CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 4 - MCQExams.com

The value of $$\sin^{-1}\left\{ \cot { \left\{ \sin ^{ -1 }{ \sqrt { \frac { 2-\sqrt { 3 }  }{ 4 }  } +\cos ^{  }{ \frac { \sqrt { 12 }  }{ 4 } +\sec ^{ -1 }{ \sqrt { 2 }  }  }  }  \right\}  }  \right\} $$ is equal to
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{6}$$
  • $$0$$
  • $$\dfrac{\pi}{2}$$
$$\tan ^{ -1 }{ \sqrt { 3 }  } -\cot ^{ -1 }{ \left( -\sqrt { 3 }  \right)  } $$ is equal to
  • $$\pi$$
  • $$-\cfrac{\pi}{2}$$
  • $$0$$
  • $$2\sqrt{3}$$
$${\cos ^{ - 1}}\left\{ {\dfrac{1}{{\sqrt 2 }}\left( {\cos \dfrac{{9\pi }}{{10}} - \sin \dfrac{{9\pi }}{{10}}} \right)} \right\} = $$
  • $$\frac{{23\pi }}{{20}}$$
  • $$\frac{{7\pi }}{{10}}$$
  • $$\frac{{7\pi }}{{20}}$$
  • $$\frac{{17\pi }}{{20}}$$
The solution set of the equation $$\sin^{-1}\sqrt{1-x^2}+\cos^{-1}x=\cot^{-1}\dfrac{\sqrt{1-x^2}}{x}-\sin^{-1}x$$ is?
  • $$[-1, 1]-\{0\}$$
  • $$(0, 1] \cup \{-1\}$$
  • $$[-1, 0)\cup \{1\}$$
  • $$[-1, 1]$$
Given that $$0 \le x\le \dfrac 12$$ the value of $$\tan \left[ { sin }^{ -1 }\left( \dfrac { x }{ \sqrt { 2 }  } +\sqrt { \dfrac { 1-{ x }^{ 2 } }{ 2 }  }  \right) -{ sin }^{ -1 }x \right]$$ is
  • $$-1$$
  • $$1$$
  • $$1\sqrt{3}$$
  • $$\sqrt{3}$$
the value of $$\cos ^{ -2 }{ \left( \cos { 10 }  \right)  }$$ is
  • $$4\pi +10$$
  • $$4\pi -10$$
  • $$2\pi +10$$
  • $$2\pi -10$$
$$2\cot ^{ -1 }{ 7 } +\cos ^{ -1 }{ \dfrac { 3 }{ 5 }  } $$ is equal to 
  • $$\cot ^{ -1 }{ \left( \dfrac { 44 }{ 117 } \right) } $$
  • $$cosec ^{ -1 }{ \left( \dfrac { 125 }{ 117 } \right) } $$
  • $$\tan ^{ -1 }{ \left( \dfrac { 4 }{ 117 } \right) } $$
  • $$\tan ^{ -1 }{ \left( \dfrac { 44 }{ 125 } \right) } $$
The value of $$\tan ^{ -1 }{ \left( 2\sin { \left( \sec ^{ -1 }{ \left( 2 \right)  }  \right)  }  \right)  } $$ is
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{2}$$
$${\tan ^{ - 1}}\left( {\frac{1}{2}\tan 2A} \right) + {\tan ^{ - 1}}\cot A + {\tan ^{ - 1}}{\cot ^3}A = $$
  • $$0;\,if\,\pi /4 < A < \pi /2$$
  • $$0;\,if\,0 < A < \pi /4$$
  • $$\pi; \,if\pi /4 < A < \pi /2$$
  • $$\pi ;\,if0 < A < \pi /2$$
The value of $$\cos ^{ -1 }{ \left\{ \frac { \sqrt { 1-\sin { x }  } +\sqrt { 1+\sin { x }  }  }{ \surd \left( 1-\sin { x }  \right) -\surd \left( 1+\sin { x }  \right)  }  \right\}  }$$ is $$\left( 0<x<2\pi  \right)$$
  • $$\pi -\frac { x }{ 2 }$$
  • $$2\pi -x$$
  • $$-\frac { x}{ 2 } $$
  • $$2\pi -\frac { X }{ 2 }$$
Annual expenses of  A and B are in the ratio  $$5:3$$. The  saving  of Aand B are in the ratio $$1:2$$. Find the expenses of A. given that the income of A is $$Rs. 8000$$  and that of B is $$Rs.9000$$.
  • $$3000$$
  • $$4000$$
  • $$4500$$
  • $$5000$$
$${ cos }^{ -1 }\left( \dfrac { 3+5\cos { x }  }{ 5+3\cos { x }  }  \right) =$$
  • $${ tan }^{ -1 }\left( \dfrac { 1 }{ 2 } \tan { \dfrac { x }{ 2 } } \right)$$
  • $$2\tan ^{ -1 }{ \left(- \dfrac { 1 }{ 2 } \tan { \dfrac { x }{ 2 } } \right) }$$
  • $$\dfrac { 1 }{ 2 } \tan ^{ -1 }{ \left( 2\tan { \dfrac { x }{ 2 } } \right) }$$
  • $$2\tan ^{ -1 }{ \left( \dfrac { 1 }{ 2 } \tan { \dfrac { x }{ 2 } } \right) }$$
If $$\sin^{-1}\left(x-\dfrac {x^{2}}{2}+\dfrac {x^{3}}{4}-...\infty \right)+\cos^{-1}\left(x^{2}-\dfrac {x^{4}}{2}+\dfrac {x^{6}}{4}-...\infty \right)=\dfrac {\pi}{2}$$ for $$0 < |x| < \sqrt {2}$$,then $$x$$ equal
  • $$\dfrac {1}{2}$$
  • $$1$$
  • $$\dfrac {-1}{2}$$
  • $$-1$$
$${ cos }^{ -1 }\left( \dfrac { \cos { \alpha  } \cos { \beta  }  }{ 1+\cos { \alpha  } \cos { \beta  }  }  \right) =2{ tan }^{ -1 }\left( \tan { \dfrac { \alpha  }{ 2 }  } \tan { \dfrac { \beta  }{ 2 }  }  \right)$$.
  • True
  • False
If $${sin}^{-1}x+{sin}^{-1}y+{sin}^{-1}z=\pi$$, then $$x\sqrt {{1}-{x}^{2}}+y\sqrt {{1}-{y}^{2}}+z \sqrt {{1}-{z}^{2}}=-2xyz$$
  • True
  • False
State true or false.
$${\tan ^{ - 1}}\left( {\dfrac{1}{3}} \right) + {\tan ^{ - 1}}\left( {\dfrac{1}{5}} \right) + {\tan ^{ - 1}}\left( {\dfrac{1}{8}} \right) + {\tan ^{ - 1}}\left( {\dfrac{1}{7}} \right) = \dfrac{\pi }{2}$$
  • True
  • False
The value of $$\cos\left\{\tan^{-1}\left(\tan \dfrac{15\pi}{4}\right)\right\}$$ is?
  • $$\dfrac{1}{\sqrt{2}}$$
  • $$-\dfrac{1}{\sqrt{2}}$$
  • $$1$$
  • None of these
If the number $$93215x2$$ is completely divisible by $$11$$, then $$x$$ is equal to  
  • $$2$$
  • $$3$$
  • $$1$$
  • $$4$$
Find the values of $$\cos^{-1}\left(\cos \dfrac {7\pi}{6}\right)$$ is equal to
  • $$\dfrac {7\pi}{6}$$
  • $$\dfrac {5\pi}{6}$$
  • $$\dfrac {\pi}{3}$$
  • $$\dfrac {\pi}{6}$$
The value of $$3\tan^{-1}\dfrac {1}{2}+2\tan^{-1}\dfrac {1}{5}+\sin^{-1}\dfrac {142}{65\sqrt {5}}$$ is
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{2}$$
  • $$\pi$$
  • $$none\ of\ these$$
Find the approximate value of $$\sin^{-1}(0.51)$$ given that $$\sqrt {3}=1.7321$$
  • $$\dfrac {\pi}{6}+0.011541$$
  • $$\dfrac {\pi}{3}+0.011541$$
  • $$\dfrac {\pi}{6}+0.011547$$
  • $$\dfrac {\pi}{3}+0.011547$$
$$\cot ^{ -1 }{ \left( \dfrac { \sqrt { 1-\sin { x }  } +\sqrt { 1+\sin { x }  }  }{ \sqrt { 1-\sin { x }  } -\sqrt { 1+\sin { x }  }  }  \right)  }$$=....$$\left( 0<x<\dfrac { \pi  }{ 2 }  \right)$$
  • $$\dfrac { x }{ 2 } $$
  • $$\dfrac { \pi }{ 2 } -2x$$
  • $$2\pi -x$$
  • $$\pi -\dfrac { x }{ 2 } $$
The value of $$\tan^{-1}\dfrac {1}{2}+\tan^{-1}\dfrac {1}{3}$$ is
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {\pi}{3}$$
  • $$0$$
Let in $$\Delta ABC, \, \angle A = \dfrac{\pi}{2}$$. Then value of $$\tan^{-1} \dfrac{b}{a + c} + \tan^{-1} \dfrac{c}{a + b}$$ equals 
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{4}$$
  • $$\dfrac{\pi}{3}$$
  • None of these
$${\sin ^{ - 1}}(\cos \left( {{{\sin }^{ - 1}}x} \right)) + {\cos ^{ - 1}}(\sin \left( {{{\cos }^{ - 1}}x} \right)){\text{is}}\;{\text{equal}}\;{\text{to}}{\text{.}}$$
  • $$\frac{\pi }{2}$$
  • $$\frac{\pi }{4}$$
  • $$\frac{{3\pi }}{4}$$
  • 0
If $${\tan ^{ - 1}}\dfrac{1}{{a - 1}} = {\tan ^{ - 1}}\dfrac{1}{x} + {\tan ^{ - 1}}\dfrac{1}{{{a^2} - x + 1}},\,then\,x\,is\,$$
  • $$a$$
  • $${a^{3\,}}$$
  • $${a^2} - a + 1\,$$
  • $${a^2} + a - 1$$
$${\cot ^{ - 1}}\left( {2 + \sqrt 3 } \right) = $$
  • $$\frac{\pi }{{12}}$$
  • $$\frac{\pi }{{15}}$$
  • $$\frac{\pi }{5}$$
  • $$\frac{{3\pi }}{{10}}$$
If minimum value of $${\left( {{{\sin }^{ - 1}}x} \right)^2} + {\left( {{{\cos }^{ - 1}}x} \right)^2} = \frac{{{\pi ^2}}}{K}$$ then the value of $$K$$ is 
  • $$4$$
  • $$8$$
  • $$6$$
  • $$16$$
The value of $${\sin ^{ - 1}}\left( {\cos \dfrac{{33\pi }}{5}} \right)$$ is 
  • $$\dfrac{{3\pi }}{5}$$
  • $$\dfrac{{7\pi }}{5}$$
  • $$\dfrac{\pi }{{10}}$$
  • $$ - \dfrac{\pi }{{10}}$$
The value of $$\cos(\tan^{-1}(\tan2))$$
  • $$\dfrac{1}{\sqrt{5}}$$
  • $$-\dfrac{1}{\sqrt{5}}$$
  • $$\cos 2$$
  • $$-\cos 2$$
$$cot^{-1}(\sqrt{cos\alpha })-tan^{-1}(\sqrt{cos\alpha })=x$$ , then sin x is equal to
  • $$\tan^{2}\dfrac{\alpha }{2}$$
  • $$\cot^{2}\dfrac{\alpha }{2}$$
  • $$\tan \alpha $$
  • $$\cot\dfrac{\alpha }{2}$$
$${\cos ^{ - 1}}\dfrac{3}{5} - {\sin ^{ - 1}}\dfrac{4}{5} = {\cos ^{ - 1}}x$$ then $$x$$ is equal to:
  • $$0$$
  • $$1$$
  • $$-1$$
  • none of these
If $$\sin ^{ -1 }{ x } =\cot ^{ -1 }{ x } $$ then
  • $${ x }^{ 2 }=\cfrac { \sqrt { 5 } -1 }{ 2 } $$
  • $${ x }^{ 2 }=\cfrac { \sqrt { 5 } +1 }{ 2 } $$
  • $${ x }^{ }=\cfrac { \sqrt { 5 } +1 }{ 2 } $$
  • $${ x }^{ }=\cfrac { \sqrt { 5 } -1 }{ 2 } $$
$$Sin^{-1}(2x(1-x^2)) = 2sin^{-1}x$$ is true if.
  • $$x\in [0,1]$$
  • [$$-\dfrac{1}{2} ,\dfrac{1}{2}$$]
  • [$$\dfrac{1}{2},- \dfrac{1}{2}$$]
  • [$$\dfrac{3}{2}, -\dfrac{3}{2}$$]
The greatest and least value of  $${\left( {{{\sin }^{ - 1}}x} \right)^2} + {\left( {{{\cos }^{ - 1}}x} \right)^2}$$ are respectively 
  • $$\dfrac{{{\pi ^2}}}{4}and0$$
  • $$\dfrac{\pi }{2}and\, - \dfrac{\pi }{2}$$
  • $$\dfrac{{5{\pi ^2}}}{4}and\dfrac{{{\pi ^2}}}{8}$$
  • $$\dfrac{{{\pi ^2}}}{4}\,and\,\dfrac{{ - \pi }}{4}$$
The number of elements in the range of
$$f(x)=\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } +\sec ^{ -1 }{ x } $$ is
  • $$1$$
  • $$2$$
  • $$3$$
  • $$4$$
The principal value of $$tan^{-1}[cot\dfrac{3\pi}{4}]$$ is :
  • $$\dfrac{-3\pi}{4}$$
  • $$\dfrac{3\pi}{4}$$
  • $$\dfrac{-\pi}{4}$$
  • $$\dfrac{\pi}{4}$$
The algebraic expression for $$\tan \left(\sin^{-1}\cos\ \tan^{-1}\dfrac {x}{2}\right)$$ is
  • $$\dfrac {2}{x}$$
  • $$\dfrac {x}{2}$$
  • $$\dfrac {1}{x}$$
  • $$\dfrac {2}{|x|}$$
If $$\sin ^{-1}x + \sin ^{-1}y + \sin ^{-1}z = \dfrac{3\pi}{2}$$, then $$(x^{500} + y^{500} + z^{500}) - (x^{501} + y^{501} + z^{501})$$ is
  • $$0$$
  • $$1$$
  • $$2$$
  • $$4$$
The value of $$\sin \left(\dfrac {1}{4}\sin^{-1}\dfrac {\sqrt {63}}{8}\right)$$ is
  • $$\dfrac {1}{2}$$
  • $$\dfrac {1}{3}$$
  • $$\dfrac {1}{2\sqrt {2}}$$
  • $$\dfrac {1}{5}$$
If $$\tan^{-1}\left(\dfrac{x+1}{x-1}\right)+\tan^{-1}\left(\dfrac{x-1}{x}\right)=\pi +\tan^{-1}(-7)$$, then $$x=$$
  • $$2$$
  • $$-2$$
  • $$1$$
  • No solution
The number of real solutions of $$\cos ^ { - 1 } x + \cos ^ { - 1 } 2 x = - \pi$$ is

  • $$0$$
  • $$1$$
  • $$2$$
  • None
The value of $$ \sin (\cot^{-1}x)$$ is 
  • $$\sqrt{1+x^2}$$
  • $$x$$
  • $$(1+x^2)^{-3/2}$$
  • $$(1+x^2)^{-1/2}$$
$$\cos ^ { - 1 } \left( \cos \left( \dfrac { - 17 \pi } { 5 } \right) \right)$$ is equal to

  • $$- \dfrac { 17 \pi } { 5 }$$
  • $$\dfrac { 3 \pi } { 5 }$$
  • $$\dfrac { 2 \pi } { 5 }$$
  • none of these
$$\sinh { \left( \cosh ^{ -1 }{ x }  \right)  } =$$
  • $$\sqrt{x^{2}+1}$$
  • $$\dfrac{1}{\sqrt{x^{2}+1}}$$
  • $$\sqrt{x^{2}-1}$$
  • $$\dfrac{1}{\sqrt{x^{2}-1}}$$
$$\sin\ h^{-1}{\left(2^{3/2}\right)}$$=
  • $$\log \left(2+\sqrt{8}\right)$$
  • $$\log \left(3+\sqrt{8}\right)$$
  • $$\log \left(2-\sqrt{8}\right)$$
  • $$\log \left(\sqrt{8}+\sqrt{7}\right)$$
The value of $$\tan ^{-1} \left (\dfrac{1}{3}\right) +\tan ^{-1} \left (\dfrac{2}{9}\right) + \tan ^{-1}\left  (\dfrac{4}{33}\right) + \tan ^{-1} \left (\dfrac{8}{129}\right) +$$.......$$n$$ terms is
  • $$\tan^{-1} 2^n - \dfrac{\pi}{4}$$
  • $$\tan^{-1} 2^n$$
  • $$\cot^{-1} 2^n$$
  • $$\dfrac{\sin^{-1} 2^n}{\cos^{-1} 2^n}$$
If $$\cot^{-1} [\sqrt{\cos \alpha}] - \tan^{-1} [\sqrt{\cos \alpha}] = x$$, then $$\sin x$$ is equal to
  • $$\tan^2 \left (\dfrac{\alpha}{2}\right)$$
  • $$\cot^2 \left (\dfrac{\alpha}{2}\right)$$
  • $$\tan \alpha$$
  • $$\cot \dfrac{\alpha}{2}$$
$$\sec\ h^{-1}\left(\dfrac{1}{5}\right)$$=
  • $$\log( {\sqrt{24}+5})$$
  • $$\log {5+\sqrt{27}}$$
  • $$\log {26+\sqrt{5}}$$
  • $$\log {27+\sqrt{5}}$$
If $$\cot \dfrac {2x}{3}+\tan \dfrac {x}{3}=\csc \dfrac {x}{3}$$ then value of $$\tan^{-1(\tan k)}$$ equals
  • $$2$$
  • $$2-\pi$$
  • $$\pi-2$$
  • $$2\pi-2$$
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