CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 2 - MCQExams.com

Range of $$\sin^{-1}x-\cos^{-1}x$$ is
  • $$[\displaystyle \frac{-3\pi}{2}, \displaystyle \frac{\pi}{2}]$$
  • $$[\displaystyle \frac{-5\pi}{3}, \displaystyle \frac{\pi}{3}]$$
  • $$[\displaystyle \frac{-3\pi}{2}, \pi]$$
  • $$[0, \pi]$$
$$ABC$$ is a triangular park with $$AB=AC=100$$ cm. $$A$$ clock tower is situated at the midpoint of $$BC$$. The angles of elevation of the top of the tower from $$A$$ and $$B$$ are $$cot ^{-1}3.2$$ and $$cosec ^{-1} 2.6$$. The height of the tower is
  • $$25$$ mt
  • $$50$$ mt
  • $$100$$ mt
  • $$50\sqrt{2}$$ mt
A vertical pole more than $$100ft$$ high consists of two portions, the lower being $$ 1/3$$ of the whole. lf the upper portion subtends an angle $$\tan^{-1}(1/2)$$ at a point distant 40 ft. from the foot of the pole, the height of the pole is
  • $$ 105 $$
  • $$ 120$$
  • $$135$$
  • $$150$$
If $$\theta\equiv sin^{-1}x+cos^{-1}x-tan^{-1}x,\ 0\leq x\leq 1$$, 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 value of sin $$\sin \left\{ {{{\tan }^{ - 1}}\left( {\tan \frac{{7\pi }}{6}}

\right) + {{\cos }^{ - 1}}\left( {\cos \frac{{7\pi }}{3}} \right)}

\right\}$$ is
  • 0
  • 1
  • -1
  • None of these
If $$\alpha \epsilon \left ( 0,\dfrac{\pi}{2}\right )$$, then the value of $$\tan ^{-1}(\cot \alpha )-\cot ^{-1}(\tan \alpha )+\sin ^{-1}(\sin \alpha )-\cos ^{-1}(\cos \alpha )$$ is equal to
  • $$2\alpha $$
  • $$\pi +\alpha $$
  • $$0$$
  • $$\pi -2\alpha$$
If $$\tan ^{-1}\dfrac{\sqrt{1+x^{2}}-1}{x}=4^{\circ}$$,then
  • $$x=\tan 2^{\circ}$$
  • $$x=\tan 4^{\circ}$$
  • $$x=\tan (1/4)^{\circ}$$
  • $$x=\tan 8^{\circ}$$
If $${\cot ^{ - 1}}x + {\cot ^{ - 1}}y + {\cot ^{ - 1}}z = \dfrac{\pi }{2}$$  then $$x+y+z$$ equals
  • $$xyz$$
  • $$xy+yz+zx$$
  • $$2xyz$$
  • None of these
$$f(x)=tan^{-1}(sin  x+ cos  x) $$ is an increasing function in
  • $$\displaystyle (0, \frac{\pi}{4})$$
  • $$\displaystyle (0, \frac{\pi}{2})$$
  • $$\displaystyle (\frac{-\pi}{4}, \frac{\pi}{4})$$
  • None of these
If $$sin\left\{ \sin ^{ -1 }{ \cfrac { 1 }{ 5 }  } +\cos ^{ -1 }{ x }  \right\} =1$$, then $$x$$ is equal to
  • $$1$$
  • $$0$$
  • $$\displaystyle\frac{ 4 }{ 5 }$$
  • $$\displaystyle\frac{ 1 }{ 5 }$$
If $$0\le x\le 1$$, then $$\tan { \left\{ \cfrac { 1 }{ 2 } \sin ^{ -1 }{ \cfrac { 2x }{ 1+{ x }^{ 2 } } +\cfrac { 1 }{ 2 } \cos ^{ -1 }{ \cfrac { 2x }{ 1+{ x }^{ 2 } }  }  }  \right\}  } $$
  • 1
  • $$0$$
  • $$\cfrac { 2x }{ 1+{ x }^{ 2 } } $$
  • $$x$$
$$\tan { \left( \cot ^{ -1 }{ x }  \right)  } $$ is equal to
  • $$\cfrac { \pi }{ 2 } -x$$
  • $$\cot { \left( \tan ^{ -1 }{ x } \right) } $$
  • $$\tan { x } $$
  • none of these
If $$\displaystyle x$$ takes negative permissible value, then $$\displaystyle \sin ^{-1}x$$ is equal to
  • $$\displaystyle \cos ^{-1}\sqrt{1-x^{2}}$$
  • $$\displaystyle -\cos ^{-1}\sqrt{1-x^{2}}$$
  • $$\displaystyle \cos ^{-1}\sqrt{x^{2}-1}$$
  • $$\displaystyle \pi -\cos ^{-1}\sqrt{1-x^{2}}$$
$$\sec ^{ 2 }{ \left( \tan ^{ -1 }{ 2 }  \right) +cosec ^{ 2 }{ \left( \cot ^{ -1 }{ 3 }  \right)  }  } $$=
  • 5
  • 10
  • 15
  • 20
The range of the function, $$\displaystyle f\left ( x \right ) = \left ( 1 + \sec^{-1} x \right ) \left ( 1 + \cos^{-1} x \right )$$ is
  • $$ \displaystyle \left ( -\infty ,\: \infty \right )$$
  • $$\displaystyle \left (-\infty, \: 0 \right ] \cup \left [4, \: \infty \right )$$
  • $$\displaystyle \left \{ 0, \: \left ( 1 + \pi \right )^{2} \right \}$$
  • $$\displaystyle \left [ 1, \: \left ( 1 + \pi \right )^{2} \right ]$$
Number of real value of $$x$$ satisfying the equation, $$\displaystyle \arctan \sqrt{x \left ( x+1 \right )} + \arcsin \sqrt{x \left ( x+1 \right ) + 1} = \frac{\pi}{2}$$ is
  • 0
  • 1
  • 2
  • more than 2
Range of $$\displaystyle f\left( x \right)=\sin ^{ -1 }{ x } +\tan ^{ -1 }{ x } +\cos ^{ -1 }{ x } $$ is
  • $$\displaystyle \left[ 0,\pi  \right] $$
  • $$\displaystyle \left[ \frac { \pi  }{ 4 } ,\frac { 3\pi  }{ 4 }  \right] $$
  • $$\displaystyle \left[ -\pi ,2\pi  \right] $$
  • None of these
If $$\displaystyle f\left ( x \right ) = \sin^{-1}x + \sec^{-1} x$$ is defined, then which of the following value/values is/are in its range?
  • $$\displaystyle \dfrac{- \pi}{ 2}$$
  • $$\displaystyle \dfrac{\pi}{ 2}$$
  • $$\displaystyle \pi$$
  • $$\displaystyle \dfrac{3\pi}{ 2}$$
$$\displaystyle \alpha = \sin^{-1} \left ( \cos \left ( \sin^{-1} x \right ) \right )$$ and $$\displaystyle \beta = \cos^{-1} \left ( \sin \left ( \cos^{-1} x \right ) \right )$$ then:
  • $$\displaystyle \tan \alpha = \cot \beta$$
  • $$\displaystyle \tan \alpha = - \cot \beta$$
  • $$\displaystyle \tan \alpha = \tan \beta$$
  • $$\displaystyle \tan \alpha = - \tan \beta$$
The value of $$\displaystyle sin^{-1}(cos(cos^{-1}(cos\:x)+sin^{-1}(sin\:x)))$$, where $$\displaystyle x\:\epsilon\:\left ( \frac{\pi}{2},\pi \right )$$, is equal to 
  • $$\displaystyle \frac{\pi}{2}$$
  • $$-\pi$$
  • $$\pi$$
  • $$\displaystyle -\frac{\pi}{2}$$
There exists a positive real number $$x$$ satisfying $$\displaystyle \cos \left ( \tan^{-1} x \right ) = x$$. The value of $$\displaystyle \cos^{-1} \left ( \frac{x^{2}}{2} \right )$$ is
  • $$\displaystyle \frac{\pi}{10}$$
  • $$\displaystyle \frac{\pi}{5}$$
  • $$\displaystyle \frac{2 \pi}{5}$$
  • $$\displaystyle \frac{4 \pi}{5}$$
If $$\displaystyle \alpha , \: \beta  \left ( \alpha \: < \: \beta \right )$$ are the roots of the equation $$\displaystyle 6x^{2} + 11x + 3 = 0$$ then which of the following are real?
  • $$\displaystyle \cos^{-1} \alpha$$
  • $$\displaystyle \sin^{-1} \beta$$
  • $$\displaystyle cosec^{-1} \alpha$$
  • Both $$\displaystyle \cot^{-1} \alpha$$ and $$\displaystyle \cot^{-1} \beta$$
The value of $$\sin ^{ -1 }{ \left( \sin { 10 }  \right)  } $$ is
  • $$10$$
  • $$10-3\pi $$
  • $$3\pi -10$$
  • none of these
Which one of the following statement is meaningless?
  • $$\displaystyle \cos^{-1} \left ( ln \left ( \frac{2e + 4}{3} \right ) \right )$$
  • $$\displaystyle cosec^{-1} \left ( \frac{\pi}{3} \right )$$
  • $$ \displaystyle \cot^{-1} \left ( \frac{\pi}{2} \right )$$
  • $$\displaystyle \sec^{-1} \left ( \pi \right )$$
The value of $$\displaystyle tan(sin^{-1}(cos(sin^{-1}x)))tan(cos^{-1}(sin(cos^{-1}x)))$$, where $$x\:\epsilon\:(0,1)$$, is equal to 
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$none\:of\:these$$
$$\displaystyle sec^{2}(tan^{-1}2)+cosec^{2}(cot^{-1}3)$$ is equal to 
  • $$5$$
  • $$13$$
  • $$15$$
  • $$6$$
Which of the following is the solution set of the equation $$\displaystyle 2cos^{-1}x=cot^{-1}\left(\frac{2x^{2}-1}{2x\sqrt{1-x^{2}}}\right)$$
  • $$(0,1)$$
  • $$(-1,1)-\left\{0\right\}$$
  • $$(-1,0)$$
  • $$[-1,1]$$
The value of $$\displaystyle sin^{-1}\left[x\sqrt{1-x}-\sqrt{x}\sqrt{1-x^{2}}\right]$$ is equal to
  • $$sin^{-1}x+sin^{-1}\sqrt{x}$$
  • $$sin^{-1}x-sin^{-1}\sqrt{x}$$
  • $$sin^{-1}\sqrt {x}-sin^{-1}x$$
  • $$0$$
The number of integer $$x$$ satisfying $$\displaystyle sin^{-1}|x-2|+cos^{-1}(1-|3-x|)=\frac{\pi}{2}$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
If $$x_{ 1 }=2\: tan^{ -1 }\left( \frac { 1+x }{ 1-x }  \right) ,x_{ 2 }=sin^{ -1 }\left( \frac { 1-x^{ 2 } }{ 1+x^{ 2 } }  \right) $$, where $$x_1,x_2\:\epsilon\:(0,1)$$, then $$2\left( x_{ 1 }+x_{ 2 } \right) $$ is equal  to
  • $$0$$
  • $$2\pi$$
  • $$\pi$$
  • $$none\:of\:these$$
If $$\displaystyle f\left ( x \right ) = \left ( \sin^{-1} x \right )^{2} + \left ( \cos^{-1} x \right )^{2}$$, then
  • $$\displaystyle f\left ( x \right )$$ has the least value of $$\displaystyle \frac{\pi^{2}}{8}$$
  • $$\displaystyle f\left ( x \right )$$ has the greatest value of $$\displaystyle \frac{5 \pi^{2}}{8}$$
  • $$\displaystyle f\left ( x \right )$$ has the least value of $$\displaystyle \frac{\pi^{2}}{16}$$
  • $$\displaystyle f\left ( x \right )$$ has the greatest value of $$\displaystyle \frac{5 \pi^{2}}{4}$$
  • Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
  • Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
  • STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
  • STATEMENT 1 is FALSE and STATEMENT 2 is TRUE
$$\displaystyle \tan^{-1}\left [ \cfrac{\cos\:x}{1+\sin\:x} \right ]$$ is equal to 
  • $$\displaystyle \frac{\pi}{4}-\frac{x}{2},for\:x\:\epsilon\:\left(-\frac{\pi}{2},\frac{3\pi}{2}\right)$$
  • $$\displaystyle \frac{\pi}{4}-\frac{x}{2},for\:x\:\epsilon\:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$$
  • $$\displaystyle \frac{\pi}{4}-\frac{x}{2},for\:x\:\epsilon\:\left(\frac{3\pi}{2},\frac{5\pi}{2}\right)$$
  • $$\displaystyle \frac{\pi}{4}-\frac{x}{2},for\:x\:\epsilon\:\left(-\frac{3\pi}{2},-\frac{3\pi}{2}\right)$$
The value of $$\displaystyle sin^{-1}(x^{2}-4x+6)+cos^{-1}(x^{2}-4x+6)$$ for all $$x\:\epsilon\:R$$ is
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\pi$$
  • $$0$$
  • none of these
Number of solutions of the equation $$ \sin \left ( \displaystyle \frac{1}{3}\cos^{-1}x \right )=1 $$ are
  • only one
  • no solution
  • only three
  • at least two
The value of $$\displaystyle \sin ^{-1}\left ( \sin 2 \right )$$ is?
  • $$2+n\pi$$
  • $$2-\pi$$
  • $$-2+\pi$$
  • $$2-2n\pi$$
The equation $$\displaystyle 3cos^{-1}x-\pi\:x-\frac{\pi}{2}=0$$ has
  • one negative solution
  • one positive solution
  • no solution
  • more than one solution
If $$ \sin^{-1}\left ( x^{2}-7x+12 \right )=m\pi ,\:\forall \:n\:\:\in \:I $$, then $$ x = $$
  • -4
  • 4
  • 3
  • -3
The value of $$ \tan^{-1}\left (\displaystyle \frac{1}{2}\tan 2A \right )+\tan^{-1}\left ( \cot A\right )+\tan^{-1}\left ( \cot ^{3}A\right ) $$, for $$ 0< A< \pi /4 $$, is :
  • $$\tan^{-1}2 $$
  • $$ \tan^{-1}\left ( \cot A \right ) $$
  • $$ 4\:tan^{-1}\:\left ( 1 \right ) $$
  • $$ 2\tan^{-1}\:\left ( 2 \right ) $$
There exists a positive real number $$x$$ satisfying $$\displaystyle \cos(\tan^{-1}x)=x$$. Then the value of $$\displaystyle \cos^{-1}\left(\frac{x^{2}}{2}\right)$$ is 
  • $$\displaystyle \frac{\pi}{10}$$
  • $$\displaystyle \frac{\pi}{5}$$
  • $$\displaystyle \frac{2\pi}{5}$$
  • $$\displaystyle \frac{4\pi}{5}$$
The number of real solution of the equation $$\displaystyle \sqrt{1+cos2x}=\sqrt{2}sin^{-1}(sin\:x),-\pi< x\leq \pi$$, is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$infinite$$
The equation $$\sin ^{-1}x=2\sin ^{-1}a$$ , has a solution for
  • $$\forall \:R$$
  • $$\displaystyle \left | a \right |< \frac{1}{2}$$
  • $$\displaystyle \left | a \right |\geq \frac{-1}{2}$$
  • $$\displaystyle \frac{-1}{\sqrt{2}}\leq a\leq \frac{1}{\sqrt{2}}$$
If $$\displaystyle \sin ^{-1}x-\sin ^{-1}y= \frac{\pi }{2}$$ ,then
  • $$x^{2}+y^{2}= 1$$
  • $$y= -\sqrt{1-x^{2}}, \:0\leq x\leq 1,-1\leq y\leq 0$$
  • $$y= \sqrt{1-x^{2}}, \: \left | x \right |< 1$$
  • None of these
Let $$f(x) =e^{\displaystyle\cos^{-1}\sin \left ( x+\displaystyle\frac{\pi }{3} \right )}$$, then $$f\left (\displaystyle \frac{8\pi }{9} \right )$$  equals
  • $$e^{\displaystyle\frac{7\pi }{12}}$$
  • $$e^{\displaystyle\frac{13\pi }{18}}$$
  • $$e^{\displaystyle\frac{5\pi }{18}}$$
  • $$e^{\displaystyle\frac{\pi }{12}}$$
If $$\displaystyle \cot ^{-1}\left [ \left ( \cos \alpha  \right )^{1/2} \right ]+\left [ \tan ^{-1}\left ( \cos \alpha  \right )^{1/2} \right ]=x$$ , then $$\sin x$$ equals 
  • $$1$$
  • $$\displaystyle \cot ^{2}\left ( \frac{\alpha }{2} \right )$$
  • $$\tan \alpha $$
  • $$\displaystyle \cot\left ( \frac{\alpha }{2} \right )$$
$$\displaystyle \sin ^{-1}\left ( a-\frac{a^{2}}{3}+\frac{a^{3}}{9}\cdots \infty  \right )+\cos ^{-1}\left ( 1+b+b^{2}+b^{3}+\cdots \infty  \right )= \frac{\pi }{2}$$ , when
  • $$\displaystyle a= 1, b= -\frac{1}{3}$$
  • $$\displaystyle a= -\frac{1}{6}, b= \frac{1}{2}$$
  • $$\displaystyle a= \frac{1}{6}, b= \frac{1}{2}$$
  • None of these
The domain of $$\sin ^{-1}[x]$$, where $$[x]$$ is greatest integer function, given by
  • $$\left [ -1,1 \right ]$$
  • $$\left [ -1,2 \right )$$
  • $$\left \{ -1,0,1 \right \}$$
  • None of these
If $$\cos^{-1}x+\cos^{-1}y+\cos^{-1}z= 3\pi $$, then value of $$\displaystyle \sum xy$$ equals
  • -3
  • 0
  • 3
  • -1
The domain of $$f(x) =\displaystyle \frac{\sin ^{-1}x}{x}$$ is
  • $$\left [ -1,1 \right ]$$
  • $$\left \{ 0 \right \}$$
  • $$\left [-1,0 \right )$$
  • None of these
Number of triplets $$\left ( x, y, z \right )$$ satisfying $$\sin ^{-1}x+\sin ^{-1}y+\cos ^{-1}z=2\pi$$ is
  • $$1$$
  • $$0$$
  • $$2$$
  • $$\infty$$
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