2.5 Miscellaneous Exercise on Chapter 2

Screen Readable NCERT Class 12 Mathematics Textbook for Blind and Visually Impaired Students prepared by Professor T K Bansal.

Find the value of the following:

Question 1


\[\cos^{−1} (\cos \frac{13π}{6})\]

Answer 1


π/6

Question 2


\[\tan^{−1} (\tan \frac{7π}{6})\]

Answer 2


π/6

Prove that

Question 3


\[2 \sin^{−1} \frac{3}{5}\ =\ \tan^{−1} \frac{24}{7}\]

Question 4


\[\sin^{−1} \frac{8}{17}\ +\ \sin^{−1} \frac{3}{5}\ =\ \tan^{−1} \frac{77}{36}\]

Question 5


\[\cos^{−1} \frac{4}{5}\ +\ \cos^{−1} \frac{12}{13}\ =\ \cos^{−1} \frac{33}{65}\]

Question 6


\[\cos^{−1} \frac{12}{13}\ +\ \sin^{−1} \frac{3}{5}\ =\ \sin^{−1} \frac{56}{65}\]

Question 7


\[\tan^{−1} \frac{63}{16}\ =\ \sin^{−1} \frac{5}{13}\ +\ \cos^{−1} \frac{3}{5}\]

Prove that

Question 8


\[\tan^{−1} √x\ =\ \frac{1}{2}\ \cos^{−1} \frac{1−x}{1+x},\ x\ ∈\ [0,\ 1]\]

Question 9


\[cot^{−1} (\frac{\sqrt{1+\sin x}\ +\ \sqrt{1− \sin x}}{\sqrt{1+\sin x} − \sqrt{1−\sin x}})\ =\ \frac{x}{2},\ x ∈ (0,\ \frac{π}{4})\]

Question 10


\[tan^{−1} (\frac{\sqrt{1+x}\ −\ \sqrt{1−x}}{\sqrt{1+x}\ +\ \sqrt{1−x}})\ =\ \frac{π}{4}\ −\ \frac{1}{2 \cos^{−1} x},\ \frac{−1}{√2} ≤x≤1\]


[Hint: Put x = cos 2 θ]

Solve the following equations:

Question 11


\[2 \tan^{−1} \cos x\ =\ \tan^{−1} (2 \csc x)\]

Answer 11


x = nπ + (π/4), n ∈ Ƶ

Question 12


\[\tan^{−1} \frac{1−x}{1+x}\ =\ \frac{1}{2} \tan^{−1}x,\ x>0 \]

Answer 12


x = 1/√3

Question 13


\[\sin \tan^{−1}x,\ |x|\ <\ 1\ =\]


(A) x/√(1 − x^2)


(B) 1/√(1 − x^2)


(C) 1/√(1 + x^2)


(D) x/√(1 + x^2)

Answer 13


D

Question 14


\[\sin^{−1} (1−x)\ −\ 2\sin^{−1}x\ =\ \frac{π}{2},\]


then x is equal to


(A) 0, ½


(B) 1, ½


(C) 0


(D) ½

Answer 14


C