-
question_answer1)
The circular wire of diameter 10cm is cut and placed along the circumference of a circle of diameter 1 metre. The angle subtended by the wire at the centre of the circle is equal to [MNR 1974]
A)
\[\frac{\pi }{4}radian\] done
clear
B)
\[\frac{\pi }{3}radian\] done
clear
C)
\[\frac{\pi }{5}radian\] done
clear
D)
\[\frac{\pi }{10}radian\] done
clear
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question_answer2)
The value of \[{{\sin }^{2}}{{5}^{o}}+{{\sin }^{2}}{{10}^{o}}+{{\sin }^{2}}{{15}^{o}}+...+\] \[{{\sin }^{2}}{{85}^{o}}+{{\sin }^{2}}{{90}^{o}}\] is equal to [Karnataka CET 1999]
A)
7 done
clear
B)
8 done
clear
C)
9 done
clear
D)
\[9\frac{1}{2}\] done
clear
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question_answer3)
If \[\frac{3\pi }{4}<\alpha <\pi ,\]then \[\sqrt{\text{cose}{{\text{c}}^{2}}\alpha +2\cot \alpha }\] is equal to [Pb. CET 2000; AMU 2001; MP PET 2004]
A)
\[1+\cot \alpha \] done
clear
B)
\[1-\cot \alpha \] done
clear
C)
\[-1-\cot \alpha \] done
clear
D)
\[-1+\cot \alpha \] done
clear
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question_answer4)
If \[a\,{{\cos }^{3}}\alpha +3a\,\cos \alpha \,{{\sin }^{2}}\alpha =m\] and\[a\,{{\sin }^{3}}\alpha +3a\,{{\cos }^{2}}\alpha \sin \alpha =n,\] then \[{{(m+n)}^{2/3}}+{{(m-n)}^{2/3}}\] is equal to
A)
\[2{{a}^{2}}\] done
clear
B)
\[2{{a}^{1/3}}\] done
clear
C)
\[2{{a}^{2/3}}\] done
clear
D)
\[2{{a}^{3}}\] done
clear
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question_answer5)
If\[\cos \,(\theta -\alpha )=a,\,\,\sin \,(\theta -\beta )=b,\,\,\]then \[{{\cos }^{2}}(\alpha -\beta )+2ab\,\sin \,(\alpha -\beta )\] is equal to
A)
\[4{{a}^{2}}{{b}^{2}}\] done
clear
B)
\[{{a}^{2}}-{{b}^{2}}\] done
clear
C)
\[{{a}^{2}}+{{b}^{2}}\] done
clear
D)
\[-{{a}^{2}}{{b}^{2}}\] done
clear
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question_answer6)
If \[\sin A=n\sin B,\] then \[\frac{n-1}{n+1}\tan \,\frac{A+B}{2}=\]
A)
\[\sin \frac{A-B}{2}\] done
clear
B)
\[\tan \frac{A-B}{2}\] done
clear
C)
\[\cot \frac{A-B}{2}\] done
clear
D)
None of these done
clear
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question_answer7)
If \[x+\frac{1}{x}=2\,\cos \theta ,\] then \[{{x}^{3}}+\frac{1}{{{x}^{3}}}=\] [MP PET 2004]
A)
\[\cos \,\,3\theta \] done
clear
B)
\[2\,\cos \,3\theta \] done
clear
C)
\[\frac{1}{2}\cos \,3\theta \] done
clear
D)
\[\frac{1}{3}\cos \,3\theta \] done
clear
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question_answer8)
If \[\sin x+\text{cosec}\,x=2,\] then \[4\sin A\,\,\sin B\,\,\sin C\] is equal to [UPSEAT 2002]
A)
2 done
clear
B)
\[{{2}^{n}}\] done
clear
C)
\[{{2}^{n-1}}\] done
clear
D)
\[{{2}^{n-2}}\] done
clear
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question_answer9)
If \[\tan \theta =\frac{\sin \alpha -\cos \alpha }{\sin \alpha +\cos \alpha },\]then \[\sin \alpha +\cos \alpha \] and \[\sin \alpha -\cos \alpha \] must be equal to [WB JEE 1971]
A)
\[\sqrt{2}\cos \theta ,\,\,\sqrt{2}\sin \theta \] done
clear
B)
\[\sqrt{2}\sin \theta ,\,\,\sqrt{2}\cos \theta \] done
clear
C)
\[\sqrt{2}\sin \theta ,\,\,\sqrt{2}\sin \theta \] done
clear
D)
\[\sqrt{2}\,\cos \theta ,\,\,\sqrt{2}\,\cos \theta \] done
clear
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question_answer10)
If \[{{\cos }^{6}}\alpha +{{\sin }^{6}}\alpha +K\,{{\sin }^{2}}2\alpha =1,\] then K =
A)
\[\frac{4}{3}\] done
clear
B)
\[\frac{3}{4}\] done
clear
C)
\[\frac{1}{2}\] done
clear
D)
2 done
clear
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question_answer11)
\[\sin {{20}^{o}}\,\sin {{40}^{o}}\,\sin {{60}^{o}}\,\sin {{80}^{o}}=\] [MNR 1976, 81]
A)
\[-3/16\] done
clear
B)
\[5/16\] done
clear
C)
\[3/16\] done
clear
D)
\[-5/16\] done
clear
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question_answer12)
The value of\[\sin \frac{\pi }{14}\sin \frac{3\pi }{14}\sin \frac{5\pi }{14}\sin \frac{7\pi }{14}\sin \frac{9\pi }{14}\sin \frac{11\pi }{14}\sin \frac{13\pi }{14}\] is equal to [IIT 1991; MNR 1992]
A)
\[\frac{1}{8}\] done
clear
B)
\[\frac{1}{16}\] done
clear
C)
\[\frac{1}{32}\] done
clear
D)
\[\frac{1}{64}\] done
clear
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question_answer13)
\[\tan \alpha +2\tan 2\alpha +4\tan 4\alpha +8\cot \,8\alpha =\] [IIT 1988; MP PET 1991]
A)
\[\tan \alpha \] done
clear
B)
\[\tan 2\alpha \] done
clear
C)
\[\cot \,\alpha \] done
clear
D)
\[\cot \,2\alpha \] done
clear
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question_answer14)
\[\sqrt{3}\,\text{cosec}\,{{20}^{o}}-\sec \,{{20}^{o}}=\] [IIT 1988]
A)
2 done
clear
B)
\[\frac{2\,\sin {{20}^{o}}}{\sin {{40}^{o}}}\] done
clear
C)
4 done
clear
D)
\[\frac{4\,\sin {{20}^{o}}}{\sin {{40}^{o}}}\] done
clear
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question_answer15)
\[1+\cos \,{{56}^{o}}+\cos \,{{58}^{o}}-\cos {{66}^{o}}=\] [IIT 1964]
A)
\[2\,\cos {{28}^{o}}\,\cos \,{{29}^{o}}\,\cos \,{{33}^{o}}\] done
clear
B)
\[4\,\cos {{28}^{o}}\,\cos \,{{29}^{o}}\,\cos \,{{33}^{o}}\] done
clear
C)
\[4\,\cos {{28}^{o}}\,\cos \,{{29}^{o}}\,\sin {{33}^{o}}\] done
clear
D)
\[2\,\cos {{28}^{o}}\,\cos \,{{29}^{o}}\,\sin \,{{33}^{o}}\] done
clear
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question_answer16)
If \[x=\sin {{130}^{o}}\,\cos {{80}^{o}},\,\,y=\sin \,{{80}^{o}}\,\cos \,{{130}^{o}},\,\,z=1+xy,\]which one of the following is true [AMU 1999]
A)
\[x>0,\,\,y>0,\,\,z>0\] done
clear
B)
\[x>0,\,\,y<0,\,\,0<z<1\] done
clear
C)
\[x>0,\,\,y<0,\,\,z>1\] done
clear
D)
\[x<0,\,\,y<0,\,0<z<1\] done
clear
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question_answer17)
If \[\alpha ,\,\beta ,\,\gamma \in \,\left( 0,\,\frac{\pi }{2} \right)\], then \[\frac{\sin \,(\alpha +\beta +\gamma )}{\sin \alpha +\sin \beta +\sin \gamma }\] is
A)
< 1 done
clear
B)
>1 done
clear
C)
= 1 done
clear
D)
None of these done
clear
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question_answer18)
If \[a\,\cos 2\theta +b\,\sin 2\theta =c\]has a and b as its solution, then the value of \[\tan \alpha +\tan \beta \] is [Kurukshetra CEE 1998]
A)
\[\frac{c+a}{2b}\] done
clear
B)
\[\frac{2b}{c+a}\] done
clear
C)
\[\frac{c-a}{2b}\] done
clear
D)
\[\frac{b}{c+a}\] done
clear
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question_answer19)
If \[\tan x=\frac{2b}{a-c}(a\ne c),\]\[y=a\,{{\cos }^{2}}x+2b\,\sin x\cos x+c\,{{\sin }^{2}}x\]and \[z=a{{\sin }^{2}}x-2b\sin x\cos x+c{{\cos }^{2}}x,\] then
A)
\[y=z\] done
clear
B)
\[y+z=a+c\] done
clear
C)
\[y-z=a+c\] done
clear
D)
\[y-z={{(a-c)}^{2}}+4{{b}^{2}}\] done
clear
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question_answer20)
If \[\text{cosec}\theta =\frac{p+q}{p-q},\] then \[\cot \,\left( \frac{\pi }{4}+\frac{\theta }{2} \right)=\] [EAMCET 2001]
A)
\[\sqrt{\frac{p}{q}}\] done
clear
B)
\[\sqrt{\frac{q}{p}}\] done
clear
C)
\[\sqrt{pq}\] done
clear
D)
\[pq\] done
clear
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question_answer21)
If \[a{{\sin }^{2}}x+b{{\cos }^{2}}x=c,\,\,\]\[b\,{{\sin }^{2}}y+a\,{{\cos }^{2}}y=d\] and \[a\,\tan x=b\,\tan y,\]then \[\frac{{{a}^{2}}}{{{b}^{2}}}\] is equal to
A)
\[\frac{(b-c)\,\,(d-b)}{(a-d)\,\,(c-a)}\] done
clear
B)
\[\frac{(a-d)\,\,(c-a)}{(b-c)\,\,(d-b)}\] done
clear
C)
\[\frac{(d-a)\,\,(c-a)}{(b-c)\,\,(d-b)}\] done
clear
D)
\[\frac{(b-c)\,\,(b-d)}{(a-c)\,\,(a-d)}\] done
clear
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question_answer22)
\[{{\left( \frac{\cos A+\cos B}{\sin A-\sin B} \right)}^{n}}+{{\left( \frac{\sin A+\sin B}{\cos A-\cos B} \right)}^{n}}\](n even or odd) =
A)
\[2{{\tan }^{n}}\frac{A-B}{2}\] done
clear
B)
\[2{{\cot }^{n}}\frac{A-B}{2}\] done
clear
C)
\[0\] done
clear
D)
None of these done
clear
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question_answer23)
If \[\sin \alpha =1/\sqrt{5}\]and \[\sin \beta =3/5\],then \[\beta -\alpha \]lies in the interval [Roorkee Qualifying 1998]
A)
\[[0,\,\pi /4]\] done
clear
B)
\[[\pi /2,\,3\pi /4]\] done
clear
C)
\[[3\pi /4,\,\pi ]\] done
clear
D)
\[[\pi ,\,5\pi /4]\] done
clear
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question_answer24)
If \[2\sec 2\alpha =\tan \beta +\cot \beta ,\]then one of the values of \[\alpha +\beta \]is [Karnataka CET 2000]
A)
\[\frac{\pi }{4}\] done
clear
B)
\[\frac{\pi }{2}\] done
clear
C)
\[\pi \] done
clear
D)
\[2\pi \] done
clear
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question_answer25)
If \[\frac{x}{\cos \theta }=\frac{y}{\cos \left( \theta -\frac{2\pi }{3} \right)}=\frac{z}{\cos \left( \theta +\frac{2\pi }{3} \right)},\]then \[x+y+z=\]
A)
\[1\] done
clear
B)
\[0\] done
clear
C)
\[-1\] done
clear
D)
None of these done
clear
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question_answer26)
If \[\sin 6\theta =32{{\cos }^{5}}\theta \sin \theta -32{{\cos }^{3}}\theta \sin \theta +3x,\] then \[x=\] [EAMCET 2003]
A)
\[\cos \theta \] done
clear
B)
\[\cos 2\theta \] done
clear
C)
\[\sin \theta \] done
clear
D)
\[\sin 2\theta \] done
clear
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question_answer27)
\[{{\sin }^{4}}\frac{\pi }{4}+{{\sin }^{4}}\frac{3\pi }{8}+{{\sin }^{4}}\frac{5\pi }{8}+{{\sin }^{4}}\frac{7\pi }{8}=\] [Roorkee 1980]
A)
\[\frac{1}{2}\] done
clear
B)
\[\frac{1}{4}\] done
clear
C)
\[\frac{3}{2}\] done
clear
D)
\[\frac{3}{4}\] done
clear
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question_answer28)
\[\left( 1+\cos \frac{\pi }{8} \right)\,\left( 1+\cos \frac{3\pi }{8} \right)\,\left( 1+\cos \frac{5\pi }{8} \right)\,\left( 1+\cos \frac{7\pi }{8} \right)=\] [IIT 1984; WB JEE 1992]
A)
\[\frac{1}{2}\] done
clear
B)
\[\frac{1}{4}\] done
clear
C)
\[\frac{1}{8}\] done
clear
D)
\[\frac{1}{16}\] done
clear
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question_answer29)
If A lies in the third quadrant and \[3\,\tan A-4=0,\] then \[5\,\sin 2A+3\,\sin A+4\,\cos A=\] [EAMCET 1994]
A)
0 done
clear
B)
\[\frac{-24}{5}\] done
clear
C)
\[\frac{24}{5}\] done
clear
D)
\[\frac{48}{5}\] done
clear
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question_answer30)
\[\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}\] is equal to [IIT 1966, 1975]
A)
\[\cot 7\frac{{{1}^{o}}}{2}\] done
clear
B)
\[\sin 7\frac{{{1}^{o}}}{2}\] done
clear
C)
\[\sin \,{{15}^{o}}\] done
clear
D)
\[\cos \,\,{{15}^{o}}\] done
clear
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question_answer31)
If \[\tan \,(A+B)=p,\,\,\tan \,(A-B)=q,\] then the value of \[\tan \,2A\] in terms of p and q is [MP PET 1995, 2002]
A)
\[\frac{p+q}{p-q}\] done
clear
B)
\[\frac{p-q}{1+pq}\] done
clear
C)
\[\frac{p+q}{1-pq}\] done
clear
D)
\[\frac{1+pq}{1-p}\] done
clear
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question_answer32)
\[2\,{{\sin }^{2}}\beta +4\,\,\cos \,(\alpha +\beta )\,\,\sin \,\alpha \,\sin \,\beta +\cos \,2\,(\alpha +\beta )=\] [MNR 1993; IIT 1977]
A)
\[\sin \,\,2\alpha \] done
clear
B)
\[\cos \,\,2\beta \] done
clear
C)
\[\cos \,\,2\alpha \] done
clear
D)
\[\sin \,\,2\beta \] done
clear
View Solution play_arrow
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question_answer33)
The value of \[\sin \theta +\cos \theta \] will be greatest when [MNR 1977, 1983; RPET 1995]
A)
\[\theta ={{30}^{o}}\] done
clear
B)
\[\theta ={{45}^{o}}\] done
clear
C)
\[\theta ={{60}^{o}}\] done
clear
D)
\[\theta ={{90}^{o}}\] done
clear
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question_answer34)
If \[f(x)={{\cos }^{2}}x+{{\sec }^{2}}x,\] then [MNR 1986]
A)
\[f(x)<1\] done
clear
B)
\[f(x)=1\] done
clear
C)
\[1<f(x)<2\] done
clear
D)
\[f(x)\ge 2\] done
clear
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question_answer35)
The value of \[\frac{\tan x}{\tan \,3x}\]whenever defined never lie between [Kurukshetra CEE 1998; IIT 1992]
A)
1/3 and 3 done
clear
B)
1/4 and 4 done
clear
C)
1/5 and 5 done
clear
D)
5 and 6 done
clear
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question_answer36)
\[\cos \,\,2\theta +2\,\,\cos \theta \] is always
A)
Greater than \[-\frac{3}{2}\] done
clear
B)
Less than or equal to \[\frac{3}{2}\] done
clear
C)
Greater than or equal to \[-\frac{3}{2}\] and less than or equal to 3 done
clear
D)
None of these done
clear
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question_answer37)
Let A, B and C are the angles of a plain triangle and \[\tan \frac{A}{2}=\frac{1}{3},\,\,\tan \frac{B}{2}=\frac{2}{3}\]. Then \[\tan \frac{C}{2}\] is equal to [Orissa JEE 2003]
A)
7/9 done
clear
B)
2/9 done
clear
C)
1/3 done
clear
D)
2/3 done
clear
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question_answer38)
If \[A+B+C=\pi \] and \[\cos A=\cos B\,\cos C,\] then \[\tan B\,\,\tan C\] is equal to [AMU 2001]
A)
\[\frac{1}{2}\] done
clear
B)
2 done
clear
C)
1 done
clear
D)
\[-\frac{1}{2}\] done
clear
View Solution play_arrow
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question_answer39)
If \[A+C=B,\] then \[\tan A\,\tan B\,\tan C=\] [EAMCET 1986]
A)
\[\tan A\,\tan B+\tan \,C\] done
clear
B)
\[\tan \,B-\tan \,C-\tan \,A\] done
clear
C)
\[\tan A+\tan C-\tan B\] done
clear
D)
\[-\,(\tan A\tan B+\tan C)\] done
clear
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question_answer40)
If \[\left| \cos \,\theta \,\left\{ \sin \theta +\sqrt{{{\sin }^{2}}\theta +{{\sin }^{2}}\alpha } \right\}\, \right|\,\le k,\] then the value of k is
A)
\[\sqrt{1+{{\cos }^{2}}\alpha }\] done
clear
B)
\[\sqrt{1+{{\sin }^{2}}\alpha }\] done
clear
C)
\[\sqrt{2+{{\sin }^{2}}\alpha }\] done
clear
D)
\[\sqrt{2+{{\cos }^{2}}\alpha }\] done
clear
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