Mathematics - Trigonometry
Exam Duration: 45 Mins Total Questions : 30
If sin A = sin B and cos A = cos B, then
- (a)
\(sin{1\over2}(A+B)=0\)
- (b)
\(sin{1\over2}(A-B)=0\)
- (c)
\(cos{1\over2}(A+B)=0\)
- (d)
\(cos{1\over2}(A-B)=0\)
The value of the expression \(1-\frac { { \sin { } }^{ 2 }y }{ 1+\cos { \quad y } } +\frac { 1+\cos { \quad y } }{ sin\quad y } -\frac { sin\quad y }{ 1-\cos { \quad y } } \),is
- (a)
0
- (b)
1
- (c)
sin y
- (d)
cos y
The period of the function 4cos3 \(\theta\)-3cos \(\theta\) is
- (a)
\(\pi\over3\)
- (b)
\(\pi\over4\)
- (c)
\(2\pi\over3\)
- (d)
None of these
In a \( \triangle \) ABC ,\({1\over a+c }+{1\over b+c }={3\over a+b+c }\) ,then angle equal to
- (a)
30°
- (b)
60°
- (c)
90°
- (d)
None of these
If \(r_1,r_2,r_3,\) are in A.P.then (a-b)(s-c) equals
- (a)
(b-c)(s-a)
- (b)
(c-a)(s-b)
- (c)
(a-b)(s-c)
- (d)
None of these
The expression \(3\left[ \sin ^{ 4 }{ \left( \frac { 3\pi }{ 2 } -\alpha \right) } +{ sin }^{ 4 }(3\pi +\alpha ) \right] -2\left[ \sin ^{ 6 }{ \left( \frac { \pi }{ 2 } +\alpha \right) } +{ sin }^{ 6 }(5\pi -\alpha ) \right] \) is equal to
- (a)
0
- (b)
1
- (c)
3
- (d)
sin 4\(\alpha\) +cos 6x
If cos-1 \(({x\over 2})\) +cos-1 \(({y\over 3})\) =\(\theta\),then 9x2+12xy cos\(\theta\)+4y2 , equals
- (a)
sin2\(\theta\)
- (b)
9 sin2\(\theta\)
- (c)
36 sin2\(\theta\)
- (d)
36 cos2\(\theta\)
The value of tan 9° - tan 27°- tan 63° + tan 81° is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If \({ U }_{ n }=\sin { n\theta } \sec ^{ n }{ \theta } ,{ V }_{ n }=\cos { n\theta } \sec ^{ n }{ \theta } \neq 1,\) then \(\frac { { V }_{ n }-{ V }_{ n-1 } }{ { U }_{ n-1 } } +\frac { 1 }{ n } \frac { { U }_{ n } }{ { V }_{ n } } \) is equal to
- (a)
0
- (b)
\(\tan { \theta } \)
- (c)
\(-\tan { \theta } +\frac { \tan { n\theta } }{ n } \)
- (d)
\(\tan { \theta } +\frac { \tan { n\theta } }{ n } \)
In a quadrilateral if
\(\sin { \left( \frac { A+B }{ 2 } \right) } +\cos { \left( \frac { A-B }{ 2 } \right) } +\sin { \left( \frac { C+D }{ 2 } \right) } +\cos { \left( \frac { C-D }{ 2 } \right) } =2\), then \(\sum { \cos { \frac { A }{ 2 } } } \cos { \frac { B }{ 2 } } \) is equal to
- (a)
0
- (b)
6
- (c)
3
- (d)
2
If \(\tan { \alpha } ,\tan { \beta } ,\tan { \gamma } \) are the roots of the equation x3 - px2 - r = 0, then the value of \(\left( 1+\tan ^{ 2 }{ \alpha } \right) \left( 1+\tan ^{ 2 }{ \beta } \right) \left( 1+\tan ^{ 2 }{ \gamma } \right) \)is equal to
- (a)
(p - r)2
- (b)
1 + (p - r)2
- (c)
1 - (p - r)2
- (d)
none of these
The least value of cosec2x + 25 sec2x is
- (a)
0
- (b)
26
- (c)
28
- (d)
36
\(\tan { 7\frac { 1 }{ 2 } ^{ ° } } \) is equal to
- (a)
\(\frac { 2\sqrt { 2 } -\left( 1+\sqrt { 3 } \right) }{ \sqrt { 3 } -1 } \)
- (b)
\(\frac { 1+\sqrt { 3 } }{ 1-\sqrt { 3 } } \)
- (c)
\(\frac { 1 }{ \sqrt { 3 } } +\sqrt { 3 } \)
- (d)
\(2\sqrt { 2 } +\sqrt { 3 } \)
\(\left( 1+\cos { \frac { \pi }{ 8 } } \right) \left( 1+\cos { \frac { 3\pi }{ 8 } } \right) \left( 1+\cos { \frac { 5\pi }{ 8 } } \right) \left( 1+\cos { \frac { 7\pi }{ 8 } } \right) \) is equal to
- (a)
1/2
- (b)
\(\cos { { \pi }/{ 8 } } \)
- (c)
1/8
- (d)
\(\frac { 1+\sqrt { 2 } }{ 2\sqrt { 2 } } \)
If \(x=y\cos { \frac { 2\pi }{ 3 } } =z\cos { \frac { 4\pi }{ 3 } } \) , then xy + yz + zx is equal to
- (a)
- 1
- (b)
0
- (c)
1
- (d)
2
If \(\alpha ,\beta ,\gamma \epsilon \left( 0,\frac { \pi }{ 2 } \right) \), then the value of \(\frac { \sin { \left( \alpha +\beta +\gamma \right) } }{ \sin { \alpha } +\sin { \beta } +\sin { \gamma } } \) is
- (a)
< 1
- (b)
> 1
- (c)
= 1
- (d)
none of these
If \(\frac { x }{ \cos { \alpha } } =\frac { y }{ \cos { \left( \alpha -\frac { 2\pi }{ 3 } \right) } } =\frac { z }{ \cos { \left( \alpha +\frac { 2\pi }{ 3 } \right) } } \), then x + y + z is equal to
- (a)
1
- (b)
0
- (c)
- 1
- (d)
none of these
If \(\cos { \alpha } +\cos { \beta } =\sin { \alpha } +\sin { \beta } \), then \(\cos { 2\alpha } +\cos { 2\beta } \) is equal to
- (a)
\(-2\sin { \left( \alpha +\beta \right) } \)
- (b)
\(-2\cos { \left( \alpha +\beta \right) } \)
- (c)
\(2\sin { \left( \alpha +\beta \right) } \)
- (d)
\(2\cos{ \left( \alpha +\beta \right) } \)
If \(\sin { \alpha } =-{ 3 }/{ 5 }\) and lies in the third quadrant, then the value of \(\cos { { \alpha }/{ 2 } } \) is
- (a)
1/5
- (b)
\({ -1 }/{ \sqrt { 10 } }\)
- (c)
-1/5
- (d)
\({ 1 }/{ \sqrt { 10 } }\)
If \(\cos { x } +\sin { x } =a\left( -\frac { \pi }{ 2 } <x<-\frac { \pi }{ 4 } \right) \), then cos 2x is equal to
- (a)
a2
- (b)
\(a\sqrt { \left( 2-a \right) } \)
- (c)
\(a\sqrt { \left( 2+a \right) } \)
- (d)
\(a\sqrt { \left( 2-{ a }^{ 2 } \right) } \)
If Pn = sin n \(\theta \)+ cos n \(\theta \) where n\(\epsilon\) W (whole number) and (real number)
On the basis of above information, answer the following questions:
If Pn-2-Pn = sin2 \(\theta\) cos2 \(\theta\) P, then the value of is = sin2 cos2 P\(\lambda\), then the value of \(\lambda\) is
- (a)
n-1
- (b)
n-2
- (c)
n-3
- (d)
n-4
If Pn = sin n \(\theta \)+ cos n \(\theta \) where n\(\epsilon\) W (whole number) and (real number)
On the basis of above information, answer the following questions:
The value of \(\frac { { P }_{ 7 }-{ P }_{ 5 } }{ { P }_{ 5 }-{ P }_{ 3 } } \) is
- (a)
\(\frac { { P }_{ 7 } }{ { P }_{ 5 } } \)
- (b)
\(\frac { { P }_{ 5 } }{ { P }_{ 3 } } \)
- (c)
\(\frac { { P }_{ 3 } }{ { P }_{ 1 } } \)
- (d)
\(\frac { { P }_{ 3 } }{ { P }_{ 5 } } \)
If \(\sin { \left( \theta +\alpha \right) } =a\) and \(\sin { \left( \theta +\beta \right) } =b\), then \(\cos { 2\left( \alpha -\beta \right) } -4ab\cos { \left( \alpha -\beta \right) } \)is equal to
- (a)
1 - a2 - b2
- (b)
1 - 2a2 - 2b2
- (c)
2 + a2 - b2
- (d)
2 - a2 - b2
Minimum value of \({ 4x }^{ 2 }-4x\left| \sin { \theta } \right| -\cos ^{ 2 }{ \theta } \) is equal to
- (a)
- 2
- (b)
- 1
- (c)
-1/2
- (d)
0
If \(\left| \tan { A } \right| <1,\) and \(\left| A \right| \) is acute, then \(\frac { \sqrt { \left( 1+\sin { 2A } \right) } +\sqrt { \left( 1-\sin { 2A } \right) } }{ \sqrt { \left( 1+\sin { 2A } \right) } -\sqrt { \left( 1-\sin { 2A } \right) } } \) is equal to
- (a)
tan A
- (b)
- tan A
- (c)
cot A
- (d)
- cot A
If \(3\sin { \beta =\sin { \left( 2\alpha +\beta \right) } } \), then
- (a)
\(\left[ \cot { \alpha } +\cot { \left( \alpha +\beta \right) } \right] \left[ \cot { \beta } -3\cot { \left( 2\alpha +\beta \right) } \right] =6\)
- (b)
\(\sin { \beta } =\cos { \left( \alpha +\beta \right) } \sin { \alpha } \)
- (c)
\(2\sin { \beta } =\sin { \left( \alpha +\beta \right) } \cos { \alpha } \)
- (d)
\(\tan { \left( \alpha +\beta \right) =2\tan { \alpha } } \)
If \(\tan { \theta } =\frac { \sin { \alpha } -\cos { \alpha } }{ \sin { \alpha } +\cos { \alpha } } \), then
- (a)
\(\sin { \alpha } -\cos { \alpha } =\pm \sqrt { 2 } \sin { \theta } \)
- (b)
\(\sin { \alpha } +\cos { \alpha } =\pm \sqrt { 2 } \cos { \theta } \)
- (c)
\(\cos { 2\theta } =\sin { 2\alpha } \)
- (d)
\(\sin { 2\theta } +\cos { 2\alpha } =0\)
If x = a cos3 \(\theta \) sin2 \(\theta \), y =a sin3\(\theta \) cos2 \(\theta \) and \(\frac { \left( { x }^{ 2 }+{ y }^{ 2 } \right) ^{ p } }{ \left( { xy } \right) ^{ q } } \left( p,q\epsilon N \right) \) is independent of \(\theta \), then
- (a)
p = 4
- (b)
p = 5
- (c)
q = 4
- (d)
q = 5
If \(\left( \frac { sin\ \phi }{ sin \phi } \right) ^{ 2 }=\frac { tan\ \theta }{ tan\ \phi } =3\), then
- (a)
\(tan\phi =1/\sqrt { 3 } \)
- (b)
\(tan\phi =-1\sqrt { 3 } \)
- (c)
\(tan\theta =\sqrt { 3 } \)
- (d)
\(tan\theta =-\sqrt { 3 } \)