NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions Miscellaneous Exercise
Page No 51:
Question 1:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1d636bf5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1d636bf5.gif)
Answer:
We know that cos−1 (cos x) = x if
, which is the principal value branch of cos −1x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1db67982.gif)
Here,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1ebbd55c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1ebbd55c.gif)
Now,
can be written as:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1d636bf5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_17a89613.gif)
Question 2:
Find the value of ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_m698c2fd2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_m698c2fd2.gif)
Answer:
We know that tan−1 (tan x) = x if
, which is the principal value branch of tan −1x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_m448424b2.gif)
Here,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_3410c8b6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_3410c8b6.gif)
Now,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_m698c2fd2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_7484334d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_5e7fa954.gif)
Question 3:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8259/Chapter%202_html_146e32d2.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8259/Chapter%202_html_146e32d2.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8259/Chapter%202_html_m1942f788.gif)
Now, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8259/Chapter%202_html_m2d9a523.gif)
Question 4:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8260/Chapter%202_html_52305fd8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8260/Chapter%202_html_52305fd8.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8260/Chapter%202_html_m71982799.gif)
Now, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8260/Chapter%202_html_63c5330e.gif)
Question 5:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8261/Chapter%202_html_m4c4f3bd6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8261/Chapter%202_html_m4c4f3bd6.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8261/Chapter%202_html_m6bdf9784.gif)
Now, we will prove that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8261/Chapter%202_html_m496bbc3f.gif)
Question 6:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8262/Chapter%202_html_m3083220a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8262/Chapter%202_html_m3083220a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8262/Chapter%202_html_5f64b108.gif)
Now, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8262/Chapter%202_html_67b96f40.gif)
Question 7:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8263/Chapter%202_html_me2b90e4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8263/Chapter%202_html_me2b90e4.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8263/Chapter%202_html_3f1d20cd.gif)
Using (1) and (2), we have
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8263/Chapter%202_html_m2ac7a6ca.gif)
Question 8:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8264/Chapter%202_html_153122fd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8264/Chapter%202_html_153122fd.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8264/Chapter%202_html_m27a307.gif)
Page No 52:
Question 9:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8265/Chapter%202_html_m30182ff8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8265/Chapter%202_html_m30182ff8.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8265/Chapter%202_html_7c715840.gif)
Question 10:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8266/Chapter%202_html_m1364ded.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8266/Chapter%202_html_m1364ded.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8266/Chapter%202_html_6c2ebb3e.gif)
Question 11:
Prove
[Hint: putx = cos 2θ]
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8267/Chapter%202_html_231bb4b8.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8267/Chapter%202_html_1b7b23c.gif)
Question 12:
Prove ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8268/Chapter%202_html_m36facb96.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8268/Chapter%202_html_m36facb96.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8268/Chapter%202_html_m340b058c.gif)
Question 13:
Solve![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8269/Chapter%202_html_m20f6d595.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8269/Chapter%202_html_m20f6d595.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8269/Chapter%202_html_714d8c95.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8269/Chapter%202_html_m7c0a8251.gif)
Question 14:
Solve![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8270/Chapter%202_html_m5748e9ce.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8270/Chapter%202_html_m5748e9ce.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8270/Chapter%202_html_18fb2583.gif)
Question 15:
Solve
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_2afe913f.gif)
(A)
(B)
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_26b35df5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m141b5ac3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m5a4dc0be.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_2b0981f8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_26b35df5.gif)
Answer:
Let tan−1 x = y. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m4ed03241.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m4ed03241.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_3eea93b9.gif)
The correct answer is D.
Question 16:
Solve
, then x is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_3ca204da.gif)
(A)
(B)
(C) 0 (D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_m5a4d85ce.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_2ba9b133.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_410097d8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_m5a4d85ce.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_410715d0.gif)
Therefore, from equation (1), we have
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_5d4e8a32.gif)
Put x = sin y. Then, we have:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_3573abd3.gif)
But, when
, it can be observed that:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_24eee63.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_1e798d11.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_6299d24f.gif)
Thus, x = 0.
Hence, the correct answer is C.
Question 17:
Solve
is equal to
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_72c8e820.gif)
(A)
(B).
(C)
(D) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_4945c1c5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m4c4df2e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m4e8d241e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m6201c3cd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_4945c1c5.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_7eaf97d1.gif)
Hence, the correct answer is C.