CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 1 - MCQExams.com

Statement I :  The equation $$(sin^{-1}x)^3+(cos^{-1}x)^3-a\pi^3=0$$ has a solution for all $$a\geqslant \dfrac {1}{32}.$$
Statement II : For any $$x\epsilon R, sin^{-1}x+cos^{-1}x=\dfrac {\pi}{2}$$ and $$0\leq (sin^{-1}x-\dfrac {\pi}{4})^2\leq \dfrac {9\pi^2}{16}$$.
  • Both statements I and II are true.
  • Both statements I and II are true but I is not an explanation of II
  • Statement I is true and statement II is false
  • Statement I is false and statement II is true.
If $$\cos ^{ -1 }{ x } -\cos ^{ -1 }{ \cfrac { y }{ 2 }  } =\alpha $$, then $$4{ x }^{ 2 }-4xy\cos { \alpha  } +{ y }^{ 2 }\quad $$ is equal to
  • $$2\sin { 2\alpha } $$
  • $$4$$
  • $$4\sin ^{ 2 }{ \alpha } $$
  • $$-4\sin ^{ 2 }{ \alpha } $$
If $$\sin^{-1}\left (\dfrac {2a}{1 + a^{2}}\right ) - \cos^{1}\left (\dfrac {1 - b^{2}}{1 + b^{2}}\right ) = \tan^{-1}\left (\dfrac {2x}{1 - x^{2}}\right )$$, then what is the value of $$x$$?
  • $$\dfrac {a}{ b}$$
  • $$ab$$
  • $$\dfrac {b}{a}$$
  • $$\dfrac {a - b}{1 + ab}$$
The value of $$\tan^{-1}(1)+\cos^{-1}\left(\dfrac{-1}{2}\right)+\sin^{-1}\left(\dfrac{-1}{2}\right)$$ is equal to 
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{5\pi}{12}$$
  • $$\dfrac{3\pi}{4}$$
  • $$\dfrac{13\pi}{12}$$
$${cos}^{-1}\left \{ \frac{1}{2}x^2+\sqrt{1-x^2}.\sqrt{1-\frac{x^2}{4}} \right \}={cos}^{-1}\frac{x}{2}-{cos}^{-1}x$$ holds for

  • $$|x|\leq 1$$
  • $$x\epsilon R$$
  • $$0\leq x\leq 1$$
  • $$-1\leq x\leq 0$$
Solve: $$\displaystyle { \tan }^{ -1 }\left( \frac { x }{ y }  \right) -{ \tan }^{ -1 }\frac { x-y }{ x+y } $$ is equal to
  • $$\displaystyle \frac { \pi }{ 2 } $$
  • $$\displaystyle \frac { \pi }{ 3 } $$
  • $$\displaystyle \frac { \pi }{ 4 } $$
  • $$\displaystyle \frac { -3\pi }{ 4 } $$
Value of $$\tan^{-1}\begin{Bmatrix}\dfrac{\sin\,2-1}{\cos\,2}\end{Bmatrix}$$ is
  • $$\dfrac{\pi}{2}-1$$
  • $$1-\dfrac{\pi}{4}$$
  • $$2-\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}-1$$
The number of triplets $$\left ( x,y,z \right )$$ satisfies the equation $$\displaystyle f\left ( x,y,z \right )= \sin ^{-1}x+\sin ^{-1}y+\sin ^{-1}z= \dfrac{3\pi }{2}$$ is
  • $$1$$
  • $$2$$
  • $$0$$
  • Infinite
Sin-1 $$(\displaystyle \frac{3}{5})+\mathrm{S}\mathrm{i}\mathrm{n}^{-1}(\frac{5}{13})=\sin_{X}^{-1}$$ then$$\mathrm{x}=$$
  • $$(^{\frac{63}{65})}$$
  • $$\displaystyle \frac{56}{65}$$
  • $$\displaystyle \frac{56}{63}$$
  • $$\displaystyle \frac{16}{63}$$
If $$\sec^{-1} x+ \sec^{-1}y + \sec^{-1}z = 3\pi$$, then $$xy + yz + zx =$$ _______.
  • $$0$$
  • $$-3$$
  • $$3$$
  • $$1$$
Range of $$f(x)=\tan^{-1}\Bigg[\dfrac{2}{\pi}(2\tan^{-1}x-\sin^{-1}x+\cot^{-1}x-\cos^{-1}x)\Bigg]$$ contains
  • Only one integer
  • More than 2 integers
  • Only two integers
  • No integer
The value of $$\sin^{-1} \left( \dfrac{2 \sqrt 2}{3} \right ) + \sin^{-1} \left( \dfrac{1}{3}\right )$$ is equal to
  • $$\dfrac {\pi}{ 6}$$
  • $$\dfrac {\pi}{ 4}$$
  • $$\dfrac {\pi}{ 2}$$
  • $$\dfrac {2\pi}{ 3}$$
  • $$0$$
$${ tan }^{ -1 }x+{ tan }^{ -1 }y={ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $$,      $$xy<1$$
                                    $$=\pi +{ tan }^{ -1 }\dfrac { x+y }{ 1-xy } $$,      $$xy>1$$.

 Evaluate:  $${ tan }^{ -1 }\dfrac { 3sin2\alpha  }{ 5+3cos2\alpha  } +{ tan }^{ -1 }\left( \dfrac { tan\alpha  }{ 4 }  \right) $$
                                  where $$-\dfrac { \pi  }{ 2 } <\alpha <\dfrac { \pi  }{ 2 } $$
  • $$\alpha$$
  • $$2\alpha$$
  • $$3\alpha$$
  • $$4\alpha$$
Simplify $${\cot ^{ - 1}}\dfrac{1}{{\sqrt {{x^2} - 1} }}$$ for $$x <  - 1$$
  • $$\cos ^{-1}x$$
  • $$\sec^{-1}x$$
  • $$\text{cosec}^{-1}x$$
  • $$\tan ^{-1}x$$
Find value of $$\cos^{-1}\left(-\dfrac {1}{2}\right)$$.
  • $$\dfrac {7\pi}{3}$$
  • $$\dfrac {2\pi}{3}$$
  • $$\dfrac {5\pi}{3}$$
  • $$\dfrac {\pi}{3}$$
if f(x)=$${\tan ^{ - 1}}$$(x)than f(x)+f(y) is equal to:
  • $${{\tan }^{-1}}x-{{\tan }^{-1}}y$$=$${{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)$$
  • $${{\tan }^{-1}}x+{{\tan }^{-1}}y$$=$${{\tan }^{-1}}\left( \dfrac{x-y}{1+xy} \right)$$
  • $${{\tan }^{-1}}x+{{\tan }^{-1}}y$$=$${{\tan }^{-1}}\left( \dfrac{x+y}{1-xy} \right)$$
  • None of these
State True or False
$$\sin^{-1}2+\cos^{-1} 2=\dfrac{\pi}{2}$$.
  • True
  • False
For $$x\in(0,\pi/2) $$
$${\sin ^{ - 1}}(\cos x)=?$$
  • $$\pi-x$$
  • $$\dfrac {\pi}{2}-x$$
  • $$\pi-\dfrac{x}{2}$$
  • $$\pi-2x$$
$$\cos ^{-1}\left ( \cos \left ( \frac{5\pi}{4} \right ) \right )$$ is given by 
  • $$5\pi/4$$
  • $$3\pi/4$$
  • $$-\pi/4$$
  • None of these
If $$\sin^{-1}\dfrac{1}{3} + \sin^{-1}\dfrac{2}{3} = \sin^{-1}x$$, then $$x$$ is equal to-
  • $$0$$
  • $$\dfrac{\sqrt{5} - 4\sqrt{2}}{9}$$
  • $$\dfrac{\sqrt{5} + 4\sqrt{2}}{9}$$
  • $$\dfrac{\pi}{2}$$
$$\tan^{-1} \dfrac{1}{2} + \tan^{-1} \dfrac{1}{3} $$ equals
  • $$-\cfrac{\pi }{4}$$
  • $$\cfrac{\pi }{6}$$
  • $$\cfrac{\pi }{4}$$
  • None of these 
The value of $$\cos^{-1} (\cos 12) - \sin^{-1} (\sin 12)$$ is 
  • $$0$$
  • $$\pi$$
  • $$8 \pi - 24$$
  • none of these
If $$f(x)=sin^{-1}(sinx)+cos^{-1}(sinx)$$ and $$\phi (x)=f(f(f(x))),$$ then $$\phi '(x)=$$ 
  • 1
  • sin x
  • 0
  • None of these
$$\tan^{-1}\dfrac {1}{5}+\tan^{-1}\dfrac {1}{7}+\tan^{-1}\dfrac {1}{3}+\tan^{-1}\dfrac {1}{8}$$ is equal to
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{2 }$$
  • $$\pi$$
If $$n - 1\sum\limits_{}^\infty  {{{\cot }^{ - 1}}\left( {{{{n^2}} \over 8}} \right) = \pi .} $$ where $${a \over b}$$ is rational number in its lowest, then correct option is/are 
  • $$a - b = 3$$
  • $$a + b = 11$$
  • $$a + b = 10$$
  • $$a - b = 4$$
$$\tan(\cot^{-1}x)$$ is equal to :
  • $$\dfrac{\pi}{2}-x$$
  • $$\cot(\tan^{-1}x)$$
  • $$\tan x$$
  • $$\dfrac1x$$
If $$\sin (\sin^{-1}\dfrac{1}{5}+\cos ^{-1}x)=1$$, then find the value of x.
  • $$-1$$
  • $$\dfrac { 1 }{ 3 } $$
  • $$\dfrac { 1 }{ 5 } $$
  • $$\dfrac { 1 }{ 2 } $$
The value of the expression $$2\sec^{-1} 2 +\sin^{-1} \dfrac{1}{2}$$ is 
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{5\pi}{6}$$
  • $$\dfrac{7\pi}{6}$$
  • $$1$$
Number of solutions of the equation $$3{\tan ^{ - 1}}x + {\cot ^{ - 1}}x = \pi $$ is
  • Zero
  • $$2$$
  • $$3$$
  • $$1$$
If $$x,y,z \in [-1,1]$$ such that $$\cos^{-1}x +\cos^{-1}y +\cos^{-1}z=0$$, find $$x+y+z$$.
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
Find the value of :
$$\sec^2 (\tan^{-1} 2) +\csc^2 (\cot^{-1} 3)$$
  • $$11$$
  • $$15$$
  • $$17$$
  • $$21$$
Find the value of $$\sin^{-1}(2\cos^{2}x-1)+\cos^{-1}(1-2\sin^{2}x)$$.
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{6}$$
Find the value of $$\cot (\tan^{-1} a +\cot^{-1} a)$$.
  • $$0$$
  • $$-1$$
  • $$2$$
  • $$1$$
$$\sin^{-1}{0}$$ is equal to:
  • $$0$$
  • $$\dfrac{\pi }{6}$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{3}$$
The value of $$x$$ for which $$\sin { \left( \cot ^{ -1 }{ \left( 1+x \right)  }  \right)  } =\cos { \left( \tan ^{ -1 }{ x }  \right)  } $$ is
  • $$\displaystyle \frac{1}{2}$$
  • $$1$$
  • $$0$$
  • $$\displaystyle -\frac{1}{2}$$
Assertion (A) : The maximum value of $$f(x)=\sin^{-1}x+\cos^{-1}x+\tan^{-1}x$$ is $$\displaystyle \frac{3\pi}{4}$$
Reason (R) : $$\sin ^{-1} x>\cos^{-1}x$$ for all $$x$$ in $$R$$
  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true and R is not correct explanation of A
  • A is true but R is false
  • A is false but R is true
The number of solutions of the equation

$$2(Sin^{-1}x)^{2}-5Sin^{-1}x+2=0$$ is



  • 0
  • 1
  • 2
  • 3
The number of triplets $$(x,y,z)$$ satisfying $$\sin^{-1}x+\sin^{-1}y+\cos^{-1}z=2\pi$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$\infty$$
Assertion ($$A$$) lf $$0<\displaystyle x<\frac{\pi}{2}$$ then $$\sin^{-1}(cosx)+\cos^{-1}(sinx)=\pi-2x$$
Reason (R) $$\displaystyle \cos^{-1}x=\frac{\pi}{2}-\sin^{-1}x\forall x\in[0,1]$$


  • Both A and R are true and R is the correct explanation of A
  • Both A and R are true but R is not correct explanation of A
  • A is true but Ris false
  • A is false but R is true
The smallest and the largest values of
$$\displaystyle \tan^{-1}\left (\dfrac{1-x}{1+x}\right)$$ , $$0\leq x\leq 1$$ are.
  • $$ 0,\pi$$
  • $$0,\displaystyle \dfrac{\pi}{4}$$
  • $${-\dfrac{\pi}{4},\dfrac{\pi}{4}}$$
  • $$\displaystyle \dfrac{\pi}{4},\dfrac{\pi}{2}$$
The value of $$x$$ where $$x>0$$ $$\displaystyle \tan(\sec^{-1}\frac{1}{x})=\sin(\tan^{-1}2)$$ is
  • $$\sqrt{5}$$
  • $$\displaystyle \frac{\sqrt{5}}{3}$$
  • $$1$$
  • $$\displaystyle \frac{2}{3}$$
The ascending order of $$A=\sin^{-1}(\log_{3}{2})$$ , $$B=\displaystyle \cos^{-1}\left(\log_{3}\left(\frac{1}{2}\right)\right)$$ , and $$C=\tan^{-1}\left(\log_{1/3} 2 \right)$$ is
  • $$\text{C, B, A}$$
  • $$\text{B, A, C}$$
  • $$\text{C, A, B}$$
  • $$ \text {B, C, A}$$
The equation 2$$\displaystyle \cos^{-1}x+\sin^{-1}x=\frac{11\pi}{6}$$ has
  • No solution
  • One solution
  • Two solutions
  • Three solutions
If $$x$$ takes negative permissible value, then  $$\sin^{-1}x=$$
  • $$\cos^{-1}\sqrt{1-x^{2}}$$
  • $${-\cos^{-1}}\sqrt{1-x^{2}}$$
  • $${\cos^{-1}}\sqrt{x^{2}-1}$$
  • $${\pi-\cos^{-1}}\sqrt{1-x^{2}}$$
There is flag-staff at the top of $$10$$ metres high tower. lf the flag-staff makes an angle $$\tan ^{ -1 }{ \left( 1/8 \right)  }$$ at a point $$24$$ metres away from the tower, then the height of the flag staff in metres is
  • $$26/7$$
  • $$27/8$$
  • $$27/6$$
  • $$26/3$$
If $$\tan(cos^{-1}x)=\sin (\sec^{-1} (\sqrt{5}) )$$, then $$x=$$
  • $$\sqrt{\dfrac{5}{3}}$$
  • $$\displaystyle \dfrac{\sqrt{5}}{3}$$
  • $$\sqrt{\dfrac{5}{2}}$$
  • $$\displaystyle \dfrac{\sqrt{5}}{2}$$
A tower stands at the top of a hill whose height is three times the height of the tower. The tower is found to subtend an angle of\$$\tan ^{ -1 }{ \left( { 1 }/{ 7 } \right)  } $$ at a point $$2km$$ away on the horizontal throught the foot of the hill. Then the height of the tower is
  • $$\displaystyle \frac{1}{2}km$$ or $$\displaystyle \frac{1}{3}km$$
  • $$\displaystyle \frac{1}{3}km \space or \space \frac{2}{3}km$$
  • $$\displaystyle \frac{2}{3}km \space or \space\frac{1}{2}km$$
  • $$\displaystyle \frac{3}{4}km \space or \space  \frac{1}{2}km$$
If $$\theta = sin^{-1}x+cos^{-1}x+tan^{-1}x,\ 0\leq x\leq 1$$, then the smallest interval in which $$\theta$$ lies is given by
  • $$\displaystyle \frac{\pi}{4}\leq\theta\leq\frac{\pi}{2}$$
  • $$-\displaystyle \frac{\pi}{4}\leq\theta\leq 0$$
  • $$0\displaystyle \leq\theta\leq\frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{2}\leq\theta\leq\frac{3\pi}{4}$$
The domain of $$\sin^{-1}[\log_{2}(x^{2}/2)]$$ is
  • $$[2, 1]$$
  • $$[1, 2]$$
  • $$[-2,-1]\cup [1,2]$$
  • $$[-2,0]$$
$$A$$ vertical pole subtends an angle $$\displaystyle \tan^{-1}(\frac{1}{2})$$ at a point $$P$$ on the ground. The angle subtended by the upper half of the pole at $$P$$ is
  • $$\displaystyle \tan^{-1}(\frac{1}{4})$$
  • $$\displaystyle \tan^{-1}(\frac{1}{8})$$
  • $$\displaystyle \tan^{-1}(\frac{2}{3})$$
  • $$\displaystyle \tan^{-1}(\frac{2}{9})$$
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