CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 10 - MCQExams.com

If for $$x < -1,\ cos^{-1}{\left(\dfrac{x^{2}-1}{x^{2}+1}\right)}+sin^{-1}{\left(\dfrac{2x}{1+x^{2}}\right)}-tan^{-1}{\left(\dfrac{2x}{x^{2}-1}\right)}=\dfrac{\pi}{3}$$, then $$x=$$
  • $$-\sqrt{3}$$
  • $$-\sqrt{2}$$
  • $$-2$$
  • $$-2\sqrt{3}$$
The value of a for which 
$$ax+sec^{ -1 }\sqrt { { 2x }^{ 2 }-{ x }^{ 4 } } +cose^{ -1 }\sqrt { { 2x }^{ 2 }-{ x }^{ 4 } } =0$$
  • $$\frac { \pi }{ 2 } $$
  • $$-\frac { \pi }{ 2 } $$
  • $$\frac { 2 }{ \pi } $$
  • $$-\frac { 2 }{ \pi } $$
The sum of the solution of the equation $$2\sin^{-1}\sqrt {x^{2}+x+1}+\cos^{-1}\sqrt {x^{2}+x}\dfrac {3\pi}{2}$$ is
  • $$0$$
  • $$-1$$
  • $$3$$
  • $$2$$
If $$\tan^{-1}{\left(\dfrac{3a^{2}x^{3}}{a^{3}-3ax^{2}}\right)}=k\tan^{-1}{\left(\dfrac{x}{a}\right)}$$, then $$k=$$
  • $$2$$
  • $$3$$
  • $$-2$$
  • $$4$$
If $${ cos }^{ -1 }\sqrt { p } +{ cos }^{ -1 }\sqrt { 1-p } +{ cos }^{ -1 }\sqrt { 1-q } =\frac { 3\pi  }{ 4 } $$ then the value of q is equal to:
  • 1
  • $$\dfrac { 1 }{ \sqrt { 2 } } $$
  • $$\dfrac { 1 }{ 3 } $$
  • $$\dfrac { 1 }{ 2 } $$
The value of $$\sin ^{-1}[\cot{(\sin ^{-1}\sqrt{(\frac{2 - \sqrt 3}{4})} + \cos ^{-1}\frac{\sqrt {12}}{4})} + \sec ^{-1}\sqrt 2]$$
  • 0
  • $$\frac{\pi}{4}$$
  • $$\frac{\pi}{6}$$
  • $$\frac{\pi}{2}$$
If $$tan^{ -1 }\left( { sin }^{ 2 }\theta +2sin+2 \right) $$+$${ cot }^{ -1 }\left( { 4 }^{ { sec }^{ 2 } }+1 \right) =\frac { \pi  }{ 2 } $$ has solution for some $$\theta $$ and $$\phi $$ then
  • $$sin\theta =-1$$
  • $$cos\phi =-1$$
  • $$cos\phi =1$$
  • $$sin\theta =1$$
The sum of roots of the equation $$\tan^{-1}\dfrac{1}{1+2x}+\tan^{-1}\dfrac{1}{1+4x}=\tan^{-1}\dfrac{2}{x^{2}}$$ is 
  • $$\dfrac{7}{3}$$
  • $$3$$
  • $$4$$
  • $$5$$
If $$\sin^{-1} x+\sin^{-1} y=\dfrac{\pi}{3}$$ then the value of $$\cos^{-1}x+\cos^{-1}y$$ is equal to which of the following :
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{\pi}{3}$$
  • $$\dfrac{2\pi}{3}$$
  • $$\pi$$
$$ \tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1}\left(\frac{x-y}{x+y}\right)=\dots $$
  • $$ \frac{\pi}{2} $$
  • $$ \frac{\pi}{3} $$
  • $$ \frac{\pi}{2} $$
  • $$ \frac{\pi}{4} $$ or $$ \frac{3 \pi}{4} $$
If $${\sin ^{ - 1}}x + {\cos ^{ - 1}}y = \frac{{2\pi }}{5},$$ then $${\cos ^{ - 1}}x + {\sin ^{ - 1}}y$$ is 
  • $$\frac{{2\pi }}{5}$$
  • $$\frac{{3\pi }}{5}$$
  • $$\frac{{4\pi }}{5}$$
  • $$\frac{{3\pi }}{10}$$
The value of x satisfying $$ sin^{-1}x+sin^{-1}(1-x)=cos^{-1}x$$ are
  • 0
  • 1/2
  • 1
  • 2
Solve $${ \tan }^{ -1 }\left(\cfrac { \cos x }{ 1+\sin x }\right) $$.
  • $$ \dfrac { \pi }{ 4 } + \dfrac { x }{ 2 } $$
  • $$ \dfrac { \pi }{ 4 } - \dfrac { x }{ 2 } $$
  • $$ \dfrac { \pi }{ 4 } + { x }$$
  • $$ \dfrac { \pi }{ 4 } -  { x } $$
If $$x = {\sin ^{ - 1}}\left( {\sin 10} \right)\,\,and\,\,y = {\cos ^{ - 1}}\left( {\cos 10} \right),\,\,then\,\,y - x$$ is equal to:
  • $$\pi $$
  • $$0$$
  • $$7\pi $$
  • $$10$$
If $$x = {\sin ^{ - 1}}\left( {\sin 10} \right)\,\,and\,\,y = {\cos ^{ - 1}}\left( {\cos 10} \right)$$
  • 0
  • 10
  • $$7\pi $$
  • $$\pi $$
If $$\theta \epsilon [4\pi ,5\pi ]$$, then $${ cos }^{ -1 }(cos\theta )$$ equals 
  • $$-4\pi +\theta $$
  • $$5\pi -\theta $$
  • $$4\pi -\theta $$
  • $$\theta -5\pi $$
The solution set of the equation $${ sin }^{ -1 }\sqrt { 1-{ x }^{ 2 } } +{ cos }^{ -1 }x={ cot }^{ -1 }\left( \frac { \sqrt { 1-{ x }^{ 2 } }  }{ x }  \right) -{ sin }^{ -1 }x$$ is 
  • [-1,1] - {0}
  • (0,1] U {-1]
  • [-1,0) U {1}
  • [-1,1]
If $$[\sin^{-1}(\cos ^{-1}(\sin ^{-1}(\tan ^{-1}x)))]=1$$, where $$\left[ \bullet  \right] $$ denotes the greatest integer function, then $$x\in $$
  • $$[\tan \sin \cos 1,\tan \sin \cos \sin 1]$$
  • $$(\tan \sin \cos 1,\tan \sin \cos \sin 1)$$
  • [-1, 1]
  • None of these
Number of solutions of the equation
$$tan^{ -1 } \left( \dfrac { 1 }{ a-1 }\right) = tan^{ -1 } \left(\dfrac { 1 }{ x }\right) + tan^{ -1 } \left( \dfrac { 1 }{ a^{ 2 } x + 1}\right)$$
  • $$one$$
  • $$Two$$
  • $$Three$$
  • $$Zero$$
Solution of $${\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}} + {\cot ^{ - 1}}\frac{{1 - {x^2}}}{{2x}} = \frac{{2\pi }}{3}\,\,$$ are
  • $$\frac{1}{{\sqrt 3 }}$$
  • $$ - \sqrt 3 $$
  • $$\sqrt 3 + 2$$
  • $$\sqrt 3 - 2$$
$${\cos ^{ - 1}}\left( {\cos \;x} \right) = [x],[.]\;denotes\;the\;greatest\;\operatorname{int} eger\;function,\;is\;$$
  • $$2\pi + 3$$
  • $$\pi + 3$$
  • $$\pi - 3$$
  • $$2\pi - 3$$
If $$\alpha$$ is the only real root of the equation $$\displaystyle x^3 + bx^2 + c = 0 $$ ($$b$$ < $$c$$) then the value of $$\displaystyle Tan^{1}\alpha + Tan^{1}\left ( \dfrac{1}{\alpha} \right  ) $$ = 
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{2}$$
  • $$0$$
  • $$\pi$$
The number of solution for the equation $$\cos^{-1}(1-x)+m\ \cos^{-1}x=\dfrac {n\pi}{2}$$ where $$m > 0,n \le 0$$ is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$infinite$$
The value of $${ sin }^{ -1 }\left[ cot\left( { sin }^{ -1 }\sqrt { \left( \cfrac { 2-\sqrt { 3 }  }{ 4 }  \right)  } +{ cos }^{ -1 }\left( \cfrac { \sqrt { 12 }  }{ 4 }  \right) +{ sec }^{ -1 }\sqrt { 2 }  \right)  \right] $$ is :
  • $$0$$
  • $$\cfrac { \pi }{ 4 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\cfrac { \pi }{ 2 } $$
If $$x$$ = $$Tan^{1}(1)$$ + $$Cos^{1}(\dfrac{1}{2})$$ + $$Sin^{1}(\dfrac{1}{2})$$ and $$y$$ = $$ cos\left [ \dfrac{1}{2}Cos^{1}(1/8) \right ] $$ then 
  • $$x$$ = $$2\, \pi\, y$$
  • $$y$$ = $$3\, \pi\, x$$
  • $$x$$ = $$\pi\, y$$
  • $$y$$ = $$\pi\, x$$
Solve the equation : $$4 \tan ^ { - 1 } \dfrac { 1 } { 5 } - \tan ^ { - 1 } \dfrac { 1 } { 239 } =?$$
  • $$\pi$$
  • $$\pi / 2$$
  • $$\pi / 3$$
  • $$\pi / 4$$
Let $$f( x ) = \sin ^ { - 1 } x + \cos ^ { - 1 } x \cdot$$ Then $$\frac { \pi } { 2 }$$ is equal to:
  • f(-2)
  • $$f \left( k ^ { 2 } - 2 k + 3 \right) , k \in R$$
  • $$\mathrm { f } \left( \frac { 1 } { 1 + \mathrm { k } ^ { 2 } } \right) , \mathrm { k } \in \mathrm { R }$$
  • none
If $${ sin }^{ -1 }x+{ sin }^{ -1 }y=\cfrac { 2\pi  }{ 3 } $$, then $${ cos }^{ -1 }x+{ cos }^{ -1 }y=$$
  • $$\cfrac { 2\pi }{ 3 } $$
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\pi $$
If $$tan^{-1}2x + tan^{-1}3x = \dfrac{\pi}{4}$$ then $$x$$ =
  • $$-1$$
  • $$\dfrac{1}{6}$$
  • $$-1, \dfrac{1}{6}$$
  • None of these
If $$A=\tan {-1}(\frac {x\sqrt 3}{2k-x})$$ and $$B=\tan ^{-1}(\frac {2x-k}{k\sqrt 3})$$, then the value of A-B is
  • $$0^o$$
  • $$45^o$$
  • $$60^o$$
  • $$30^o$$
If $$\displaystyle Sin^{1}(x - \dfrac{x^2}{2} + \dfrac{x^3}{4} - .......) + Cos^{1}(x^2 - \dfrac{x^4}{2} + \dfrac{x^6}{4} - ......) = \dfrac{\pi}{2} $$ for 0 < $$\left | x \right |$$ < $$\sqrt{2}$$ then $$x$$ = 
  • 1/2
  • 1
  • 1/2
  • 1
$$ \sin \cot ^{-1} \tan \cos ^{-1} x  $$ is always equal to
  • x
  • $$
    \sqrt{1-x^{2}}
    $$
  • $$
    \frac{1}{x}
    $$
  • None of these
$$3tan^{ -1 }x\quad =\quad tan^{ -1 }\{ \dfrac { 3x-x^{ 3 } }{ 1-3x^{ 2 } } \} ,\quad then\quad x\quad belong\quad to\quad $$
  • [-1,1]
  • (-1,1)
  • $$\{ -\dfrac { 1 }{ \sqrt { 3 } } ,\dfrac { 1 }{ \sqrt { 3 } } \} $$
  • None of these
If $$A=\tan ^{-1}(\frac {x\sqrt 3}{2k-x})$$ and $$B=\tan {-1}(\frac {2x-k}{k\sqrt 3})$$, then the value of $$A-B$$ is
  • $$0^o$$
  • $$45^o$$
  • $$60^o$$
  • $$30^o$$
Value of $$\sin ^{ -1 }{ \frac { 3 }{ \sqrt { 13 }  } +\cos ^{ -1 }{ \frac { 11 }{ \sqrt { 146 }  }  } +\cot ^{ -1 }{ \sqrt { 3 }  }  } $$ is 
  • $$\pi$$
  • $$\pi$$/2
  • 5$$\pi$$/12
  • $$\pi$$/3
If $$A=2{\tan}^{-1}\left(2\sqrt{2}-1\right)$$ and $$B=3{\sin}^{-1}\left(1/3\right)+{sin}^{-1}\left(3/5\right)$$, then 
  • $$A=B$$
  • $$A<B$$
  • $$A>B$$
  • $$none \of \these$$
The least positive integer n for which $$\left( \frac { 1+i }{ 1-i }  \right) ^{ n }=\frac { 2 }{ \pi  } \left( \sec ^{ -1 }{ \frac { 1 }{ x } +\sin ^{ -1 }{ x }  }  \right) \left( where,\quad x\neq 0,-1\le x\ge 1\quad and\quad i=\sqrt { -1 }  \right) $$, is 
  • 2
  • 4
  • 6
  • 8
If $$2sin^{-1}( \frac {3}{5})-cos^{-1}(\frac{5}{13})=cos^{-1} (\lambda)$$, then $$\lambda$$ is eual to
  • 323/225
  • 223/325
  • 323/325
  • 123/125
the number o real soluttion of $${ tan }^{ -1 }\sqrt { x(x+1) } +{ sin }^{ -1 }\sqrt { { x }^{ 2 }+x+1 } =\frac { \pi  }{ 2 } is$$
  • zero
  • one
  • two
  • infinit
The numerical value of $$cot\left( { 2sin }^{ -1 }\dfrac { 3 }{ 5 } +{ cos }^{ -1 }\dfrac { 3 }{ 5 }  \right) $$ is
  • $$\dfrac { -4 }{ 3 } $$
  • $$\dfrac { -3 }{ 4 } $$
  • $$\dfrac { 3 }{ 4 } $$
  • $$\dfrac { 4 }{ 3 } $$
The value of $${\sin}^{-1}\left( \sin12 \right )+{\sin}^{-1}\left( \cos12 \right )=$$
  • $$0$$
  • $$24-2\pi$$
  • $$4\pi-24$$
  • $$8\pi$$
$$\cot^{ -1 } 3 + \cot^{ -1 } 7 + \cot^{ -1 } 13 + n terms =$$
  • $$\tan^{ -1 }\left(\dfrac { n }{ n + 2 }\right)$$
  • $$\tan^{ -1 }\left(\dfrac { n + 2 }{ n }\right)$$
  • $$\tan^{ -1 }\left(\dfrac { 3n }{ 4n + 1 }\right)$$
  • $$\tan^{ -1 }\left(\dfrac { 4n + 1 }{ 3n }\right)$$
If $$\theta =\cot ^{ -1 }{ \sqrt { \cos { x }  } -\tan ^{ -1 }{ \sqrt { \cos { x }  }  }  } $$, then $$\sin { \theta  } =$$
  • $$\tan { \frac { 1 }{ 2 } } x$$
  • $$\tan ^{ 2 }{ \left( \frac { x }{ 2 } \right) } $$
  • $$\frac { 1 }{ 2 } \tan ^{ -1 }{ \left( \frac { x }{ 2 } \right) } $$
  • none of these
If $$\sin^{ -1 }\left(\dfrac { 2a }{ 1 + a^{ 2 } }\right) + \cos^{ -1 }\left(\dfrac { 1-a^{ 2 } }{ 1 + a^{ 2 } }\right) = \tan^{ -1 } \left(\dfrac { 2x }{ 1 x^{ 2 } }\right)$$ where, $$a, x \epsilon \left( 0, 1\right)$$ then the value of $$x$$ is
  • $$0$$
  • $$\dfrac { a }{ 2 }$$
  • $$a$$
  • $$\dfrac { 2a }{ 1 a^{ 2 } }$$
If $$\tan^{-1}{\left(\cfrac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)}=\alpha,\left(\alpha\in\left[0,\cfrac{\pi}{4}\right)\right]$$ yhen $$x^2$$ is equal to
  • $$\cos{2\alpha}$$
  • $$\tan{2\alpha}$$
  • $$\sin{2\alpha}$$
  • None of these
If If $$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = r ^ { 2 } ,$$ then $$tan^ { - 1 } \left( \frac { x y } { z r } \right) + \tan ^ { - 1 } \left( \frac { y z } { x r } \right) + \tan ^ { - 1 } \left( \frac { x z } { y r } \right)$$ =
  • $$\pi$$
  • $$\frac { \pi } { 2 }$$
  • 0
  • none of these.
If $$cos^{-1}x - cos^{-1}\dfrac{y}{2} = \alpha$$ , then $$4x^2 - 4xy\, cos\, \alpha + y^2$$ is equal to -
  • $$2\, sin\, 2\alpha$$
  • $$4$$
  • $$4\, sin^2\alpha$$
  • $$-4\, sin^2\alpha$$
If a is a real of the equation $${ x }^{ 3 }+3x-tan2=0$$ then $${ cot }^{ -1 }a+{ cot }^{ -1 }\dfrac { 1 }{ a } -\dfrac { x }{ 2 } $$ can be equal to
  • 0
  • $$\dfrac { \pi }{ 2 } $$
  • $$\pi $$
  • $$\dfrac { 3\pi }{ 2 } $$
If  $$\cos ^ { - 1 } ( x / a ) + \cos ^ { - 1 } ( y / b ) = \alpha ,$$  Then  $$x ^ { 2 } / a ^ { 2 } + y ^ { 2 } / b ^ { 2 }$$  is equal to:
  • $$( 2 x y / a b ) \cos \alpha + \sin ^ { 2 } \alpha$$
  • $$( 2 x y / a b ) \sin \alpha + \cos ^ { 2 } \alpha$$
  • $$( 2 x y / a b ) \cos ^ { 2 } \alpha + \sin \alpha$$
  • $$( 2 x y / a b ) \sin ^ { 2 } \alpha + \cos \alpha$$
If $${ sin }^{ -1 }\left( \dfrac { \sqrt { x }  }{ 2 }  \right) +{ sin }^{ -1 }\left( \sqrt { 1-\dfrac { x }{ 4 }  }  \right) +tan^{ -1 }y=\dfrac { 2\pi  }{ 3 } $$ then
  • maximum value of $${ x }^{ 2 }+{ y }^{ 2 }$$ is $$\dfrac { 49 }{ 3 } $$
  • maximum value of $${ x }^{ 2 }+{ y }^{ 2 }$$ is 4
  • maximum value of $${ x }^{ 2 }+{ y }^{ 2 }$$ is $$\quad \dfrac { 1 }{ 2 } $$
  • maximum value of $${ x }^{ 2 }+{ y }^{ 2 }$$ is 3
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