Inverse Trigonometric Functions Jee Main and Advanced Questions
Multiple Choice Questions
Question 1
if $sin^{-1} x + sin^{-1} y + sin^{-1} z = \frac {3 \pi}{2}$, then the value of $x^{21} +y^{21} + z^{21} - 3xyz$ is
(a) 0
(b) 1
(c) -1
(d) None of these
Question 2
if $x \geq 1$, then the value $2tan^{-1}x + sin^{-1} \frac {2x}{1+x^2}$ is ?
(a) $\pi$
(b) $2 \pi$
(c) $0 $
(d) $-\pi$
Question 3
Two angles of triangle are $tan^{-1} 1/2$ and $cot^{-1} 3$, then the third angle is ?
(a) $\pi/2$
(b) $3\pi/4$
(c) $\pi/6$
(d) $\pi/3$
Question 4
The equation $2sin^{-1} x + cos^{-1} =\frac {11 \pi}{6}$ has ?
(a) 1 solution
(b) 2 solution
(c) Infinite solutions
(d) No solutions
Question 5
The value of $cot (\sum_{n=1}^{23}cot^{-1}\left ( 1+\sum_{k=1}^{n} 2k\right ))$ ?
(a) 23/25
(b) 25/23
(c) 24/23
(d) 23/24
Question 6
The number of real solutions of $tan^{-1} \sqrt {x(x+1)} + sin^{-1} \sqrt {x^2 +x + 1} = \frac {\pi}{2}$ is
(a)1
(b)0
(c)2
(d) 3
Question 7
If $sin^{-1} \frac {x}{5} + cosec^{-1} \frac {5}{4} = \frac {\pi}{2}$ , then value of x is
(a) 1
(b) 3
(d) 4
(d) 2
Question 8
The value of $sin^{-1} sin12 + cos^{-1} cos 12$ then
(a) 0
(b) 12
(c) 24
(d) -12
Question 9
The greatest and least value of the function $f(x) = (sin^{-1}x)^2 + (cos^{-1}x)^2$ are
(a)$\frac{\pi ^2}{4}$, 0
(b) $\frac{5\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(c) $\frac{\pi ^2}{4}$, $\frac{\pi ^2}{8}$
(d) $\frac{3\pi ^2}{4}$, $\frac{\pi ^2}{8}$
Question 10
The value of $sin^{-1} ( cos \frac {33\pi}{5})$ is
(a) $\frac {3\pi}{5}$
(b) $-\frac {\pi}{10}$
(c) $\frac {\pi}{10}$
(d) $\frac {7\pi}{5}$ Answers 1-10
1. a
as $sin^{-1} x \leq \frac {\pi}{2}$
so, $sin^{-1} x=\frac {\pi}{2}$, $sin^{-1} y=\frac {\pi}{2}$,$sin^{-1} z=\frac {\pi}{2}$
or x=y=z=1
$x^{21} +y^{21} + z^{21} - 3xyz=0$
2. a
3. b
4. d
5. b
$cot (\sum_{n=1}^{23}cot^{-1}\left ( 1+\sum_{k=1}^{n} 2k\right ))= cot(\sum_{n=1}^{23}cot^{-1} ( 1+ 2\frac {n(n+1)}{2} ))$
$=cot(\sum_{n=1}^{23}cot^{-1} [ 1+ n(n+1)])= cot(\sum_{n=1}^{23}tan^{-1} \frac {1}{ 1+ n(n+1)})$
$=cot(\sum_{n=1}^{23} [ tan^{-1} (n+1) - tan^{-1} n]) =cot(tan^{-1} 24 - tan^{-1} 1) = cot tan^{-1} \frac {23}{25} =\frac {25}{23}$
6. c
7. b
$sin^{-1} \frac {x}{5} + cosec^{-1} \frac {5}{4} = \frac {\pi}{2}$
$sin^{-1} \frac {x}{5} + sin^{-1} \frac {4}{5} = \frac {\pi}{2}$
$sin^{-1} \frac {x}{5}= \frac {\pi}{2} - sin^{-1} \frac {4}{5}$
$sin^{-1} \frac {x}{5}=cos^{-1} \frac {4}{5}$
$sin^{-1} \frac {x}{5}=sin^{-1} \frac {3}{5}$
x=3
8. a
$sin^{-1} sin12 + cos^{-1} cos 12= 12 - 3 \pi + 3 \pi -12=0$
9. b
$f(x) = (sin^{-1}x)^2 + (cos^{-1}x)^2 = (sin^{-1} x + cos^{-1} x) ^2 - 2 sin^{-1} x cos^{-1} x$
$= \frac {\pi^2}{4} - 2 sin^{-1} x ( \frac {\pi}{2} - sin^{-1} x)= \frac {\pi^2}{4} - \pi sin^{-1} x +2 (sin^{-1} x)^2$
$= 2((sin^{-1} x)^2 - \frac {\pi }{2} sin^{-1} x \frac {\pi^2}{8})= 2[(sin^{-1} x - \frac {\pi}{4})^2 + \frac {\pi^2}{16}]$
So greatest and least value of the function are $\frac{5\pi ^2}{4}$, $\frac{\pi ^2}{8}$
10. b
$ sin^{-1} ( cos \frac {33\pi}{5})= sin^{-1} [ cos (6\pi + \frac {3\pi}{5})]= sin^{-1} cos( \frac {3\pi}{5}) = sin^{-1} sin(\frac {\pi}{2}- \frac {3\pi}{5}) $
$=sin^{-1} sin(\frac {-\pi}{10})=\frac {-\pi}{10} $
Numerical values questions
Question 11
(i) Considering only the principal valeus of the inverse trigonometric function , the value of the expression
$\frac {3}{2} cos^{-1} \sqrt {\frac {2}{2 + \pi ^2}} +\frac {1}{4}sin^{-1} \frac {2\sqrt 2 \pi}{2 + \pi^2} + tan^{-1} \frac {\sqrt 2}{\pi}$ is ________
(ii) The principal value of $tan^{-1} \sqrt 3$ is _____
(iii) The number of real solution of the equation
$sin^{-1}\left ( \sum_{i=1}^{\infty }x^{i+1} - x \sum_{i=1}^{\infty } (x/2)^i\right )=\frac {\pi}{2} - cos^{-1}\left ( \sum_{i=1}^{\infty }(-x/2)^{i} - \sum_{i=1}^{\infty } (-x)^i\right )$
lying in the interval (-1/2,1/2) is ________
(iv) The value of $sec^{-1} \left ( \frac {1}{4} \sum_{k=0}^{10} sec \left ( \frac {7\pi}{12} + \frac {k \pi}{2} \right ) sec \left ( \frac {7\pi}{12} + \frac {(k+1)\pi}{2} \right ) \right )$ in the interval $[-\frac {\pi}{4}, \frac {3 \pi}{4}]$ is _____ Answers
