JEE Main & Advanced Mathematics Inverse Trigonometric Functions Question Bank

done Self Evaluation Test - Inverse Trigonometric Functions Total Questions - 70

• question_answer1)
  The value of \[\sin {{\cot }^{-1}}\tan {{\cos }^{-1}}x,\] is
  A)
  \[x\] done clear
  B)
  \[\frac{1}{x}\] done clear
  C)
  \[1\] done clear
  D)
  \[0\] done clear
  View Solution play_arrow

• question_answer2)
  The domain of the function\[f(x)=si{{n}^{-1}}\left\{ {{\log }_{2}}\left( \frac{1}{2}{{x}^{2}} \right) \right\}\] is
  A)
  \[[-2,-1)\cup [1,2]\] done clear
  B)
  \[(-2,-1]\cup [1,2]\] done clear
  C)
  \[[-2,-1]\cup [1,2]\] done clear
  D)
  \[(-2,-1)\cup (1,2)\] done clear
  View Solution play_arrow

• question_answer3)
  Simplified form of \[\tan \left( \frac{\pi }{4}+\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right)+\tan \left( \frac{\pi }{4}-\frac{1}{2}{{\cos }^{-1}}\frac{a}{b} \right)\] is
  A)
  0 done clear
  B)
  \[\frac{2a}{b}\] done clear
  C)
  \[\frac{2b}{a}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer4)
  If \[x\in [\pi /2,\pi ]\] then\[{{\cot }^{-1}}\left( \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right)=\]
  A)
  \[\frac{x-\pi }{2}\] done clear
  B)
  \[\frac{\pi -x}{2}\] done clear
  C)
  \[\frac{3\pi -x}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  The limit \[\underset{x\to \infty }{\mathop{\lim }}\,x\left[ {{\tan }^{-1}}\left( \frac{x+1}{x+2} \right)-{{\tan }^{-1}}\left( \frac{x}{x+2} \right) \right]\]is equal to
  A)
  2 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[-\frac{1}{3}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  The sum to the n term of the series \[\cos e{{c}^{-1}}\sqrt{10}+\cos e{{c}^{-1}}\sqrt{50}+\cos e{{c}^{-1}}\sqrt{170}+...\]\[+\cos e{{c}^{-1}}\sqrt{({{n}^{2}}+1)({{n}^{2}}+2n+2)}\]
  A)
  \[{{\tan }^{-1}}(n+1)-\pi /4\] done clear
  B)
  \[\pi /4\] done clear
  C)
  \[{{\tan }^{-1}}(n+1)\] done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer7)
  Let \[a,b,c>0\] and \[x={{\tan }^{-1}}\sqrt{\frac{a}{bc}(a+b+c),}\] \[y={{\tan }^{-1}}\sqrt{\frac{b}{ca}(a+b+c)}\] and\[z={{\tan }^{-1}}\sqrt{\frac{c}{ab}(a+b+c)}\], then
  A)
  \[\Sigma \tan x\tan y=1\] done clear
  B)
  \[\Sigma \cot x\cot y=1\] done clear
  C)
  \[\Sigma x=\frac{\pi }{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  If \[{{\cos }^{-1}}x+{{\cos }^{-1}}y+{{\cos }^{-1}}z=\pi ,\]then
  A)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+xyz=0\] done clear
  B)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xyz=0\] done clear
  C)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+xyz=1\] done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xyz=1\] done clear
  View Solution play_arrow

• question_answer9)
  If \[{{\cos }^{-1}}\lambda +{{\cos }^{-1}}\mu +{{\cos }^{-1}}\gamma =3\pi ,\] then the value of \[\lambda \mu +\mu \gamma +\gamma \lambda \] is
  A)
  0 done clear
  B)
  1 done clear
  C)
  3 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer10)
  If \[{{\sin }^{-1}}(x-1)+co{{s}^{-1}}(x-3)+ta{{n}^{-1}}\left( \frac{x}{2-{{x}^{2}}} \right)\]\[={{\cos }^{-1}}k+\pi ,\] Then the value of k is
  A)
  1 done clear
  B)
  \[-\frac{1}{\sqrt{2}}\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  The equation \[{{\tan }^{-1}}(1+x)+ta{{n}^{-1}}(1-x)=\frac{\pi }{2}\] is satisfied by
  A)
  \[x=1\] done clear
  B)
  \[x=-1\] done clear
  C)
  \[x=0\] done clear
  D)
  \[x=\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer12)
  What is \[\sin [co{{t}^{-1}}\{cos(ta{{n}^{-1}}x)]\] where \[x>0\], equal to?
  A)
  \[\sqrt{\frac{({{x}^{2}}+1)}{({{x}^{2}}+2)}}\] done clear
  B)
  \[\sqrt{\frac{({{x}^{2}}+2)}{({{x}^{2}}+1)}}\] done clear
  C)
  \[\frac{({{x}^{2}}+1)}{({{x}^{2}}+2)}\] done clear
  D)
  \[\frac{({{x}^{2}}+2)}{({{x}^{2}}+1)}\] done clear
  View Solution play_arrow

• question_answer13)
  The range of the function \[f(x)=si{{n}^{-1}}(log[x])+log(si{{n}^{-1}}[x]);\] (Where [.] denotes the greatest integer function) is
  A)
  \[R\] done clear
  B)
  \[[1,2)\] done clear
  C)
  \[\left\{ \log \frac{\pi }{2} \right\}\] done clear
  D)
  \[\{-sin1\}\] done clear
  View Solution play_arrow

• question_answer14)
  The solutions set of the equations \[{{\sin }^{-1}}x=2{{\tan }^{-1}}x\] is
  A)
  \[\left\{ 1,2 \right\}\] done clear
  B)
  \[\left\{ -1,2 \right\}\] done clear
  C)
  \[\left\{ -1,1,0 \right\}\] done clear
  D)
  \[\left\{ 1,1/2,0 \right\}\] done clear
  View Solution play_arrow

• question_answer15)
  If \[{{\sin }^{-1}}\frac{1}{x}={{\sin }^{-1}}\frac{1}{a}+{{\sin }^{-1}}\frac{1}{b},\] then the value of x is
  A)
  \[\frac{ab}{\sqrt{{{a}^{2}}-1}+\sqrt{{{b}^{2}}-1}}\] done clear
  B)
  \[\frac{ab}{\sqrt{{{a}^{2}}-1}-\sqrt{{{b}^{2}}-1}}\] done clear
  C)
  \[\frac{2ab}{\sqrt{{{a}^{2}}-1}+\sqrt{{{b}^{2}}-1}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  If \[\sum\limits_{i=1}^{2n}{{{\cos }^{-1}}{{x}_{i}}=0}\] then \[\sum\limits_{i=1}^{2n}{{{x}_{i}}}\]is
  A)
  n done clear
  B)
  2n done clear
  C)
  \[\frac{n\left( n+1 \right)}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer17)
  The set of values of x for which the identity\[{{\cos }^{-1}}x+{{\cos }^{-1}}\left( \frac{x}{2}+\frac{1}{2}\sqrt{3-3{{x}^{2}}} \right)=\frac{\pi }{3}\] holds good is
  A)
  \[\left[ 0,1 \right]\] done clear
  B)
  \[\left[ 0,\frac{1}{2} \right]\] done clear
  C)
  \[\left[ \frac{1}{2},1 \right]\] done clear
  D)
  \[\left\{ -1,0,1 \right\}\] done clear
  View Solution play_arrow

• question_answer18)
  What is the value of \[\tan \left( {{\tan }^{-1}}x+{{\tan }^{-1}}y+{{\tan }^{-1}}z \right)-\cot (co{{t}^{-1}}x+co{{t}^{-1}}y+co{{t}^{-1}}z)?\]
  A)
  0 done clear
  B)
  \[2(x+y+z)\] done clear
  C)
  \[\frac{3\pi }{2}\] done clear
  D)
  \[\frac{3\pi }{2}+x+y+z\]   done clear
  View Solution play_arrow

• question_answer19)
  If \[{{\sin }^{-1}}a+{{\sin }^{-1}}b+{{\sin }^{-1}}c=\pi ,\] then find the value of\[a\sqrt{1-{{a}^{2}}}+b\sqrt{1-{{b}^{2}}}+c\sqrt{1-{{c}^{2}}}.\]
  A)
  \[abc\] done clear
  B)
  \[a+b+c\] done clear
  C)
  \[\frac{1}{a}\times \frac{1}{b}\times \frac{1}{c}\] done clear
  D)
  \[2abc\] done clear
  View Solution play_arrow

• question_answer20)
  The value of \[3{{\tan }^{-1}}\frac{1}{2}+2{{\tan }^{-1}}\frac{1}{5}+{{\sin }^{-1}}\frac{142}{65\sqrt{5}}\]is
  A)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[\pi \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer21)
  The range of \[y=(co{{t}^{-1}}x)(co{{t}^{-1}}(-x))\] is
  A)
  \[(\left. 0,\frac{{{\pi }^{2}}}{4} \right]\] done clear
  B)
  \[(0,\pi )\] done clear
  C)
  \[(0,2\pi ]\] done clear
  D)
  \[(0,1]\] done clear
  View Solution play_arrow

• question_answer22)
  \[\sum\limits_{r=1}^{n}{{{\sin }^{-1}}}\left( \frac{\sqrt{r}-\sqrt{r-1}}{\sqrt{r(r+1)}} \right)\] is equal to
  A)
  \[{{\tan }^{-1}}(\sqrt{n})-\frac{\pi }{4}\] done clear
  B)
  \[{{\tan }^{-1}}(\sqrt{n+1})-\frac{\pi }{4}\] done clear
  C)
  \[{{\tan }^{-1}}(\sqrt{n})\] done clear
  D)
  \[{{\tan }^{-1}}(\sqrt{n+1})\] done clear
  View Solution play_arrow

• question_answer23)
  What is the value of x that satisfies the equation\[{{\cos }^{-1}}x=2{{\sin }^{-1}}x\]?
  A)
  \[\frac{1}{2}\] done clear
  B)
  \[-1\] done clear
  C)
  \[1\] done clear
  D)
  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer24)
  If \[[si{{n}^{-1}}co{{s}^{-1}}si{{n}^{-1}}ta{{n}^{-1}}x]=1,\] where \[[.]\] denotes the greatest integer function, then x belongs to the interval
  A)
  \[[tan\,sin\,cos1,tan\,sin\,cos\,sin1]\] done clear
  B)
  \[(tan\,sin\,cos1,tan\,sin\,cos\,sin1)\] done clear
  C)
  \[[-1,1]\] done clear
  D)
  \[[sin\,cos\,tan1,sin\,cos\,tan1]\] done clear
  View Solution play_arrow

• question_answer25)
  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
  A)
  1 done clear
  B)
  \[\frac{1}{\sqrt{2}}\] done clear
  C)
  \[\frac{1}{3}\] done clear
  D)
  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer26)
  Which of the following is the principal value branch of \[\cos e{{c}^{-1}}x?\]
  A)
  \[\left( \frac{-\pi }{2},\frac{\pi }{2} \right)\] done clear
  B)
  \[(0,\pi )-\left[ \frac{\pi }{2} \right]\] done clear
  C)
  \[\left[ \frac{-\pi }{2},\frac{\pi }{2} \right]\] done clear
  D)
  \[\left[ \frac{-\pi }{2},\frac{\pi }{2} \right]-\{0\}\] done clear
  View Solution play_arrow

• question_answer27)
  If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=\pi /2\] and\[{{\cos }^{-1}}x-{{\cos }^{-1}}y=0.\] then values x and y are respectively
  A)
  \[\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\] done clear
  B)
  \[\frac{1}{2},\frac{1}{2}\] done clear
  C)
  \[\frac{1}{2},-\frac{1}{2}\] done clear
  D)
  \[\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer28)
  \[\theta ={{\tan }^{-1}}(2ta{{n}^{2}}\theta )-ta{{n}^{-1}}\left( \frac{1}{3}\tan \theta  \right)\] then \[\tan \theta =\]
  A)
  \[-2\] done clear
  B)
  \[-1\] done clear
  C)
  \[2/3\] done clear
  D)
  \[2\] done clear
  View Solution play_arrow

• question_answer29)
  Total number of positive integral value ?n? so that the equations \[{{\cos }^{-1}}x+{{(si{{n}^{-1}}y)}^{2}}=\frac{n{{\pi }^{2}}}{4}\] and \[{{(si{{n}^{-1}}y)}^{2}}-{{\cos }^{-1}}x=\frac{{{\pi }^{2}}}{16}\] are consistent, is equal to
  A)
  1 done clear
  B)
  4 done clear
  C)
  3 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer30)
  If \[u={{\cot }^{-1}}\sqrt{\tan \alpha }-{{\tan }^{-1}}\sqrt{\tan \alpha },\] then  \[\tan \left( \frac{\pi }{4}-\frac{u}{2} \right)\] is equal to
  A)
  \[\sqrt{\tan \alpha }\] done clear
  B)
  \[\sqrt{\cot \alpha }\] done clear
  C)
  \[\tan \alpha \] done clear
  D)
  \[\cot \alpha \] done clear
  View Solution play_arrow

• question_answer31)
  The solution set of the equation\[{{\tan }^{-1}}x-{{\cot }^{-1}}x={{\cos }^{-1}}(2-x)\] will lie in the interval
  A)
  \[[0,1]\] done clear
  B)
  \[[-1,1]\] done clear
  C)
  \[[1,3]\]  done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  Complete solution set of \[{{\tan }^{2}}(si{{n}^{-1}}x)>1\]is
  A)
  \[\left( -1,-\frac{1}{\sqrt{2}} \right)\cup \left( \frac{1}{\sqrt{2}},1 \right)\] done clear
  B)
  \[\left( -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}} \right)\tilde{\ }\{0\}\] done clear
  C)
  \[(-1,1)\tilde{\ }\{0\}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[{{\sin }^{-1}}x={{\tan }^{-1}}y,\] what is the value of\[\frac{1}{{{x}^{2}}}-\frac{1}{{{y}^{2}}}?\]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer34)
  If \[ax+b(sec(ta{{n}^{-1}}x))=c\] and \[ay+b\]\[(sec\,.\,(ta{{n}^{-1}}y))=c,\] then \[\frac{x+y}{1-xy}=\]
  A)
  \[\frac{ac}{{{a}^{2}}+{{c}^{2}}}\] done clear
  B)
  \[\frac{2ac}{a-c}\] done clear
  C)
  \[\frac{2ac}{{{a}^{2}}-{{c}^{2}}}\] done clear
  D)
  \[\frac{a+c}{1-ac}\] done clear
  View Solution play_arrow

• question_answer35)
  \[\sum\limits_{r=1}^{\infty }{{{\tan }^{-1}}\left( \frac{1}{1+r+{{r}^{2}}} \right)=....}\]
  A)
  \[\frac{\pi }{2}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{2\pi }{3}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer36)
  Let \[x\in (0,1).\]The set of all x such that \[{{\sin }^{-1}}x>{{\cos }^{-1}}x,\] is the interval:
  A)
  \[\left( \frac{1}{2},\frac{1}{\sqrt{2}} \right)\] done clear
  B)
  \[\left( \frac{1}{\sqrt{2}},1 \right)\] done clear
  C)
  \[(0,1)\] done clear
  D)
  \[\left( 0,\frac{\sqrt{3}}{2} \right)\] done clear
  View Solution play_arrow

• question_answer37)
  The range of \[f(x)=si{{n}^{-1}}x+{{\tan }^{-1}}x+{{\sec }^{-1}}x\] is
  A)
  \[\left( \frac{\pi }{4},\frac{3\pi }{4} \right)\] done clear
  B)
  \[\left[ \frac{\pi }{4},\frac{3\pi }{4} \right]\] done clear
  C)
  \[\left\{ \frac{\pi }{4},\frac{3\pi }{4} \right\}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer38)
  If \[0<a<b<c,\] then\[{{\cot }^{-1}}\left( \frac{ab+1}{a-b} \right)+{{\cot }^{-1}}\left( \frac{bc+1}{b-c} \right)+{{\cot }^{-1}}\left( \frac{ca+1}{c-a} \right)=\]
  A)
  0 done clear
  B)
  \[\pi \] done clear
  C)
  \[2\pi \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  The equation \[{{\sin }^{-1}}(3x-4{{x}^{3}})=3si{{n}^{-1}}(x)\]is true for all values of x lying in which one of the following intervals?
  A)
  \[\left[ -\frac{1}{2},\frac{1}{2} \right]\] done clear
  B)
  \[\left[ \frac{1}{2},1 \right]\] done clear
  C)
  \[\left[ -1,-\frac{1}{2} \right]\] done clear
  D)
  \[[-1,1]\] done clear
  View Solution play_arrow

• question_answer40)
  Two angles of a triangle are \[{{\cot }^{-1}}2\] and \[{{\cot }^{-1}}3.\]then the third angle is
  A)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{3\pi }{4}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[\frac{\pi }{3}\] done clear
  View Solution play_arrow

• question_answer41)
  \[\tan \left\{ \frac{1}{2}{{\sin }^{-1}}\frac{2x}{1+{{x}^{2}}}+\frac{1}{2}{{\cos }^{-1}}\frac{1-{{y}^{2}}}{1+{{y}^{2}}} \right\}=\]
  A)
  \[\frac{x-y}{1+xy}\] done clear
  B)
  \[\frac{x+y}{1-xy}\] done clear
  C)
  \[\frac{x-y}{x+y}\] done clear
  D)
  \[\frac{1-xy}{1+xy}\] done clear
  View Solution play_arrow

• question_answer42)
  If \[{{\tan }^{-1}}\frac{x}{\pi }<\frac{\pi }{3},x\in N,\] than the maximum value of  x is
  A)
  2 done clear
  B)
  5 done clear
  C)
  7 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  \[{{\sin }^{-1}}\left( a-\frac{{{a}^{2}}}{3}+\frac{{{a}^{3}}}{9}+... \right)+{{\cos }^{-1}}(1+b+{{b}^{2}}+...)=\frac{\pi }{2}\]when
  A)
  \[a=-3\] and \[b=1\] done clear
  B)
  \[a=1\] and \[b=-\frac{1}{3}\] done clear
  C)
  \[a=\frac{1}{6}\] and \[b=\frac{1}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  The sum of the infinite series \[{{\cot }^{-1}}2+{{\cot }^{-1}}8+{{\cot }^{-1}}18+{{\cot }^{-1}}32+...\] is,
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[\frac{\pi }{4}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  The \[+ve\]integral solution of \[{{\tan }^{-1}}x+{{\cos }^{-1}}\frac{y}{\sqrt{1+{{y}^{2}}}}={{\sin }^{-1}}\frac{3}{\sqrt{10}}\] is
  A)
  \[x=1,y=2;x=2,y=7\] done clear
  B)
  \[x=1,y=3;x=2,y=4\] done clear
  C)
  \[x=0,y=0;x=3,y=4\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer46)
  The complete solution set of \[{{[co{{t}^{-1}}x]}^{2}}-6[co{{t}^{-1}}x]+9\le 0,\] Where [.] denotes the greatest integer function, is
  A)
  \[(-\infty ,\cot 3]\] done clear
  B)
  \[[\cot 3,\cot 2)\] done clear
  C)
  \[[\cot 3,\infty )\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  Solving \[2{{\cos }^{-1}}x={{\sin }^{-1}}(2x\sqrt{1-{{x}^{2}}}),\]we get
  A)
  \[x\in \left[ \frac{\sqrt{2}}{2},1 \right]\] done clear
  B)
  \[x=3\] done clear
  C)
  \[x\in [3,4]\] done clear
  D)
  \[x=0\] done clear
  View Solution play_arrow

• question_answer48)
  If \[{{\tan }^{-1}}(2x)+ta{{n}^{-1}}(3x)=\frac{\pi }{4}\]then x is equal to
  A)
  -1 done clear
  B)
  -2 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer49)
  If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y=\frac{\pi }{2},\] then \[\frac{1+{{x}^{4}}+{{y}^{4}}}{{{x}^{2}}-{{x}^{2}}{{y}^{2}}+{{y}^{2}}}\] is equal to
  A)
  1 done clear
  B)
  2 done clear
  C)
  ½ done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  What is the value of:\[\cos \left[ {{\tan }^{-1}}\left\{ \tan \left( \frac{15\pi }{4} \right) \right\} \right]?\]
  A)
  \[-\frac{1}{\sqrt{2}}\] done clear
  B)
  0 done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  \[\frac{1}{2\sqrt{2}}\] done clear
  View Solution play_arrow

• question_answer51)
  If the equation \[{{(si{{n}^{-1}}x)}^{3}}+{{(co{{s}^{-1}}x)}^{3}}=a{{\pi }^{2}}\] has no real root then
  A)
  \[a>0\] done clear
  B)
  \[a<\frac{1}{32}\] done clear
  C)
  \[a<3\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  The value of\[{{\tan }^{-1}}\left( \frac{1}{2}\tan 2A)+{{\tan }^{-1}}(cotA)+ta{{n}^{-1}}(co{{t}^{3}}A) \right)\] is
  A)
  0 if \[\frac{\pi }{4}<A<\frac{\pi }{2}\] done clear
  B)
  \[\pi \], if \[0<A<\frac{\pi }{4}\] done clear
  C)
  Both a and b done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer53)
  If \[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha ,\] then \[4{{x}^{2}}-4xy\cos \alpha +{{y}^{2}}\] is equal to
  A)
  \[2\sin 2\alpha \] done clear
  B)
  \[4\] done clear
  C)
  \[4{{\sin }^{2}}\alpha \] done clear
  D)
  \[-4{{\sin }^{2}}\alpha \] done clear
  View Solution play_arrow

• question_answer54)
  The value of \[{{\sin }^{-1}}\left\{ \cot \left( {{\sin }^{-1}}\sqrt{\left( \frac{2-\sqrt{3}}{4} \right)}+{{\cos }^{-1}}\frac{\sqrt{12}}{4}+{{\sec }^{-1}}\sqrt{2} \right) \right\}\]is
  A)
  0 done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer55)
  What is the value of \[{{\sec }^{2}}{{\tan }^{-1}}\left( \frac{5}{11} \right)?\]
  A)
  \[121/96\] done clear
  B)
  \[211/921\] done clear
  C)
  \[146/121\] done clear
  D)
  \[267/121\] done clear
  View Solution play_arrow

• question_answer56)
  The sum of the infinite series \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{2}} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{2}-1}{\sqrt{6}} \right)+{{\sin }^{-1}}\left( \frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}} \right)+...\]\[+...+{{\sin }^{-1}}\left( \frac{\sqrt{n}-\sqrt{(n-1)}}{\sqrt{\{n(n+1)\}}} \right)+...\] is
  A)
  \[\frac{\pi }{8}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  \[\pi \] done clear
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• question_answer57)
  If \[{{\sin }^{-1}}\left( x-\frac{{{x}^{2}}}{2}+\frac{{{x}^{3}}}{4}-... \right)\]\[+{{\cos }^{-1}}\left( {{x}^{2}}-\frac{{{x}^{4}}}{2}+\frac{{{x}^{6}}}{4}-... \right)=\frac{\pi }{2}\] for \[0<\left| x \right|<\sqrt{2},\] then x equals
  A)
  ½ done clear
  B)
  1 done clear
  C)
  -1/2 done clear
  D)
  -1 done clear
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• question_answer58)
  The value of \[{{\cot }^{-1}}7+{{\cot }^{-1}}8+{{\cot }^{-1}}18\] is
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[{{\cot }^{-1}}5\] done clear
  D)
  \[{{\cot }^{-1}}3\] done clear
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• question_answer59)
  If \[{{\sin }^{-1}}\left( \frac{2a}{1+{{a}^{2}}} \right)-{{\cos }^{-1}}\left( \frac{1-{{b}^{2}}}{1+{{b}^{2}}} \right)={{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right),\]then what is the value of x?
  A)
  \[a/b\] done clear
  B)
  \[ab\] done clear
  C)
  \[b/a\]   done clear
  D)
  \[\frac{a-b}{1+ab}\] done clear
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• question_answer60)
  If \[{{\sin }^{-1}}1+{{\sin }^{-1}}\frac{4}{5}={{\sin }^{-1}}x,\] then what is x equal to?
  A)
  3/5 done clear
  B)
  4/5 done clear
  C)
  1 done clear
  D)
  0 done clear
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• question_answer61)
  Let \[-1\le x\le 1.\] If \[\cos (si{{n}^{-1}}x)=\frac{1}{2},\] then how many value does \[\tan (co{{s}^{-1}}x)\] assume?
  A)
  One done clear
  B)
  Two done clear
  C)
  Four      done clear
  D)
  Infinite done clear
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• question_answer62)
  If \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{r}^{2}},\] then\[{{\tan }^{-1}}\frac{xy}{zr}+{{\tan }^{-1}}\frac{yz}{xr}+{{\tan }^{-1}}\frac{xz}{yr}=\]
  A)
  \[\pi \] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  0 done clear
  D)
  None of these done clear
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• question_answer63)
  If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y+{{\sin }^{-1}}z=\pi ,\] then \[{{x}^{4}}+{{y}^{4}}+{{z}^{4}}+4{{x}^{2}}{{y}^{2}}{{z}^{2}}=k({{x}^{2}}{{y}^{2}}+{{y}^{2}}{{z}^{2}}+{{z}^{2}}{{x}^{2}}).\]where k =
  A)
  1 done clear
  B)
  2 done clear
  C)
  4 done clear
  D)
  None of these done clear
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• question_answer64)
  If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},....{{a}_{n}}\] is an \[A.P.\] with common difference d; \[(d>0)\] then \[\tan \left[ {{\tan }^{-1}}\left( \frac{d}{1+{{a}_{1}}{{a}_{2}}} \right)+{{\tan }^{-1}}\left( \frac{d}{1+{{a}_{2}}{{a}_{3}}} \right)+...+ta{{n}^{-1}}\left( \frac{d}{1+{{a}_{n-1}}{{a}_{n}}} \right) \right]\]is equal to
  A)
  \[\frac{(n-1)d}{{{a}_{1}}+{{a}_{n}}}\] done clear
  B)
  \[\frac{(n-1)d}{1+{{a}_{1}}{{a}_{n}}}\] done clear
  C)
  \[\frac{nd}{1+{{a}_{1}}{{a}_{n}}}\] done clear
  D)
  \[\frac{{{a}_{n}}-{{a}_{1}}}{{{a}_{n}}+{{a}_{1}}}\] done clear
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• question_answer65)
  The formula \[{{\sin }^{-1}}\{2x(1-{{x}^{2}})\}=2si{{n}^{-1}}x\] is true for all values of x lying in the interval
  A)
  \[[-1,1]\] done clear
  B)
  \[[0,1]\] done clear
  C)
  \[[-1,0]\] done clear
  D)
  \[\left[ -1/\sqrt{2},1/\sqrt{2} \right]\] done clear
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• question_answer66)
  If \[k\le {{\sin }^{-1}}x+{{\cos }^{-1}}x+{{\tan }^{-1}}x\le K,\] then
  A)
  \[k=-\pi ,K=\pi \] done clear
  B)
  \[k=0,K=\frac{\pi }{2}\] done clear
  C)
  \[k=\frac{\pi }{4},K=\frac{3\pi }{4}\] done clear
  D)
  \[k=0,K=\pi \] done clear
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• question_answer67)
  The set of values of k for which \[{{x}^{2}}-kx+{{\sin }^{-1}}(sin4)>0\] for all real x is
  A)
  \[\phi \] done clear
  B)
  \[(-2,2)\] done clear
  C)
  \[R\] done clear
  D)
  \[(-\infty ,-2)\cup (2,\infty )\] done clear
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• question_answer68)
  In a triangle ABC. If \[A={{\tan }^{-1}}2\] and \[B={{\tan }^{-1}}3,\]then C is equal to
  A)
  \[\frac{\pi }{3}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
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• question_answer69)
  If \[\sin \left( {{\sin }^{-1}}\frac{1}{5}+{{\cos }^{-1}}x \right)=1\], then what is x equal to?
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[\frac{4}{5}\] done clear
  D)
  \[\frac{1}{5}\] done clear
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• question_answer70)
  The value of \[{{\cos }^{-1}}x+{{\cos }^{-1}}\left( \frac{x}{2}+\frac{1}{2}\sqrt{3-3{{x}^{2}}} \right);\frac{1}{2}\le x\le 1\] is
  A)
  \[-\frac{\pi }{3}\]            done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[\frac{3}{\pi }\] done clear
  D)
  \[-\frac{3}{\pi }\] done clear
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