JEE Main & Advanced Mathematics Trigonometric Identities Question Bank

done Self Evaluation Test - Trigonometric Function Total Questions - 70

• question_answer1)
   If \[0<x<\pi \]it and \[\cos x+\sin x=\frac{1}{2},\] then tan x is
  A)
  \[\frac{(1-\sqrt{7})}{4}\]  done clear
  B)
  \[\frac{(4-\sqrt{7})}{3}\] done clear
  C)
  \[-\frac{(4+\sqrt{7})}{3}\]          done clear
  D)
  \[\frac{(1+\sqrt{7})}{4}\] done clear
  View Solution play_arrow

• question_answer2)
  For which real values of x and y, the equation \[{{\sec }^{2}}\theta =\frac{4xy}{{{(x+y)}^{2}}}\]is possible ?
  A)
  \[x=y\]              done clear
  B)
  \[x>y\] done clear
  C)
  \[x<y\]              done clear
  D)
   None of these done clear
  View Solution play_arrow

• question_answer3)
  The value of \[{{\tan }^{2}}\theta {{\sec }^{2}}\theta ({{\cot }^{2}}\theta -{{\cos }^{2}}\theta )\] is
  A)
  \[0\]     done clear
  B)
  \[1\]     done clear
  C)
  \[-1\]     done clear
  D)
  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer4)
  The expression \[{{\left( \frac{\cos A+\cos B}{\sin A-\sin B} \right)}^{n}}+\left( \frac{\sin A+\sin B}{\cos A-\cos B} \right)=\]
  A)
  \[2{{\cot }^{n}}\left( \frac{A-B}{2} \right)\] if n is even done clear
  B)
  0 if n is even done clear
  C)
  \[2{{\cot }^{n}}\left( \frac{A-B}{2} \right)\]if n is odd done clear
  D)
  3 if n is odd done clear
  View Solution play_arrow

• question_answer5)
  If \[0\le x\le \pi \]and \[{{81}^{{{\sin }^{2}}x}}+{{81}^{{{\cos }^{2}}x}}=30,\] then x=
  A)
  \[\pi /6\]             done clear
  B)
  \[\pi /2\] done clear
  C)
  \[\pi /4\]             done clear
  D)
  \[3\pi /4\] done clear
  View Solution play_arrow

• question_answer6)
  The difference of two angles is \[1{}^\circ ;\] the circular measure of their sum is 1. What is the smaller angle in circular measure?
  A)
  \[\left[ \frac{180}{\pi }-1 \right]\]  done clear
  B)
  \[\left[ 1-\frac{\pi }{180} \right]\] done clear
  C)
  \[\frac{1}{2}\left[ 1-\frac{\pi }{180} \right]\]         done clear
  D)
  \[\frac{1}{2}\left[ \frac{180}{\pi }-1 \right]\] done clear
  View Solution play_arrow

• question_answer7)
  If \[(\sec \alpha +\tan \alpha )(\sec \beta +\tan \beta )(\sec \gamma +\tan \gamma )\] \[=\tan \,\alpha \tan \beta \tan \gamma ,\] then expression \[(\sec \alpha -\tan \alpha )\,(sec\beta -tan\beta )(sec\gamma -tan\gamma )\]is equal to
  A)
  \[\cot \alpha \,\,\cot \beta \,\,\cot \gamma \] done clear
  B)
  \[\tan \alpha \,\,tan\beta \,\,tan\gamma \] done clear
  C)
  \[\cot \alpha +\cot \beta +\cot \gamma \] done clear
  D)
  \[tan\alpha +tan\beta +tan\gamma \] done clear
  View Solution play_arrow

• question_answer8)
  If \[m=\cos \sec \theta -\sin \theta \] and \[n=\sec \theta -\cos \theta ,\]then \[{{m}^{2/3}}+{{n}^{2/3}}=\]
  A)
  \[{{(mn)}^{-2/3}}\] done clear
  B)
  \[{{(mn)}^{2/3}}\] done clear
  C)
  \[{{(mn)}^{-1/3}}\]                  done clear
  D)
  \[{{(mn)}^{1/3}}\] done clear
  View Solution play_arrow

• question_answer9)
  Which pairs of function is identical?
  A)
  \[f(x)=\sqrt{{{x}^{2}}},\] \[g(x)=x\] done clear
  B)
  \[f(x)={{\sin }^{2}}x+{{\cos }^{2}}x;\,g(x)=1\] done clear
  C)
  \[f(x)=\frac{x}{x},\,\,g(x)=1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer10)
  What is the angle (in circular measure) between the hour hand and the minute hand of a clock when the time is half past 4?
  A)
  \[\frac{\pi }{3}\]             done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{6}\]             done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  If \[(1+\sin \alpha )(1+\sin \beta )(1+\sin \gamma )=(1-\sin \alpha )\] \[(1-\sin \beta )(1-\sin \gamma )=k,\] then k is equal to:
  A)
  \[2\cos \alpha \cos \beta \cos \gamma \] done clear
  B)
  \[-\cos \alpha \cos \beta \cos \gamma \] done clear
  C)
  \[+\cos \alpha \cos \beta \cos \gamma \] done clear
  D)
  \[+2sin\alpha sin\beta sin\gamma \] done clear
  View Solution play_arrow

• question_answer12)
  If \[{{p}_{n}}={{\cos }^{n}}\theta +{{\sin }^{n}}\theta ,\] then \[^{pn-p}n-2{{=}^{kp}}n-4\] where:
  A)
  \[k=1\]              done clear
  B)
  \[k=-{{\sin }^{2}}\theta {{\cos }^{2}}\theta \] done clear
  C)
  \[k={{\sin }^{2}}\theta \]      done clear
  D)
  \[k={{\cos }^{2}}\theta \] done clear
  View Solution play_arrow

• question_answer13)
  If \[(\sin \theta =3\sin (\theta +2\alpha ),\] then the value of \[\tan (\theta +\alpha )+2\tan \alpha \]is
  A)
  3                      done clear
  B)
  2   done clear
  C)
  \[-1\]    done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer14)
  What is \[\cos 20{}^\circ +\cos 100{}^\circ +\cos 140{}^\circ \] equal to?
  A)
  2                      done clear
  B)
  1   done clear
  C)
  \[1/2\]    done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer15)
  The value of \[\tan A+\tan (60{}^\circ +A)-\tan (60{}^\circ -A)\] is
  A)
  \[\tan \,3A\]   done clear
  B)
  \[2\,\,tan\,3A\] done clear
  C)
  \[3\text{ }tan\text{ }3A\]        done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  Find the value of \[\cot 5{}^\circ \cot 10........cot85{}^\circ \].
  A)
  1                      done clear
  B)
  -1    done clear
  C)
  2                      done clear
  D)
  -2 done clear
  View Solution play_arrow

• question_answer17)

  \[sin\text{ }A+2\text{ }sin\,2A+sin\text{ }3A\]is equal to which of the following?

  1. \[4\sin 2A{{\cos }^{2}}\left( \frac{A}{2} \right)\]

  2. \[2\sin 2A{{\left( \sin \frac{A}{2}+\cos \frac{A}{2} \right)}^{2}}\]

  3. \[8\sin A\cos A{{\cos }^{2}}\left( \frac{A}{2} \right)\]

  Select the correct answer using the code given below:

  A)
  1 and 2 only     done clear
  B)
  2 and 3 only done clear
  C)
  1 and 3 only     done clear
  D)
  1, 2 and 3 done clear
  View Solution play_arrow

• question_answer18)
  What is the value of \[\left( 1+\cos \frac{\pi }{8} \right)\left( 1+\cos \frac{3\pi }{8} \right)\left( 1+\cos \frac{5\pi }{8} \right)\left( 1+\cos \frac{7\pi }{8} \right)?\]
  A)
  \[\frac{1}{2}\]               done clear
  B)
  \[\frac{1}{2}+\frac{1}{2\sqrt{2}}\] done clear
  C)
  \[\frac{1}{2}-\frac{1}{2\sqrt{2}}\]          done clear
  D)
  \[\frac{1}{8}\] done clear
  View Solution play_arrow

• question_answer19)
  What is \[\frac{\cot 224{}^\circ -\cot 134{}^\circ }{\cot 226{}^\circ +\cot 316{}^\circ }\] equal to?
  A)
  \[-\text{cosec }88{}^\circ \]     done clear
  B)
  \[-\text{cosec 2}{}^\circ \] done clear
  C)
  \[-\text{cosec 44}{}^\circ \]      done clear
  D)
  \[-\text{cosec 46}{}^\circ \] done clear
  View Solution play_arrow

• question_answer20)
  If an angle B is complement of an angle A, what are the greatest and least values of \[cos\text{ }A\text{ }cos\text{ }B\] respectively?
  A)
  \[0,-\frac{1}{2}\]    done clear
  B)
  \[\frac{1}{2},-1\] done clear
  C)
  \[1,0\]               done clear
  D)
  \[\frac{1}{2},-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer21)
  Value of \[2({{\sin }^{6}}\theta +{{\cos }^{6}}\theta )\] \[-3({{\sin }^{4}}\theta +{{\cos }^{4}}\theta )+1\] is
  A)
  2                      done clear
  B)
  0     done clear
  C)
  4                      done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer22)
  If \[\alpha +\beta +\gamma =\pi \]then the minimum value of \[cos\text{ }A+cos\text{ }B+cos\text{ }C\]
  A)
  is zero done clear
  B)
  is positive done clear
  C)
  lies between \[-2\] and \[-3\] done clear
  D)
  is \[-3\] done clear
  View Solution play_arrow

• question_answer23)
  If \[x\cos \theta +y\sin \theta =z,\] then what is the value of \[{{(x\sin \theta -y\cos \theta )}^{2}}\]?
  A)
  \[{{x}^{2}}+{{y}^{2}}-{{z}^{2}}\]    done clear
  B)
  \[{{x}^{2}}-{{y}^{2}}-{{z}^{2}}\] done clear
  C)
  \[{{x}^{2}}-{{y}^{2}}+{{z}^{2}}\]   done clear
  D)
  \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\] done clear
  View Solution play_arrow

• question_answer24)
  \[\frac{\sin x-\sin 3x}{{{\sin }^{2}}x-{{\cos }^{2}}x}\] is equal to
  A)
  \[\sin 2x\]    done clear
  B)
  \[\cos 2x\] done clear
  C)
  \[\tan 2x\]           done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  \[{{(1-\sin A+\cos A)}^{2}}\]is equal to
  A)
  \[2(1-\cos A)(1+\sin A)\] done clear
  B)
  \[2(1-sinA)(1+\cos A)\] done clear
  C)
  \[2(1-cos\,A)(1-sinA)\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer26)
  The expression \[\frac{\cos 6x+6\cos 4x+15\cos 2x+10}{\cos 5x+5\cos 3+10\cos x}\] is equal to
  A)
  \[cos\text{ }2x\]        done clear
  B)
  \[2\,cos\,x\] done clear
  C)
  \[co{{s}^{2}}x\]            done clear
  D)
  \[1+\cos x\] done clear
  View Solution play_arrow

• question_answer27)
  If \[\sin {{18}^{0}}=\frac{\sqrt{5}-1}{4},\] then what is the value of \[\sin 18{}^\circ \]?
  A)
  \[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\] done clear
  B)
  \[\frac{\sqrt{3+\sqrt{5}}+\sqrt{5+\sqrt{5}}}{4}\] done clear
  C)
  \[\frac{\sqrt{3-\sqrt{5}}+\sqrt{5-\sqrt{5}}}{4}\] done clear
  D)
  \[\frac{\sqrt{3+\sqrt{5}}-\sqrt{5-\sqrt{5}}}{4}\] done clear
  View Solution play_arrow

• question_answer28)
  If A and B are positive acute angles satisfying \[3{{\cos }^{2}}A+2{{\cos }^{2}}B=4\] and \[\frac{3\sin A}{\sin B}=\frac{2\cos B}{\cos A}\] Then the value of \[A+2B\]is equal to:
  A)
  \[\frac{\pi }{6}\]             done clear
  B)
  \[\frac{\pi }{2}\]    done clear
  C)
  \[\frac{\pi }{3}\]             done clear
  D)
  \[\frac{\pi }{4}\] done clear
  View Solution play_arrow

• question_answer29)
  What is \[{{\sin }^{2}}(3\pi )+{{\cos }^{2}}(4\pi )+{{\tan }^{2}}(5\pi )\] equal to?
  A)
  0                      done clear
  B)
  1     done clear
  C)
  2                      done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer30)
  The value of \[\frac{\sin 8x+7\sin 6x+18\sin 4x+12\sin 2x}{\sin 7x+6\sin 5x+12\sin 3x}\] equal to?
  A)
  \[2\cos x\]    done clear
  B)
  \[\cos x\] done clear
  C)
  \[2sinx\]             done clear
  D)
  \[sinx\] done clear
  View Solution play_arrow

• question_answer31)
  What is \[\frac{1-\tan {{2}^{0}}\cot {{62}^{0}}}{\tan {{152}^{0}}-\cot {{88}^{0}}}\] equal to?
  A)
  \[\sqrt{3}\]                     done clear
  B)
  \[-\sqrt{3}\] done clear
  C)
  \[\sqrt{2}-1\]                  done clear
  D)
  \[1-\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer32)

  Three expressions are given below:

  \[{{Q}_{1}}=\sin (A+B)+\sin (B+C)+\sin (C+A)\]

  \[{{Q}_{2}}=\cos (A-B)+\cos (B-C)+\cos (C-A)\]

  \[{{Q}_{3}}=\sin A(\cos B+\cos C)+\sin B(\cos C+\cos A)+\]\[\sin C(\cos A+\cos B)\]

  Which one of the following is correct?

  A)
  \[{{Q}_{1}}={{Q}_{2}}\]        done clear
  B)
  \[{{Q}_{2}}={{Q}_{3}}\] done clear
  C)
  \[{{Q}_{1}}={{Q}_{3}}\] done clear
  D)
  All the expressions are different done clear
  View Solution play_arrow

• question_answer33)
  If \[A=(\cos 12{}^\circ -\cos 36{}^\circ )(\sin 96{}^\circ +\sin 24{}^\circ )\] and \[B=(\sin 60{}^\circ -\sin 12{}^\circ )(\cos 48{}^\circ -\cos 72{}^\circ ),\] then what is \[\frac{A}{B}\]equal to?
  A)
  -1                     done clear
  B)
  \[0\]     done clear
  C)
  \[1\]                  done clear
  D)
  \[2\] done clear
  View Solution play_arrow

• question_answer34)
  If \[\sin \theta =\frac{12}{13}\left( 0<\theta <\frac{\pi }{2} \right)\] and \[\cos \phi =-\frac{3}{5},\left( \pi <\phi <\frac{3\pi }{2} \right)\] Then \[\sin (\theta +\phi )\]will be
  A)
  \[\frac{-56}{61}\]           done clear
  B)
  \[\frac{-56}{65}\] done clear
  C)
  \[\frac{1}{65}\]              done clear
  D)
  \[-56\] done clear
  View Solution play_arrow

• question_answer35)
  The value of \[{{\sin }^{2}}5{}^\circ +{{\sin }^{2}}10{}^\circ +{{\sin }^{2}}15{}^\circ +{{\sin }^{2}}\]\[20{}^\circ +.....+{{\sin }^{2}}90{}^\circ \] is
  A)
  7                      done clear
  B)
  8     done clear
  C)
  9                      done clear
  D)
  \[\frac{19}{2}\] done clear
  View Solution play_arrow

• question_answer36)
  \[f(x)=(\sin \,{{x}^{7}}).{{e}^{{{x}^{5}}\,sgn {{x}^{9}}}}\] is
  A)
  an even function        done clear
  B)
  an odd function done clear
  C)
  neither even nor odd done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  Period of the function \[\left| {{\sin }^{3}}\frac{x}{2} \right|+\left| {{\cos }^{5}}\frac{x}{5} \right|\]is:
  A)
  \[2\pi \]              done clear
  B)
  \[10\pi \] done clear
  C)
  \[8\pi \]                          done clear
  D)
  \[5\pi \] done clear
  View Solution play_arrow

• question_answer38)
  On simplifying \[\frac{{{\sin }^{3}}A+\sin 3A}{\sin A}+\frac{{{\cos }^{3}}A-\cos 3A}{\cos A},\] we get
  A)
  \[\sin 3A\]           done clear
  B)
  \[\cos 3A\] done clear
  C)
  \[\sin A+\cos A\]    done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer39)
  Which of the following functions has period \[2\pi \]?
  A)
  \[y=\sin \left( 2\pi t+\frac{\pi }{3} \right)+2\sin \left( 3\pi t+\frac{\pi }{4} \right)+3\sin 5\pi t\] done clear
  B)
  \[y=\sin \frac{\pi }{3}t+\sin \frac{\pi }{4}t\] done clear
  C)
  \[y\sin t+\cos 2t\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
   If \[\sin x+\sin y=a\] and \[cos\text{ }x+cos\,y=b,\]then \[{{\tan }^{2}}\left( \frac{x+y}{2} \right)+{{\tan }^{2}}\left( \frac{x-y}{2} \right)\] is equal to            
  A)
  \[\frac{{{a}^{4}}+{{b}^{4}}+4{{b}^{2}}}{{{a}^{2}}{{b}^{2}}+{{b}^{4}}}\] done clear
  B)
  \[\frac{{{a}^{4}}-{{b}^{4}}+4{{b}^{2}}}{{{a}^{2}}{{b}^{2}}+{{b}^{4}}}\] done clear
  C)
  \[\frac{{{a}^{4}}-{{b}^{4}}+4{{a}^{2}}}{{{a}^{2}}{{b}^{2}}+{{a}^{4}}}\] done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer41)
  What is \[{{\sin }^{2}}66\frac{1{}^\circ }{2}-{{\sin }^{2}}23\frac{1{}^\circ }{2}\] equal to?
  A)
  \[sin\text{ }47{}^\circ \]   done clear
  B)
  \[\text{cos }47{}^\circ \] done clear
  C)
  \[\text{2}\,\sin 47{}^\circ \]          done clear
  D)
  \[\text{2}\,\cos 47{}^\circ \] done clear
  View Solution play_arrow

• question_answer42)
  What is \[\frac{\cos 7x-\cos 3x}{\sin 7x-2\sin 5x+\sin 3x}\] equal to?
  A)
  \[\tan x\]             done clear
  B)
  \[\cot x\] done clear
  C)
  \[tan2x\]           done clear
  D)
  \[\cot 2x\] done clear
  View Solution play_arrow

• question_answer43)
  Let \[x+y=3-\cos 4\theta \] and \[x-y=4sin2\theta \] then the greatest of \[xy\]is
  A)
  \[\frac{3}{4}\]               done clear
  B)
  \[1\]     done clear
  C)
  \[\frac{1}{2}\]                           done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer44)
  In a triangle ABC, \[\sin A-\cos B=\cos C,\] then what is B equal to?
  A)
  \[\pi \]                done clear
  B)
  \[\pi /3\]  done clear
  C)
  \[\pi /2\]             done clear
  D)
  \[\pi /4\] done clear
  View Solution play_arrow

• question_answer45)
  If \[\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b},\] then what is \[\frac{\tan x}{\tan y}\] equal to?
  A)
  \[\frac{b}{a}\]               done clear
  B)
  \[\frac{a}{b}\]     done clear
  C)
  \[ab\]                 done clear
  D)
  \[1\] done clear
  View Solution play_arrow

• question_answer46)
  If \[\sin A\,(60{}^\circ -A)\,\sin (60{}^\circ +A)=k\sin 3A,\] then what is k equal to?
  A)
  \[1/4\]   done clear
  B)
  \[1/2\]   done clear
  C)
  \[1\]                  done clear
  D)
  \[4\] done clear
  View Solution play_arrow

• question_answer47)
  The line \[y=\sqrt{3}\]meets the graph \[y=tan\text{ }x,\]where \[x\in \left( 0,\frac{\pi }{2} \right),\] in k points. What is k equal to?
  A)
  One                  done clear
  B)
  Two done clear
  C)
  Three                done clear
  D)
  Infinity done clear
  View Solution play_arrow

• question_answer48)
  If \[\sin (\pi \cos x)=cos(\pi sinx),\] then what is one of the values of\[sin\text{ }2x\]?
  A)
  \[-\frac{1}{4}\] done clear
  B)
  \[-\frac{1}{2}\] done clear
  C)
  \[-\frac{3}{4}\]              done clear
  D)
  \[-1\] done clear
  View Solution play_arrow

• question_answer49)
  The number of solution of \[\tan x+\sec x=2\cos x\] in \[(0,2\pi )\]is
  A)
  2                      done clear
  B)
  3    done clear
  C)
  0          done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer50)
  If \[\cos \theta +\cos 2\theta +\cos 3\theta =0,\] then the general value of \[\theta \] is :
  A)
  \[\theta =2m\pi \pm 2\pi /3\] done clear
  B)
  \[\theta =2m\pi \pm \pi /4\] done clear
  C)
  \[\theta =m\pi +{{(-1)}^{n}}2\pi /3\] done clear
  D)
  \[\theta =m\pi +{{(-1)}^{n}}\pi /3\] done clear
  View Solution play_arrow

• question_answer51)
  Which one of the following is one of the solutions of the equation of the equation\[\tan 2\theta .\tan \theta =1\]?
  A)
  \[\pi /12\]           done clear
  B)
  \[\pi /6\] done clear
  C)
   \[\pi /4\]            done clear
  D)
  \[\pi /3\] done clear
  View Solution play_arrow

• question_answer52)
  General solution of the equation \[(\sqrt{3}-1)\sin \theta +(\sqrt{3}+1)\cos \theta =2\] is
  A)
  \[2n\pi \pm \frac{\pi }{4}+\frac{\pi }{12}\]    done clear
  B)
  \[n\pi +{{(-1)}^{n}}\frac{\pi }{2}\] done clear
  C)
  \[2n\pi \pm \frac{\pi }{4}-\frac{\pi }{12}\]  done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer53)
  For what values of x is the equation \[2\,sin\,\theta =x+\frac{1}{x}\]valid?
  A)
  \[x=\pm 1\]          done clear
  B)
  All real values of x done clear
  C)
  \[-1<x<1\]        done clear
  D)
  \[x>1\]and \[x<-1\] done clear
  View Solution play_arrow

• question_answer54)
  The solution set of the system of equation \[x+y=2\pi /3,\] \[\cos x+\cos y=3/2,\] where x and y are real, is
  A)
  \[x=\frac{\pi }{3}-n\pi ,y=n\pi \] done clear
  B)
  \[\phi \] done clear
  C)
  \[x=n\pi ,y=\frac{\pi }{3}-n\pi \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  The equation \[{{\sin }^{4}}x-(k+2){{\sin }^{2}}x-(k+3)=0\]possesses a solution if
  A)
  \[k>-3\] done clear
  B)
  \[k<-2\] done clear
  C)
  \[-3\le k\le -2\] done clear
  D)
  k is any positive integer done clear
  View Solution play_arrow

• question_answer56)
  Let n be a fixed positive integer such that \[\sin \left( \frac{\pi }{2n} \right)+\cos \left( \frac{\pi }{2n} \right)=\frac{\sqrt{n}}{2},\] then:
  A)
  \[n=4\]     done clear
  B)
  \[n=5\] done clear
  C)
  \[n=6\]              done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer57)
  \[2{{\sin }^{2}}x+{{\sin }^{2}}2x=2,\] \[-\pi <x<\pi ,\] then; \[x=\]
  A)
  \[\pm \frac{\pi }{6}\]               done clear
  B)
  \[\pm \frac{\pi }{4}\] done clear
  C)
  \[\pm \frac{3\pi }{2}\]                done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer58)
  The number of solutions of the equation \[\sin \left( \frac{\pi x}{2\sqrt{3}} \right)={{x}^{2}}-2\sqrt{3}\,\,x+4\]
  A)
  forms an empty set done clear
  B)
  is only one done clear
  C)
  is only two      done clear
  D)
  is more than 2 done clear
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• question_answer59)
  If \[0\le x\le 2\pi ,\] then number of roots of equation \[{{e}^{\sin x}}-{{e}^{-\sin x}}=4\]is                                
  A)
  0                      done clear
  B)
  1     done clear
  C)
  2                      done clear
  D)
  4 done clear
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• question_answer60)
  The least positive non-integral solution of the equation \[\sin \pi ({{x}^{2}}+x)=\sin \pi {{x}^{2}}\] is
  A)
  rational done clear
  B)
  irrational of the form \[\sqrt{p}\] done clear
  C)
  irrational of the form \[\frac{\sqrt{p}-1}{4},\] where p is an odd integer done clear
  D)
  irrational of the form \[\frac{\sqrt{p}+1}{4},\] where p is an even integer done clear
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• question_answer61)
  \[{{\left| \cos x \right|}^{{{\sin }^{2}}x-\frac{3}{2}\sin x+\frac{1}{2}}}=1,\] then possible values of x :
  A)
  \[n\,\pi \] or \[2n\,\pi +\frac{\pi }{2}\] done clear
  B)
  \[n\,\pi \]or \[2n\,\,\pi +\frac{\pi }{2}\] or \[n\,\,\pi +{{(-1)}^{n}}\frac{\pi }{6},\] \[n\in I\] done clear
  C)
  \[n\,\,\pi +{{(-1)}^{n}}\frac{\pi }{6},\,n\in I\] done clear
  D)
  None of these done clear
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• question_answer62)
  The equation \[2{{\cos }^{2}}\left( \frac{x}{2} \right).{{\sin }^{2}}x={{x}^{2}}+\frac{1}{{{x}^{2}}},\]\[0\le x\le \frac{\pi }{2}\] has
  A)
  one real solution done clear
  B)
  no solution done clear
  C)
  more than one real solution done clear
  D)
  None of these done clear
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• question_answer63)
  The number of solutions of the equation \[\cos (\pi \sqrt{x-4})\,\cos (\pi \sqrt{x})=1\] is
  A)
  \[>2\]   done clear
  B)
  2     done clear
  C)
  1          done clear
  D)
  0 done clear
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• question_answer64)
  The least difference between the roots, in the first quadrant \[\left( 0\le x\le \frac{\pi }{2} \right),\] of the equation \[4\cos x(2-3{{\sin }^{2}}x)+(\cos 2x+1)=0\]is
  A)
  \[\frac{\pi }{6}\]             done clear
  B)
  \[\frac{\pi }{4}\]    done clear
  C)
  \[\frac{\pi }{3}\]             done clear
  D)
  \[\frac{\pi }{2}\] done clear
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• question_answer65)
  General solution of the equation \[2{{\cot }^{2}}\theta +2\sqrt{3}\cot \theta +4\operatorname{cosec}+8=0\] is
  A)
  \[\theta =n\pi \pm \frac{\pi }{6},n\in I\] done clear
  B)
  \[n\pi +\frac{\pi }{6},n\in I\] done clear
  C)
  \[2n\pi +\frac{\pi }{6},n\in I\] done clear
  D)
  \[2n\pi +\frac{11\pi }{6},n\in I\] done clear
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• question_answer66)
  The general solution of the equation \[{{\sin }^{50}}x-{{\cos }^{50}}x=1\]is
  A)
  \[2n\pi +\frac{\pi }{2}\]         done clear
  B)
  \[2n\pi +\frac{\pi }{3}\] done clear
  C)
  \[n\pi +\frac{\pi }{2}\]          done clear
  D)
  \[n\pi +\frac{\pi }{3}\] done clear
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• question_answer67)
  If \[\cos 7\theta =\cos \theta -\sin 4\theta ,\] then the general value of \[\theta \] is
  A)
  \[\frac{n\pi }{6},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18}\] done clear
  B)
  \[\frac{n\pi }{3},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18}\] done clear
  C)
  \[\frac{n\pi }{4},\frac{n\pi }{3}\pm \frac{\pi }{18}\]         done clear
  D)
  \[\frac{n\pi }{4},\frac{n\pi }{3}+{{(-1)}^{n}}\frac{\pi }{18}\] done clear
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• question_answer68)
  Number of values of x which lie in \[[0,2\pi ]\] and satisfy the equation \[\left( \cos \frac{x}{4}-2\sin x \right)\sin x+\left( 1+\sin \frac{x}{4}-2\cos x \right)\cos x=0\]
  A)
  1                      done clear
  B)
  2    done clear
  C)
  3                      done clear
  D)
  4 done clear
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• question_answer69)
  The number of solutions of the equation \[{{\sin }^{5}}x-{{\cos }^{5}}x=\frac{1}{\cos x}-\frac{1}{\operatorname{sinx}}(\sin x\ne \cos x)\]is
  A)
  0                      done clear
  B)
  1 done clear
  C)
  infinite               done clear
  D)
  None of these done clear
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• question_answer70)
  Domain of the function \[f(x)=\sqrt{\frac{1}{\sin x}-1},\] is
  A)
  \[\underset{n\in I}{\mathop{\bigcup }}\,\left( 2n\pi ,2n\pi +\frac{\pi }{2} \right)\]      done clear
  B)
  \[\underset{n\in I}{\mathop{\bigcup }}\,[2n\pi ,\,(2n+1)\pi ]\] done clear
  C)
  \[\underset{n\in I}{\mathop{\bigcup }}\,[(2n-1)\pi ,2n\pi ]\] done clear
  D)
  None of these done clear
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