JEE Mathematics - Trigonometry Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
If tan \(\theta\) = \(cos 9°+sin 9°\over cos 9°-sin 9°\) ,then \(\theta\) is
- (a)
9°
- (b)
54°
- (c)
18°
- (d)
None of these
If tan \(\alpha ={1\over7}\) ,tan \(\beta ={1\over3}\) ,then cos 2\(\alpha\) is
- (a)
\(1\over21\)
- (b)
\(3\over7\)
- (c)
\(24\over25\)
- (d)
None of these
The value of \(({1+cos{\pi\over8}})({1+cos{3\pi\over8}})({1+cos{5\pi\over8}})({1+cos{7\pi\over8}})\) is
- (a)
\(1\over2\)
- (b)
\(cos {\pi\over8}\)
- (c)
\(1\over8\)
- (d)
\(1+\sqrt2\over2\sqrt2\)
The general solution of the equation of the equation sin x + cos x =1 is
- (a)
\(x=2n\pi \)
- (b)
\(x=2n\pi +\frac { \pi }{ 2} \)
- (c)
\(x=nx +(-1)^n\frac { \pi }{ 4 } -\frac { \pi }{ 4 } \)
- (d)
None of these
The equation (a+b)2 sec2\(\theta\) =4ab is possible only when
- (a)
a=0
- (b)
a+b=0
- (c)
a=b
- (d)
b=0
If y=sec2 \(\theta\) + cos2 \(\theta\), (\(\theta\)=0),then
- (a)
y = 0
- (b)
y \(\le\) 2
- (c)
y \(\ge\) 2
- (d)
y \(\ne\) 2
The value of sin (2 sin-1 0.8) is
- (a)
sin (1.2)
- (b)
sin(1.6)
- (c)
0.48
- (d)
0.96
The numerical value of tan \(\left[ { 2tan }^{ -1 }\left( \frac { 1 }{ 5 } \right) -\frac { \pi }{ 4 } \right] \) is
- (a)
\(7\over17\)
- (b)
\(-7\over17\)
- (c)
\(17\over7\)
- (d)
\(-17\over7\)
The sides a,b,c of the triangle ABC are in A.P., then cot \(A\over 2\) ,cot \(B\over 2\) ,cot \(C\over 2\) ,are in
- (a)
A.P
- (b)
G.P
- (c)
H.P
- (d)
None of these
In the \(\triangle\) ABC \({b-c\over r_1} +{c-a\over r_2}+{a-b\over r_3}\) is equal to
- (a)
0
- (b)
1
- (c)
3
- (d)
None of these
The sides of an euilateral triangle, a square and a regular hexagon circumscribed in a circle are in
- (a)
A.P
- (b)
G.P
- (c)
H.P
- (d)
None of these
The number of real solutions of sin(ex )=5x +5-x is
- (a)
0
- (b)
1
- (c)
2
- (d)
infinitely many
If in a triangle PQR, sin P,sin Q,sin R are in A.P then
- (a)
the altitudes are in A.P
- (b)
the altitudes are in H.P
- (c)
the medians are in G.P
- (d)
the medians are in H.P
Let n be an odd integer.If \(sin\quad n\theta \quad =\sum _{ r=0 }^{ n }{ { b }_{ r } } { sin }^{ r }\theta \) ,for every value of \(\theta\) ,then
- (a)
\(b_0=1,b_1=3\)
- (b)
\(b_0=0,b_1=n\)
- (c)
\(b_0=-1,b_1=n\)
- (d)
\(b_0=0,b_1=n^{2}+n+3\)
If \({tan 3A\over tan A} = {k}\), then \(sinn 3A\over sin A \),is equal to
- (a)
\({2k\over k-1} ,k \in R\)
- (b)
\({2k\over k-1} ,k \in ({1\over 3},3)\)
- (c)
\({2k\over k-1} ,k \notin ({1\over 3},3)\)
- (d)
\({k-1\over 2k} ,k \notin ({1\over 3},3)\)
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)
\(({{3\pi\over2},{2\pi}})\)
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
The period of the function \(f(x)={sin nx\over sin(x/n)}\), is 4\(\pi\); the least value of n is
- (a)
1
- (b)
2
- (c)
4
- (d)
None of these
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\(\theta\)
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)
\(\sqrt3\over4+\sqrt3\)
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
If \(\frac { \sec ^{ 8 }{ \theta } }{ a } +\frac { \tan ^{ 8 }{ \theta } }{ b } =\frac { 1 }{ a+b } \), then for every real value of \(\sin ^{ 2 }{ \theta } \)
- (a)
\(ab\le 0\)
- (b)
\(ab\ge 0\)
- (c)
a + b = 0
- (d)
none of these
If \({ P }_{ n }=\cos ^{ n }{ \theta } +\sin ^{ n }{ \theta } \), then \({ P }_{ n }-{ P }_{ n-2 }={ kP }_{ n-4 }\), where
- (a)
k = 1
- (b)
\(k=-\sin ^{ 2 }{ \theta } \cos ^{ 2 }{ \theta } \)
- (c)
\(k=\sin ^{ 2 }{ \theta }\)
- (d)
\(k=\cos ^{ 2 }{ \theta } \)
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)
\(\tan { \theta } +\frac { \tan { n\theta } }{ n } \)
Let n be a fixed positive integer such that \(\sin { \frac { \pi }{ 2n } } +\cos { \frac { \pi }{ 2n } } =\frac { \sqrt { n } }{ 2 } \), then
- (a)
n = 4
- (b)
n = 5
- (c)
n = 6
- (d)
none of these
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
If \(a\sec { \alpha } -c\tan { \alpha } =d\) and \(b\sec { \alpha } +d\tan { \alpha } =c\) then
- (a)
a2 + c2 = b2 + d2
- (b)
a2 + d2 = b2 + c2
- (c)
a2 + b2 = c2 + d2
- (d)
ab = cd
\(\left( 1+\cos { \frac { \pi }{ 8 } } \right) \left( 1+\cos { \frac { 3\pi }{ 8 } } \right) \left( 1+\cos { \frac { 5\pi }{ 8 } } \right) \left( 1+\cos { \frac { 7\pi }{ 8 } } \right) \) is equal to
- (a)
1/2
- (b)
\(\cos { { \pi }/{ 8 } } \)
- (c)
1/8
- (d)
\(\frac { 1+\sqrt { 2 } }{ 2\sqrt { 2 } } \)
The value of \(\cos { \frac { 2\pi }{ 7 } } +\cos { \frac { 4\pi }{ 7 } } +\cos { \frac { 6\pi }{ 7 } } \) is equal to
- (a)
1
- (b)
- 1
- (c)
1/2
- (d)
-1/2
If \(\pi <\alpha <\frac { 3\pi }{ 2 } \), then the expression \(\sqrt { 4\sin ^{ 4 }{ \alpha +\sin ^{ 2 }{ 2\alpha } } } +4\cos ^{ 2 }{ \left( \frac { \pi }{ 4 } -\frac { \alpha }{ 2 } \right) } \)is equal to
- (a)
\(2+4\sin { \alpha } \)
- (b)
\(2-4\sin { \alpha } \)
- (c)
2
- (d)
none of these
The value of \(\cos { \frac { \pi }{ 15 } } \cos { \frac { 2\pi }{ 15 } } \cos { \frac { 3\pi }{ 15 } } \cos { \frac { 4\pi }{ 15 } } \cos { \frac { 5\pi }{ 15 } } \cos { \frac { 6\pi }{ 15 } } \cos { \frac { 7\pi }{ 15 } } \) is equal to
- (a)
1/26
- (b)
1/27
- (c)
1/28
- (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)
\({ 1 }/{ \sqrt { 10 } }\)
The values of \(\theta \left( 0<\theta <{ 360 }^{ ° } \right) \) satisfying \(cosec\theta +2=0\) are
- (a)
210°, 300°
- (b)
240°, 300°
- (c)
210°, 240°
- (d)
210°, 330°
If \(4n\alpha =\pi \), then the numerical value of \(\tan { \alpha } \tan { 2\alpha } \tan { 3\alpha } ....\tan { \left( 2n-1 \right) } \alpha \) is equal to
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
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
The ratio of the greatest value of 2 - cos x + sin2 x to its least value is
- (a)
1/4
- (b)
9/4
- (c)
13/4
- (d)
none of these
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 in a triangle ABC, cos 3A + cos 3B + cos 3C = 1, then one angle must be exactly equal to
- (a)
\(\frac { \pi }{ 3 } \)
- (b)
\(\frac {2 \pi }{ 3 } \)
- (c)
\(\ { \pi }\)
- (d)
\(\frac { 4\pi }{ 3 } \)
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:
If \(\alpha =\frac { \pi }{ 13 } \) then the value of \(\overset { 6 }{ \underset { r=1 }{ \Pi } } cosr\alpha \) is
- (a)
\(\frac { 1 }{ 64 } \)
- (b)
\(-\frac { 1 }{ 64 } \)
- (c)
\(\frac { 1 }{ 32 } \)
- (d)
\(-\frac { 1 }{ 8 } \)
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 \(sin\left( \frac { \pi }{ 18 } \right) sin\left( \frac { 5\pi }{ 18 } \right) sin\left( \frac { 7\pi }{ 18 } \right) \) is
- (a)
\(\frac { 1 }{ 16 } \)
- (b)
\(\frac { 1 }{ 8 } \)
- (c)
\(-\frac { 1 }{ 8 } \)
- (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 value of cos 1c is
- (a)
\(<\frac{1}{2}\)
- (b)
\(\frac{1}{2}\)
- (c)
\(>\frac{1}{2}\)
- (d)
0
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
The set of values of \(\lambda \epsilon R\) such that \(\tan ^{ 2 }{ \theta } +\sec { \theta } =\lambda \) holds for some \(\theta \) is
- (a)
\((-\infty ,1]\)
- (b)
\((-\infty ,-1]\)
- (c)
\(\phi \)
- (d)
\([1,\infty )\)
If \(\tan { \theta } =n\tan { \phi } \), then maximum value of \(\tan ^{ 2 }{ \left( \theta -\phi \right) } \) is equal to
- (a)
\(\frac { { \left( n+1 \right) }^{ 2 } }{ 4n } \)
- (b)
\(\frac { { \left( n-1 \right) }^{ 2 } }{ 4n } \)
- (c)
\(\frac { { \left( 2n+1 \right) }^{ 2 } }{ 4n } \)
- (d)
\(\frac { { \left( 2n-1 \right) }^{ 2 } }{ 4n } \)
Let \({ f }_{ n }\left( \theta \right) =\tan { \frac { \theta }{ 2 } } \left( 1+\sec { \theta } \right) \left( 1+\sec { 2\theta } \right) \left( 1+\sec { 4\theta } \right) ...\left( 1+\sec { { 2 }^{ n }\theta } \right) \), then
- (a)
\({ f }_{ 2 }\left( \frac { \pi }{ 16 } \right) =1\)
- (b)
\({ f }_{ 3 }\left( \frac { \pi }{ 32 } \right) =1\)
- (c)
\({ f }_{ 4 }\left( \frac { \pi }{ 64 } \right) =1\)
- (d)
\({ f }_{ 5 }\left( \frac { \pi }{ 128} \right) =1\)
If \(\frac { x }{ y } =\frac { \cos { A } }{ \cos { B } } \), where \(A\neq B\) , then
- (a)
\(\tan { \left( \frac { A+B }{ 2 } \right) } =\frac { x\tan { A } +y\tan { B } }{ x+y } \)
- (b)
\(\tan { \left( \frac { A-B }{ 2 } \right) } =\frac { x\tan { A } -y\tan { B } }{ x+y } \)
- (c)
\(\frac { \sin { \left( A+B \right) } }{ \sin { \left( A-B \right) } } =\frac { y\sin { A } +x\sin { B } }{ y\sin { A } -x\sin { B } } \)
- (d)
x cos A + y cos B = 0
In a \(\triangle ABC\)
- (a)
\(\sin { A } \sin { B } \sin { C } \le \frac { 3\sqrt { 3 } }{ 8 } \)
- (b)
\(\sin ^{ 2 }{ A } +\sin ^{ 2 }{ B } +\sin ^{ 2 }{ C } \le \frac { 9 }{ 4 } \)
- (c)
sin A sin B sin C is always positive
- (d)
\(\sin ^{ 2 }{ A } +\sin ^{ 2 }{ B } +\le 1+\cos { C } \)
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 sec\(\theta\) + tan\(\theta\)=1, then one root of equation a(b-c)x2+b(c-a)x+c(a-b)=0 is
- (a)
tan \(\theta\)
- (b)
sec \(\theta\)
- (c)
cos \(\theta\)
- (d)
sin \(\theta\)
If sin3 x sin3x=\(\overset { n }{ \underset { m=0 }{ \Sigma } } \) cm cos mx where c0,c1,c2,......cn are constants and cn≠0, then the value of n is
- (a)
2
- (b)
3
- (c)
6
- (d)
8