CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 3 - MCQExams.com

The number of solutions of the equation  $$ 2\sin^{-1}\left ( \sqrt{x^{2}-x+1} \right )+\cos^{-1}\left ( \sqrt{x^{2}-x} \right )= \displaystyle \frac{3\pi }{2} $$ is
  • $$0$$
  • Infinite
  • $$2$$
  • $$4$$
The value of $$\displaystyle \tan^{-1} \left( \frac{\sin{  2} - 1}{\cos {2}} \right )$$ is equal to
  • $$\displaystyle \frac{\pi}{2} - 1$$
  • $$\displaystyle 2 - \frac{\pi}{2}$$
  • $$\displaystyle 1- \frac{\pi}{4}$$
  • $$\displaystyle \frac{\pi}{4} - 1$$
The value of $$\cos^{-1} (\cos  12) - \sin^{-1} (\sin  14)$$ is
  • $$0$$
  • $$8 \pi - 26$$
  • $$4 \pi + 2$$
  • None of these
Find the value of $$x$$ if $$\sin (arc \sin x) = \dfrac {\sqrt {2}}{4}$$
  • $$\dfrac {\sqrt {2}}{4}$$
  • $$\dfrac {\sqrt {7}}{7}$$
  • $$\dfrac {\sqrt {2}}{2}$$
  • $$\dfrac {2\sqrt {2}}{3}$$
If $$\displaystyle \tan^{-1}{\left(\frac{a}{x}\right)}+\tan^{-1}{\left(\frac{b}{x}\right)}=\frac{\pi}{2}$$, then $$x$$ is equal to :
  • $$ab$$
  • $$\dfrac{a}{b}$$
  • $$\dfrac{b}{a}$$
  • $$\sqrt{ab}$$
Given $$0 \leq x \leq \frac{1}{2}$$, then the value of $$\tan\left [ \sin^{-1}\left \{ \dfrac{x}{\sqrt{2}}+\dfrac{\sqrt{1-x^2}}{\sqrt{2}} \right \} \right ]$$ is.
  • $$\dfrac{\sqrt{1-x^2}+x}{\sqrt{1-x^2}-x}$$
  • $$\dfrac{\sqrt{1-x^2}-x}{\sqrt{1-x^2}+x}$$
  • $$\dfrac{x+\sqrt{1-x^2}}{x-\sqrt{1-x^2}}$$
  • $$\dfrac{x-\sqrt{1-x^2}}{x+\sqrt{1-x^2}}$$
The trigonometric equation $$\sin^{-1}x = 2\sin^{-1}2a$$ has a real solution if
  • $$|a| > \dfrac {1}{\sqrt {2}}$$
  • $$\dfrac {1}{2\sqrt {2}} < |a| < \dfrac {1}{\sqrt {2}}$$
  • $$|a| > \dfrac {1}{2\sqrt {2}}$$
  • $$|a| \leq \dfrac {1}{2\sqrt {2}}$$
The domain of the function $$f(x)=\sqrt{cos^{-1}\begin{pmatrix}\dfrac{1-|x|}{2}\end{pmatrix}}$$ is
  • $$(-3,\,3)$$
  • $$[-3,\,3]$$
  • $$(-\infty,\,-3)\cup(-3,\,\infty)$$
  • $$(-\infty,\,-3)\cup(3,\,\infty)$$
The trigonometric equation $$\sin^{-1}x=2\, \sin^{-1}2a$$ has a real solution, if
  • $$|a| > \dfrac{1}{\sqrt{2}}$$
  • $$\dfrac{1}{2\sqrt{2}} < |a| < \dfrac{1}{\sqrt{2}}$$
  • $$|a| > \dfrac{1}{2\sqrt{2}}$$
  • $$|a| \le \dfrac{1}{2\sqrt{2}}$$
If $$\theta =\sin ^{ -1 }{ x } +\cos ^{ -1 }{ x } -\tan ^{ -1 }{ x } ,1\le x<\infty $$, then the smallest interval in which $$\theta $$ lies is
  • $$\cfrac { \pi }{ 2 } \le \theta \le \cfrac { 3\pi }{ 4 } $$
  • $$0\le \theta \le \cfrac { \pi }{ 4 } $$
  • $$-\cfrac { \pi }{ 4 } \le \theta \le 0$$
  • $$\cfrac { \pi }{ 4 } \le \theta \le \cfrac { \pi }{ 2 } $$
$$\tan \left [3\tan^{-1}\left (\dfrac {1}{5}\right ), -\dfrac {\pi}{4}\right ]$$ is equal to
  • $$-\dfrac {13}{46}$$
  • $$-\dfrac {11}{46}$$
  • $$-\dfrac {7}{46}$$
  • $$-\dfrac {4}{23}$$
  • $$-\dfrac {9}{46}$$
If $$\tan ^{ -1 }{ x } +\tan ^{ -1 }{ y } =\cfrac { 2\pi  }{ 3 } $$, then $$\cot ^{ -1 }{ x } +\cot ^{ -1 }{ y } $$ is equal to
  • $$\cfrac { \pi }{ 2 } $$
  • $$\cfrac { 1 }{ 2 } $$
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \sqrt { 3 } }{ 2 } $$
  • $$\pi $$
If $$ 2 sin h^{-1}\left ( \dfrac{a}{\sqrt{1-a^{2}}} \right ) = log\left ( \dfrac{1+X}{1-X} \right )$$ then X=
  • a
  • $$\dfrac{1}{a}$$
  • $$\sqrt{1-a^{2}}$$
  • $$\dfrac{1}{\sqrt{1-a^{2}}} $$
The value of $$x$$ satisfying the equation
$$\tan ^{ -1 }{ x } +\tan ^{ -1 }{ \left( \cfrac { 2 }{ 3 }  \right)  } =\tan ^{ -1 }{ \left( \cfrac { 7 }{ 4 }  \right)  } $$ is equal to
  • $$\cfrac { 1 }{ 2 } $$
  • $$-\cfrac { 1 }{ 2 } $$
  • $$\cfrac { 3 }{ 2 } $$
  • $$-\cfrac { 1 }{ 3 } $$
  • $$\cfrac { 1 }{ 3 } $$
The value of $$\tan { \left\{ \dfrac { 1 }{ 2 } \cos ^{ -1 }{ \left( \dfrac { \sqrt { 5 }  }{ 3 }  \right)  }  \right\}  } $$ is
  • $$\dfrac { 3+\sqrt { 5 } }{ 2 } $$
  • $$3+\sqrt { 5 } $$
  • $$\dfrac { 1 }{ 2 } \left( 3-\sqrt { 5 } \right) $$
  • None of these
If $$ab > -1, bc > -1$$ and $$ca > -1$$, then the value of $$\cot^{-1}\left (\dfrac {ab + 1}{a - b}\right ) + \cot^{-1}\left (\dfrac {bc + 1}{b - c}\right ) + \cot^{-1}\left (\dfrac {ca + 1}{c - a}\right )$$ is
  • $$-1$$
  • $$\cot^{-1}(a + b + c)$$
  • $$\cot^{-1}(abc)$$
  • $$0$$
  • $$\tan^{-1}(a + b + c)$$
If $$\cos^{-1}\left (\dfrac {1 - x^{2}}{1 + x^{2}}\right ) + \cos^{-1}\left (\dfrac {1 - y^{2}}{1 + y^{2}}\right ) = \dfrac {\pi}{2}$$, where $$xy < 1$$, then
  • $$x - y - xy = 1$$
  • $$x - y + xy = 1$$
  • $$x + y - xy = 1$$
  • $$x + y + xy = 1$$
  • $$y - x - xy = 1$$
If $$\tan^{-1} (-x) + \cos^{-1}\left (\dfrac {-1}{2}\right ) = \dfrac {\pi}{2}$$, them the value of $$x$$ is
  • $$\sqrt {3}$$
  • $$\dfrac {-1}{\sqrt {3}}$$
  • $$\dfrac {1}{\sqrt {3}}$$
  • $$-\sqrt {3}$$
  • $$1$$
$$\cos^{-1}\left (\cos \left (\dfrac  {7\pi}{5}\right)\right)$$ is equal to
  • $$\dfrac{3 \pi}{5}$$
  • $$\dfrac{2 \pi}{5}$$
  • $$\dfrac{-7 \pi}{5}$$
  • $$\dfrac{7 \pi}{5}$$
  • $$\dfrac{-2 \pi}{5}$$
If the non-zero numbers $$x,y,z$$ are $$AP$$ and $$\tan ^{ -1 }{ x } ,\tan ^{ -1 }{ y } ,\tan ^{ -1 }{ z } $$ are also in $$AP$$, then
  • $$xy=yz$$
  • $${z}^{2}=xy$$
  • $$x=y=z$$
  • $${ x }^{ 2 }=yz$$
The value of $$ \cos \left( \sin^{-1} \left( \dfrac {2}{3} \right) \right) $$ is equal to :
  • $$ \dfrac {\sqrt3}{5} $$
  • $$ \dfrac {5}{3} $$
  • $$ \dfrac {5}{\sqrt3} $$
  • $$ \sqrt { \dfrac {5}{3} } $$
  • $$ \dfrac {\sqrt5}{3} $$
If two angles of a triangle are $$\tan ^{ -1 }{ (2) } $$ and $$\tan ^{ -1 }{ (3) } $$, then the third angle is
  • $$\cfrac { \pi }{ 4 } $$
  • $$\cfrac { \pi }{ 6 } $$
  • $$\cfrac { \pi }{ 3 } $$
  • $$\cfrac { \pi }{ 2 } $$
If $$\sin { \left[ \cot ^{ -1 }{ (x+1) }  \right]  } =\cos { \left[ \tan ^{ -1 }{ x }  \right]  } $$, then $$x$$ is equal to
  • $$\cfrac { -1 }{ 2 } $$
  • $$\cfrac { 1 }{ 2 } $$
  • $$0$$
  • $$\cfrac { 9 }{ 2 } $$
The sum of $$\cot ^{ -1 }{ 2 } +\cot ^{ -1 }{ 8 } +\cot ^{ -1 }{ 18 } ....\infty =\cfrac { \pi  }{ \lambda  } $$, then $$\lambda $$ is
  • $$2$$
  • $$4$$
  • $$6$$
  • $$8$$
The domain of function $$f\left( x \right) =\sin ^{ -1 }{ 5x } $$ is
  • $$\left( -\dfrac { 1 }{ 5 } ,\dfrac { 1 }{ 5 } \right) $$
  • $$\left[ -\dfrac { 1 }{ 5 } ,\dfrac { 1 }{ 5 } \right] $$
  • $$R$$
  • $$\left( 0,\dfrac { 1 }{ 5 } \right) $$
$$\tan^{-1}(x + \sqrt{1 + x^{2}})$$ =
  • $$\dfrac{\pi}{4} - \dfrac{1}{2} \tan^{-1} x$$
  • $$\dfrac{1}{2} \tan^{-1} x$$
  • $$\dfrac{\pi}{2} - \dfrac{1}{2} \tan^{-1} x$$
  • $$\dfrac{\pi}{4} + \dfrac{1}{2} \tan^{-1} x$$
If $$2\sinh ^{ -1 }{ \left( \dfrac { a }{ \sqrt { 1-{ a }^{ 2 } }  }  \right)  } =\log { \left( \dfrac { 1+x }{ 1-x }  \right)  }$$, then $$x=$$
  • $$a$$
  • $$\dfrac { 1 }{ a } $$
  • $$\sqrt { 1-{ a }^{ 2 } } $$
  • $$\dfrac { 1 }{ \sqrt { 1-{ a }^{ 2 } } } $$
The value of $$\sec ^{ 2 }{ \left( \tan ^{ -1 }{ 2 }  \right)  } +\text{cosec} ^{ 2 }{ \left( \cot ^{ -1 }{ 3 }  \right)  }=$$ ____
  • $$6$$
  • $$15$$
  • $$13$$
  • $$25$$

The value of $$\cot^{-1} \left[ \dfrac{\sqrt{1 - \sin x} +\sqrt{1 + \sin x}}{\sqrt{(1 - \sin x)} - \sqrt{(1 + \sin x)}} \right]$$ is
  • $$\pi - x$$
  • $$2 \pi - x$$
  • $$x/2$$
  • $$\pi - \dfrac{1}{2} x$$

$$\sin^{-1} \sqrt{2 - x} = \cos^{-1} \sqrt{x-1}$$ holds for all real x.
  • True
  • False
The value of $$\cos^{-1} (-1) - \sin^{-1} (1)$$ is
  • $$\pi$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{3 \pi}{2}$$
  • $$- \dfrac{3 \pi}{2}$$
Range of the function $$f(x)=4{ tan }^{ -1 }x+3{ sin }^{ -1 }x+{ sec }^{ -1 }x$$ is
  • $$\left\{ \frac { -3\pi }{ 2 } ,\frac { -5\pi }{ 2 } \right\} $$
  • $$\left\{ \frac { 3\pi }{ 2 } ,\frac { 5\pi }{ 2 } \right\} $$
  • $$\left\{ \frac { 3\pi }{ 2 } ,\frac { -5\pi }{ 2 } \right\} $$
  • $$\left\{ \frac { -3\pi }{ 2 } ,\frac { 5\pi }{ 2 } \right\} $$
If $$y= \sec(\tan^{-1} x) $$ then, $$\dfrac{dy}{dx} $$ at $$x=1$$ is equal to
  • $$\dfrac{1}{2}$$
  • $$1$$
  • $$\sqrt2$$
  • $$\frac{1}{\sqrt2}$$
Number of solution of the equation $$\cos^{-1}(1-x)-2\cos^{-1}x=\dfrac{\pi}{2}$$ is 
  • $$3$$
  • $$2$$
  • $$1$$
  • $$0$$
 $${\tan ^{ - 1}}x + {\cot ^{ - 1}}x = {\pi  \over 2},x \in \mathop R\limits^ \bullet  $$
  • True
  • False
The value of $$k$$ if the equation $$kx + \ \sin^{-1}(x^2-8x+17) + \ \cos^{-1}(x^2-8x+17) = \frac{9\pi}{2}$$ has atleast one solution is 
  • $$2\pi$$
  • $$\pi$$
  • $$1$$
  • $$\frac{\pi}{2}$$
The value of $$\cot \left( \dfrac{\pi}{4} - 2 \cot^{-1} 3\right)$$, is:
  • $$1$$
  • $$7$$
  • $$-1$$
  • None of these
$$\tan^{-1}(\dfrac{1}{4}) + \tan^{-1}(\dfrac{2}{9})$$ is equal to
  • $$\dfrac{1}{2} \ \cos^{-1}(\dfrac{3}{5})$$
  • $$\dfrac{1}{2} \sin^{-1}(\dfrac{4}{5})$$
  • $$\tan^{-1}(\dfrac{1}{2})$$
  • $$ \cos^{-1}(\dfrac{8}{9})$$
If $$\sin^{-1} x + \sin ^{-1} y + \sin ^{-1} z = \pi $$, then the value of $$x\sqrt{1-x^2}+ y\sqrt{1-y^2}+ z\sqrt{1- z^2}$$ will be:
  • $$2 xyz$$
  • $$xyz$$
  • $$\dfrac{1}{2} xyz$$
  • $$\dfrac{1}{3} xyz$$
$$\sum^n_{m = 1} \tan^{-1} \left(\dfrac{2m}{m^4 + m^2 + 2} \right)$$ is equal to
  • $$\tan^{-1} \left({n^2 + n + 1}\right)-\dfrac {\pi}{4}$$
  • $$\tan^{-1} \left({n^2 + n + 1}\right)+\dfrac {\pi}{4}$$
  • $$\tan^{-1} \left({n^2 + n - 1}\right)-\dfrac {\pi}{4}$$
  • $$\tan^{-1} \left({n^2 - n - 1}\right)-\dfrac {\pi}{4}$$
If $$ a sin^{-1} x -b cos^{-1}x =c$$, then the value of $$ a sin ^{-1} x + b cos^{-1} x$$ (whenever exists) is equal to
  • $$0$$
  • $$\frac{\pi ab + c(b-a)}{a+b}$$
  • $$\frac{\pi}{2}$$
  • $$\frac{\pi ab + c(a-b)}{a+b}$$
If $$\tan^{-1}\dfrac{a+x}{a}+\tan^{-1}\dfrac{a-x}{a}=\dfrac{\pi}{6}$$, then $$x^2=$$?
  • $$2\sqrt{3}a$$
  • $$\sqrt{3}a$$
  • $$2\sqrt{3}a^2$$
  • None of these
The value of $$\displaystyle {\sin ^{ - 1}}\left( {\frac{{12}}{{13}}} \right)+ {\cos ^{ - 1}}\left( {\frac{4}{5}} \right) + {\tan ^{ - 1}}\left( {\frac{{63}}{{16}}} \right) $$
  • $$\dfrac {2\pi }{3}$$
  • $$\pi$$
  • $$2\pi$$
  • $$3\pi$$
Solve: $${\sin ^{ - 1}}\dfrac{4}{5} + {\sin ^{ - 1}}\dfrac{5}{{13}} + {\sin ^{-1}}\dfrac{{16}}{{65}}$$
  • $$\dfrac {\pi}{2}$$
  • $$\dfrac {\pi}{4}$$
  • $$\dfrac {\pi}{6}$$
  • $$\dfrac {\pi}{8}$$
If $$\sin ^{ -1 }{ \left( x-\displaystyle\frac { { x }^{ 2 } }{ 2 } +\displaystyle\frac { { x }^{ 3 } }{ 4 } -....\infty  \right)  } +\cos ^{ -1 }{ \left( { x }^{ 2 }-\displaystyle\frac { { x }^{ 4 } }{ 2 } +\displaystyle\frac { { x }^{ 6 } }{ 4 } -....\infty  \right)  } =\displaystyle\frac { \pi  }{ 2 } $$ and $$0<x<\sqrt { 2 } $$ then $$x$$ =
  • $$\displaystyle\frac { 1 }{ 2 } $$
  • $$1$$
  • $$\displaystyle\frac{- 1 }{ 2 } $$
  • $$-1$$
If $$x=\cos^{ - 1}\left( \dfrac{2}{3} \right) + \tan^{ - 1}\left( \dfrac{1}{7} \right)$$ then $$x$$=
  • $$ {{\cos }^{-1}}\left\{ \dfrac{14-\sqrt{5}}{3\sqrt{50}} \right\} $$
  • $$ {{\cos }^{-1}}\left\{ \dfrac{10-\sqrt{5}}{3\sqrt{50}} \right\} $$
  • $$ {{\cos }^{-1}}\left\{ \dfrac{14-\sqrt{15}}{3\sqrt{50}} \right\} $$
  • None of these
If $$\sin ^{ -1 }{ x } =\cfrac { \pi  }{ 6 } $$ then find $$'x'$$
  • $$\dfrac {\sqrt 3}{2}$$
  • $$\dfrac {1}{2}$$
  • $$1$$
  • None of these
If $$\cos^{ - 1}x + {\cos ^{ - 1}}y = \dfrac{\pi }{2}$$ and $$\tan^{ - 1}x - \tan^{ - 1}y = 0$$ then  value of $${x^2} + ax + {y^2} $$ is where a= $$\dfrac{1}{\sqrt{2}}$$ 
  • $$0$$
  • $$ - \dfrac{1}{2}$$
  • $$\dfrac{1}{2}$$
  • $$\dfrac{3}{2}$$
$$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {{{\tan }^{ - 1}}\left( {\dfrac{{2r}}{{1 - {r^2} + {r^4}}}} \right)} $$ is equal to 
  • $$\dfrac{\pi} {4}$$
  • $$\dfrac{\pi} {2}$$
  • $$\dfrac{{3\pi }}{4}$$
  • none of these
$$\tan \left[\dfrac{\pi}{4} + \dfrac{1}{2} \cos^{-1} \left(\dfrac{5}{7} \right) \right] + \cot \left[\dfrac{\pi}{4} + \dfrac{1}{2} \cos^{-1} \left(\dfrac{5}{7}\right) \right]$$ is equal to 
  • $$\dfrac{5}{7}$$
  • $$\dfrac{10}{7}$$
  • $$\dfrac{14}{5}$$
  • $$\dfrac{7}{5}$$
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