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 period of the function 4cos3 \(\theta\)-3cos \(\theta\) is
- (a)
\(\pi\over3\)
- (b)
\(\pi\over4\)
- (c)
\(2\pi\over3\)
- (d)
None of these
The value of tan 3x-tan 2x-tan x is
- (a)
tan x tan 2x tan 3x
- (b)
- tan x tan 2x tan 3x
- (c)
tan x tan 2x - tan 2x tan 3x -tan 3x tan x
- (d)
None of these
The value of cot \(({\pi\over4}+\theta)\) cot \(({\pi\over4}-\theta)\) is
- (a)
0
- (b)
1
- (c)
-1
- (d)
cot\(\pi\over8\)
If tan2 \(\theta\) = 2 tan2 \(\phi \) + 1 then cos 2 \(\theta\)+sin2 \(\varphi \) equals
- (a)
-1
- (b)
0
- (c)
1
- (d)
None of these
If \(r_1=r_2+r_3+r\), then the triangles is? \(\)
- (a)
acute angled
- (b)
right angled
- (c)
obtuse angled
- (d)
isosceles
The number of distinct real roots of the euation \(\begin{vmatrix} sinx\quad & cosx & cosx \\ cosx & sinx & cosx \\ cosx & cosx & sinx \end{vmatrix}=0\) ,in \([{-\pi\over 4},{\pi\over 4}]\) is
- (a)
0
- (b)
2
- (c)
1
- (d)
3
The number of integral values of k for which the equation 7cosx+5 sinx=2k+1,has a real solution,are
- (a)
4
- (b)
8
- (c)
10
- (d)
12
Which of the following number(s) is /are rational?
- (a)
sin 15°
- (b)
cos 15°
- (c)
sin 15° cos 15°
- (d)
sin 15° cos 75°
There exists a triangle ABC satisfying the conditons
- (a)
b sin A=a,A<\(\pi\over2\)
- (b)
b sin A\(\pi\over2\) ,b>a
- (c)
b sin A>a,A>\(\pi\over2\)
- (d)
b sin A>a,A<\(\pi\over2\)
If 1+sin x +sin2 x+....\(\infty \) =4+2\(\sqrt3\) ,then x equals
- (a)
\({2\pi\over3 } or {\pi\over3}\)
- (b)
\(7\pi\over6\)
- (c)
\(\pi\over6\)
- (d)
\(\pi\over4\)
In a triangle ABC ,the line joining the circumcentre and the in centre is parallel to BC ,then cos B+ cos C,is equal to
- (a)
\(3\over2\)
- (b)
1
- (c)
\(3\over4\)
- (d)
\(1\over2\)
If \({3\pi\over4}<\theta<\pi\), then \(\sqrt{2cot\theta+{1\over sin^{2}\theta}}\), is equal to
- (a)
1+cot \(\theta\)
- (b)
(1-cot \(\theta\))
- (c)
-(1+cot\(\theta\))
- (d)
- 1+cot\(\theta\)
The number of roots of the equation sin x +sin 5x = sin 3x,in the interval [0,\(\pi\)] is
- (a)
0
- (b)
2
- (c)
6
- (d)
10
Value of sin 12° sin 48° sin 54° is
- (a)
\(1\over8\)
- (b)
\(1\over16\)
- (c)
\(1\over32\)
- (d)
\(1\over64\)
If the mapping f(x) = ax + b, a < 0 maps [-1,1] onto [0,2], then for all values of \(\theta ,A=\cos ^{ 2 }{ \theta } +\sin ^{ 4 }{ \theta } \) is such that
If \(\tan { { \alpha }/{ 2 } } \) and \(\tan { { \beta}/{ 2 } } \) are the roots of the equation 8x2 - 26x + 15 = 0, then \(\cos { \left( \alpha +\beta \right) } \) is equal to
- (a)
\(-\frac{627}{725}\)
- (b)
\(\frac{627}{725}\)
- (c)
- 1
- (d)
none of these
If A lies in the third quadrant and 3 tan A - 4 = 0, then 5 sin 2A + 3 sin A + 4 cos A is equal to
- (a)
0
- (b)
\(-\frac{24}{5}\)
- (c)
\(\frac{24}{5}\)
- (d)
\(\frac{48}{5}\)
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 \(\sin { \alpha } =-{ 3 }/{ 5 }\) 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)
\({ 1 }/{ \sqrt { 10 } }\)
For what and only what values of \(\alpha\) lying between 0 and \(\pi\) is the inequality \(\sin { \alpha } \cos ^{ 3 }{ \alpha } >\sin ^{ 3 }{ \alpha } \cos { \alpha } \) valid?
- (a)
\(\alpha \epsilon \left( 0,{ \pi }/{ 4 } \right) \)
- (b)
\(\alpha \epsilon \left( 0,{ \pi }/{ 2} \right) \)
- (c)
\(\alpha \epsilon \left( \frac { \pi }{ 4 } ,\frac { \pi }{ 2 } \right) \)
- (d)
None of these
Expression \({ 2 }^{ \sin { \theta } }+{ 2 }^{ -\cos { \theta } }\) is minimum when \(\theta =...\) and its minimum value is
- (a)
\(2n\pi +\frac { \pi }{ 4 } ,n\epsilon I,{ 2 }^{ 1+1/\sqrt { 2 } }\)
- (b)
\(2n\pi +\frac { 7\pi }{ 4 } ,n\epsilon I,{ 2 }^{ 1-1/\sqrt { 2 } }\)
- (c)
\(n\pi \pm { \pi }/{ 4 },n\epsilon I,{ 2 }^{ 1-1/\sqrt { 2 } }\)
- (d)
none of these
If \(cos\frac { \pi }{ 7 } ,\ cos\frac { 3\pi }{ 7 } ,\ cos\frac { 5\pi }{ 7 } \) are the roots of the equation 8x3-4x2-4x+1=0
On the basis of above information, answer the following questions:
The value of \(\sum _{ r=1 }^{ 3 }{ { tan }^{ 2 } } \left( \frac { 2r-1 }{ 7 } \right) \sum _{ r=1 }^{ 3 }{ { cot }^{ 2 } } \left( \frac { 2r-1 }{ 7 } \right) \) is
- (a)
15
- (b)
105
- (c)
21
- (d)
147
The value of \({ e }^{ \log _{ 10 }{ \tan { { 1 }^{ ° } } +\log _{ 10 }{ \tan { { 2 }^{ ° }+\log _{ 10 }{ \tan { { 3 }^{ ° } } } +...+\log _{ 10 }{ \tan { { 89 }^{ ° } } } } } } }\) is equal to
- (a)
0
- (b)
e
- (c)
1/e
- (d)
none of these
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)
\(\tan { \left( \alpha +\beta \right) =2\tan { \alpha } } \)
For \(0<\phi \le { \pi }/{ 2 }\) , if \(x=\sum _{ n=0 }^{ \infty }{ \cos ^{ 2n }{ \phi } } ,y=\sum _{ n=0 }^{ \infty }{ \sin ^{ 2n }{ \phi } } z=\sum _{ n=0 }^{ \infty }{ \cos ^{ 2n }{ \phi } } \sin ^{ 2n }{ \phi } \) , then
- (a)
xyz = xz + y
- (b)
xyz = xy + z
- (c)
xyz = x + y + z
- (d)
xyz = yz + x
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 (a-b)sin\(\left( \theta +\phi \right) \)=(a+b)sin\(\left( \theta -\phi \right) \)and a tan\(\frac { \theta }{ 2 } \)-b tan \(\frac { \phi }{ 2 } \) = c, then
- (a)
b tan\(\phi\) = a tan \(\theta\)
- (b)
a tan \(\phi\) = b tan \(\theta\)
- (c)
\(sin\ \phi =\frac { 2bc }{ { a }^{ 2 }-{ b }^{ 2 }-{ c }^{ 2 } } \)
- (d)
\(sin\ \theta =\frac { 2ac }{ { a }^{ 2 }-{ b }^{ 2 }+{ c }^{ 2 } } \)