Question 1: Palette 1/60

JEE Mains 2026 MCQ

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{\sin ^2 \mathrm{C}}{\sin ^2 \mathrm{~A}}=1, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \in\left(0, \frac{\pi}{2}\right)$, then

A

$\tan \mathrm{A}, \tan \mathrm{C}, \tan \mathrm{B}$ are in A.P.

B

$\tan \mathrm{A}, \tan \mathrm{C}, \tan \mathrm{B}$ are in G.P.

C

$\tan A, \tan B, \tan C$ are in G.P.

D

$\tan A, \tan B, \tan C$ are in A.P.

Question 2: Palette 2/60

JEE Mains 2026 MCQ

The value of $\frac{\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}}{\cos 20^{\circ} \cos 40^{\circ} \cos 60^{\circ} \cos 80^{\circ}}$ is equal to

A

32

B

64

C

12

D

16

Question 3: Palette 3/60

JEE Mains 2026 MCQ

If $\cot x=\frac{5}{12}$ for some $x \in\left(\pi, \frac{3 \pi}{2}\right)$, then $\sin 7 x\left(\cos \frac{13 x}{2}+\sin \frac{13 x}{2}\right)+\cos 7 x\left(\cos \frac{13 x}{2}-\sin \frac{13 x}{2}\right)$ is equal to

A

$\frac{5}{\sqrt{13}}$

B

$\frac{6}{\sqrt{26}}$

C

$\frac{4}{\sqrt{26}}$

D

$\frac{1}{\sqrt{13}}$

Question 4: Palette 4/60

JEE Mains 2026 MCQ

Let $\frac{\pi}{2}<\theta<\pi$ and $\cot \theta=-\frac{1}{2 \sqrt{2}}$. Then the value of

$ \sin \left(\frac{15 \theta}{2}\right)(\cos 8 \theta+\sin 8 \theta)+\cos \left(\frac{15 \theta}{2}\right)(\cos 8 \theta-\sin 8 \theta) $

is equal to :

A

$\frac{\sqrt{2}-1}{\sqrt{3}}$

B

$\frac{\sqrt{2}}{\sqrt{3}}$

C

$\frac{1-\sqrt{2}}{\sqrt{3}}$

D

$-\frac{\sqrt{2}}{\sqrt{3}}$

Question 5: Palette 5/60

JEE Mains 2026 MCQ

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to :

A

2

B

6

C

8

D

4

Question 6: Palette 6/60

JEE Mains 2026 MCQ

Let $\tan A, \tan B$, where $A, B \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, be the roots of the quadratic equation $x^2-2 x-5=0$. Then $20 \sin ^2\left(\frac{A+B}{2}\right)$ is equal to:

A

$10+\sqrt{10}$

B

$10-2 \sqrt{10}$

C

10-3 $\sqrt{10}$

D

$10-\sqrt{10}$

Question 7: Palette 7/60

JEE Mains 2026 MCQ

Let $P = \{ \theta \in [0, 4\pi] : \tan^2 \theta \neq 1 \}$ and $S = \{ a \in \mathbb{Z} : 2(\cos^8 \theta - \sin^8 \theta) \sec 2 \theta = a^2, \theta \in P \}$. Then $n(S)$ is:

A

0

B

1

C

2

D

3

Question 8: Palette 8/60

JEE Mains 2026 MCQ

If $\sin\left(\frac{\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \sin\left(\frac{7\pi}{18}\right) = K$, then the value of $\sin\left(\frac{10K\pi}{3}\right)$ is:

A

$\frac{\sqrt{3}+1}{2\sqrt{2}}$

B

$\frac{\sqrt{3}-1}{\sqrt{2}}$

C

$\frac{\sqrt{3}}{2}$

D

$\frac{1}{2}$

Question 9: Palette 9/60

JEE Mains 2025 MCQ

If for $\theta \in\left[-\frac{\pi}{3}, 0\right]$, the points $(x, y)=\left(3 \tan \left(\theta+\frac{\pi}{3}\right), 2 \tan \left(\theta+\frac{\pi}{6}\right)\right)$ lie on $x y+\alpha x+\beta y+\gamma=0$, then $\alpha^2+\beta^2+\gamma^2$ is equal to :

A

75

B

96

C

80

D

72

Question 10: Palette 10/60

JEE Mains 2025 MCQ

If $10 \sin ^4 \theta+15 \cos ^4 \theta=6$, then the value of $\frac{27 \operatorname{cosec}^6 \theta+8 \sec ^6 \theta}{16 \sec ^8 \theta}$ is

A

$\frac{2}{5}$

B

$\frac{3}{5}$

C

$\frac{1}{5}$

D

$\frac{3}{4}$

Question 11: Palette 11/60

JEE Mains 2025 MCQ

If $\sin x + \sin^2 x = 1$, $x \in \left(0, \frac{\pi}{2}\right)$, then

$(\cos^{12} x + \tan^{12} x) + 3(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to:

A

3

B

4

C

2

D

1

Question 12: Palette 12/60

JEE Mains 2025 MCQ

If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+\frac{r \pi}{6}\right)}\right\}=a \sqrt{3}+b, a, b \in Z$, then $a^2+b^2$ is equal to :

A

10

B

4

C

8

D

2

Question 13: Palette 13/60

JEE Mains 2025 MCQ

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

A

11

B

10

C

8

D

9

Question 14: Palette 14/60

JEE Mains 2025 MCQ

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

A

0

B

2/3

C

1

D

3/2

Question 15: Palette 15/60

JEE Mains 2024 MCQ

If the value of $\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}$ is $\frac{a \sqrt{5}-b}{c}$, where $a, b, c$ are natural numbers and $\operatorname{gcd}(a, c)=1$, then $a+b+c$ is equal to :

A

54

B

52

C

50

D

40

Question 16: Palette 16/60

JEE Mains 2024 MCQ

If $\sin x=-\frac{3}{5}$, where $\pi< x <\frac{3 \pi}{2}$, then $80\left(\tan ^2 x-\cos x\right)$ is equal to

A

109

B

108

C

19

D

18

Question 17: Palette 17/60

JEE Mains 2024 MCQ

Suppose $\theta \in\left[0, \frac{\pi}{4}\right]$ is a solution of $4 \cos \theta-3 \sin \theta=1$. Then $\cos \theta$ is equal to :

A

$\frac{6-\sqrt{6}}{(3 \sqrt{6}-2)}$

B

$\frac{4}{(3 \sqrt{6}+2)}$

C

$\frac{6+\sqrt{6}}{(3 \sqrt{6}+2)}$

D

$\frac{4}{(3 \sqrt{6}-2)}$

Question 18: Palette 18/60

JEE Mains 2024 MCQ

If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and

$\tan \mathrm{C}=\left(x^{-3}+x^{-2}+x^{-1}\right)^{1 / 2}, 0<\mathrm{A}, \mathrm{B}, \mathrm{C}<\frac{\pi}{2}$, then $\mathrm{A}+\mathrm{B}$ is equal to :

A

$\mathrm{C}$

B

$\pi-C$

C

$2 \pi-C$

D

$\frac{\pi}{2}-\mathrm{C}$

Question 19: Palette 19/60

JEE Mains 2024 MCQ

The number of solutions, of the equation $e^{\sin x}-2 e^{-\sin x}=2$, is :

A

0

B

1

C

2

D

more than 2

Question 20: Palette 20/60

JEE Mains 2024 MCQ

For $\alpha, \beta \in(0, \pi / 2)$, let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha=k \tan \beta$. Then, the value of $k$ is equal to

A

5

B

$-$2/3

C

$-$5

D

2/3

Question 21: Palette 21/60

JEE Mains 2023 MCQ

$96\cos {\pi \over {33}}\cos {{2\pi } \over {33}}\cos {{4\pi } \over {33}}\cos {{8\pi } \over {33}}\cos {{16\pi } \over {33}}$ is equal to :

A

4

B

2

C

1

D

3

Question 22: Palette 22/60

JEE Mains 2023 MCQ

The value of $36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)$ is :

A

18

B

36

C

54

D

27

Question 23: Palette 23/60

JEE Mains 2023 MCQ

If $\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a$, then the value of $\left( {a + {1 \over a}} \right)$ is :

A

$5 - {3 \over 2}\sqrt 3 $

B

$4 - 2\sqrt 3 $

C

2

D

4

Question 24: Palette 24/60

JEE Mains 2023 MCQ

The set of all values of $\lambda$ for which the equation ${\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda $ has a real solution $x$, is :

A

$\left[ { - 2, - 1} \right]$

B

$\left[ { - {3 \over 2}, - 1} \right]$

C

$\left[ { - 2, - {3 \over 2}} \right]$

D

$\left[ { - 1, - {1 \over 2}} \right]$

Question 25: Palette 25/60

JEE Mains 2023 MCQ

Let $f(\theta ) = 3\left( {{{\sin }^4}\left( {{{3\pi } \over 2} - \theta } \right) + {{\sin }^4}(3\pi + \theta )} \right) - 2(1 - {\sin ^2}2\theta )$ and $S = \left\{ {\theta \in [0,\pi ]:f'(\theta ) = - {{\sqrt 3 } \over 2}} \right\}$. If $4\beta = \sum\limits_{\theta \in S} \theta $, then $f(\beta )$ is equal to

A

$\frac{9}{8}$

B

$\frac{3}{2}$

C

$\frac{5}{4}$

D

$\frac{11}{8}$

Question 26: Palette 26/60

JEE Mains 2022 MCQ

$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$ is equal to :

A

$\frac{3}{16}$

B

$\frac{1}{16}$

C

$\frac{1}{32}$

D

$\frac{9}{32}$

Question 27: Palette 27/60

JEE Mains 2022 MCQ

If cot$\alpha$ = 1 and sec$\beta$ = $ - {5 \over 3}$, where $\pi < \alpha < {{3\pi } \over 2}$ and ${\pi \over 2} < \beta < \pi $, then the value of $\tan (\alpha + \beta )$ and the quadrant in which $\alpha$ + $\beta$ lies, respectively are :

A

$ - {1 \over 7}$ and IV^th quadrant

B

7 and I^st quadrant

C

$-$7 and IV^th quadrant

D

$ {1 \over 7}$ and I^st quadrant

Question 28: Palette 28/60

JEE Mains 2022 MCQ

$\alpha = \sin 36^\circ $ is a root of which of the following equation?

A

$16{x^4} - 10{x^2} - 5 = 0$

B

$16{x^4} + 20{x^2} - 5 = 0$

C

$16{x^4} - 20{x^2} + 5 = 0$

D

$4{x^4} - 10{x^2} + 5 = 0$

Question 29: Palette 29/60

JEE Mains 2022 MCQ

The value of $\cos \left( {{{2\pi } \over 7}} \right) + \cos \left( {{{4\pi } \over 7}} \right) + \cos \left( {{{6\pi } \over 7}} \right)$ is equal to :

A

$-$1

B

$-$${1 \over 2}$

C

$-$${1 \over 3}$

D

$-$${1 \over 4}$

Question 30: Palette 30/60

JEE Mains 2022 MCQ

$16\sin (20^\circ )\sin (40^\circ )\sin (80^\circ )$ is equal to :

A

$\sqrt 3 $

B

2$\sqrt 3 $

C

3

D

4$\sqrt 3 $

Question 31: Palette 31/60

JEE Mains 2022 MCQ

The value of 2sin (12$^\circ$) $-$ sin (72$^\circ$) is :

A

${{\sqrt 5 (1 - \sqrt 3 )} \over 4}$

B

${{1 - \sqrt 5 } \over 8}$

C

${{\sqrt 3 (1 - \sqrt 5 )} \over 2}$

D

${{\sqrt 3 (1 - \sqrt 5 )} \over 4}$

Question 32: Palette 32/60

JEE Mains 2021 MCQ

The value of

$2\sin \left( {{\pi \over 8}} \right)\sin \left( {{{2\pi } \over 8}} \right)\sin \left( {{{3\pi } \over 8}} \right)\sin \left( {{{5\pi } \over 8}} \right)\sin \left( {{{6\pi } \over 8}} \right)\sin \left( {{{7\pi } \over 8}} \right)$ is :

A

${1 \over {4\sqrt 2 }}$

B

${1 \over 4}$

C

${1 \over 8}$

D

${1 \over {8\sqrt 2 }}$

Question 33: Palette 33/60

JEE Mains 2021 MCQ

If $\tan \left( {{\pi \over 9}} \right),x,\tan \left( {{{7\pi } \over {18}}} \right)$ are in arithmetic progression and $\tan \left( {{\pi \over 9}} \right),y,\tan \left( {{{5\pi } \over {18}}} \right)$ are also in arithmetic progression, then $|x - 2y|$ is equal to :

A

4

B

3

C

0

D

1

Question 34: Palette 34/60

JEE Mains 2021 MCQ

If $\sin \theta + \cos \theta = {1 \over 2}$, then 16(sin(2$\theta$) + cos(4$\theta$) + sin(6$\theta$)) is equal to :

A

23

B

$-$27

C

$-$23

D

27

Question 35: Palette 35/60

JEE Mains 2021 MCQ

The value of $\cot {\pi \over {24}}$ is :

A

$\sqrt 2 + \sqrt 3 + 2 - \sqrt 6 $

B

$\sqrt 2 + \sqrt 3 + 2 + \sqrt 6 $

C

$\sqrt 2 - \sqrt 3 - 2 + \sqrt 6 $

D

$3\sqrt 2 - \sqrt 3 - \sqrt 6 $

Question 36: Palette 36/60

JEE Mains 2021 MCQ

If 15sin⁴$\alpha$ + 10cos⁴$\alpha$ = 6, for some $\alpha$$\in$R, then the value of

27sec⁶$\alpha$ + 8cosec⁶$\alpha$ is equal to :

A

500

B

400

C

250

D

350

Question 37: Palette 37/60

JEE Mains 2021 MCQ

If for x $\in$ $\left( {0,{\pi \over 2}} \right)$, log₁₀sinx + log₁₀cosx = $-$1 and log₁₀(sinx + cosx) = ${1 \over 2}$(log₁₀ n $-$ 1), n > 0, then the value of n is equal to :

A

16

B

9

C

12

D

20

Question 38: Palette 38/60

JEE Mains 2021 MCQ

If 0 < x, y < $\pi$ and cosx + cosy $-$ cos(x + y) = ${3 \over 2}$, then sinx + cosy is equal to :

A

${{1 + \sqrt 3 } \over 2}$

B

${{1 \over 2}}$

C

${{\sqrt 3 } \over 2}$

D

${{1 - \sqrt 3 } \over 2}$

Question 39: Palette 39/60

JEE Mains 2021 MCQ

If ${e^{\left( {{{\cos }^2}x + {{\cos }^4}x + {{\cos }^6}x + ...\infty } \right){{\log }_e}2}}$ satisfies the equation t² - 9t + 8 = 0, then the value of
${{2\sin x} \over {\sin x + \sqrt 3 \cos x}}\left( {0 < x < {\pi \over 2}} \right)$ is :

A

$\sqrt 3 $

B

${3 \over 2}$

C

2$\sqrt 3 $

D

${1 \over 2}$

Question 40: Palette 40/60

JEE Mains 2020 MCQ

If L = sin²$\left( {{\pi \over {16}}} \right)$ - sin²$\left( {{\pi \over {8}}} \right)$ and
M = cos²$\left( {{\pi \over {16}}} \right)$ - sin²$\left( {{\pi \over {8}}} \right)$, then :

A

L = $ - {1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$

B

M = ${1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$

C

M = ${1 \over {4\sqrt 2 }} + {1 \over 4}\cos {\pi \over 8}$

D

L = ${1 \over {4\sqrt 2 }} - {1 \over 4}\cos {\pi \over 8}$

Question 41: Palette 41/60

JEE Mains 2020 MCQ

If the equation cos⁴ $\theta $ + sin⁴ $\theta $ + $\lambda $ = 0 has real solutions for $\theta $, then $\lambda $ lies in the interval :

A

$\left[ { - {3 \over 2}, - {5 \over 4}} \right]$

B

$\left( { - {1 \over 2}, - {1 \over 4}} \right]$

C

$\left( { - {5 \over 4}, - 1} \right]$

D

$\left[ { - 1, - {1 \over 2}} \right]$

Question 42: Palette 42/60

JEE Mains 2020 MCQ

If $x = \sum\limits_{n = 0}^\infty {{{\left( { - 1} \right)}^n}{{\tan }^{2n}}\theta } $ and $y = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } $

for 0 < $\theta $ < ${\pi \over 4}$, then :

A

x(1 + y) = 1

B

y(1 – x) = 1

C

y(1 + x) = 1

D

x(1 – y) = 1

Question 43: Palette 43/60

JEE Mains 2020 MCQ

The value of
${\cos ^3}\left( {{\pi \over 8}} \right)$${\cos}\left( {{3\pi \over 8}} \right)$+${\sin ^3}\left( {{\pi \over 8}} \right)$${\sin}\left( {{3\pi \over 8}} \right)$
is :

A

${1 \over {\sqrt 2 }}$

B

${1 \over 2}$

C

${1 \over 4}$

D

${1 \over 2{\sqrt 2 }}$

Question 44: Palette 44/60

JEE Mains 2019 MCQ

The equation y = sinx sin (x + 2) – sin² (x + 1) represents a straight line lying in :

A

first, second and fourth quadrants

B

first, third and fourth quadrants

C

second and third quadrants only

D

third and fourth quadrants only

Question 45: Palette 45/60

JEE Mains 2019 MCQ

The value of sin 10º sin30º sin50º sin70º is :-

A

${1 \over {36}}$

B

${1 \over {16}}$

C

${1 \over {32}}$

D

${1 \over {18}}$

Question 46: Palette 46/60

JEE Mains 2019 MCQ

The value of cos²10° – cos10°cos50° + cos²50° is

A

${3 \over 2} + \cos {20^o}$

B

${3 \over 4}$

C

${3 \over 2}(1 + \cos {20^o})$

D

${3 \over 2}$

Question 47: Palette 47/60

JEE Mains 2019 MCQ

If cos($\alpha $ + $\beta $) = 3/5 ,sin ( $\alpha $ - $\beta $) = 5/13 and 0 < $\alpha , \beta$ < $\pi \over 4$, then tan(2$\alpha $) is equal to :

A

21/16

B

63/52

C

33/52

D

63/16

Question 48: Palette 48/60

JEE Mains 2019 MCQ

The maximum value of 3cos$\theta $ + 5sin $\left( {\theta - {\pi \over 6}} \right)$ for any real value of $\theta $ is :

A

$\sqrt {34} $

B

$\sqrt {31} $

C

$\sqrt {19} $

D

${{\sqrt {79} } \over 2}$

Question 49: Palette 49/60

JEE Mains 2019 MCQ

The value of $\cos {\pi \over {{2^2}}}.\cos {\pi \over {{2^3}}}\,.....\cos {\pi \over {{2^{10}}}}.\sin {\pi \over {{2^{10}}}}$ is -

A

${1 \over {256}}$

B

${1 \over {2}}$

C

${1 \over {1024}}$

D

${1 \over {512}}$

Question 50: Palette 50/60

JEE Mains 2019 MCQ

For any $\theta \in \left( {{\pi \over 4},{\pi \over 2}} \right)$, the expression

$3{(\cos \theta - \sin \theta )^4}$$ + 6{(\sin \theta + \cos \theta )^2} + 4{\sin ^6}\theta $

equals :

A

13 – 4 cos²$\theta $ + 6sin²$\theta $cos²$\theta $

B

13 – 4 cos⁶$\theta $

C

13 – 4 cos²$\theta $ + 6cos²$\theta $

D

13 – 4 cos⁴$\theta $ + 2sin²$\theta $cos²$\theta $

Question 51: Palette 51/60

JEE Mains 2017 MCQ

If $5\left( {{{\tan }^2}x - {{\cos }^2}x} \right) = 2\cos 2x + 9$,

then the value of $\cos 4x$ is :

A

${1 \over 3}$

B

${2 \over 9}$

C

$ - {7 \over 9}$

D

$ - {3 \over 5}$

Question 52: Palette 52/60

JEE Mains 2016 MCQ

If  m and M are the minimum and the maximum values of

4 + ${1 \over 2}$ sin² 2x $-$ 2cos⁴ x, x $ \in $ R, then M $-$ m is equal to :

A

${{15} \over 4}$

B

${{9} \over 4}$

C

${{7} \over 4}$

D

${{1} \over 4}$

Question 53: Palette 53/60

JEE Mains 2014 MCQ

Let $f_k\left( x \right) = {1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$ where $x \in R$ and $k \ge \,1.$
Then ${f_4}\left( x \right) - {f_6}\left( x \right)\,\,$ equals :

A

${1 \over 4}$

B

${1 \over 12}$

C

${1 \over 6}$

D

${1 \over 3}$

Question 54: Palette 54/60

JEE Mains 2013 MCQ

The expression ${{\tan {\rm A}} \over {1 - \cot {\rm A}}} + {{\cot {\rm A}} \over {1 - \tan {\rm A}}}$ can be written as:

A

$\sin {\rm A}\,\cos {\rm A} + 1$

B

$\,\sec {\rm A}\,\cos ec{\rm A} + 1$

C

$\tan {\rm A} + \cot {\rm A}$

D

$\sec {\rm A} + \cos ec{\rm A}$

Question 55: Palette 55/60

JEE Mains 2011 MCQ

If $A = {\sin ^2}x + {\cos ^4}x,$ then for all real $x$:

A

${{13} \over {16}} \le A \le 1$

B

$1 \le A \le 2$

C

${3 \over 4} \le A \le {{13} \over {16}}$

D

${{3} \over {4}} \le A \le 1$

Question 56: Palette 56/60

JEE Mains 2010 MCQ

Let $\cos \left( {\alpha + \beta } \right) = {4 \over 5}$ and $\sin \,\,\,\left( {\alpha - \beta } \right) = {5 \over {13}},$ where $0 \le \alpha ,\,\beta \le {\pi \over 4}.$
Then $tan\,2\alpha $ =

A

${56 \over 33}$

B

${19 \over 12}$

C

${20 \over 7}$

D

${25 \over 16}$

Question 57: Palette 57/60

JEE Mains 2009 MCQ

Let A and B denote the statements

A: $\cos \alpha + \cos \beta + \cos \gamma = 0$

B: $\sin \alpha + \sin \beta + \sin \gamma = 0$

If $\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3 \over 2},$ then:

A

A is false and B is true

B

both A and B are true

C

both A and B are false

D

A is true and B is false

Question 58: Palette 58/60

JEE Mains 2006 MCQ

If $0 < x < \pi $ and $\cos x + \sin x = {1 \over 2},$ then $\tan x$ is :

A

${{\left( {1 - \sqrt 7 } \right)} \over 4}$

B

${{\left( {4 - \sqrt 7 } \right)} \over 3}$

C

$ - {{\left( {4 + \sqrt 7 } \right)} \over 3}$

D

${{\left( {1 + \sqrt 7 } \right)} \over 4}$

Question 59: Palette 59/60

JEE Mains 2004 MCQ

If $u = \sqrt {{a^2}{{\cos }^2}\theta + {b^2}{{\sin }^2}\theta } + \sqrt {{a^2}{{\sin }^2}\theta + {b^2}{{\cos }^2}\theta } $

then the difference between the maximum and minimum values of ${u^2}$ is given by :

A

${\left( {a - b} \right)^2}$

B

$2\sqrt {{a^2} + {b^2}} $

C

${\left( {a + b} \right)^2}$

D

$2\left( {{a^2} + {b^2}} \right)$

Question 60: Palette 60/60

JEE Mains 2004 MCQ

Let $\alpha ,\,\beta $ be such that $\pi < \alpha - \beta < 3\pi $.
If $sin{\mkern 1mu} \alpha + \sin \beta = - {{21} \over {65}}$ and $\cos \alpha + \cos \beta = - {{27} \over {65}}$ then the value of $\cos {{\alpha - \beta } \over 2}$ :

A

${{ - 6} \over {65}}\,\,$

B

${3 \over {\sqrt {130} }}$

C

${6 \over {65}}$

D

$ - {3 \over {\sqrt {130} }}$