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

If sin1x+sin1y+sin1z=3π2 and f(1)=2,f(x+y)=f(x)f(y)  for all  x,yR. Then xf(1)+yf(2)+zf(3)x+y+zxf(1)+yf(2)+zf(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
ctan^{-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
0:0:1


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