \(\int_0^2\)f(t) dt and \(\int_4^2\)f(u) du = 1 then \(\int_1^4\)f(x) dx = ______.
(a) 0
(b) 1
(c) 3
(d) -3
Solution:
(d) -3

Question 26.
I(f) = \(\int_a^x\)f(t) dt and Df = f'(x) then (ID – DI) f = ______.
(a) -f(a)
(b) 2f(a)
(c) f(a)
(d) None of the above
Solution:
(a) -f(a)

Question 27.
\(\int_0^\pi\)cos¹⁰¹ x dx = ______.
(a) 0
(b) 1
(c) -1
(d) 101
Solution:
(a) 0

Question 28.
Let f satisfies all the conditions of Rolle’s theorem in [1, 6] then \(\int_1^6\)f'(x) dx = ______.
(a) 0
(b) 1
(c) -1
(d) 6
Solution:
(a) 0

Question 29.
\(\int_{-2}^2\)|x| dx = ______.
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
(d) 4

Question 30.
Integrate ∫log x dx
(a) x. log x + x + C
(b) x. log x – x + C
(c) log x – x + C
(d) None of these
Solution:
(b) x. log x – x + C

Question 31.
Evaluate \(\int_0^2\)[x – 1] dx
(a) 0
(b) 1
(c) -1
(d) 2
Solution:
(b) 1

Question 32.
What is the value of: ∫\(\frac{f^{\prime}(x)-f(x)}{e^x}\) dx?
(a) e^x f(x) + C.
(b) e^2x f(x) + C.
(c) e^-x f(x) + C.
(d) None of the above
Solution:
(c) e^-x f(x) + C.

Question 33.
What is the value of \(\int_0^1\)x(1 – x)⁹⁹ dx?
(a) \(\frac{1}{100}\)
(b) \(\frac{1}{10}\)
(c) \(\frac{1}{1010}\)
(d) \(\frac{1}{10100}\)
Solution:
(d) \(\frac{1}{10100}\)

Question 34.
Solution of \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = xy + x + y + 1 is ______.
(a) 2x + \(\frac{x^2}{2}\) + C
(b) x + \(\frac{x}{2}\) + C
(c) x + \(\frac{2 x^2}{2}\) + C
(d) x + \(\frac{x^2}{2}\) + C
Solution:
(d) x + \(\frac{x^2}{2}\) + C

Question 35.
f(x) = \(\int_0^x\)t sin t dt then f ‘(x) = ______.
(a) x cos x
(b) x sin t
(c) x sin x
(d) x tan x
Solution:
(c) x sin x

Question 36.
What is the value of the integral \(\int_a^b \frac{|x|}{x}\)dx?
(a) |b| – |a|
(b) |a| – |b|
(c) |b| + |a|
(d) |a| + |b|
Solution:
(a) |b| – |a|

Question 37.
What is the value of ∫x^x (1 + ln x) dx?
(a) x^2x + C
(b) x^x + C
(c) 2x^x + C
(d) x² + C
Solution:
(b) x^x + C

Question 38.
Evaluate: \(\int_0^{\mathrm{p} / 2}\)ln(cot x) dx.
(a) 0
(b) 1
(c) cot x
(d) sin x
Solution:
(a) 0

Question 39.
Evaluate: \(\int_{-3}^4\)|x| dx
(a) \(\frac{2}{25}\)
(b) \(\frac{25}{2}\)
(c) \(\frac{25}{4}\)
(d) \(\frac{25}{-3}\)
Solution:
(b) \(\frac{25}{2}\)

Question 40.
Evaluate: \(\int_0^{\frac{\pi}{2}}\)(cos x – sin x) dx
(a) 0
(b) 1
(c) -1
(d) π
Solution:
(a) 0

Question 41.
Evaluate: \(\int_0^{\frac{\pi}{2}}\)log tan x dx.
(a) 1
(b) -1
(c) 0
(d) π
Solution:
(c) 0

Question 42.
Integrate: \(\frac{d x}{3 e^x-1}\)
(a) \(\ln \left(\frac{e^{3 x}-1}{e^x}\right)\) + C
(b) \(\ln \left(\frac{3 e^x+1}{e^x}\right)\) + C
(c) \(\ln \left(\frac{3 e^x-1}{e^x}\right)\) + C
(d) \(\ln \left(\frac{3 e^x+1}{e^{3 x}}\right)\) + C
Solution:
(c) \(\ln \left(\frac{3 e^x-1}{e^x}\right)\) + C

Question 43.
Evaluate: \(\int_0^1 \ln \left(\frac{1}{x}-1\right)\)dx
(a) 1
(b) 2
(c) 0
(d) -1
Solution:
(c) 0

Question 44.
Evaluate: ∫e^x\(\left(\frac{1-\sin x}{1-\cos x}\right)\)dx
(a) -e^x cot\(\frac{x}{2}\) + C
(b) e^x tan\(\frac{x}{2}\) + C
(c) e^x cot\(\frac{x}{2}\) + C
(d) -e^x sin\(\frac{x}{2}\) + C
Solution:
(a) -e^x cot\(\frac{x}{2}\) + C

Question 45.
Evaluate: \(\int_0^1\)x log(1 + x) dx
(a) \(\frac{1}{2}\)
(b) \(\frac{1}{4}\)
(c) \(\frac{1}{3}\)
(d) \(\frac{2}{3}\)
Solution:
(b) \(\frac{1}{4}\)

Question 46.
What is the integrating factor of the equation y’ + y cot x = cosec x?
(a) cot x
(b) sin x
(c) cos x
(d) cosec x
Solution:
(b) sin x

(B) Very Short Type Questions With Answers

Question 1.
Write the order of the differential equation whose solution is given by
y = (c₁ + c₂) cos (x + c₃) + c₄\(e^{x+c_5}\) where c₁, c₂, c₄ and c₅ are arbitrary constants.
Solution:
y = (c₁ + c₂) cos (x + c₃) + c₄\(e^{x+c_5}\)
y = (c₁ + c₂) cos (x + c₃) + c₄\(e^{c_5}\).e^x
= A cos(x + c₃) + Be^x
Where c₁ + c₂ = A, c₄\(e^{c_5}\) = B
As there are 3 independent constants the order of the differential equation is 3.

Question 2.
If p and q are respectively degree and order of the differential equation y = e^dy/dx, then write the relation between p and q.
Solution:
Given differential equation is
y = \(e^{\frac{d y}{d x}}\) ⇒ \(\frac{d y}{d x}\) = ln y
Whose order = 1 = p
Degree = 1 = q
∴ p = q

Question 3.
Write the value of \(\int_0^1\){x} dx where {x} stands for fractional part of x.
Solution:

Question 4.
Write the order of the differential equation of the family of circles
ar² + ay² + 2gx + 2fy + c = 0
ax² + ay² + 2gx + 2fy + c = 0
Solution:
As there are 3 independent constants, the order of the differential equation is 3.

Question 5.
If p and q are the order and degree of the differential equation
y\(\left(\frac{d y}{d x}\right)^2\) + x² \(\frac{d^2 y}{d x^2}\) + xy = sin x, then choose the correct statement out of (i) p > q, (ii) p = q, (iii) p < q.
Solution:
Order of the given differential = p = 2
Degree of the given differential equation = q = 1
∴ p > q

Question 6.
Write the order of the differential equation of the system of ellipses:
\(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1
Solution:
As there are two unknown constants in the system of ellipses \(\frac{x^2}{a^2}\) + \(\frac{y^2}{b^2}\) = 1 the order of the differential equation is 2.

Question 7.
What do you mean by integration? Write your answer in one sentence.
Solution:
Integration is the antiderivative of a function.

Question 8.
Write the differential equation of the family of straight lines parallel to the y-axis.
Solution:
\(\frac{d x}{d y}\) = 0 is the differential equation of family of lines parallel to y-axis.

Question 9.
Write the value of ∫\(\int_{-\pi / 4}^{\pi / 4}\)sin⁵ x cos x dx.
Solution:
Let f(x) = sin⁵ x cos x
f(-x) = sin⁵ (-x) cos (-x)
= -sin⁵ x cos x = -f(x)
i.e. f is an odd function.
Thus \(\int_{-\pi / 4}^{\pi / 4}\)sin⁵ x cos x dx = 0

Question 10.
Write the degree of the differential equation ln\(\left(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\right)\) = y
Solution:
The degree of the differential equation ln\(\left(\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}\right)\) = y is 1.

Question 11.
What is F'(t) if F(t) = \(\int_a^t\)e^3x .cos 2x dx ?
Solution:
F(t) = \(\int_a^t\)e^3x .cos 2x dx
⇒ F'(t) = e^3x cos 2t

Question 12.
Write the order and degree of the following differential equation:
\(\frac{d^2 y}{d x^2}\) = \(\frac{2 y^3+\left(\frac{d y}{d x}\right)^4}{\sqrt{\frac{d^2 y}{d x^2}}}\)
Solution:
Order = 2, Degree = 3

Question 13.
∫\(\frac{\cot x d x}{\ln \sin x}\) = ?
Solution:
∫\(\frac{\cot x d x}{\ln \sin x}\) = ln(ln sin x) + C

Question 14.
What is F'(x) if F(x) = \(\int_0^{\mathbf{x}}\)e^2t sin 3t dt?
Solution:
If F(x) = \(\int_0^{\mathbf{x}}\)e^2t sin 3t dt then F'(x) = e^2x sin 3x

Question 15.
∫\(\frac{d x}{\cos ^2 x \sin ^2 x}\) = ?
Solution:
∫\(\frac{d x}{\cos ^2 x \sin ^2 x}\) = 4∫\(\frac{d x}{\sin ^2 2 x}\)
= 4∫cosec² 2x dx = -2 cot 2x + C

Question 16.
What is the value of ∫\(\frac{d}{d x}\)f(x) dx – \(\frac{d}{d x}\)(∫f(x) dx)?
Solution:
∫\(\frac{d}{d x}\)f(x) dx – \(\frac{d}{d x}\)(∫f(x) dx)
= f(x) + C – f(x) = C (constant)

Question 17.
If \(\int_1^2\)f(x) dx = λ, then what is the value \(\int_1^2\)f(3 – x) dx?
Solution:
If \(\int_1^2\)f(x) dx = λ, then \(\int_1^2\)f(3 – x) dx = λ

Question 18.
What is the value of \(\int_{-1}^1 \frac{d x}{1+x^2}\)?
Solution:
\(\int_{-1}^1 \frac{d x}{1+x^2}\) = \(\left[\tan ^{-1} x\right]_{-1}^1\)
= tan^-1 1 – tan^-1 (-1)
= tan^-1 1 + tan^-1 1
= 2tan^-1 (1) = 2 . \(\frac{\pi}{4}\) = \(\frac{\pi}{2}\)

Question 19.
Write the order and the degree of the following differential equation:
\(\frac{d^3 y}{d x^3}\) = \(\left(\frac{d^2 y}{d x^2}\right)^2\) + \(\left(\frac{d y}{d x}\right)^4\) + y
Solution:
Order = 3
Degree = 1

Question 20.
Write the particular solution of \(\frac{d y}{d x}\) = (1 + x)⁴, y = 0 when x = -1.
Solution:
\(\frac{d y}{d x}\) = (1 + x)⁴ ⇒ \(\frac{(1+x)^5}{5}\) + C
Given y = 0 for x = -1
⇒ o = o + c ⇒ c = o
∴ The particular solution is y = \(\frac{(1+x)^5}{5}\)

(C) Short Type Questions With Answers

Question 1.
Evaluate: ∫\(\frac{2 x+1}{\sqrt{x^2+10 x+29}}\)dx
Solution:

Question 2.
Evaluate: \(\int_0^{\pi / 2} \frac{\cos x d x}{(2-\sin x)(3+\sin x)}\)
Solution:

Question 3.
Evaluate: ∫\(\frac{d x}{(1+x) \sqrt{1-x^2}}\)
Solution:

Question 4.
Solve: cosec x \(\frac{d^2 y}{d x^2}\) = x.
Solution:
cosec x \(\frac{d^2 y}{d x^2}\) = x => \(\frac{d^2 y}{d x^2}\) = x sin x
⇒ \(\frac{d y}{d x}\) = ∫x sin x dx + A
= x (-cos x) – ∫(-cos x) dx + A
= -x cos x + sin x + A
⇒ y = -∫x cos x dx + ∫sin x dx + A∫dx + B
= [x sin x – ∫sin x dx] – cos x + Ax = B
⇒ y = -x sin x – 2 cos x + Ax + B is the solution.

Question 5.
Find the particular solution of the following differential equation:
\(\frac{d y}{d x}\) = \(\frac{1+y^2}{1+x^2}\) given that y = √3 when x = 1
Solution:

Question 6.
Evaluate: \(\int_0^a x^2\left(a^2-x^2\right)^{5 / 2}\) dx
Solution:

Question 7.
Evaluate: \(\int_0^a \frac{d x}{e^{4 x}-5}\)
Solution:

Question 8.
Evaluate: ∫x² tan^-1 x dx.
Solution:

Question 9.
If f(x) = e^x + \(\frac{1}{1+x^2}\) and f(0) = 1, then find f(x).
Solution:
f(x) = e^x + \(\frac{1}{1+x^2}\)
⇒ f(x) = ∫\(\left(e^x+\frac{1}{1+x^2}\right)\)dx + C
= e^x + tan^-1 x + C
f(0) = 1
⇒ 1 = 1 + 0 + C => C = 0
Thus f(x) = e^x + tan^-1 x

Question 10.
Evaluate: ∫(log x)² dx
Solution:
I = ∫(log x)² dx
= (log x)². x – 2∫(log x) . \(\frac{1}{x}\) . x . dx
= x (log x)² – 2 ∫log x. dx
= x (log x)² – 2 {(log x) x – ∫dx}
= x (log x)² – 2x log x + 2x + C

Question 11.
Evaluate: ∫\(\frac{2 x+9}{(x+3)^2}\)dx
Solution:

Question 12.
Solve: ydy + e^-y x sin x dx = 0
Solution:
ydy = e^-y x sin x dx = 0
⇒ y e^y dy + x sin x dx = 0
⇒ ∫y e^y dy + ∫x sin x dx =C
⇒ y e^y – e^y + (-x cos x) + sin x = C
⇒ e^y (y – 1) – x cos x + sin x = C is the general solution.

Question 13.
Evaluate: ∫\(\frac{d x}{x \ln x \sqrt{(\ln x)^2-4}}\)
Solution:

Question 14.
Find the particular solution of the differential equation \(\frac{d^2 y}{d x^2}\) = 6x given that y = 1 and \(\frac{d y}{d x}\) = 2 when x = 0.
Solution:
\(\frac{d^2 y}{d x^2}\) = 6x ⇒ \(\frac{d y}{d x}\) = 6 . \(\frac{x^2}{2}\) + A
\(\frac{d y}{d x}\) = 3x² + A ⇒ y = x³ + Ax + B
Using the givne conditions x = 0, \(\frac{d y}{d x}\) = 2, y = 1, we get
2 = 0 + A ⇒ A = 2
and 1 = 0 + 0 + B ⇒ B = 1
The particular solution is y = x³ + 2x + 1

Question 15.
Evaluate: \(\int_0^{\frac{3}{2}}\)[x²] dx
Solution:

Question 16.
Find the differential equation whose general solution is ax² + by = 1, where a and b are arbitrary constants.
Solution:

Question 17.
Integrate: ∫\(\frac{\sin 6 x+\sin 4 x}{\cos 6 x+\cos 4 x}\) dx.
Solution:

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(a)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) \(\vec{a}\) = î + 2ĵ + k̂, \(\vec{b}\) = 2î – 2ĵ + 2k̂ and \(\vec{c}\) = -î + 2 ĵ + k̂ then
(a) \(\vec{a}\) and \(\vec{b}\) have the same direction
(b) \(\vec{a}\) and \(\vec{c}\) have opposite directions.
(c) \(\vec{b}\) and \(\vec{c}\) have opposite directions
(d) no pair of vectors have same direction
Solution:
(d) no pair of vectors have same direction

(ii) If the vectors \(\vec{a}\) = 2î + 3ĵ – 6k̂ and \(\vec{b}\) = -α î – ĵ + 2k̂ are parallel, then α = ______.
(a) 2
(b) \(\frac{2}{3}\)
(c) –\(\frac{2}{3}\)
(d) \(\frac{1}{3}\)
Solution:
(c) –\(\frac{2}{3}\)

(iii) If the position vectors of two points A and B are 3î + k̂, and 2î + ĵ – k̂, then the vector \(\overrightarrow{BA}\) is
(a) -î + ĵ – 2k̂
(b) î + ĵ
(c) î – ĵ + 2k̂
(d) î – ĵ – 2k̂
Solution:
(c) î – ĵ + 2k̂

(iv) If \(|k \vec{a}|\) = 1, then
(a) \(\vec{a}=\frac{1}{k}\)
(b) \(\vec{a}=\frac{1}{|k|}\)
(c) \(k=\frac{1}{|\vec{a}|}\)
(d) \(k=\frac{+1}{|\vec{a}|}\)
Solution:
(d) \(k=\frac{+1}{|\vec{a}|}\)

(v) The direction cosines of the vectors \(\overrightarrow{PQ}\) where \(\overrightarrow{OP}\) = (1, 0, -2) and \(\overrightarrow{OQ}\) = (3, -2, 0) are
(a) 2, -2, 2
(b) 4, -2, -2
(c) \(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)
(d) \(\frac{2}{\sqrt{6}},-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}}\)
Solution:
(c) \(\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\)

Question 2.
Rectify the mistakes, if any
(i) \(\vec{a}-\vec{a}\) = 0
Solution:
\(\overrightarrow{0}\)

(ii) The vector \(\overrightarrow{0}\) has unique direction.
Solution:
indefinite direction

(iii) All unit vectors are equal.
Solution:
equal magnitude

(iv) \(|\vec{a}|=|\vec{b}| \Rightarrow \vec{a}=\vec{b}\)
Solution:
\(\vec{a}=\vec{b} \Rightarrow|\vec{a}|=|\vec{b}|\)

(v) Subtraction of vectors is not commutative.
Solution:
true

Question 3.
(i) If \(\vec{a}\) = (2, 1), \(\vec{b}\) = (-1, 0), find \(3 \vec{a}+2 \vec{b}\).
Solution:
\(3 \vec{a}+2 \vec{b}\) = 3 (2, 1) + 2 (-1, 0)
= (6 – 2, 3 + 0)
= (4, 3 )

(ii) If \(\vec{a}\) = (1, 1, 1) , \(\vec{b}\) = (-1, 3, 0) and \(\vec{c}\) =(2, 0, 2), find \(\vec{a}+2 \vec{b}-\frac{1}{2} \vec{c}\).
Solution:
\(\vec{a}+2 \vec{b}-\frac{1}{2} \vec{c}\)
= (1, 1, 1) + 2 (-1, 3, 0) – \(\frac{1}{2}\)(2, 0, 2)
= (1 – 2 – 1, 1 + 6 – 0, 1 + 0 – 1)
= (-2, 7, 0)

Question 4.
If A, B, C and D are the vertices of a square, find \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CD}+\overrightarrow{DA}\).
Solution:
Let ABCD be a square.

Question 5.
The given points A, B, C are the vertices of a triangle. Determine the vectors \(\overrightarrow{A B}, \overrightarrow{B C} \text { and } \overrightarrow{C A}\) and the lengths of these vectors in the following cases.
(i) A (4, 5, 5), B (3, 3, 3), C (1, 2, 5)
Solution:

(ii) A (8, 6, 1), B (2, 0, 1), C (-4, 0, -5)
Solution:

Question 6.
Find the vector from origin to the midpoint of the vector \(\overrightarrow{{P}_1 {P}_2}\) joining the points P₁(4, 3) and P₂(8, -5).
Solution:
P₁ = (4, 3) and P₂ = (8, -5)
If P is the mid-point of P₁P₂ then P = (6, -1).
Position vector of P = \(\overrightarrow{{OP}}\) = 6î – ĵ

Question 7.
Find the vectors from the origin to the points of trisection the vector \(\overrightarrow{{P}_1 {P}_2}\) joining P₁ (-4, 3) and P₂ (5, -12).
Solution:

Question 8.
Find the vector from the origin to the intersection of the medians of the triangle whose vertices are A (5, 2, 1), B(-4, 7, 0) and C (5, -3, 5).
Solution:

Question 9.
Prove that the sum of all the vectors drawn from the centre of a regular octagon to its vertices is the null vector.
Solution:

Question 10.
Prove that the sum of the vectors represented by the sides of a closed polygon taken in order is a zero vector.
Solution:
Consider a closed polygon ABCDEFA.

Question 11.
(a) Prove that:
(i) \(|\overrightarrow{a}+\overrightarrow{{b}}| \leq|\overrightarrow{a}|+|\overrightarrow{b}|\)
State when the equality will hold;
Solution:

(ii) \(|\overrightarrow{a}-\overrightarrow{b}| \geq|\overrightarrow{a}|-|\overrightarrow{b}|\)
Solution:

(b) What is the geometrical significance of the relation \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\)?
Solution:

Question 12.
Find the magnitude of the vector \(\overrightarrow{PQ}\), its scalar components and the component vectors along the coordinate axes, if P and Q have the coordinates.
(i) P (-1, 3), Q (1, 2)
Solution:

(ii) P (-1, -2), Q (-5, -6)
Solution:

(iii) P (1, 4, -3), Q (2, -2, -1).
Solution:

Question 13.
In each of the following find the vector \(\overrightarrow{PQ}\), its magnitude and direction cosines, if P and Q have co-ordinates.
(i) P (2, -1, -1), Q (-1, -3, 2);
Solution:

(ii) P (3, -1, 7), Q (4, -3, -1).
Solution:

Question 14.
If \(\vec{a}\) = (2, -2, 1), \(\vec{b}\) = (2, 3, 6) and \(\vec{c}\) = (-1, 0, 2), find the magnitude and direction of
\(\vec{a}-\vec{b}+2 \vec{c}\).
Solution:

Question 15.
Determine the unit vector having the direction of the given vector in each of the following problems:
(i) 5î – 12ĵ
Solution:

(ii) 2î + ĵ
Solution:

(iii) 3î + 6ĵ – k̂
Solution:

(iv) 3î + ĵ – 2k̂
Solution:

Question 16.
Find the unit vector in the direction of the vector \(\overrightarrow{r_1}-\overrightarrow{r_2}\), where \(\vec{r}_1\) = î + 2ĵ + k̂ and \(\vec{r}_2\) = 3î + ĵ – 5k̂.
Solution:

Question 17.
Find the unit vector parallel to the sum of the vectors \(\vec{a}\) = 2î + 4ĵ – 5k̂ and \(\vec{b}\) = î + 2ĵ + 3k̂. Also find its direction cosines.
Solution:

Question 18.
If the sum of two unit vectors is a unit vector, show that the magnitude of their difference is √3.
Solution:

Question 19.
The position vectors of the points A, B, C and D are 4î + 3ĵ – k̂, 5î + 2ĵ + 2k̂, 2î – 2ĵ – 3k̂ and 4î – 4ĵ + 3k̂ respectively. Show that AB and CD are parallel.
Solution:
Given that the
position vector of A = 4î + 3ĵ – k̂
position vector of B = 5î + 2ĵ + 2k̂
position vector of C = 2î – 2ĵ – 3k̂
position vector of D = 4î – 4ĵ + 3k̂

Question 20.
In each of the following problems, show by vector method that the given points are collinear.
(i) A (2, 6, 3), B (1, 2, 7) and C (3, 10, -1)
Solution:
Given that A = (2, 6, 3), B = (1, 2, 7) and C = (3, 10, -1)
Then

(ii) P (2, -1, 3), Q (3, -5, 1) and R (-1, 11, 9).
Solution:
Given that P = (2, -1, 3) Q = (3, -5, 1) and R = (-1, 11, 9)

Hence the points P, Q, R are collinear. (Proved)

Question 21.
Prove that the vectors 2î – ĵ + k̂, î – 3ĵ – 5k̂, 3î – 4ĵ – 4k̂ are the sides of a right angled triangle.
Solution:
Let A, B and C be the points whose position vectors are 2î – ĵ – k̂, î – 3ĵ – 5k̂ and 3î – 4ĵ – 4k̂ respectively.

Question 22.
Prove by vector method that:
(a) the medians of a triangle are concurrent;
Solution:

The symmetry of the result shows that the point G also lies on the other two medians.
Hence the medians are concurrent. (Proved)

(b) the diagonals of a parallelogram bisect each other;
Solution:

(c) the line segment joining the midpoints of two sides of a triangle is parallel to the third and half of it;
Solution:

(d) the lines joining the midpoints of consecutive sides of a quadrilateral is a parallelogram;
Solution:

⇒ SR = PQ and SR || PQ
Hence PQRS is a parallelogram.
(Proved)

(e) in any triangle ABC, the point P being on the side \(\overrightarrow{B C} \text {; if } \overrightarrow{P Q}\) is the resultant of the vectors \(\overrightarrow{A P}, \overrightarrow{P B}\) and \(\overrightarrow{P C}\) then ABQC is a parallelogram;
Solution:

Hence ABQC is parallelogram. (Proved)

(f) In a parallelogram, the line joining a vertex to the midpoint of an opposite side trisects the other diagonal.
Solution:

⇒ P divides BD into the ratio 1 : 2.
Similarly we can show that Q divides BD into the ratio 2 : 1.
Hence P, Q are the points of trisection of the diagonal BD. (Proved)

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(b) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(b)

Solve the following differential equations.
Question 1.
\(\frac{d y}{d x}\) + y = e^-x
Solution:
Given equation is \(\frac{d y}{d x}\) + y = e^-x … (1)
This is a linear differential equation.
Here P = 1, Q = e^-x
So the integrating factor
I.F. = e^{∫P dx} = e^∫dx = e^x
The solution of (1) is given by
ye^x = ∫e^-x . e^x dx = ∫dx = x + C
⇒ y – xe^-x + Ce^-x

Question 2.
(x² – 1)\(\frac{d y}{d x}\) + 2xy = 1
Solution:
Given equation is (x² – 1)\(\frac{d y}{d x}\) + 2xy = 1

Question 3.
(1 – x²)\(\frac{d y}{d x}\) + 2xy = x \(\sqrt{1-x^2}\)
Solution:
Given equation is

Question 4.
x log x \(\frac{d y}{d x}\) + y = 2 log x
Solution:
Given equation is

Question 5.
(1 + x²)\(\frac{d y}{d x}\) + 2xy = cos x
Solution:

Question 6.
\(\frac{d y}{d x}\) + y sec x = tan x
Solution:
Given equation is
\(\frac{d y}{d x}\) + y sec x = tan x
This is a linear equation where
P = sec x, Q = tan x
I.F. = e^{∫sec dx}
= e^{(sec x + tan x)} = sec x + tan x
The solution is y . (sec x + tan x)
= ∫(sec x + tan x) tan x dx
= ∫(sec x tan x + tan² x) dx
= ∫(sec x . tan x + sec² x – 1) dx
= ∫(sec x + tan x) – x + C
⇒ (y – 1) (sec x + tan x) + x = C

Question 7.
(x + tan y) dy = sin 2y dx
Given equation can be written as
Solution:

Question 8.
(x + 2y³)\(\frac{d y}{d x}\) = y
Solution:
Given equation can be written as

Question 9.
sin x\(\frac{d y}{d x}\)+ 3y = cos x
Solution:

Question 10.
(x + y + 1)\(\frac{d y}{d x}\) = 1
Solution:

Question 11.
(1 + y²) dx + (x – \(e^{-\tan ^{-1} y}\)) dy = 0
Solution:
Given equation can be written as

Question 12.
x\(\frac{d y}{d x}\) + y = xy²
Solution:
Given equation can be written as

⇒ z = -x ln x + Cx
⇒ \(\frac{1}{y}\) = -x ln x + Cx
⇒ 1 = -xy ln x + Cxy
∴ The solution is (C – ln x) xy = 1

Question 13.
\(\frac{d y}{d x}\) + y = y² log x
Solution:

Question 14.
(1 + x²)\(\frac{d y}{d x}\) = xy – y²
Solution:
The given equation can be written as

Question 15.
\(\frac{d y}{d x}\) + \(\frac{y}{x-1}\) = \(x y^{\frac{1}{2}}\)
Solution:
The given equation can be written as

Question 16.
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x², y(1) = 1
Solution:
The given equation can be written as
\(\frac{d y}{d x}\) + \(\frac{y}{x}\) = x², y(1) = 1 … (1)
This is a linear equation.

Question 17.
\(\frac{d y}{d x}\) + 2y tan x = sin x, y\(\left(\frac{\pi}{3}\right)\) = 0.
Solution:

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 11 Differential Equations Ex 11(a) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 11 Differential Equations Exercise 11(a)

Question 1.
Determine the order and degree of each of the following differential equations.
(i) y sec² x dx + tan x dy = 0
Solution:
Order: 1, Degree: 1

(ii) \(\left(\frac{d y}{d x}\right)^4\) + y⁵ = \(\frac{d^3 y}{d x^3}\)
Solution:
Order: 3, Degree: 1

(iii) a\(\frac{d^2 y}{d x^2}\) = \(\left\{1+\left(\frac{d y}{d x}\right)^2\right\}^{\frac{3}{2}}\)
Solution:
Order: 2, Degree: 2

(iv) tan^-1\(\sqrt{\frac{d y}{d x}}\) = x
Solution:
Order: 1, Degree: 1

(v) ln\(\left(\frac{d^2 y}{d x^2}\right)\) = y
Solution:
Order: 2, Degree: 1

(vi) \(\frac{\frac{d y}{d t}}{y+\frac{d y}{d t}}\) = \(\frac{y t}{d y}\)
Solution:
Order: 1, Degree: 2

(vii) \(\frac{d^2 y}{d u^2}\) = \(\frac{3 y+\frac{d y}{d u}}{\sqrt{\frac{d^2 y}{d u^2}}}\)
Solution:
Order: 2, Degree: 3

(viii) \(e^{\frac{d z}{d x}}\) = x²
Solution:
Order: 1, Degree: 1

Question 2.
Form the differential equation by eliminating the arbitrary constants in each of the following cases.
(i) y = A sec x
Solution:
y = A sec x
Then \(\frac{d y}{d x}\) = A sec x tan x = y tan x

(ii) y = C tan^-1 x
Solution:

(iii) y = Ae^t + Be^2t
Solution:

(iv) y = Ax² + Bx
Solution:

(v) y = -acos x + b sin x
Solution:

(vi) y = a sin^-1 x + b cos^-1 x
Solution:

(vii) y = at + be^t
Solution:

(viii) y = a sin t + be^t
Solution:

(ix) ax² + by = 1
Solution:

Question 3.
Find the general solution ofthe following differential equations.
(i) \(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
Solution:
\(\frac{d y}{d x}\) = \(\frac{e^{2 x}+1}{e^x}\)
⇒ y = ∫(e^x + e^-x) dx = e^x – e^-x + C

(ii) \(\frac{d y}{d x}\) = x cos x
Solution:
\(\frac{d y}{d x}\) = x cos x
⇒ y = ∫x cos x dx
= x . sin x – ∫sin x dx – x sin x + cos x + C

(iii) \(\frac{d y}{d x}\) = t⁵ log t
Solution:

(iv) \(\frac{d y}{d x}\) = 3t² + 4t + sec² t
Solution:
\(\frac{d y}{d x}\) = 3t² + 4t + sec² t
⇒ y = t³ + 2t² + tan t + C

(v) \(\frac{d y}{d x}\) = \(\frac{1}{x^2-7 x+12}\)
Solution:

(vi) \(\frac{d y}{d u}\) = \(\frac{u+1}{\sqrt{3 u^2+6 u+5}}\)
Solution:

(vii) (x² + 3x + 2) dy – dx = 0
Solution:

(viii) \(\frac{d y}{d t}\) = \(\frac{\sin ^{-1} t e^{\sin ^{-1} t}}{\sqrt{1-t^2}}\)
Solution:

Question 4.
Solve the following differential equations.
(i) \(\frac{d y}{d x}\) = y + 2
Solution:

(ii) \(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
Solution:
\(\frac{d y}{d t}\) = \(\sqrt{1-y^2}\)
⇒ \(\frac{d y}{\sqrt{1-y^2}}\) = dt
⇒ sin^-1 y = t + C

(iii) \(\frac{d y}{d z}\) = sec y
Solution:
\(\frac{d y}{d z}\) = sec y
⇒ cos y dy = dz
⇒ sin y = z + C

(iv) \(\frac{d y}{d x}\) = e^y
Solution:

(v) \(\frac{d y}{d x}\) = y² + 2y
Solution:

(vi) dy + (y² + 1) dx = 0
Solution:

(vii) \(\frac{d y}{d x}\) + \(\frac{e^y}{y}\) = 0
Solution:

(viii) dx + cot x dt = 0
Solution:
dx + cot x dt = 0
⇒ tan x dx + dt = 0
⇒ ∫tan x dx + ∫dt = C₁
⇒ In sec x + t = C₁
⇒ In sec x = C₁ – t
⇒ sec x = \(e^{C_1}\) . e^-t
⇒ cos x = \(e^{-C_1}\) . e^t
⇒ cos x = Ce^t where C = \(e^{-C_1}\)

Question 5.
Obtain the general solution of the following differential equations.
(i) \(\frac{d y}{d x}\) = (x² + 1) (y² + 1)
Solution:

(ii) \(\frac{d y}{d t}\) = e^2t+3y
Solution:

⇒ 2e^-3y + 3e^2t + 6C₁ = 0
⇒ 2e^-3y + 3e^2t = C
where C = -6C₁

(iii) \(\frac{d y}{d z}\) = \(\frac{\sqrt{1-y^2}}{\sqrt{1-z^2}}\)
Solution:

(iv) \(\frac{d y}{d z}\) = \(\frac{x \log x}{3 y^2+4 y}\)
Solution:

(v) x²\(\sqrt{y^2+3}\) dx + y\(\sqrt{x^3+1}\) dy = 0
Solution:

(vi) tan y dx + cot x dy = 0
Solution:
tan y dx + cot x dy = 0
⇒ tan x . dx + cot y dy = 0
⇒ ∫tan x dx + ∫cot y dy = 0
⇒ -ln cos x + ln siny = ln C
⇒ ln\(\frac{\sin y}{\cos x}\) = ln C
⇒ \(\frac{\sin y}{\cos x}\) = C
⇒ sin y = C cos x

(vii) (x² + 7x + 12) dy + (y² – 6y + 5) dx = 0
Solution:

(viii) y dy + e^-y x sin x dx = 0
Solution:
y dy + e^-y x sin x dx = 0
⇒ ye^y dy + x sin x dx = 0
⇒ ∫ye^y dy + ∫x sin dx = C
[Integrating by parts.
⇒ ye^y – ∫e^y dy + x(-cos x) – ∫(-cos x) dx = C
⇒ ye^y – e^y – x cos x + sin x = C
⇒ (y – 1) e^y – x cos x + sin x = C

Question 6.
Solve the following second order equations.
(i) \(\frac{d^2 y}{d x^2}\) = 12x² + 2x
Solution:

(ii) \(\frac{d^2 y}{d t^2}\) =e^2t +e^-t
Solution:

(iii) \(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec² υ
Solution:
\(\frac{d^2 y}{d \vartheta^2}\) = -sin υ + cos υ + sec² υ
Integrating we get
\(\frac{d y}{d υ}\) = ∫sin υ dυ + ∫cos υ dυ + ∫sec² υ dυ
= cos υ + sin υ + tan υ + A
Again integratingwe get
y = ∫(cos υ + sin υ + tan υ + A)dυ + B
where A, B are arbritrary constants.
⇒ y = sin υ – cos υ + ln |sec υ| + A.υ. + B

(iv) cosec x \(\frac{d^2 y}{d x^2}\) = x
Solution:
cosec x \(\frac{d^2 y}{d x^2}\) = x
\(\frac{d^2 y}{d x^2}\) = x sin x
Integrating we get
\(\frac{d y}{d x}\) = ∫x sin x dx + A
= x . (-cos x) – ∫(-cos x) dx + A
= -x cos x + ∫cos x dx + A
= -x cos x + sin x + A
Again integrating we get
y = -∫x cos x dx + ∫sin x + ∫A dx + B
= -{x sin x -∫1 . sin x dx} – cos x + Ax + B
= -x sin x – 2cos x + Ax + B

(v) x²\(\frac{d^2 y}{d x^2}\) + 2 = 0
Solution:

(vi) sec x \(\frac{d^2 y}{d x^2}\) = sec 3x
Solution:

(vii) \(\frac{d^2 y}{d x^2}\) = sec² x + cos² x
Solution:

(viii) e^-x\(\frac{d^2 y}{d x^2}\) = x
Solution:
e^–x\(\frac{d^2 y}{d x^2}\) = x
⇒ \(\frac{d^2 y}{d x^2}\) = xe^x
Integrating we get
\(\frac{d y}{d x}\) = ∫xe^x dx = ∫e^x dx + Ax + B
= xe^x – e^x – e^x + Ax + B
= (x – 2)e^x + Ax + B

Question 7.
Find the particular solutions of the following equations subject to the given conditions.
(i) \(\frac{d y}{d x}\) = cos x, given that y = 2 when x = 0.
Solution:
\(\frac{d y}{d x}\) = cos x
Integrating we get
y = ∫cos x dx = sin x + C
Given that when x = 0, y = 2
So 2 = C
∴ The particular solution is y = sin x + 2

(ii) \(\frac{d y}{d t}\) = cos² y subject to y = \(\frac{\pi}{4}\) when t = 0.
Solution:
\(\frac{d y}{d t}\) = cos² y
⇒ sec² y dy = dt
∫sec² dy = ∫dt
⇒ tan y = t + C
When t = 0, y = \(\frac{\pi}{4}\)
So tan \(\frac{\pi}{4}\) = C ⇒ C = 1
∴ The particular solution is tan y = t + 1

(iii) \(\frac{d y}{d x}\) = \(\frac{1+y^2}{1+x^2}\) given that y = √3 when x = 1.
Solution:

(iv) \(\frac{d^2 y}{d x^2}\) = 6x given that y = 1 and \(\frac{d y}{d x}\) = 2 when x = 0.
Solution:
\(\frac{d^2 y}{d x^2}\) = 6x ⇒ \(\frac{d y}{d x}\) = 3x² + 2
When x = 0, \(\frac{d y}{d x}\) = 2
So 2 = A
∴ \(\frac{d y}{d x}\) = 3x² + 2
Again integrating we get
y = x³ + 2x + B
When x = 0, y = 1
So B = 1.
∴ The particular solution is y = x³ + 2x + 1

Question 8.
(i) Solve : \(\frac{d y}{d x}\) = sec (x + y)
Solution:

(ii) Solve : \(\frac{d y}{d x}\) = sin(x + y) + cos(x + y)
Solution:

(iii) Solve : \(\frac{d y}{d x}\) = cos (x + y)
Solution:

(iv) Solve : \(\frac{d y}{d x}\) + 1 = e^x+y
Solution:

Odisha State Board CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Ex 12(d) Textbook Exercise Questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 12 Vectors Exercise 12(d)

Question 1.
Each question given below has four possible answers out of which only one is correct. Choose the correct one.
(i) \(\vec{a} \cdot \vec{b} \times \vec{a}\) = _______.
(a) \(\overrightarrow{0}\)
(b) 0
(c) 1
(d) \(\vec{a}^2 \vec{b}\)
Solution:
\(\vec{a} \cdot(\vec{b} \times \vec{a})\) = \((\vec{b} \times \vec{a}) \cdot \vec{a}\)
= \(\vec{b} \cdot(\vec{a} \times \vec{a})\) = \(\vec{b} \cdot \overrightarrow{0}\)
= 0 [∴ Dot product is commutative and in the scalar triple product the dot and cross can be interchanged.]

(ii) \((-\vec{a}) \cdot \vec{b} \times(-\vec{c}))\) = _______.
(a) \(\vec{a} \times \vec{b} \cdot \vec{c}\)
(b) \(-\vec{a} \cdot(\vec{b} \times \vec{c})\)
(c) \(\vec{a} \times \vec{c} \cdot \vec{b}\)
(d) \(\vec{a} \cdot(\vec{c} \times \vec{b})\)
Solution:
\((-\vec{a}) \cdot \vec{b} \times(-\vec{c})\) = \(\vec{a} \cdot(\vec{b} \times \vec{c})\)

(iii) For the non-zero vectors \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{a} \cdot(\vec{b} \times \vec{c})\) = 0 if
(a) \(\vec{b} \perp \vec{c}\)
(b) \(\vec{a} \perp \vec{b}\)
(c) \(\vec{a} \| \vec{c}\)
(d) \(\vec{a} \perp \vec{c}\)
Solution:
\(\vec{a} \cdot(\vec{b} \times \vec{c})\) = \((\vec{a} \times \vec{b}) \cdot \vec{c}\)
\(\vec{c} \perp(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})\)
but \(\vec{a} \times \vec{b}\) is perpendicular to \(\vec{a}\) and \(\vec{b}\)
∴ \(\vec{a} \| \vec{b}\)

Question 2.
Find the scalar triple product \(\vec{b} \cdot(\vec{c} \times \vec{a})\) where \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are respectively.
(i) î + ĵ, î – ĵ, 5î + 2ĵ + 3k̂
Solution:

= 1 (0 – 3) + 1 (0 – 3) + 0 (5 – 2)
= 3 – 3 = -6

(ii) 5î – ĵ + 4k̂, 2î + 3ĵ + 5k̂, 5î – 2ĵ
Solution:

= 5 (18 + 10) + 1 (12 – 25) + 4 (- 4 – 15)
= 140 – 13 – 76 = 140 – 89 = 51

Question 3.
Find the volume of the parallelopiped whose sides are given by the vectors.
(i) î + ĵ + k̂, k̂, 3î – ĵ + 2k̂
Solution:

= 1 (0 + 1) – 1 (0 – 3) + 1 (0 – 0)
= 1 + 3 = 4 cube units.

(ii) (1, 0, 0), (0, 1, 0), (0, 0, 1).
Solution:

Question 4.
Show that the following vector are co-planar
(i) î – 2ĵ + 2k̂, 3î + 4ĵ + 5k̂, -2î + 4ĵ – 4k̂
Solution:

(ii) î + 2ĵ + 3k̂, -2î – 4ĵ + 5k̂, 3î + 6
Solution:

Question 5.
Find the value of λ so that the three vectors are co-planar.
(i) î + 2ĵ + 3k̂, 4î + ĵ + λk̂ and λî – 4ĵ + k̂
Solution:

(ii) (2, -1, 1), (1, 2, -3) and (3, λ, 5)
Solution:

⇒ 2 (10 + 3λ) + 1 (5 + 9) + 1 (λ – 6) = 0
⇒ 20 + 6λ +14 + λ – 6 = 0
⇒ 7λ + 28 = 0 ⇒ λ = -4

Question 6.
If \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) mutually perpendiculars, show that \([\vec{a} .(\vec{b} \times \vec{c})]^2\) = a²b²c²
Solution:

Question 7.
Show that \([\vec{a}+\vec{b} \vec{b}+\vec{c} \vec{c}+\vec{a}]\) = 2\([\vec{a} \vec{b} \vec{c}]\)
Solution:

Question 8.
Prove that \([\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]\) = \([\vec{a} \vec{b} \vec{c}]^2\)
Solution:

Question 9.
For \(\vec{a}\) = î + ĵ, \(\vec{b}\) = -î + 2k̂, \(\vec{c}\) = ĵ + k̂ obtain \(\vec{a} \times(\vec{b} \times \vec{c})\) and also verify the formula \(\vec{a} \times(\vec{b} \times \vec{c})\) = \((\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}\).
Solution:

Question 10.
Prove that \(\vec{a} \times(\vec{b} \times \vec{c})+\vec{b} \times(\vec{c} \times \vec{a})+\vec{c} \times(\vec{a} \times \vec{b})\) and hence prove that \(\vec{a} \times(\vec{b} \times \vec{c}), \vec{b} \times(\vec{c} \times \vec{a}), \vec{c} \times(\vec{a} \times \vec{b})\) are coplanar.
Solution:

Question 11.
If \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) unit vectors and \(\hat{a} \times(\hat{b} \times \hat{c})=\frac{1}{2} \hat{b}\) find the angles that â makes with b̂ and ĉ, where b̂, ĉ are not parallel.
Solution:

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Additional Exercise Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Additional Exercise

Question 1.
∫\(\sqrt{1-\sin 2 x}\) dx
Solution:
I = ∫\(\sqrt{1-\sin 2 x}\) dx
= ∫\(\sqrt{(\cos x-\sin x)^2}\) dx
= ∫(cos x – sin x) dx
= sin x + cos x + c

Question 2.
∫\(\frac{d x}{1+\sin x}\)
Solution:
I = ∫\(\frac{d x}{1+\sin x}\)
= ∫\(\frac{1-\sin x}{\cos ^2 x}\)
= ∫sec² x – sec x tan x dx
= tan x – sec x + c

Question 3.
∫\(\frac{\sin x}{1+\sin x}\) dx
Solution:

Question 4.
∫\(\frac{\sec x}{\sec x+\tan x}\) dx
Solution:
I = ∫\(\frac{\sec x}{\sec x+\tan x}\) dx
= ∫\(\frac{\sec x(\sec x-\tan x)}{\sec ^2 x-\tan ^2 x}\) dx
= ∫sec² x – sec x tan x dx
= tan x – sec x + c

Question 5.
∫\(\frac{1+\sin x}{1-\sin x}\) dx
Solution:
I = ∫\(\frac{1+\sin x}{1-\sin x}\) dx
= ∫\(\frac{(1+\sin x)^2}{\cos ^2 x}\) dx
= ∫[sec² x+ tan² x+ 2sec x tan x) dx
= ∫[2sec² x – 1 + 2sec x tan x) dx
= 2tan x – x + 2sec x + c

Question 6.
∫tan^-1 (sec x + tan x) dx
Solution:

Question 7.
∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
Solution:
I = ∫\(\frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha}\) dx
= ∫\(\frac{\left(2 \cos ^2 x-1\right)-\left(2 \cos ^2 \alpha-1\right)}{\cos x-\cos \alpha}\) dx
= 2 ∫(cos x + cos α) dx
= 2 sin x + 2x cos α + c

Question 8.
∫tan-1\(\sqrt{\frac{1-\cos 2 x}{1+\cos 2 x}}\) dx
Solution:

Question 9.
∫\(\frac{d x}{\sqrt{x+1+} \sqrt{x+2}}\)
Solution:

Question 10.
∫\(\frac{2+3 x}{3-2 x}\) dx
Solution:

Question 11.
∫\(\frac{d x}{\sqrt{x}+x}\)
Solution:

Question 12.
∫\(\frac{d x}{1+\tan x}\)
Solution:

Question 13.
∫\(\frac{x+\sqrt{x+1}}{x+2}\) dx (Hints put : \(\sqrt{x+1}\) = t)
Solution:

Question 14.
∫sin^-1\(\sqrt{\frac{x}{a+x}}\) dx (Hints put : x = a tan² t)
Solution:

Question 15.
∫e^x\(\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right)\) dx
Solution:

Question 16.
∫\(\frac{\left(x^2+1\right) e^x}{(x+1)^2}\) dx
Solution:

Question 17.
∫\(\frac{x^2-1}{x^4+x^2+1}\) dx
Solution:

Question 18.
∫\(\frac{x^2 d x}{x^4+x^2+1}\)
Solution:

Question 19.
∫\(\sqrt{\cot x}\) dx
Solution:

Question 20.
∫\((\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:

Question 21.
∫\(\frac{\mathrm{dx}}{x\left(x^4+1\right)}\)
Solution:

Question 22.
∫\(\frac{\mathrm{dx}}{e^x-1}\)
Solution:

Question 23.
∫\(\frac{(x-1)(x-2)(x-3)}{(x+4)(x-5)(x-6)}\) dx
Solution:

Question 24.
∫\(\frac{d x}{\left(e^x-1\right)^2}\)
Solution:

Question 25.
∫\(\frac{d x}{\sin x \cos ^2 x}\)
Solution:

Question 26.
\(\int_2^4 \frac{\left(x^2+x\right) d x}{\sqrt{2 x+1}}\)
Solution:

Question 27.
\(\int_{-a}^a \sqrt{\frac{a-x}{a+x}}\) dx
Solution:

Let a² – x² = t²
-2x dx = 2t dt
x = -a ⇒ 0 t = 0
x = a ⇒ t = 0
= 0
I = aI₁ – I₂ = aπ

Question 28.
\(\int_0^{\pi / 2}(\sqrt{\tan x}+\sqrt{\cot x})\) dx
Solution:

Question 29.
\(\int_0^{\pi / 2} \frac{\cos x d x}{1+\cos x+\sin x}\)
Solution:

Question 30.
\(\int_0^1\)x (1 – x)ⁿ dx
Solution:

Question 31.
\(\int_0^{\pi / 2}\)sin 2x log (tan x) dx
Solution:

Question 32.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{\sin x+\cos x}\)
Solution:

Question 33.
\(\int_0^{\pi / 2} \frac{\sin ^2 x d x}{1+\sin x \cos x}\)
Solution:

Question 34.
\(\int_0^{\pi / 2} \frac{x d x}{\sin x+\cos x}\)
Solution:

Question 35.
Prove that \(\int_0^\pi\) x sin³ x dx = \(\frac{2 \pi}{3}\)
Solution:

Question 36.
\(\int_{\pi / 5}^{3 \pi / 10} \frac{\sin x d x}{\sin x+\cos x}\)
Solution:

Question 37.
\(\int_0^\pi\)|cos x| dx
Solution:

Question 38.
\(\int_1^4\)(|x – 1| + |x – 2| + |x – 3|) dx
Solution:

Question 39.
\(\int_{-\pi / 2}^{\pi / 2}\)(sin |x| + cos |x|) dx
Solution:

Question 40.
\(\int_0^\pi\)log (1 + cos x) dx
Solution:

Odisha State Board Elements of Mathematics Class 12 Solutions CHSE Odisha Chapter 9 Integration Ex 9(l) Textbook Exercise questions and Answers.

CHSE Odisha Class 12 Math Solutions Chapter 9 Integration Exercise 9(l)

Question 1.
\(\int_0^{\frac{\pi}{2}}\)sin¹⁰ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin¹⁰ θ dθ = \(\frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{405 \pi}{7680}\)

Question 2.
\(\int_0^{\frac{\pi}{2}}\)cos¹² θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos¹² θ dθ = \(\frac{11}{12} \cdot \frac{9}{10} \cdot \frac{7}{8} \cdot \frac{5}{6} \cdot \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}\) = \(\frac{4455 \pi}{92160}\)

Question 3.
\(\int_0^{\frac{\pi}{2}}\)sin¹¹ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)sin¹¹ θ dθ = \(\frac{10}{11} \cdot \frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{3840}{4455}\)

Question 4.
\(\int_0^{\frac{\pi}{2}}\)cos⁹ θ dθ
Solution:
\(\int_0^{\frac{\pi}{2}}\)cos⁹ θ dθ = \(\frac{8}{9} \cdot \frac{6}{7} \cdot \frac{4}{5} \cdot \frac{2}{3}\) = \(\frac{384}{405}\)

Question 5.
\(\int_0^1 \frac{x^7}{\sqrt{1-x^2}}\) dx
Solution:

Question 6.
\(\int_0^1 \frac{x^5\left(4-x^2\right)}{\sqrt{1-x^2}}\) dx
Solution:

Question 7.
\(\int_0^a x^3\left(a^2-x^2\right)^{\frac{5}{2}}\) dx
Solution:

Question 8.
\(\int_0^1 x^5 \sqrt{\frac{1+x^2}{1-x^2}}\) dx
Solution:

Question 9.
\(\int_0^{\infty} \frac{x^2}{\left(1+x^6\right)^n}\) dx
Solution:

Question 10.
\(\int_0^\pi\)sin⁸ θ dθ
Solution: