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question_answer1)
The sum of the solutions of the equation\[2{{\sin }^{-1}}\sqrt{{{x}^{2}}+x+1}+{{\cos }^{-1}}\sqrt{{{x}^{2}}+x}=\frac{3\pi }{2}\]is
A)
0 done
clear
B)
-1 done
clear
C)
1 done
clear
D)
2 done
clear
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question_answer2)
If \[\alpha \in \left( -\frac{3\pi }{2},-\pi \right)\]then the value of \[{{\tan }^{-1}}(cot\alpha )\]-\[{{\cot }^{-1}}(tan\alpha )+si{{n}^{-1}}(sin\alpha )+co{{s}^{-1}}(cos\alpha )\]is equal to
A)
\[2\pi +a\] done
clear
B)
\[\pi +a\] done
clear
C)
0 done
clear
D)
\[\pi -a\] done
clear
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question_answer3)
The value of the expression \[{{\sin }^{-1}}\left( \sin \frac{22\pi }{7} \right)\]\[{{\cos }^{-1}}\left( \cos \frac{5\pi }{3} \right)\]+\[{{\tan }^{-1}}\left( \tan \frac{5\pi }{3} \right)\]+\[{{\sin }^{-1}}(cos2)\]is
A)
\[\frac{17\pi }{42}-2\] done
clear
B)
\[-\,2\] done
clear
C)
\[\frac{-\pi }{21}-2\] done
clear
D)
none of these done
clear
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question_answer4)
There exists a positive real number x satisfying \[\cos (ta{{n}^{-1}}x)=x\], Then the value of \[{{\cos }^{-1}}\left( \frac{{{x}^{2}}}{2} \right)\]is
A)
\[\frac{\pi }{10}\] done
clear
B)
\[\frac{\pi }{5}\] done
clear
C)
\[\frac{2\pi }{5}\] done
clear
D)
\[\frac{4\pi }{5}\] done
clear
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question_answer5)
If\[a{{\sin }^{-1}}x-b{{\cos }^{-1}}x=c,\]then\[a{{\sin }^{-1}}x+b{{\cos }^{-1}}x\]is equal to
A)
0 done
clear
B)
\[\frac{\pi ab+c(b-c)}{a+b}\] done
clear
C)
\[\frac{\pi }{2}\] done
clear
D)
\[\frac{\pi ab+c(a-b)}{a+b}\] done
clear
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question_answer6)
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)
\[\frac{1}{2}\] done
clear
D)
none of these done
clear
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question_answer7)
If \[\left| {{\sin }^{-1}}x \right|+\left| {{\cos }^{-1}}x \right|=\frac{\pi }{2}\], then x\[\in \]
A)
R done
clear
B)
\[\left[ -1,1 \right]\] done
clear
C)
\[\left[ 0,1 \right]\] done
clear
D)
\[\phi \] done
clear
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question_answer8)
The number of integer x satisfying \[{{\sin }^{-1}}\left| x-2 \right|+{{\cos }^{-1}}(1-\left| 3-x \right|)=\frac{\pi }{2}\] is
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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question_answer9)
\[f(x)=ta{{n}^{-1}}x+{{\tan }^{-1}}\left( \frac{1}{x} \right);g(x)=si{{n}^{-1}}x+co{{x}^{-1}}x\]are identical functions if
A)
\[x\in R\] done
clear
B)
\[x>0\] done
clear
C)
\[x\in [-1,1]\] done
clear
D)
\[x\in (0,1]\] done
clear
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question_answer10)
If \[{{\tan }^{-1}}(si{{n}^{2}}\theta -2sin\theta +3)+co{{t}^{-1}}({{5}^{{{\sec }^{2}}y}}+1)=\frac{\pi }{2}\], then the value of \[{{\cos }^{2}}\theta -\sin \theta \]is equal to
A)
0 done
clear
B)
-1 done
clear
C)
1 done
clear
D)
none of these done
clear
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question_answer11)
If \[{{\cot }^{-1}}(\sqrt{\cos \alpha )}-{{\tan }^{-1}}(\sqrt{\cos \alpha })=x\], then sin x is
A)
\[{{\tan }^{2}}\frac{\alpha }{2}\] done
clear
B)
\[{{\cot }^{2}}\frac{\alpha }{2}\] done
clear
C)
\[\tan \alpha \] done
clear
D)
\[\cot \frac{\alpha }{2}\] done
clear
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question_answer12)
If \[3{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)-4{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)+2{{\tan }^{-1}}\left( \frac{2x}{1-{{x}^{2}}} \right)=\frac{\pi }{3}\]where \[\left| x \right|<1,\]then x is equal to
A)
\[\frac{1}{\sqrt{3}}\] done
clear
B)
\[-\frac{1}{\sqrt{3}}\] done
clear
C)
\[\sqrt{3}\] done
clear
D)
\[-\frac{\sqrt{3}}{4}\] done
clear
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question_answer13)
The value \[2{{\tan }^{-1}}\left[ \sqrt{\frac{a-b}{a+b}}\tan \frac{\theta }{2} \right]\]is equal to
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)
\[{{\cos }^{-1}}\left( \frac{b\cos \theta }{a\cos \theta +b} \right)\] done
clear
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question_answer14)
If \[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha ,\] then \[4{{x}^{2}}-4xy\,\cos \,\alpha +{{y}^{2}}\] is equal to
A)
4 done
clear
B)
\[2{{\sin }^{2}}\alpha \] done
clear
C)
\[-4{{\sin }^{2}}\alpha \] done
clear
D)
\[4{{\sin }^{2}}\alpha \] done
clear
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question_answer15)
The value of \[{{\sin }^{-1}}[cos\{co{{s}^{-1}}(cosx)+si{{n}^{-1}}(sinx)\}]\]where \[x\in \left( \frac{\pi }{2},\pi \right)\] is equal to
A)
\[\frac{\pi }{2}\] done
clear
B)
\[-\pi \] done
clear
C)
\[\pi \] done
clear
D)
\[-\frac{\pi }{2}\] done
clear
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question_answer16)
The number of real solutions of the equation\[\sqrt{1+\cos 2x}=\sqrt{2}{{\sin }^{-1}}(sinx),-\pi \le x\le \pi \] is
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
infinite done
clear
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question_answer17)
\[{{\cot }^{-1}}(\sqrt{\cos \alpha })-ta{{n}^{-1}}(\sqrt{\cos \alpha })=x\], then sin x is equal to
A)
\[{{\tan }^{2}}\frac{\alpha }{2}\] done
clear
B)
\[{{\cot }^{2}}\frac{\alpha }{2}\] done
clear
C)
\[\tan \alpha \] done
clear
D)
\[\cot \frac{\alpha }{2}\] done
clear
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question_answer18)
If \[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha \] then \[4{{x}^{2}}-2xy\,\,\cos \alpha +{{y}^{2}}\] is equal to
A)
2 sin \[\alpha \] done
clear
B)
4 done
clear
C)
\[4{{\sin }^{2}}\alpha \] done
clear
D)
\[-\,4{{\sin }^{2}}\alpha \] done
clear
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question_answer19)
The trigonometric equation \[{{\sin }^{-1}}x=2{{\sin }^{-1}}a\] has a solution for
A)
\[\frac{1}{2}<\left| a \right|<\frac{1}{\sqrt{2}}\] done
clear
B)
All real values of a done
clear
C)
\[\left| a \right|<1/2\] done
clear
D)
\[\left| a \right|\ge \frac{1}{\sqrt{2}}\] done
clear
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question_answer20)
The function \[f(x)={{\tan }^{-1}}(\sin x+\cos x)\] is an increasing function in
A)
\[\left( \frac{\pi }{4},\frac{\pi }{2} \right)\] done
clear
B)
\[\left( -\frac{\pi }{2},\frac{\pi }{4} \right)\] done
clear
C)
\[\left( 0,\frac{\pi }{2} \right)\] done
clear
D)
\[\left( -\frac{\pi }{2},\frac{\pi }{2} \right)\] done
clear
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question_answer21)
If \[{{\sin }^{-1}}\left( \frac{5}{x} \right)+{{\sin }^{-1}}\left( \frac{12}{x} \right)=\frac{\pi }{2}\], then x is equal to _______.
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question_answer22)
The value of such that \[{{\sin }^{-1}}\frac{2}{\sqrt{5},}{{\sin }^{-1}}\frac{3}{\sqrt{10}},{{\sin }^{-1}}\alpha \] are the angles of a triangle is _______.
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question_answer23)
The sum of roots of the equation\[{{\tan }^{-1}}\frac{1}{1+2x}+{{\tan }^{-1}}\frac{1}{1+4x}={{\tan }^{-1}}\frac{2}{{{x}^{2}}}\] is ______.
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question_answer24)
If \[y={{\tan }^{-1}}\frac{1}{2}+{{\tan }^{-1}}b,(0<b<1)\] and \[0<y\le \frac{\pi }{4}\], then the maximum value of b is ______.
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question_answer25)
If\[{{\sin }^{-1}}\frac{x}{5}+{{\operatorname{cosec}}^{-1}}\frac{5}{4}=\frac{\pi }{2}\], then a value of x is _______.
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