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On January 25, 2024, 11:35 AM

Miscellaneous Exercise - Chapter 2 Inverse Trigonometry class 12 ncert solutions Maths - SaraNextGen [2024]


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

Answer:

We know that cos−1 (cos x) = x ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1db67982.gif, which is the principal value branch of cos −1x.

Here,https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1ebbd55c.gif

Now, https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8257/Chapter%202_html_1d636bf5.gifcan be written as:

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

Answer:

We know that tan−1 (tan x) = x ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8258/Chapter%202_html_m448424b2.gif, which is the principal value branch of tan −1x.

Here,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.gifcan be written as:

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

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

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

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

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

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

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8264/Chapter%202_html_m27a307.gif

Question 9:

Prove 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

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8266/Chapter%202_html_6c2ebb3e.gif

Question 11:

Prove https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8267/Chapter%202_html_231bb4b8.gif [Hint: putx = cos 2θ]

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

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8268/Chapter%202_html_m340b058c.gif

Question 13:

Solvehttps://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:

Solvehttps://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:

Solvehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_2afe913f.gifis equal to

(A) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m141b5ac3.gif (B) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_m5a4dc0be.gif (C) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8271/Chapter%202_html_2b0981f8.gif (D) 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_3eea93b9.gif

The correct Answer is D.

Question 16:

Solvehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_3ca204da.gifthen x is equal to

(A) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_2ba9b133.gif (B) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_410097d8.gif (C) 0 (D) 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, whenhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8272/Chapter%202_html_24eee63.gif, it can be observed that:

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 is not the solution of the given equation.

Thus, x = 0.

Hence, the correct Answer is C.

Question 17:

Solvehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_72c8e820.gifis equal to

(A) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m4c4df2e.gif (B). https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m4e8d241e.gif (C) https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/231/8273/Chapter%202_html_m6201c3cd.gif (D) 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.

Also Read : Exercise-3.1-Chapter-3-Matrices-class-12-ncert-solutions-Maths

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