JEE Main Mathematics - Trigonometry
Exam Duration: 60 Mins Total Questions : 30
If sin3 x sin 3x =\(\sum _{ m=0 }^{ n }{ = } { c }_{ m }\) cos mx ,where c0,c1,c2,......cn \(\neq \) 0, is an identify in x then n =
- (a)
5
- (b)
6
- (c)
7
- (d)
8
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 equation (a+b)2 sec2\(\theta\) =4ab is possible only when
- (a)
a=0
- (b)
a+b=0
- (c)
a=b
- (d)
b=0
In the triangle ABC, a=2b and \(\angle A=3\angle B\) ;then angle A is
- (a)
30°
- (b)
60°
- (c)
90°
- (d)
None of these
In a \(\triangle\)ABC ,angle A is greater than the angle B.If measures of angle A and B satisfy the equation, 3sin x-4 sin3 x-k=0, 0
- (a)
\(\pi\over3\)
- (b)
\(\pi\over2\)
- (c)
\(2\pi\over3\)
- (d)
\(5\pi\over6\)
The value of \(\theta\) lying between \(\theta\)=0 and \(\theta\)=\(\pi\over 2\) and satisfying the equation \(\begin{vmatrix} 1+{ sin }^{ 2 }\theta \quad & { cos }^{ 2 }\theta & 4sin4\theta \\ { sin }^{ 2 }\theta & { 1+cos }^{ 2 }x & 4sin4\theta \\ { sin }^{ 2 }\theta & { cos }^{ 2 }x & 1+4sin4\theta \end{vmatrix}=0\) ,is
- (a)
\({\pi\over24},{7\pi\over24}\)
- (b)
\(5\pi\over24\)
- (c)
\(7\pi\over24\)
- (d)
\(11\pi\over24\)
If \(sin^{-1}({x}-{x^{3}\over2}+{x^{3}\over4}-...)+cos^{-1}({x^{2}}-{x^{4}\over2}+{x^{6}\over4}+...)={\pi\over2}\) for 0<|x|<2,then x equals
- (a)
\(1\over2\)
- (b)
1
- (c)
\(-1\over2\)
- (d)
-1
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°
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 A = cos (cos x) + sin (cos x) the least and greatest value of A are
- (a)
0 and 2
- (b)
- 1 and 1
- (c)
\(-\sqrt { 2 } \quad and\quad \sqrt { 2 } \)
- (d)
\(0 \quad and\quad \sqrt { 2 } \)
If in a \(\triangle ABC\), tan A + tan B + tan C > 0, then
- (a)
\(\triangle \) is always obtuse angled triangle
- (b)
\(\triangle \) is always equilateral triangle
- (c)
\(\triangle \) is always acute angled triangle
- (d)
nothing can be said about the type of triangle
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 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 \(x=y\cos { \frac { 2\pi }{ 3 } } =z\cos { \frac { 4\pi }{ 3 } } \) , then xy + yz + zx is equal to
- (a)
- 1
- (b)
0
- (c)
1
- (d)
2
If \(\cos { \alpha } +\cos { \beta } =\sin { \alpha } +\sin { \beta } \), 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 the expression \(\frac { A\cos { \left( \theta +\alpha \right) +B\sin { \left( \theta +\beta \right) } } }{ { A }^{ \prime }\sin { \left( \theta +\alpha \right) } +{ B }^{ \prime }\cos { \left( \theta +\beta \right) } } \) retain the same value for all \('\theta '\), then
- (a)
\(\left( A{ A }^{ \prime }-B{ B }^{ \prime } \right) \sin { \left( \alpha -\beta \right) } =\left( { A }^{ \prime }B-A{ B }^{ \prime } \right) \)
- (b)
\(A{ A }^{ \prime }+B{ B }^{ \prime }=\left( { A }^{ \prime }B+A{ B }^{ \prime } \right) \sin { \left( \alpha -\beta \right) } \)
- (c)
\(A{ A }^{ \prime }-B{ B }^{ \prime }=\left( { A }^{ \prime }B-A{ B }^{ \prime } \right) \sin { \left( \alpha -\beta \right) } \)
- (d)
none of the above
The minimum and maximum values of \(ab\sin { x+b } \sqrt { \left( 1-{ a }^{ 2 } \right) } \cos { x } +c\left( \left| a \right| <1,b>0 \right) \) respectively are
- (a)
{b - c, b + c}
- (b)
{b + c, b - c}
- (c)
{c - b, b + c}
- (d)
none of these
The value of \(\sum _{ r=1 }^{ 18 }{ \cos ^{ 2 }{ \left( 5r \right) } ^{ ° } } \), where x° denotes the x degrees, is equal to
- (a)
0
- (b)
7/2
- (c)
17/2
- (d)
25/2
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 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)
\(\frac { { P }_{ 3 } }{ { P }_{ 5 } } \)
The method of eliminating \('\theta '\) from two given equations involving trigonometrical functions of \('\theta '\) . By using given equations If involving \('\theta '\) and trigonometrical identities, we shall obtain an equation not involving\('\theta '\).
On the basis of above information, answer the following questions:
If sin\(\theta\) + cos\(\theta\) = a and sin3\(\theta\) + cos3\(\theta\) = b, then we get \(\lambda { a }^{ 3 }+\mu b+vc=0\)when \(\lambda ,\mu ,v\)are independent of \(\theta\), then the value of \({ \lambda }^{ 3 }+{ \mu }^{ 3 }+{ v }^{ 3 }\) is
- (a)
-6
- (b)
-18
- (c)
-36
- (d)
-98
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
Which of the following statements are possible , a, b, m and n being non-zero real numbers?
- (a)
\(4\sin ^{ 2 }{ \theta } =5\)
- (b)
\(\left( { a }^{ 2 }+{ b }^{ 2 } \right) \cos { \theta } =2ab\)
- (c)
\(\left( { m }^{ 2 }+{ n }^{ 2 } \right) \cos { ec\theta } ={ m }^{ 2 }-{ n }^{ 2 }\)
- (d)
\(\sin { \theta = } 2.375\)
If \(\cos { \theta } =\frac { a\cos { \phi +b } }{ a+b\cos { \phi } } \), then \(\tan { { \theta }/{ 2 } } \) is equal to
- (a)
\(\sqrt { \left( \frac { a-b }{ a+b } \right) } \tan { { \phi }/{ 2 } } \)
- (b)
\(\sqrt { \left( \frac { a+b }{ a-b } \right) } \cos { { \phi }/{ 2 } } \)
- (c)
\(\sqrt { \left( \frac { a-b }{ a+b } \right) } \sin{ { \phi }/{ 2 } } \)
- (d)
none of these
For any real \(\theta\), the maximum value \(\cos ^{ 2 }{ \left( \cos { \theta } \right) } +\sin ^{ 2 }{ \left( \sin { \theta } \right) } \) is
- (a)
1
- (b)
1 + sin2 1
- (c)
1 + cos2 1
- (d)
does not exist
If \(\cos { \alpha } =\frac { 3 }{ 5 } \) and \(\cos { \beta } =\frac { 5 }{ 13 } \), then
- (a)
\(\cos { \left( \alpha +\beta \right) } =\frac { 33 }{ 65 } \)
- (b)
\(\sin { \left( \alpha +\beta \right) } =\frac {56 }{ 65 } \)
- (c)
\(\sin ^{ 2 }{ \left( \frac { \alpha -\beta }{ 2 } \right) } =\frac { 1 }{ 65 } \)
- (d)
\(\cos { \left( \alpha -\beta \right) } =\frac { 63 }{ 65 } \)
If \(\tan { x } =\frac { 2b }{ a-c } ,\left( a\neq c \right) \)
y = a cos2 x + 2b sin x cos x + c sin2 x and
z = a sin2 x - 2b sin x cos x + c cos2 x, then
- (a)
y = z
- (b)
y + z = a + c
- (c)
y - z = a - c
- (d)
y - z = (a - c)2 + 4b2
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)
\(tan\theta =-\sqrt { 3 } \)