IISER Mathematics - Trigonometry
Exam Duration: 45 Mins Total Questions : 30
The value of sin \(\pi\over10\)sin \(3\pi\over10\)is equal to
- (a)
\(1\over2\)
- (b)
\(-1\over2\)
- (c)
\(-1\over4\)
- (d)
1
If \(y=\frac { tan\quad \left( \frac { \pi }{ 4 } +\theta \right) +tan\quad \left( \frac { \pi }{ 4 } -\theta \right) }{ tan\quad \left( \frac { \pi }{ 4 } +\theta \right) -tan\quad \left( \frac { \pi }{ 4 } -\theta \right) } \),then y is equal to
- (a)
sec 2\(\theta \)
- (b)
cot 2\(\theta \)
- (c)
sin 2\(\theta \)
- (d)
cosec 2\(\theta \)
If x2+y2=1 and P=(3x-4x3)2 + (3y-4y3)2 ,then value of P is
- (a)
1
- (b)
3
- (c)
6
- (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
If \(p_1,p_2,p_3\) are the lengths of the perpendiculars from the angular points of a traingle on the opposite sides, then value of \({1\over p_1}+{1\over p_2}+{1\over p_3}\) is
- (a)
r
- (b)
R
- (c)
\(1\over R\)
- (d)
\(1\over r\)
The equation (cos p-1)x2 +(cos p)x+sin p=0 in the variable x has a real root .Then p can take any value in the interval.
- (a)
(0,2\(\pi\))
- (b)
(-\(\pi\), 0)
- (c)
\(({-\pi\over2},{\pi\over 2})\)
- (d)
(0, \(\pi\))
The smallest positive real root of the equation tan x -x =0 in (0,2\(\pi\)) lies in the interval
- (a)
\(({{0},{\pi\over2}})\)
- (b)
\(({{\pi\over2},{\pi}})\)
- (c)
\(({{\pi},{3\pi\over2}})\)
- (d)
\(({{3\pi\over2},{2\pi}})\)
A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of 30° .After some time ,the angle of dependence becomes 60° .The distance (in metres) travelled by the car during this time ,is
- (a)
100\(\sqrt3\)
- (b)
\(200\sqrt3\over3\)
- (c)
\(100\sqrt3\over3\)
- (d)
\(200\sqrt3\)
Which of the following pieces of data does not uniquely determine an acute-angled triangle ABC (R-being the radius of the circumcircle)?
- (a)
a,sin A,sin B
- (b)
a,b,c
- (c)
a,sin B,R
- (d)
a,sin A,R
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)
\(\pi\over4\)
If the angles of a triangles are in the ratio:4:1:1,then the ratio of the longest side and perimiter of triangle,is
- (a)
\(\sqrt3\over1+\sqrt3\)
- (b)
\(\sqrt3\over2+\sqrt3\)
- (c)
\(\sqrt3\over5+\sqrt3\)
- (d)
\(\sqrt3\over4+\sqrt3\)
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 equation sin x=\(\pi\over2\) +x=0 has a real root in the interval
- (a)
\(({0,{\pi\over2}})\)
- (b)
\(({\pi\over2},\pi)\)
- (c)
\(({\pi,{3\pi\over2}})\)
- (d)
None of these
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
The set of values of x for which tan 3x -tan 2x - tan 3x tan 2x =1, is
- (a)
\(\phi \)
- (b)
\(\pi\over4\)
- (c)
\(n\pi+{\pi\over4}\)
- (d)
\(2n\pi\pm{\pi\over4}\)
If \(\theta\) is an acute angle and \(\tan { \theta } =\frac { 1 }{ \sqrt { 7 } } \), then the value of \(\frac { { cosec }^{ 2 }\theta -\sec ^{ 2 }{ \theta } }{ { cosec }^{ 2 }\theta +\sec ^{ 2 }{ \theta } } \) is
- (a)
3/4
- (b)
1/2
- (c)
2
- (d)
5/4
If \(\tan { \alpha } \) is an integral solution of the equation 4x2 - 16x + 15 < 0 and \(\cos { \beta } \) is the slope of the bisector of the angle in the first quadrant between the x and y axes, then the value of \(\sin { \left( \alpha +\beta \right) } :\sin { \left( \alpha -\beta \right) } \) is equal to
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
Increasing product with angles are in GP \(cos\alpha \ cos2\alpha \ cos{ 2 }^{ 2 }\alpha ....cos{ 2 }^{ n-1 }\alpha \)
\(=\begin{cases} \frac { sin\quad { 2 }^{ n }\alpha }{ { 2 }^{ n }sin\quad \alpha } , \\ \frac { 1 }{ { 2 }^{ n } } , \\ -\frac { 1 }{ { 2 }^{ n } } , \end{cases}\begin{matrix} if\quad \alpha \neq n\pi \\ if\quad \alpha =\frac { \pi }{ { 2 }^{ n }+1 } \\ if\quad \alpha =\frac { \pi }{ { 2 }^{ n }-1 } \end{matrix}\)
Where, n\(\epsilon \) I (Integer)
On the basis of above information, answer the following questions:
The value of \(64\sqrt { 3 } sin\frac { \pi }{ 48 } cos\frac { \pi }{ 48 } cos\frac { \pi }{ 24 } cos\frac { \pi }{ 12 } cos\frac { \pi }{ 6 } \)is
- (a)
8
- (b)
6
- (c)
4
- (d)
-1
The measure of an angle in degrees, grades and radius be D, G and C respectively, then the relation between them
\(\frac { D }{ 90 } =\frac { G }{ 100 } =\frac { 2C }{ \pi } but{ 1 }^{ c }=\left( \frac { 180 }{ \pi } \right) ^{ \circ }\)
\(=57^{ \circ },17',44.8'\ '\ or\ 206265'\ '\)
\(\approx 57^{ \circ }\)
On the basis of above information, answer the following questions:
The angles of triangle are in AP and the number of grades in the least is to the number if radius in the greatest as 40:\(\pi\). Then the angles in degrees are
- (a)
\(45^{ \circ },60^{ \circ },75^{ \circ }\)
- (b)
\(20^{ \circ },60^{ \circ },100^{ \circ }\)
- (c)
\(30^{ \circ },60^{ \circ },90^{ \circ }\)
- (d)
\(40^{ \circ },60^{ \circ },80^{ \circ }\)
If \(\frac { \sin ^{ 3 }{ \theta } -\cos ^{ 3 }{ \theta } }{ \sin { \theta } -\cos { \theta } } -\frac { \cos { \theta } }{ \sqrt { \left( 1+\cot ^{ 2 }{ \theta } \right) } } -2\tan { \theta } \cot { \theta } =-1,\theta \epsilon \left[ 0,2\pi \right] \), then
- (a)
\(\theta \epsilon \left( 0,{ \pi }/{ 2 } \right) -\left\{ { \pi }/{ 4 } \right\} \)
- (b)
\(\theta \epsilon \left( \frac { \pi }{ 2 } ,\pi \right) -\left\{ { 3\pi }/{ 4 } \right\} \)
- (c)
\(\theta \epsilon \left( \pi ,\frac { 3\pi }{ 2 } \right) -\left\{ { 5\pi }/{ 4 } \right\} \)
- (d)
\(\theta \epsilon \left( 0,\pi \right) -\left\{ { \pi }/{ 4,{ \pi }/2 } \right\} \)
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 \(sin\frac { \pi }{ 14 } ,\ sin\frac { 3\pi }{ 14 } ,\ sin\frac { 5\pi }{ 14 } \) is
- (a)
\(\frac { 1 }{ 4 } \)
- (b)
\(\frac { 1 }{ 8 } \)
- (c)
\(\frac { \sqrt { 7 } }{ 4 } \)
- (d)
\(\frac { \sqrt { 7 } }{ 8 } \)
If \(\cos ^{ 4 }{ \theta } \sec ^{ 2 }{ \alpha } ,\frac { 1 }{ 2 } \) and \(\sin ^{ 4 }{ \theta } { cosec }^{ 2 }\alpha \) are in AP, then \(\cos ^{ 8 }{ \theta } \sec ^{ 6 }{ \alpha } ,\frac { 1 }{ 2 } \) and \(\sin ^{ 8 }{ \theta } { cosec }^{ 6 }\alpha \) are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
none of these
If \({ a }_{ n+1 }=\sqrt { \frac { 1 }{ 2 } \left( 1+{ a }_{ n } \right) } \), then \(\cos { \left( \frac { \sqrt { 1-{ a }_{ 0 }^{ 2 } } }{ { a }_{ 1 }{ a }_{ 2 }{ a }_{ 3 }...\infty } \right) } \) is equal to
- (a)
1
- (b)
- 1
- (c)
a0
- (d)
\(\frac { 1 }{ { a }_{ 0 } } \)
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 \(\sin { \theta } +\sin { \phi } =a\) and \(\cos { \theta } +\cos { \phi } =b\), then
- (a)
\(\cos { \left( \frac { \theta -\phi }{ 2 } \right) } =\pm \frac { 1 }{ 2 } \sqrt { \left( { a }^{ 2 }+{ b }^{ 2 } \right) } \)
- (b)
\(\cos { \left( \frac { \theta -\phi }{ 2 } \right) } =\pm \frac { 1 }{ 2 } \sqrt { \left( { a }^{ 2 }-{ b }^{ 2 } \right) } \)
- (c)
\(\tan { \left( \frac { \theta -\phi }{ 2 } \right) } =\pm \sqrt { \left( \frac { 4-{ a }^{ 2 }{ - }{ b }^{ 2 } }{ { a }^{ 2 }{ + }{ b }^{ 2 } } \right) } \)
- (d)
\(\cos { \left( \theta -\phi \right) } =\frac { { a }^{ 2 }{ + }{ b }^{ 2 }-2 }{ 2 } \)
Let \(0\le \theta \le { \pi }/{ 2 }\) and \(x=X\cos { \theta } +Y\sin { \theta } ,y=X\sin { \theta } -Y\cos { \theta } \), where a, b are constants. Then
- (a)
a = - 3, b = 3
- (b)
\(\theta ={ \pi }/{ 4 }\)
- (c)
a = 3, b = - 1
- (d)
\(\theta ={ \pi }/{3 }\)
If \(A+B=\frac { \pi }{ 3 } \) and cos A + cos B = 1, then which of the following is/are true?
- (a)
\(\cos { \left( A-B \right) } =\frac { 1 }{ 3 } \)
- (b)
\(\left| \cos { A } -\cos { B } \right| =\sqrt { \frac { 2 }{ 3 } } \)
- (c)
\(\cos { \left( A-B \right) } =-\frac { 1 }{ 3 } \)
- (d)
\(\left| \cos { A } -\cos { B } \right| =\frac { 1 }{ 2\sqrt { 3 } } \)
If \(x=\sec { \phi } -\tan { \phi } \) and \(y=co\sec { \phi } +\cot { \phi } \), then
- (a)
\(x=\frac { y+1 }{ y-1 } \)
- (b)
\(x=\frac { y-1 }{ y+1 } \)
- (c)
\(y=\frac { 1+x }{ 1-x } \)
- (d)
xy + x - y + 1 = 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