CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 8 - MCQExams.com

If $$\sin ^{ -1 }{ x } +\sin ^{ -1 }{ y } +\sin ^{ -1 }{ z } =\cfrac { 3\pi  }{ 2 } $$ and $$f(1)=2,f(x+y)=f(x)f(y)$$  for all  $$x,y\in R$$. Then $${ x }^{ { f(1) } }+{ y }^{ f(2) }+{ z }^{ f(3) }-\cfrac { x+y+z }{ { x }^{ f(1) }+{ y }^{ f(2) }+{ z }^{ f(3) } } $$ is equal to
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
If $$\displaystyle \cot ^{-1}x+\cot ^{-1}y+\cot ^{-1}z=\frac{\pi }{2}, x,y,z> 0$$ and $$\displaystyle xy< 1$$, then $$\displaystyle x+y+z$$ is also equal to
  • $$\displaystyle \frac{1}{x}+\frac{1}{y}+\frac{1}{z}$$
  • $$\displaystyle xyz$$
  • $$\displaystyle xy+yz+zx$$
  • none of these
Exhaustive set of values of parameter $$a$$ so that $$\sin ^{ -1 }{ x } -\tan ^{ -1 }{ x } =a\quad $$ has a solution is
  • $$\left[ -\cfrac { \pi }{ 6 } ,\cfrac { \pi }{ 6 } \right] $$
  • $$\left[ -\cfrac { \pi }{ 4 } ,\cfrac { \pi }{ 4 } \right] $$
  • $$\left[ -\cfrac { \pi }{ 2 } ,\cfrac { \pi }{ 2 } \right] $$
  • none of these
The value of $$a$$ for which $$\displaystyle ax^{2}+sin^{-1}(x^{2}-2x+2)+cos^{-1}(x^{2}-2x+2)=0$$ has a real solution is 
  • $$\displaystyle \frac{\pi}{2}$$
  • $$\displaystyle -\frac{\pi}{2}$$
  • $$\displaystyle \frac{2}{\pi}$$
  • $$\displaystyle -\frac{2}{\pi}$$
The number of solution of the equation $$ 1+x^{2}+2x\:\sin \left ( \cos^{-1}y \right )= 0 $$ is :
  • 1
  • 2
  • 3
  • 4
The set of values of parameter $$a$$ so that the equation $$\displaystyle (\sin^{-1}x)^{3}+(\cos^{-1}x)^{3}=a\pi^{3}$$ has a solution. 
  • $$\displaystyle \left [ \frac{-1}{32},\frac{7}{8} \right ]$$
  • $$\displaystyle \left [ \frac{1}{32},,\frac{9}{8} \right ]$$
  • $$\displaystyle \left [ 0,\frac{7}{8} \right ]$$
  • $$\displaystyle \left [ \frac{1}{32},\frac{7}{8} \right ]$$
If $$\displaystyle sin^{-1}x+sin^{-1}y+sin^{-1}z=\pi$$, then $$x^{4}+y^{4}+z^{4}+4x^{2}y^{2}z^{2}=k(x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2})$$, where $$k$$ is equal to
  • $$1$$
  • $$2$$
  • $$4$$
  • $$none\:of\:these$$
The value of p for which system has a solution is
  • $$1$$
  • $$2$$
  • $$0$$
  • $$-1$$
If the equation $$\sin^{-1}\left ( x^{2}+x+1 \right )+\cos^{-1}\left ( ax+1 \right )=\displaystyle\frac{\pi }{2}$$ has exactly two distinct solutions then value of $$a$$ could not be
  • -1
  • 0
  • 1
  • 2
If $$ \sin^{-1}a +\sin^{-1}b+\sin^{-1}c= \displaystyle \frac{3\pi }{2} $$  and  $$ f\left ( 2 \right )=2,{f\left ( x+y \right )}= f\left ( x \right )\:f\left ( y \right )\:\:\forall \:\:x,\:y\:\epsilon \:R $$ then $$ a^{f\left ( 2 \right )}+\:b^{f\left ( 4 \right )}+\:c^{f\left ( 6 \right )}-\:\displaystyle \frac{3\left ( a^{f\left ( 2 \right )}. \:b^{f\left ( 4 \right )}.\:c^{f\left ( 6 \right )}\right )}{a^{f\left ( 2 \right )} +\:b^{f\left ( 4 \right )}+\:c^{f\left ( 6 \right )}} $$  equals
  • 2
  • 4
  • 6
  • 8
Indicate the relation which can hold in their respective domain for infinite values of $$\displaystyle x$$.
  • $$\displaystyle \tan \left | \tan^{-1}x \right | = \left | x \right |$$
  • $$\displaystyle \cot \left | \cot^{-1}x \right | = \left | x \right |$$
  • $$\displaystyle \tan^{-1} \left | \tan x \right | = \left | x \right |$$
  • $$\displaystyle \sin \left | \sin^{-1}x \right | = \left | x \right |$$
If $$\displaystyle \sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3\pi }{2}
$$ and $$f\left ( 2 \right )= 2, f\left ( a+b \right )=f\left ( a \right )f\left ( b \right ), \:\forall \:a,\:b \:\epsilon  \:R$$, then $$x^{f\left ( 2 \right )},y^{f\left ( 4 \right )},z^{f\left ( 6 \right )}$$ are in
  • A.P.
  • G.P
  • H.P
  • None
Let $$\displaystyle f:A\rightarrow B$$ be a function defined by $$\displaystyle y=f(x)$$ where f is a bijective function, means f is injective (one-one) as well as surjective (onto), then there exist a unique mapping $$\displaystyle g:B\rightarrow A$$ such that $$\displaystyle f(x)=y$$ if and only if $$\displaystyle g(y)=x\forall x \epsilon A,y \epsilon B $$ Then function g is said to be inverse of f and vice versa so we write $$\displaystyle g=f^{-1}:B\rightarrow A[\left \{ f(x),x \right \}:\left \{ x,f(x) \right \}\epsilon f^{-1}] $$when branch of an inverse function is not given (define) then we consider its principal value branch.

If $$\displaystyle -1<x<0$$,then $$\displaystyle \tan^{-1}x $$ equals?
  • $$\displaystyle \pi-\cos^{-1}(\sqrt{1-x^{2}})$$
  • $$\displaystyle \sin ^{-1}\left(\frac{x}{\sqrt{1+x^{2}}}\right )$$
  • $$\displaystyle -\cos^{-1}\left(\frac {\sqrt{1-x^{2}}}{x}\right)$$
  • $$\displaystyle \ cosec ^{-1}x$$
The number of solutions of $$\sin^{-1}\left ( 1+b+b^{2}+\cdots \infty \right )+\cos^{-1}\left ( a-\displaystyle\frac{a^{2}}{3}+\frac{a^{2}}{9}\cdots \infty  \right )= \displaystyle\frac{\pi }{2}$$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$\infty$$
Express in terms of an inverse function the angle formed at the intesection of the diagonals of a cube.
  • $$sin^{1} 2/3$$
  • $$cos^{1} 1/3$$
  • $$tan^{1} 1/3$$
  • $$sin^{1} 1/3$$
If $$\tan^{-1}\left(\displaystyle\tan\frac{5\pi}{4}\right)=\alpha$$ and $$\tan^{-1}\left(\displaystyle - \tan\frac{2\pi}{3}\right)=\beta$$ then.
  • $$\displaystyle\alpha -\beta =\frac{7\pi}{12}$$
  • $$\displaystyle \alpha +\beta =\frac{7\pi}{12}$$
  • $$\displaystyle 2\alpha +3\beta =\frac{7\pi}{12}$$
  • $$\displaystyle 4\alpha +3\beta =\frac{7\pi}{12}$$
$$\displaystyle \sin^{-1}\frac{3}{5}+\sin^{-1}\frac{4}{5}$$ is equal to
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{6}$$
If $$0 < x_{1} < x_{2}$$ which of following is true for $$y = \sec^{-1}x$$.
  • $$\sec^{-1}x_{1} + \sec^{-1}x_{2} > \sec^{-1} \left (\dfrac {x_{1} + x_{2}}{2}\right )$$
  • $$\sec^{-1}x_{1} + \sec^{-1}x_{2} < 2\sec^{-1} \left (\dfrac {x_{1} + x_{2}}{2}\right )$$
  • $$\sec^{-1}x_{1} > \sec^{-1}x_{2}$$
  • $$\sec^{-1}x_{1} = \sec^{-1}x_{2}$$
If $$3\cos ^{ -1 }{ x } +\sin ^{ -1 }{ x } =\pi $$, then $$x=.....$$
  • $$\cfrac { \sqrt { 3 } }{ 2 } $$
  • $$-\cfrac { 1 }{ \sqrt { 2 } } $$
  • $$\cfrac { 1 }{ \sqrt { 2 } } $$
  • $$\cfrac { 1 }{ 2 } $$
The domain of the function $${\sin ^{ - 1}}2x$$ is:
  • $$\left[ {0,\,1} \right]$$
  • $$\left[ {-1,\,1} \right]$$
  • $$\left[ {-2,\,2} \right]$$
  • $$\left[ {\dfrac{{ - 1}}{2},\,\dfrac{1}{2}} \right]$$
Domain of $$f(x)=\cot ^{ -1 }{ x } +\cos ^{ -1 }{ x } +co\sec ^{ -1 }{ x } $$ is
  • $$\left[ -1,1 \right] $$
  • $$R$$
  • $$(-\infty ,-1]\cup [1,\infty )$$
  • $$\left\{ -1,1 \right\} $$
Let $$E_{1} = \left \{x \epsilon \mathbb {R} : x\neq 1\ and\ \dfrac {x}{x - 1} > 0\right \}$$
and $$E_{2} = \left \{x \epsilon E_{1} : \sin^{-1} \left (\log_{e} \left (\dfrac {x}{x - 1}\right )\right )\text {is a real number}\right \}$$.
(Here, the inverse trigonometric function $$\sin^{-1}x$$ assumes values in $$\left [-\dfrac {\pi}{2}, \dfrac {\pi}{2}\right ]$$)
Let $$f : E_{1} \rightarrow \mathbb {R}$$ be the function define by $$f(x) = \log_{e} \left (\dfrac {x}{x -1}\right )$$ and $$g : E_{2}\rightarrow \mathbb{R}$$ be the function defined by $$g(x) = \sin^{-1} \left (\log_{e} \left (\dfrac {x}{x - 1}\right )\right )$$.
LIST - ILIST - II
P. The range of $$f$$ is$$\left (-\infty, \dfrac {1}{1 - e}\right ] \cup \left [\dfrac {e}{e - 1}, \infty \right )$$
Q. The range of $$g$$ contins$$(0, 1)$$
R. The domain of $$f$$ contains$$\left [-\dfrac {1}{2}, \dfrac {1}{2}\right ]$$
S. The domain of $$g$$ is$$(-\infty, 0)\cup (0, \infty)$$
$$\left (-\infty, \dfrac {e}{e - 1}\right ]$$
$$(-\infty, 0)\cup \left (\dfrac {1}{2}, \dfrac {e}{e - 1}\right ]$$
The correct option is
  • $$P\rightarrow 4; Q \rightarrow 2; R\rightarrow 1; S\rightarrow 1$$
  • $$P\rightarrow 3; Q \rightarrow 3; R\rightarrow 6; S\rightarrow 5$$
  • $$P\rightarrow 4; Q \rightarrow 2; R\rightarrow 1; S\rightarrow 6$$
  • $$P\rightarrow 4; Q \rightarrow 3; R\rightarrow 6; S\rightarrow 5$$
The value of $$sin^{-1} x + cos^{-1} x (|x| \geq 1)$$ is
  • 1
  • $$\pi$$
  • $$\pi / 2$$
  • $$ - \pi / 2$$
Match the entries of Column - I and Column - II.
Column - IColumn - II
aIf 4 $$sin^{-1} x + cos^{-1} x = \pi$$, then x equals1ab
bIf $$\angle C = 90^{0}$$, then the value of $$tan^{-1}$$ $$\dfrac{a}{b + c}$$ + $$tan^{-1}$$ $$\dfrac{b}{c +a}$$ is 2$$\pi$$
c$$tan^{-1}$$ 1 + $$tan^{-1}$$ 2 + $$tan^{-1}$$ 3 is3$$\pi$$/4
dIf $$sec^{-1}$$ $$\dfrac{x}{a}$$ - $$sec^{-1}$$ $$\dfrac{x}{b}$$ = $$sec^{-1}$$ b - $$sec^{-1}$$ a, then x equals41/2
  • a-4, b-3, c-2, d-1
  • a-1, b-3, c-2, d-4
  • a-4, b-3, c-1, d-2
  • a-3, b-4, c-2, d-1
Let $$a={ (\sin ^{ -1 }{ x) }  }^{ \sin ^{ -1 }{ x }  },\quad b={ \left( \sin ^{ -1 }{ x }  \right)  }^{ \cos ^{ -1 }{ x }  },\quad c={ \left( \cos ^{ -1 }{ x }  \right)  }^{ \sin ^{ -1 }{ x }  },\quad d={ \left( \cos ^{ -1 }{ x }  \right)  }^{ \cos ^{ -1 }{ x }  }$$ and if $$x\in (0,1)$$then 
  • $$a>b>d>c$$
  • $$b>a>d>c$$
  • $$d>c>a>b$$
  • none of these
If $$0< x < 1$$ , then $$\tan^{-1}(\cfrac{\sqrt{1-x^{2}}}{1+x})$$ is equal to
  • $$\cfrac{1}{2}\cos^{-1}x$$
  • $${\cos ^{ - 1}}{{\sqrt {1 + x} } \over 2}$$
  • $${\sin ^{ - 1}}\sqrt {{{1 - x} \over 2}} $$
  • $${1 \over 2}\sqrt {{{1 + x} \over {1 - x}}} $$
Let $$a, b, c$$ be a positive real numbers $$\theta = \tan^{-1} \sqrt{\dfrac{a(a + b +c)}{bc}} + \tan^{-1} \sqrt{\dfrac{b(a + b+ c)}{ca}} + \tan^{-1} \sqrt{\dfrac{c(a + b + c)}{ab}}$$, then $$\tan \theta$$
  • $$0$$
  • $$3 \pi$$
  • $$1$$
  • $$4 \pi$$
 $$\\ \cos^{-1}\left[\dfrac {\sqrt {1+x}+\sqrt {1-x}}{2}\right]=\dfrac {\pi}{2}-\dfrac {1}{2}\cos^{-1}x$$
  • True
  • False
If $$\cos ^{ -1 }{ \cfrac { x }{ 2 }  } +\cos ^{ -1 }{ \cfrac { y }{ 3 }  } =\theta $$, then $$9{x}^{2}-12xy\cos{\theta}+4{y}^{2}$$ is equal to
  • $$36 \sin ^{ 2 }{ \theta } $$
  • $$36 \cos ^{ 2 }{ \theta } $$
  • $$36 \tan ^{ 2 }{ \theta } $$
  • None of these
The range of the function $$f(x)=\sin ^{ -1 }{ \left( { x }^{ 2 }-2x+2 \right)  } $$
  • $$\phi$$
  • $$\left[ -\cfrac { \pi }{ 2 } ,\cfrac { \pi }{ 2 } \right] $$
  • $$\cfrac { \pi }{ 2 } $$
  • none of these
$$2{\tan ^{ - 1}}\left[ {\sqrt {\dfrac{{a - b}}{{a + b}}} \tan \dfrac{\theta }{2}} \right] = $$
  • $${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta + b}}{{a + b\cos \theta }}} \right)$$
  • $${\cos ^{ - 1}}\left( {\dfrac{{a + b\cos \theta }}{{a\cos \theta + b}}} \right)$$
  • $${\cos ^{ - 1}}\left( {\dfrac{{a\cos \theta }}{{a + b\cos \theta }}} \right)$$
  • $${\cos ^{ - 1}}\left( {\dfrac{{b\cos \theta }}{{a\cos \theta + b}}} \right)$$
$$\sin { ^{ -1 }\dfrac { 3 }{ 5 } +\tan { ^{ -1 } }  } \dfrac { 1 }{ 7 } =\dfrac { \pi  }{ 2 }$$
  • True
  • False
$$\cos [\tan^{-1}\{ \sin(\cot^{-1}x)\}]$$ is equal to:
  • $$\sqrt{\dfrac{x^2 + 2}{x^2 + 3}}$$
  • $$\sqrt{\dfrac{x^2 + 2}{x^2 + 1}}$$
  • $$\sqrt{\dfrac{x^2 + 1}{x^2 + 2}}$$
  • none of these
$$\sin^{-1}\left(a-\dfrac{a^2}{3}+\dfrac{a^3}{9}+...\right)+\cos^{-1}(1+b+b^2+...)=\dfrac{\pi}{2}$$ when?
  • $$a=-3$$ and $$b=1$$
  • $$a=1$$ and $$b=-\dfrac{1}{3}$$
  • $$a=\dfrac{1}{6}$$ and $$b=\dfrac{1}{2}$$
  • None of these
The range of $$ f\left( x \right) =\sin { =^{ -1 }x+ } \cos { =^{ -1 } } x+\tan { ^{ -1 } } x$$ is ?
  • $$ \left( 0,\pi \right)$$
  • $$ \left[ \dfrac { \pi }{ 4 } ,\dfrac { 3\pi }{ 4 } \right]$$
  • $$ \left[ \dfrac { -\pi }{ 4 } ,\dfrac { \pi }{ 4 } \right]$$
  • $$ \left[ 0,\dfrac { 3\pi }{ 4 } \right]$$
If $${ x }_{ 1 },{ x }_{ 2},{ x }_{ 3}$$ are positive roots of $$ x^{ 3 }-6x^{ 2 }+3px-2p=0\quad (p\in R)$$, then the value of $$\sin ^{ -1 } \left( \frac { 1 }{ { x }_{ 1 } } +\frac { 1 }{ { x }_{ 2 } }  \right) +\cos ^{ -1 }{ \left( \frac { 1 }{ { x }_{ 2 } } +\frac { 1 }{ { x }_{ 3 } }  \right)  } -\tan ^{ -1 }{ \left( \frac { 1 }{ { x }_{ 3 } } +\frac { 1 }{ { x }_{ 1 } }  \right)  } $$ is equal to
  • $$\frac { \pi }{ 8 } $$
  • $$\frac { \pi }{ 6 } $$
  • $$\frac { \pi }{ 4 } $$
  • $$\pi $$
The value of $$\sin ^{ -1 }{ \left[ \cot { \left( \sin ^{ -1 }{ \sqrt { \frac { 2-\sqrt { 3 }  }{ 4 }  }  } +\cos ^{ -1 }{ \frac { \sqrt { 2 }  }{ 4 } +\sec ^{ -1 }{ \sqrt { 2 }  }  }  \right)  }  \right]  } $$ is equal to
  • $$1$$
  • $$0$$
  • $$2$$
  • $$3$$
$$cos({ cos }^{ -1 }cos(\frac { 8\pi  }{ 7 } )+{ tan }^{ -1 }tan(\frac { 8\pi  }{ 7 } ))$$ has the value equal to -
  • $$1$$
  • $$-1$$
  • $$cos\dfrac { \pi }{ 7 } $$
  • $$0$$
If $$\left( \tan ^ { - 1 } x \right) ^ { 2 } + \left( \cot ^ { - 1 } x \right) ^ { 2 } = \dfrac { 5 \pi ^ { 2 } } { 8 } $$, then $$x$$ equals to
  • $$- 1$$
  • $$1$$
  • $$0$$
  • None Of These
The exhaustive set of values of $$'a'$$ such that $$x^{2}+ax+sing^{-1}\ (x^{2}-4x+5)+\ cos^{-1}(x^{2}-4x+5)=0$$ has at least one solution is
  • $$\left \{ -2-\frac{\pi }{4} \right \}$$
  • $$\left ( -\propto,-2-\frac{\pi}{4} \right )$$
  • $$(-\propto,-2-\frac{\pi}{4}]$$
  • $$(-2-\frac{\pi}{4},+\propto]$$
If $$\left |{\cos ^{ - 1}}\left( {\dfrac{{1 - {x^2}}}{{1 + {x^2}}}} \right)\right |\,\, < \,\dfrac{\pi }{3},\,then\,:$$ 
  • $$x \in \,\left[ { - \dfrac{1}{3},\dfrac{1}{{\sqrt 3 }}} \right]$$
  • $$x \in \,\left[ { - \dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}} \right]$$
  • $$x \in \,\left[ {0,\dfrac{1}{{\sqrt 3 }}} \right]$$
  • none of these
$$\tanh^{1}\left(\dfrac {1}{3}\right)+\coth^{1}(3)=$$..... 
  • $$\log 2$$
  • $$\log 3$$
  • $$\log \sqrt {3}$$
  • $$\log \sqrt {2}$$
If $$y=\dfrac{1}{2}\csc\ h^{-1}{\left(\dfrac{1}{2x\sqrt{1+x^{2}}}\right)}$$ then $$x$$=
  • coshy
  • sinhy
  • tanhy
  • cothy
$$3\cot^{-1}{\left(\dfrac{1}{2+\sqrt{3}}\right)}-\cot^{-1}\left(\dfrac{1}{x}\right)=\cot^{-1}\left(\dfrac{1}{3}\right)+\dfrac{\pi}{2}$$ then $$x=$$?
  • $$1$$
  • $$2$$
  • $$3$$
  • $$\sqrt{3}$$
$$Sin h (cos h ^{-1} x) =$$
  • $$\sqrt{x^2 +1}$$
  • $$\dfrac{1}{\sqrt{x^2 +1}}$$
  • $$\sqrt{x^2 -1}$$
  • $$\dfrac{1}{\sqrt{x^2 -1}}$$
The value of $$e^{sinh^{-1} (tan \theta)}$$ is equal to
  • $$cosec \theta + cot \theta$$
  • $$sec \theta + tan \theta$$
  • $$cosec \theta + sec \theta$$
  • $$tan \theta + cot \theta$$
If $$f(x)={ sin }^{ -1 }\left( \frac { \sqrt { 3 }  }{ 2 } x-\frac { 1 }{ 2 } \sqrt { 1-{ x }^{ 2 } }  \right) -\frac { 1 }{ 2 } \le x\le 1$$, then f(x) is equal to :
  • $${ sin }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ sin }^{ -1 }(x)$$
  • $${ sin }^{ -1 }x-\frac { \pi }{ 6 } $$
  • $${ sin }^{ -1 }x+\frac { \pi }{ 6 } $$
  • None of these
The value of $$\tan{\left\{\dfrac{\pi}{4}+\dfrac{1}{2}\cos^{-1}{(\dfrac{x}{y})}\right\}}+\tan{\left\{\dfrac{\pi}{4}-\dfrac{1}{2}\cos^{-1}{(\dfrac{x}{y})}\right\}}$$
  • $$\dfrac{x}{y}$$
  • $$\dfrac{y}{x}$$
  • $$\dfrac{2y}{x}$$
  • $$\dfrac{2x}{y}$$
For which value of x, $$\sin [\cot^{-1}(x+1)]=\cos (\tan^{-1}x)$$.
  • $$\dfrac{1}{2}$$
  • $$0$$
  • $$1$$
  • $$\dfrac{-1}{2}$$
If $$cot^{-1} x - cot^{-1} (x+2) = 15^0$$ then x is equal to 
  • $$\sqrt{3}$$
  • $$-\sqrt{3}$$
  • $$\sqrt{3} +2$$
  • $$-\sqrt{3} +2$$
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