CBSE Questions for Class 12 Commerce Maths Inverse Trigonometric Functions Quiz 6 - MCQExams.com

The value of $$\tan^{-1}\dfrac{1}{3}+\tan^{-1}\dfrac{1}{5}+\tan^{-1}\dfrac{1}{7}+\tan^{-1}\dfrac{1}{8}$$ is ___________.
  • $$\dfrac{11\pi}{5}$$
  • $$\dfrac{\pi}{4}$$
  • $$\pi$$
  • $$\dfrac{3\pi}{4}$$
$$\tan \left [2\tan^{-1}\dfrac {1}{5}-\dfrac {\pi}{4}\right]=$$ ?
  • $$\dfrac {7}{17}$$
  • $$\dfrac {-7}{17}$$
  • $$\dfrac {7}{12}$$
  • $$\dfrac {-7}{12}$$
$$\sin \left (\cos^{-1}\dfrac {3}{5}\right)=$$ ?
  • $$\dfrac {3}{4}$$
  • $$\dfrac {4}{5}$$
  • $$\dfrac {3}{5}$$
  • $$none\ of\ these$$
$$\cos \left(\tan^{-1} \dfrac {3}{4}\right)=$$?
  • $$\dfrac {3}{5}$$
  • $$\dfrac {4}{5}$$
  • $$\dfrac {4}{9}$$
  • $$none\ of\ these$$
Evaluate : $$\tan \dfrac {1}{2}\left (\cos^{-1}\dfrac {\sqrt 5}{3}\right)$$ 
  • $$\dfrac {(3-\sqrt 5)}{2}$$
  • $$\dfrac {(3+\sqrt 5)}{2}$$
  • $$\dfrac {(5-\sqrt 3)}{2}$$
  • $$\dfrac {(5+\sqrt 3)}{2}$$
If $$x\neq 0$$ then $$\cos (\tan^{-1}x+\cot^{-1}x)=$$?
  • $$-1$$
  • $$1$$
  • $$0$$
  • $$none\ of\ these$$
The value of $$\sin \left (\sin^{-1}\dfrac {1}{2}+\cos^{-1}\dfrac {1}{2}\right)=$$?
  • $$0$$
  • $$1$$
  • $$-1$$
  • $$none\ of\ these$$
The value of $$\sin \left(\cos^{-1}\dfrac {3}{5}\right)$$ is
  • $$\dfrac {2}{5}$$
  • $$\dfrac {4}{5}$$
  • $$\dfrac {-2}{5}$$
  • $$none\ of\ these$$
$$\sin \left [2\tan^{-1}\dfrac {5}{8}\right]$$
  • $$\dfrac {25}{64}$$
  • $$\dfrac {80}{89}$$
  • $$\dfrac {75}{128}$$
  • $$none\ of\ these$$
$$\sin \left [2\sin^{-1}\dfrac{4}{5}\right]$$
  • $$\dfrac {12}{25}$$
  • $$\dfrac {16}{25}$$
  • $$\dfrac {24}{25}$$
  • $$none\ of\ these$$
$$\sin \left\{\dfrac {\pi}{3}-\sin^{-1}\left (\dfrac {-1}{2}\right) \right\}=$$ ?
  • $$1$$
  • $$0$$
  • $$\dfrac {-1}{2}$$
  • $$none\ of\ these$$
$$\tan \left\{\cos^{-1}\dfrac {4}{5}+\tan^{-1}\dfrac {2}{3}\right\}=$$ ?
  • $$\dfrac {13}{6}$$
  • $$\dfrac {17}{6}$$
  • $$\dfrac {19}{6}$$
  • $$\dfrac {23}{6}$$
Evaluate : $$\cot (\tan^{-1}x+\cot^{-1}x)$$ 
  • $$1$$
  • $$\dfrac {1}{2}$$
  • $$0$$
  • $$none\ of\ these$$
Evaluate : $$\cos \left(2\tan^{-1}\dfrac {1}{2}\right)$$ 
  • $$\dfrac {3}{5}$$
  • $$\dfrac {4}{5}$$
  • $$\dfrac {7}{8}$$
  • $$none\ of\ these$$
The value of 
$$\displaystyle sin^{-1} \left \{ (sin\,\pi/3) \dfrac{x}{\sqrt{x^2 + k^2 - kx}} \right \} - cos^{-1} \left \{ cos\pi /6 \,\dfrac{x}{\sqrt{x^2 + k^2 - kx}} \right \} $$
 $$ \left (where \,\dfrac{k}{2} < x < 2k , \,k > 0 \right ) $$ is 
  • $$ tan^{-1} \left ( \dfrac{2x^2 + xk - k^2}{x^2 - 2xk + k^2} \right ) $$
  • $$ tan^{-1} \left ( \dfrac{x^2 + 2xk - k^2}{x^2 - 2xk + k^2} \right ) $$
  • $$ tan^{-1} \left ( \dfrac{x^2 + 2xk - 2k^2}{2x^2 - 2xk + 2k^2} \right ) $$
  • none of these
If $$sin^{-1}\dfrac{5}{x}+sin^{-1}\dfrac{12}{x}=\dfrac{\pi}{2}$$, then x is equal to
  • $$\dfrac{7}{13}$$
  • $$\dfrac{4}{3}$$
  • $$13$$
  • $$\dfrac{13}{7}$$
$$\tan^{1}x + \tan^{1}y + \tan^{1}z = \dfrac {\pi}{2}$$, then 
  • $$xy+yz+zx-xyz=0$$
  • $$xy+yz+zx+xyz=0$$
  • $$xy+yz+zx+1=0$$
  • $$xy+yz+zx-1=0$$

$$cos^{-1}(cos(\dfrac{5\pi}{4}))$$ is given by
  • $$\dfrac{5\pi}{4}$$
  • $$\dfrac{3\pi}{4}$$
  • $$\dfrac{-\pi}{4}$$
  • none of these

The value of $$sin^{-1}\Bigg(cot\bigg(sin^{-1}\sqrt{\dfrac{2-\sqrt{3}}{4}}+cos^{-1}\dfrac{\sqrt{12}}{4}+\sec^{-1}\sqrt{2}\bigg)\Bigg)$$ is
  • $$0$$
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{\pi}{3}$$
  • none of these
If $$\tan^{-1}\dfrac{1-x}{1+x}=\dfrac{1}{2}\tan^{-1}x$$, then x is equal to
  • $$1$$
  • $$\sqrt{3}$$
  • $$\dfrac{1}{\sqrt{3}}$$
  • none of these
If  $$tan^{-1}x+2 cot^{-1}x=\dfrac{2\pi}{3}$$, then x is equal to 
  • $$\dfrac{\sqrt{3}-1}{\sqrt{3}+1}$$
  • $$3$$
  • $$\sqrt{3}$$
  • $$\sqrt{2}$$
If  $$3\sin^{-1}\bigg(\dfrac{2x}{1+x^2}\bigg)-4\cos^{-1}\bigg(\dfrac{1-x^2}{1+x^2}\bigg)+2tan^{-1}\bigg(\dfrac{2x}{1-x^2}\bigg)=\dfrac{\pi}{3}$$, where $$|x|<1$$. then x is equal to
  • $$\dfrac{1}{\sqrt{3}}$$
  • $$-\dfrac{1}{\sqrt{3}}$$
  • $${\sqrt{3}}$$
  • $$-\dfrac{\sqrt{3}}{4}$$
  • $$\dfrac{\sqrt{3}}{2}$$
The principal value of $$\sin^{-1}(\sin 10)$$ is
  • $$10$$
  • $$10-3\pi$$
  • $$3\pi-10$$
  • none of these
If  $$3\tan^{-1}(\dfrac{1}{2+\sqrt{3}})-\tan^{-1}\dfrac{1}{x}=\dfrac{1}{3}$$, then x is equal to
  • $$1$$
  • $$2$$
  • $$3$$
  • $$\sqrt{2}$$
The value $$2tan^{-1}\Bigg[\sqrt{\dfrac{a-b}{a+b}}\tan\dfrac{\theta}{2}\Bigg]$$ is equal to
  • $$\cos^{-1}\Bigg[\dfrac{a\cos\theta+b}{b\cos\theta+a}\Bigg]$$
  • $$\cos^{-1}\Bigg[\dfrac{b\cos\theta+a}{a\cos\theta+b}\Bigg]$$
  • $$\cos^{-1}\Bigg[\dfrac{a\cos\theta}{b\cos\theta+a}\Bigg]$$
  • $$\cos^{-1}\Bigg[\dfrac{b\cos\theta}{a\cos\theta+b}\Bigg]$$
 If $$x=\sec^{-1}\Bigg(x+\dfrac{1}{x}\Bigg)+\sec^{-1}\Bigg(y+\dfrac{1}{y}\Bigg)$$ where xy < 0, then the possible values of z is (are)
  • $$\dfrac{8\pi}{10}$$
  • $$\dfrac{7\pi}{10}$$
  • $$\dfrac{9\pi}{10}$$
  • $$\dfrac{21\pi}{20}$$
If $$tan^{-1}(\sin^2\theta-2sin\theta+3)+\cot^{-1}(5^{\sec^2y}+1)=\dfrac{\pi}{2}$$, then the value of $$cos^2\theta-\sin\theta$$ is equal to
  • $$0$$
  • $$-1$$
  • $$1$$
  • none of these
If  f(x)=$$sin^{-1}\Bigg(\dfrac{\sqrt{3}}{2}x-\dfrac{1}{2}\sqrt{1-x^2}\Bigg),-\dfrac{1}{2}\leq x\leq1$$, then f(x) is equal to
  • $$\sin^{-1}\bigg(\dfrac{1}{2}\bigg)-sin^{-1}x$$
  • $$\sin^{-1}x-\dfrac{\pi}{6}$$
  • $$\sin^{-1}x+\dfrac{\pi}{6}$$
  • none of these
If  $$cot^{-1}(\sqrt{\cos a})- tan^{-1}(\sqrt{\cos a})=x$$, then $$\sin x$$ is  
  • $$tan^2\dfrac{\alpha}{2}$$
  • $$cot^2\dfrac{\alpha}{2}$$
  • $$\tan \alpha$$
  • $$cot\dfrac{\alpha}{2}$$
$$2tan^{-1}(-2)$$ is equal to
  • $$-\cos^{-1}\bigg(\dfrac{-3}{5}\bigg)$$
  • $$-\pi+\cos^{-1}\bigg(\dfrac{3}{5}\bigg)$$
  • $$-\dfrac{\pi}{2}+\tan^{-1}\bigg(\dfrac{-3}{4}\bigg)$$
  • $$-\pi+\cot^{-1}\bigg(\dfrac{-3}{4}\bigg)$$
The value of $$\underset{|x|\rightarrow\infty}{lim}cos(tan^{-1}(sin(tan^{-1}x)))$$  is equal to
  • $$-1$$
  • $$\sqrt{2}$$
  • $$-\dfrac{1}{\sqrt{2}}$$
  • $$\dfrac{1}{\sqrt{2}}$$
If $$x\in\bigg(\dfrac{-\pi}{2},\dfrac{\pi}{2}\bigg)$$, then the value $$\tan^{-1}\Bigg(\dfrac{\tan x}{4}\Bigg)+\tan^{-1}\Bigg(\dfrac{3\sin 2x}{5+3\cos 2x}\Bigg)$$  is
  • $$\dfrac{x}{2}$$
  • $$2x$$
  • $$3x$$
  • $$x$$
If $$\alpha,\beta \,\,and\,\, \gamma$$ are the roots of $$\displaystyle \tan^{-1} \left ( x-1 \right ) +\tan^{-1} x + \tan^{-1} \left ( x+1 \right ) = \tan^{-1} 3x$$, then
  • $$\alpha+\beta+ \gamma=0$$
  • $$\alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{-1}{4}$$
  • $$\alpha\beta\gamma=1$$
  • $$|\alpha-\beta|_{max}=1$$
If f(x) = $$(\sin^{-1}x)^2 +(\cos^{-1}x)^2$$, then.
  • f(x) has the least value of $$\dfrac{\pi^2}{8}$$
  • f(x) has the least value of $$\dfrac{5\pi^2}{8}$$
  • f(x) has the least value of $$\dfrac{\pi^2}{16}$$
  • f(x) has the least value of $$\dfrac{5\pi^2}{4}$$
If $$sin^{-1}x+sin^{-1}y=\dfrac{\pi}{2}$$ and $$\sin 2x=\cos 2y$$, then
  • $$x=\dfrac{\pi}{8}+\sqrt{\dfrac{1}{2}-\dfrac{\pi^2}{64}}$$
  • $$y=-\dfrac{\pi}{12}+\sqrt{\dfrac{1}{2}-\dfrac{\pi^2}{64}}$$
  • $$x=\dfrac{\pi}{12}+\sqrt{\dfrac{1}{2}-\dfrac{\pi^2}{64}}$$
  • $$y=-\dfrac{\pi}{8}+\sqrt{\dfrac{1}{2}-\dfrac{\pi^2}{64}}$$
$$cos^{-1}x+cos^{-1}\Bigg[\dfrac{x}{2}+\dfrac{1}{2}\sqrt{3-3x^2}\Bigg]$$ is equal to
  • $$\dfrac{\pi}{3}$$ for $$x\in\Bigg[\dfrac{1}{2},1\Bigg]$$
  • $$\dfrac{\pi}{3}$$ for $$x\in\Bigg[0,\dfrac{1}{2}\Bigg]$$
  • $$2cos^{-1}x-cos^{-1}\dfrac{1}{2}$$ for $$x\in\Bigg[\dfrac{1}{2},1\Bigg]$$
  • $$2cos^{-1}x-cos^{-1}\dfrac{1}{2}$$ for $$x\in\Bigg[0,\dfrac{1}{2}\Bigg]$$
The domain of the function $$\cos ^{-1}(2x-1)$$ is
  • $$[0,1]$$
  • $$[-1,1]$$
  • $$(-1,1)$$
  • $$[0,\pi]$$
If $$\cos \left(\sin ^{-1}\dfrac{2}{5} + \cos ^{-1} x\right)=0$$ then $$x$$ is equal to
  • $$\dfrac{1}{5}$$
  • $$\dfrac{2}{5}$$
  • $$0$$
  • $$1$$
The domain of trigonometric function can be restricted to any one of their branch (not necessarily principle value) in order to obtain their inverse functions.
  • True
  • False
The minimum value of $$n$$ for which $$\tan^{-1}\dfrac {n}{\pi} > \dfrac {\pi}{4}, n\ \in \ N$$ is valid is $$5$$.
  • True
  • False
All trigonometric functions have inverse over their respective domains.
  • True
  • False
The value of $$\cot \left[cos^{-1}\left(\dfrac{7}{25}\right)\right]$$ is
  • $$\dfrac{25}{24}$$
  • $$\dfrac{25}{7}$$
  • $$\dfrac {24} {25}$$
  • $$\dfrac {7} {24}$$
If $$|x| \le 1$$, then $$2 \tan ^{-1}x +\sin ^{-1} \left(\dfrac{2x}{1+x^2} \right)$$ is equal to 
  • $$4\ \tan^{-1}x$$
  • $$0$$
  • $$\dfrac{\pi}{2}$$
  • $$\pi$$
The value of the expression $$2 \sec ^{-1}2+ \sin^{-1}\left(\dfrac{1}{2}\right)$$ is
  • $$\dfrac{\pi}{6}$$
  • $$\dfrac{5\pi}{6}$$
  • $$\dfrac{7\pi}{6}$$
  • $$1$$
The value of $$\sin [2 \tan ^{-1} (.75)]$$ is equal to
  • $$0.75$$
  • $$1.5$$
  • $$0.96$$
  • $$\sin 1.5$$
The value of $$\cos ^{-1} \left(\cos \dfrac{3 \pi}{2}\right)$$ is equal to
  • $$\dfrac{\pi}{2}$$
  • $$\dfrac{3\pi}{2}$$
  • $$\dfrac{5\pi}{2}$$
  • $$\dfrac{7\pi}{2}$$
The value of the expression $$(\cos^{-1}x)^2$$ is equal to $$\sec^2 x$$.
  • True
  • False
If $$\cos ^{-1} \alpha +\cos ^{-1} \beta +\cos ^{-1} \gamma = 3 \pi$$, then $$\alpha (\beta + \gamma) + \beta (\gamma + \alpha) + \gamma (\alpha + \beta)$$
  • $$0$$
  • $$1$$
  • $$6$$
  • $$12$$
The domain of the function defined by $$f(x)=\sin^{-1}x+\cos x$$ is 
  • $$[-1, 1]$$
  • $$[-1, \pi +1]$$
  • $$( -\infty, \infty)$$
  • $$\phi$$
If $$\sin^{-1}x+\sin^{-1}y=\dfrac{\pi}{2}$$, then the value of $$\cos^{-1}x+\cos^{-1}y$$ is 
  • $$\dfrac{\pi}{2}$$
  • $$\pi$$
  • $$0$$
  • $$\dfrac{2\pi}{3}$$
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