JEE Main & Advanced Mathematics Inverse Trigonometric Functions Question Bank

done Critical Thinking Total Questions - 15

• question_answer1)
  \[{{\cot }^{-1}}\left[ \frac{\sqrt{1-\sin x}+\sqrt{1+\sin x}}{\sqrt{1-\sin x}-\sqrt{1+\sin x}} \right]=\] [MNR 1986]
  A)
  \[\pi -x\] done clear
  B)
  \[2\pi -x\] done clear
  C)
  \[\frac{x}{2}\] done clear
  D)
  \[\pi -\frac{x}{2}\] done clear
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• question_answer2)
  The principal value of \[{{\sin }^{-1}}\left[ \sin \left( \frac{2\pi }{3} \right) \right]\]is [IIT 1986]
  A)
  \[-\frac{2\pi }{3}\] done clear
  B)
  \[\frac{2\pi }{3}\] done clear
  C)
  \[\frac{4\pi }{3}\] done clear
  D)
  None of these done clear
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• question_answer3)
  If \[\theta ={{\tan }^{-1}}a,\varphi ={{\tan }^{-1}}b\] and \[ab=-1,\] then \[\theta -\varphi =\]
  A)
  0 done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  None of these done clear
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• question_answer4)
  If \[\tan ({{\cos }^{-1}}x)=\sin \left( {{\cot }^{-1}}\frac{1}{2} \right)\], then x =
  A)
  \[\pm \frac{5}{3}\] done clear
  B)
  \[\pm \frac{\sqrt{5}}{3}\] done clear
  C)
  \[\pm \frac{5}{\sqrt{3}}\] done clear
  D)
  None of these done clear
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• question_answer5)
  If \[{{\sin }^{-1}}x=\frac{\pi }{5}\] for some \[x\in (-1,\,1)\], then the value of \[{{\cos }^{-1}}x\] is  [IIT 1992]
  A)
  \[\frac{3\pi }{10}\] done clear
  B)
  \[\frac{5\pi }{10}\] done clear
  C)
  \[\frac{7\pi }{10}\] done clear
  D)
  \[\frac{9\pi }{10}\] done clear
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• question_answer6)
  If\[{{\cos }^{-1}}p+{{\cos }^{-1}}q+{{\cos }^{-1}}r=\pi \]then\[{{p}^{2}}+{{q}^{2}}+{{r}^{2}}+2pqr=\] [Karnataka CET 2004]
  A)
  3 done clear
  B)
  1 done clear
  C)
  2 done clear
  D)
  -1 done clear
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• question_answer7)
   \[\tan \left[ \frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right]+\tan \left[ \frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right]=\] [MP PET 1999]
  A)
  \[\frac{2a}{b}\] done clear
  B)
  \[\frac{2b}{a}\] done clear
  C)
  \[\frac{a}{b}\] done clear
  D)
  \[\frac{b}{a}\] done clear
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• question_answer8)
  \[{{\tan }^{-1}}\,\left[ \frac{\sqrt{1+{{x}^{2}}}+\sqrt{1-{{x}^{2}}}}{\sqrt{1+{{x}^{2}}}-\sqrt{1-{{x}^{2}}}} \right]=\]
  A)
  \[\frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}{{x}^{2}}\] done clear
  B)
  \[\frac{\pi }{4}+{{\cos }^{-1}}{{x}^{2}}\] done clear
  C)
  \[\frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}x\] done clear
  D)
  \[\frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}{{x}^{2}}\] done clear
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• question_answer9)
  The equation \[{{\sin }^{-1}}x-{{\cos }^{-1}}x={{\cos }^{-1}}\left( \frac{\sqrt{3}}{2} \right)\]has
  A)
  No solution done clear
  B)
  Unique solution done clear
  C)
  Infinite number of solutions done clear
  D)
  None of these done clear
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• question_answer10)
  \[\sum\limits_{m=1}^{n}{{{\tan }^{-1}}}\left( \frac{2m}{{{m}^{4}}+{{m}^{2}}+2} \right)\]is equal to
  A)
  \[{{\tan }^{-1}}\left( \frac{{{n}^{2}}+n}{{{n}^{2}}+n+2} \right)\] done clear
  B)
  \[{{\tan }^{-1}}\left( \frac{{{n}^{2}}-n}{{{n}^{2}}-n+2} \right)\] done clear
  C)
  \[{{\tan }^{-1}}\left( \frac{{{n}^{2}}+n+2}{{{n}^{2}}+n} \right)\] done clear
  D)
   None of these done clear
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• question_answer11)
  The number of real solutions of \[{{\tan }^{-1}}\sqrt{x(x+1)}+{{\sin }^{-1}}\sqrt{{{x}^{2}}+x+1}=\frac{\pi }{2}\]is [IIT 1999]
  A)
  Zero done clear
  B)
  One done clear
  C)
  Two done clear
  D)
  Infinite done clear
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• question_answer12)
  The equation \[2{{\cos }^{-1}}x+{{\sin }^{-1}}x=\frac{11\pi }{6}\]has [AMU 1999]
  A)
  No solution done clear
  B)
  Only one solution done clear
  C)
  Two solutions done clear
  D)
  Three solutions done clear
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• question_answer13)
  If \[{{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z=\pi \], then \[x+y+z\] is equal to  [Kerala (Engg.) 2002]
  A)
  xyz done clear
  B)
  0 done clear
  C)
  1 done clear
  D)
  2xyz done clear
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• question_answer14)
  \[2{{\tan }^{-1}}\left[ \sqrt{\frac{a-b}{a+b}}\tan \frac{\theta }{2} \right]=\]  [Dhanbad Engg. 1976]
  A)
  \[{{\cos }^{-1}}\left( \frac{a\cos \theta +b}{a+b\cos \theta } \right)\] done clear
  B)
  \[{{\cos }^{-1}}\left( \frac{a+b\cos \theta }{a\cos \theta +b} \right)\] done clear
  C)
  \[{{\cos }^{-1}}\left( \frac{a\cos \theta }{a+b\cos \theta } \right)\] done clear
  D)
  \[{{\cos }^{-1}}\left( \frac{a\cos +b\theta }{a+b\cos \theta } \right)\] done clear
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• question_answer15)
  \[2{{\tan }^{-1}}(\cos x)={{\tan }^{-1}}(\text{cose}{{\text{c}}^{2}}x),\] then x = [UPSEAT 2002]
  A)
  \[\frac{\pi }{2}\] done clear
  B)
  \[\pi \] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[\frac{\pi }{3}\] done clear
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