Question 1: Palette 1/60

JEE Advanced 2026 MCQ

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
(P) The number of elements in the set $\left\{x \in [-\pi,\pi] : \sin^6 x + \cos^4 x = 1 \right\}$	(1) is 1
(Q) The number of elements in the set $\left\{x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] : \sin^2 x + \cos^6 x = 1 \right\}$	(2) is 2
(R) The number of elements in the set $\left\{x \in [-\pi,\pi] : \cos^2\left(\frac{x}{2}\right) - \sin^2 x = \frac{1}{2} \right\}$	(3) is 3
(S) The number of elements in the set $\left\{x \in [-2\pi,2\pi] : 6\sin^2\left(\frac{x}{2}\right) - \cos 3x = 3 \right\}$	(4) is 4 (5) is 5

A

(P) $\to$ (2), (Q) $\to$ (5), (R) $\to$ (3), (S) $\to$ (4)

B

(P) $\to$ (5), (Q) $\to$ (3), (R) $\to$ (2), (S) $\to$ (4)

C

(P) $\to$ (5), (Q) $\to$ (4), (R) $\to$ (1), (S) $\to$ (3)

D

(P) $\to$ (4), (Q) $\to$ (3), (R) $\to$ (2), (S) $\to$ (5)

Question 2: Palette 2/60

JEE Advanced 2024 MCQ

Let $\frac{\pi}{2} < x < \pi$ be such that $\cot x=\frac{-5}{\sqrt{11}}$. Then

$ \left(\sin \frac{11 x}{2}\right)(\sin 6 x-\cos 6 x)+\left(\cos \frac{11 x}{2}\right)(\sin 6 x+\cos 6 x) $

is equal to :

A

$\frac{\sqrt{11}-1}{2 \sqrt{3}}$

B

$\frac{\sqrt{11}+1}{2 \sqrt{3}}$

C

$\frac{\sqrt{11}+1}{3 \sqrt{2}}$

D

$\frac{\sqrt{11}-1}{3 \sqrt{2}}$

Question 3: Palette 3/60

JEE Advanced 2022 MCQ

Consider the following lists :

List-I	List-II
(I) $\left\{x \in\left[-\frac{2 \pi}{3}, \frac{2 \pi}{3}\right]: \cos x+\sin x=1\right\}$	(P) has two elements
(II) $\left\{x \in\left[-\frac{5 \pi}{18}, \frac{5 \pi}{18}\right]: \sqrt{3} \tan 3 x=1\right\}$	(Q) has three elements
(III) $\left\{x \in\left[-\frac{6 \pi}{5}, \frac{6 \pi}{5}\right]: 2 \cos (2 x)=\sqrt{3}\right\}$	(R) has four elements
(IV) $\left\{x \in\left[-\frac{7 \pi}{4}, \frac{7 \pi}{4}\right]: \sin x-\cos x=1\right\}$	(S) has five elements
	(T) has six elements

The correct option is:

A

(I) $\rightarrow(\mathrm{P})$; (II) $\rightarrow(\mathrm{S})$; (III) $\rightarrow(\mathrm{P})$; (IV) $\rightarrow(\mathrm{S})$

B

(I) $\rightarrow$ (P); (II) $\rightarrow$ (P); (III) $\rightarrow$ (T); (IV) $\rightarrow$ (R)

C

(I) $\rightarrow$ (Q); (II) $\rightarrow(\mathrm{P})$; (III) $\rightarrow$ (T); (IV) $\rightarrow$ (S)

D

(I) $\rightarrow(\mathrm{Q})$; (II) $\rightarrow(\mathrm{S}) ;$ (III) $\rightarrow(\mathrm{P})$; (IV) $\rightarrow(\mathrm{R})$

Question 4: Palette 4/60

JEE Advanced 2022 MSQ

Let $P Q R S$ be a quadrilateral in a plane, where

$Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$.

If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of

$4 \alpha \beta \sin \theta^{\circ}$ is/are

A

$(0, \sqrt{2})$

B

$(1,2)$

C

$(\sqrt{2}, 3)$

D

$(2 \sqrt{2}, 3 \sqrt{2})$

Question 5: Palette 5/60

JEE Advanced 2019 MCQ

Let f(x) = sin($\pi $ cos x) and g(x) = cos(2$\pi $ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:

X = {x : f(x) = 0}, Y = {x : f'(x) = 0}

Z = {x : g(x) = 0}, W = {x : g'(x) = 0}

List - I contains the sets X, Y, Z and W. List - II contains some information regarding these sets.



Which of the following is the only CORRECT combination?

A

(IV), (P), (R), (S)

B

(III), (P), (Q), (U)

C

(III), (R), (U)

D

(IV), (Q), (T)

Question 6: Palette 6/60

JEE Advanced 2019 MCQ

Let f(x) = sin($\pi $ cos x) and g(x) = cos(2$\pi $ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order :

X = {x : f(x) = 0}, Y = {x : f'(x) = 0}

Z = {x : g(x) = 0}, W = {x : g'(x) = 0}

List - I contains the sets X, Y, Z and W. List - II contains some information regarding these sets.


Which of the following combinations is correct?

A

(II), (Q), (T)

B

(II), (R), (S)

C

(I), (P), (R)

D

(I), (Q), (U)

Question 7: Palette 7/60

JEE Advanced 2019 MSQ

For non-negative integers n, let

$f(n) = {{\sum\limits_{k = 0}^n {\sin \left( {{{k + 1} \over {n + 2}}\pi } \right)} \sin \left( {{{k + 2} \over {n + 2}}\pi } \right)} \over {\sum\limits_{k = 0}^n {{{\sin }^2}\left( {{{k + 1} \over {n + 2}}\pi } \right)} }}$

Assuming cos^$-1$ x takes values in [0, $\pi $], which of the following options is/are correct?

A

If $\alpha $ = tan(cos^$-$1 f(6)), then $\alpha $² + 2$\alpha $ $-$1 = 0

B

$f(4) = {{\sqrt 3 } \over 2}$

C

sin(7 cos^$-$1 f(5)) = 0

D

$\mathop {\lim }\limits_{n \to \infty } \,f(n) = {1 \over 2}$

Question 8: Palette 8/60

JEE Advanced 2018 MSQ

In a $\Delta $PQR = 30$^\circ $ and the sides PQ and QR have lengths 10$\sqrt 3 $ and 10, respectively. Then, which of the following statement(s) is(are) TRUE?

A

$\angle QPR = 45^\circ $

B

The area of the $\Delta PQR$ is $25\sqrt 3 $ and $\angle QRP = 120^\circ $

C

The radius of the incircle of the $\Delta PQR$ is $10\sqrt 3 $ $-$ 15

D

The area of the circumcircle of the $\Delta PQR$ is 100$\pi $

Question 9: Palette 9/60

JEE Advanced 2017 MCQ

If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

A

$ - {3 \over 2}$

B

${3 \over 2}$

C

${5 \over 3}$

D

$ - {5 \over 3}$

Question 10: Palette 10/60

JEE Advanced 2017 MSQ

Let $\alpha $ and $\beta $ be non zero real numbers such that $2(\cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$. Then which of the following is/are true?

A

$\sqrt 3 \tan \left( {{\alpha \over 2}} \right) - \tan \left( {{\beta \over 2}} \right) = 2$

B

$\tan \left( {{\alpha \over 2}} \right) - \sqrt 3 \tan \left( {{\beta \over 2}} \right) = 0$

C

$\tan \left( {{\alpha \over 2}} \right) + \sqrt 3 \tan \left( {{\beta \over 2}} \right) = 0$

D

$\sqrt 3 \tan \left( {{\alpha \over 2}} \right) + \tan \left( {{\beta \over 2}} \right) = 2$

Question 11: Palette 11/60

JEE Advanced 2016 MCQ

The value of

$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}} $ is equal to

A

$3 - \sqrt 3 $

B

$2\left( {3 - \sqrt 3 } \right)$

C

$2\left( {\sqrt 3 - 1} \right)\,\,\,$

D

$2\left( {2 - \sqrt 3 } \right)$

Question 12: Palette 12/60

JEE Advanced 2016 MCQ

Let $S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$ The sum of all distinct solutions of the equation $\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0$ in the set S is equal to

A

$ - {{7\pi } \over 9}$

B

$ - {{2\pi } \over 9}$

C

0

D

${{5\pi } \over 9}$

Question 13: Palette 13/60

JEE Advanced 2014 MCQ

For $x \in \left( {0,\pi } \right),$ the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has

A

infinitely many solutions

B

three solutions

C

one solution

D

no solution

Question 14: Palette 14/60

JEE Advanced 2013 MCQ

The number of points in $\left( { - \infty \,\infty } \right),$ for which ${x^2} - x\sin x - \cos x = 0,$ is

A

6

B

4

C

2

D

0

Question 15: Palette 15/60

JEE Advanced 2013 MSQ

Let $f\left( x \right) = x\sin \,\pi x,\,x > 0.$ Then for all natural numbers $n,\,f'\left( x \right)$ vanishes at

A

A unique point in the interval $\left( {n,\,n + {1 \over 2}} \right)$

B

A unique point in the interval $\left( {n + {1 \over 2},n + 1} \right)$

C

A unique point in the interval $\left( {n,\,n + 1} \right)$

D

Two points in the interval $\left( {n,\,n + 1} \right)$

Question 16: Palette 16/60

JEE Advanced 2012 MSQ

Let $\theta ,\,\varphi \, \in \,\left[ {0,2\pi } \right]$ be such that
$2\cos \theta \left( {1 - \sin \,\varphi } \right) = {\sin ^2}\theta \,\,\left( {\tan {\theta \over 2} + \cot {\theta \over 2}} \right)\cos \varphi - 1,\,\tan \left( {2\pi - \theta } \right) > 0$ and $ - 1 < \sin \theta \, < - {{\sqrt 3 } \over 2},$

then $\varphi $ cannot satisfy

A

$0 < \varphi < {\pi \over 2}$

B

${\pi \over 2} < \varphi < {{4\pi } \over 3}$

C

${{4\pi } \over 3} < \varphi < {{3\pi } \over 2}$

D

${{3\pi } \over 2} < \varphi < 2\pi $

Question 17: Palette 17/60

JEE Advanced 2011 MCQ

Let $P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \} $ and $Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \} $ be two sets. Then

A

$P \subset Q$ and $Q - P \ne \emptyset $

B

$Q \not\subset P$

C

$P \not\subset Q$

D

$P = Q$

Question 18: Palette 18/60

JEE Advanced 2009 MSQ

Match the statements/expressions in Column I with the values given in Column II:

	Column I		Column II
(A)	Root(s) of the expression $2{\sin ^2}\theta + {\sin ^2}2\theta = 2$	(P)	${\pi \over 6}$
(B)	Points of discontinuity of the function $f(x) = \left[ {{{6x} \over \pi }} \right]\cos \left[ {{{3x} \over \pi }} \right]$, where $[y]$ denotes the largest integer less than or equal to y	(Q)	${\pi \over 4}$
(C)	Volume of the parallelopiped with its edges represented by the vectors $\widehat i + \widehat j + \widehat i + 2\widehat j$ and $\widehat i + \widehat j + \pi \widehat k$	(R)	${\pi \over 3}$
(D)	Angle between vectors $\overrightarrow a $ and $\overrightarrow b $ where $\overrightarrow a $, $\overrightarrow b $ and $\overrightarrow c $ are unit vectors satisfying $\overrightarrow a + \overrightarrow b + \sqrt 3 \overrightarrow c = \overrightarrow 0 $	(S)	${\pi \over 2}$
		(T)	$\pi $

A

(A)$\to$(Q), (S); (B)$\to$(P), (R), (S), (T); (C)$\to$(Q); (D)$\to$(T)

B

(A)$\to$(R), (S); (B)$\to$(P), (R), (S), (T); (C)$\to$(T); (D)$\to$(P)

C

(A)$\to$(Q), (S); (B)$\to$(P), (R), (S), (T); (C)$\to$(T); (D)$\to$(R)

D

(A)$\to$(P), (S); (B)$\to$(Q), (R), (S), (T); (C)$\to$(T); (D)$\to$(R)

Question 19: Palette 19/60

JEE Advanced 2009 MSQ

For $0 < \theta < {\pi \over 2},$ the solution (s) of $$\sum\limits_{m = 1}^6 {\cos ec\,\left( {\theta + {{\left( {m - 1} \right)\pi } \over 4}} \right)\,\cos ec\,\left( {\theta + {{m\pi } \over 4}} \right) = 4\sqrt 2 } $$ is (are)

A

$\,{\pi \over 4}$

B

$\,{\pi \over 6 }$

C

$\,{\pi \over 12}$

D

$\,{5\pi \over 12}$

Question 20: Palette 20/60

JEE Advanced 2009 MSQ

If ${{{{\sin }^4}x} \over 2} + {{{{\cos }^4}x} \over 3} = {1 \over 5},$ then

A

${\tan ^2}x = {2 \over 3}$

B

${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {1 \over {125}}$

C

${\tan ^2}x = {1 \over 3}$

D

${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {2 \over {125}}$

Question 21: Palette 21/60

JEE Advanced 2008 MCQ

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II.

	Column I		Column II
(A)	The minimum value of ${{{x^2} + 2x + 4} \over {x + 2}}$ is	(P)	0
(B)	Let A and B be 3 $\times$ 3 matrices of real numbers, where A is symmetric, B is skew-symmetric and (A + B) (A $-$ B) = (A $-$ B) (A + B). If (AB)$^t$ = ($-1$)$^k$ AB, where (AB)$^t$ is the transpose of the matrix AB, then the possible values of k are	(Q)	1
(C)	Let $a=\log_3\log_3 2$. An integer k satisfying $1 < {2^{( - k + 3 - a)}} < 2$, must be less than	(R)	2
(D)	If $\sin \theta = \cos \varphi $, then the possible values of ${1 \over \pi }\left( {\theta + \varphi - {\pi \over 2}} \right)$ are	(S)	3

A

A - iii; B - ii, iv; C - iii, iv; D - i, iii

B

A - iii; B - ii; C - iii, iv; D - i, iii

C

A - ii; B - ii, iv; C - iii, iv; D - i

D

A - ii; B - ii, iv; C - iii, iv; D - i, iii

Question 22: Palette 22/60

JEE Advanced 2007 MCQ

The number of solutions of the pair of equations $$\,2{\sin ^2}\theta - \cos 2\theta = 0$$ $$2co{s^2}\theta - 3\sin \theta = 0$$

in the interval $\left[ {0,2\pi } \right]$

A

zero

B

one

C

two

D

four

Question 23: Palette 23/60

JEE Advanced 2007 MCQ

The number of solutions of the pair of equations

$2{\sin ^2}\theta - \cos 2\theta = 0$

$2{\cos ^2}\theta - 3\sin \theta = 0$

in the interval $[0,2\pi]$ is

A

zero

B

one

C

two

D

four

Question 24: Palette 24/60

JEE Advanced 2006 MCQ

Let $\theta \in\left(0, \frac{\pi}{4}\right)$ and $t_{1}=(\tan \theta)^{\tan \theta}, t_{2}=(\tan \theta)^{\cot \theta}, t_{3}=(\cot \theta)^{\tan \theta}$ and $t_{4}=(\cot \theta)^{\cot \theta}$, then

A

$t_{1}>t_{2}>t_{3}>t_{4}$

B

$t_{4}>t_{3}>t_{1}>t_{2}$

C

$t_{3}>t_{1}>t_{2}>t_{4}$

D

$t_{2}>t_{3}>t_{1}>t_{4}$

Question 25: Palette 25/60

JEE Advanced 2006 MCQ

If $0<\theta<2 \pi$, then the intervals of values of $\theta$ for which $2 \sin ^2 \theta-5 \sin \theta+2>0$, is

A

$\left(0, \frac{\pi}{6}\right) \cup\left(\frac{5 \pi}{6}, 2 \pi\right)$

B

$\left(\frac{\pi}{8}, \frac{5 \pi}{6}\right)$

C

$\left(0, \frac{\pi}{8}\right) \cup\left(\frac{\pi}{6}, \frac{5 \pi}{6}\right)$

D

$\left(\frac{41 \pi}{48}, \pi\right)$

Question 26: Palette 26/60

JEE Advanced 2005 MCQ

$\cos \left( {\alpha - \beta } \right) = 1$ and $\,\cos \left( {\alpha + \beta } \right) = 1/e$ where $\alpha ,\,\beta \in \left[ { - \pi ,\pi } \right].$
Paris of $\alpha ,\,\beta $ which satisfy both the equations is/are

A

0

B

1

C

2

D

4

Question 27: Palette 27/60

JEE Advanced 2004 MCQ

Given both $\theta $ and $\phi $ are acute angles and $\sin \,\theta = {1 \over 2},\,$ $\cos \,\phi = {1 \over 3},$ then the value of $\theta + \phi $ belongs to

A

$\left( {{\pi \over 3},\left. {{\pi \over 2}} \right]} \right.$

B

$\left( {{\pi \over 2},{{2\pi } \over 3}} \right)$

C

$\left( {{{2\pi } \over 3},\left. {{{5\pi } \over 6}} \right]} \right.$

D

$\left( {{{5\pi } \over 6},\pi } \right]$

Question 28: Palette 28/60

JEE Advanced 2002 MCQ

The number of integral values of $k$ for which the equation $7\cos x + 5\sin x = 2k + 1$ has a solution is

A

4

B

8

C

10

D

12

Question 29: Palette 29/60

JEE Advanced 2001 MCQ

The maximum value of $\left( {\cos {\alpha _1}} \right).\left( {\cos {\alpha _2}} \right).....\left( {\cos {\alpha _n}} \right),$ under the restrictions
$0 \le {\alpha _1},{\alpha _2},....,{\alpha _n} \le {\pi \over 2}$ vand $\left( {\cot {\alpha _1}} \right).\left( {\cot {\alpha _2}} \right)....\left( {\cot {\alpha _n}} \right) = 1$ is

A

$1/{2^{n/2}}$

B

$1/{2^n}$

C

$1/2n\,$

D

1

Question 30: Palette 30/60

JEE Advanced 2001 MCQ

If $\alpha + \beta = \pi /2$ and $\beta + \gamma = \alpha ,$ then $\tan \,\alpha \,$ equals

A

$2\left( {\tan \beta + \tan \gamma } \right)$

B

$\,\tan \beta + \tan \gamma $

C

$\tan \beta + 2\tan \gamma $

D

$2\tan \beta + \tan \gamma $

Question 31: Palette 31/60

JEE Advanced 2001 MCQ

The number of distinct real roots of $\left| {\matrix{ {\sin x} & {\cos x} & {\cos x} \cr {\cos x} & {\sin x} & {\cos x} \cr {\cos x} & {\cos x} & {\sin x} \cr } } \right|\,$
$\, = 0$ in the interval $ - {\pi \over 4} \le x \le {\pi \over 4}$ is

A

0

B

2

C

1

D

3

Question 32: Palette 32/60

JEE Advanced 2000 MCQ

Let $f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right)$. Then $f\left( \theta \right)$ is

A

$ \ge 0\,\,$ only when $\theta \ge 0$

B

$ \le 0$ for all real $\theta $

C

$ \ge 0$ for all real $\theta $

D

$ \le 0$ only when $\theta \le 0$

Question 33: Palette 33/60

JEE Advanced 1999 MCQ

In a triangle $PQR,\angle R = \pi /2$. If $\,\,\tan \left( {P/2} \right)$ and $\tan \left( {Q/2} \right)$ are the roots of the equation $a{x^2} + bx + c = 0\left( {a \ne 0} \right)$ then.

A

$a + b = c$

B

$a + c = b$

C

$b + c = a$

D

$b = c$

Question 34: Palette 34/60

JEE Advanced 1999 MSQ

For a positive integer $\,n$, let
${f_n}\left( \theta \right) = \left( {\tan {\theta \over 2}} \right)\,\left( {1 + \sec \theta } \right)\,\left( {1 + \sec 2\theta } \right)\,\left( {1 + \sec 4\theta } \right).....\left( {1 + \sec {2^n}\theta } \right).$ Then

A

${f_2}\left( {{\pi \over {16}}} \right) = 1$

B

${f_3}\left( {{\pi \over {32}}} \right) = 1$

C

${f_4}\left( {{\pi \over {64}}} \right) = 1$

D

${f_5}\left( {{\pi \over {128}}} \right) = 1$

Question 35: Palette 35/60

JEE Advanced 1998 MCQ

The number of values of $x\,\,$ in the interval $\left[ {0,\,5\pi } \right]$ satisfying the equation $3\,{\sin ^2}x - 7\,\sin \,x + 2 = 0$ is

A

0

B

5

C

6

D

10

Question 36: Palette 36/60

JEE Advanced 1998 MCQ

Which of the following number(s) is /are rational?

A

$\sin {\mkern 1mu} {15^ \circ }$

B

$\cos {\mkern 1mu} {15^ \circ }$

C

$\sin {\mkern 1mu} {15^ \circ }{\mkern 1mu} \cos {\mkern 1mu} {15^ \circ }$

D

$\sin {\mkern 1mu} {15^ \circ }{\mkern 1mu} \cos {\mkern 1mu} {75^ \circ }$

Question 37: Palette 37/60

JEE Advanced 1996 MCQ

${\sec ^2}\theta = {{4xy} \over {{{\left( {x + y} \right)}^2}}}\,$ is true if and only if

A

$x + y \ne 0\,$

B

$x = y,\,x \ne 0$

C

$x = y\,$

D

$x \ne 0,\,y \ne 0$

Question 38: Palette 38/60

JEE Advanced 1995 MCQ

The minimum value of the expression $\sin \,\alpha + \sin \,\beta \, + \sin \,\gamma ,\,$ where $\alpha ,\,\beta ,\,\gamma $ are real numbers satisfying $\alpha + \beta + \gamma = \pi $ is

A

positive

B

zero

C

negative

D

-3

Question 39: Palette 39/60

JEE Advanced 1995 MCQ

The general values of $\theta $ satisfying the equation $2{\sin ^2}\theta - 3\sin \theta - 2 = 0$ is

A

$n\pi + {\left( { - 1} \right)^n}\pi /6$

B

$n\pi + {\left( { - 1} \right)^n}\pi /2 $

C

$n\pi + {\left( { - 1} \right)^n}5\pi /6$

D

$n\pi + {\left( { - 1} \right)^n}7\pi /6$

Question 40: Palette 40/60

JEE Advanced 1995 MCQ

$\,3{\left( {\sin x - \cos x} \right)^4} + 6{\left( {\sin x + \cos x} \right)^2} + 4\left( {{{\sin }^6}x + {{\cos }^6}x} \right) = $

A

11

B

12

C

13

D

14

Question 41: Palette 41/60

JEE Advanced 1994 MCQ

If $\omega \,$ is an imaginary cube root of unity then the value of $\sin \left\{ {\left( {{\omega ^{10}} + {\omega ^{23}}} \right)\pi - {\pi \over 4}} \right\}$ is

A

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

B

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

C

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

D

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

Question 42: Palette 42/60

JEE Advanced 1994 MCQ

Let $0 < x < {\pi \over 4}$ then $\left( {\sec 2x - \tan 2x} \right)$ equals

A

$\tan \left[ {x - {\pi \over 4}} \right]$

B

$\tan \left[ {{\pi \over 4} - x} \right]$

C

$\tan \left[ {x + {\pi \over 4}} \right]$

D

${\tan ^2}\left[ {x + {\pi \over 4}} \right]$

Question 43: Palette 43/60

JEE Advanced 1994 MCQ

Let $n$ be a positive integer such that $\sin {\pi \over {2n}} + \cos {\pi \over {2n}} = {{\sqrt n } \over 2}.$ Then

A

$6 \le n \le 8$

B

$4 < n \le 8$

C

$4 \le n \le 8$

D

$4 < n < 8$

Question 44: Palette 44/60

JEE Advanced 1994 MCQ

Let $2{\sin ^2}x + 3\sin x - 2 > 0$ and ${x^2} - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

A

$\left( {{\pi \over 6},\,{{5\pi } \over 6}} \right)\,\,$

B

$\left( { - 1,\,{{5\pi } \over 6}} \right)$

C

$\left( { - 1,\,2} \right)\,\,\,$

D

$\left( {{\pi \over 6},\,2} \right)$

Question 45: Palette 45/60

JEE Advanced 1993 MCQ

Number of solutions of the equation $\tan x + \sec x = 2\cos x\,$ lying in the interval $\left[ {0,2\pi } \right]$ is:

A

0

B

1

C

2

D

3

Question 46: Palette 46/60

JEE Advanced 1992 MCQ

In this questions there are entries in columns 1 and 2. Each entry in column 1 is related to exactly one entry in column 2. Write the correct letter from column 2 against the entry number in column 1 in your answer book.

${{\sin \,3\alpha } \over {\cos 2\alpha }}$ is

Column ${\rm I}$

(A) positive

(B) negative

Column ${\rm I}$${\rm I}$

(p) $\left( {{{13\pi } \over {48}},{{14\pi } \over {48}}} \right)$

(q) $\left( {{{14\pi } \over {48}},\,{{18\pi } \over {48}}} \right)$

(r) $\left( {{{18\pi } \over {48}},\,{{23\pi } \over {48}}} \right)$

(s) $\left( {0,\,{\pi \over 2}} \right)$

Options:-

A

$\left( A \right) - r,\,\left( B \right) - q$

B

$\left( A \right) - r,\,\left( B \right) - p$

C

$\left( A \right) - s,\,\left( B \right) - r$

D

$\left( A \right) - p,\,\left( B \right) - q$

Question 47: Palette 47/60

JEE Advanced 1990 MCQ

The equation $\left( {\cos p - 1} \right){x^2} + \left( {\cos p} \right)x + \sin p = 0\,$ In the variable x, has real roots. Then p can take any value in the interval

A

$\left( {0,2\pi } \right)\,$

B

$\left( { - \pi ,0} \right)\,\,\,$

C

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

D

$\left( {0,\pi } \right)$

Question 48: Palette 48/60

JEE Advanced 1989 MCQ

The general solutions of $\,\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x$ is

A

$n\pi + {\pi \over 8}$

B

${{n\pi } \over 2} + {\pi \over 8}$

C

${\left( { - 1} \right)^n}{{n\pi } \over 2} + {\pi \over 8}\,\,$

D

$2n\pi + {\cos ^{ - 1}}{3 \over 2}$

Question 49: Palette 49/60

JEE Advanced 1988 MCQ

The value of the expression $\sqrt 3 \,\cos \,ec\,{20^0} - \sec \,{20^0}$ is equal to

A

2

B

$2\sin {20^0}/\sin {40^0}$

C

4

D

$4\sin {20^0}/\sin {40^0}$

Question 50: Palette 50/60

JEE Advanced 1988 MSQ

The values of $\theta $ lying between $\theta = \theta $ and $\theta = \pi /2$ and satisfying the equation

$\left| {\matrix{ {1 + {{\sin }^2}\theta } & {{{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {1 + {{\cos }^2}\theta } & {4\sin 4\theta } \cr {{{\sin }^2}\theta } & {{{\cos }^2}\theta } & {1 + 4\sin 4\theta } \cr } } \right| = 0$ are

A

$7\pi /24$

B

$5\pi /24$

C

$11\pi /24$

D

$\pi /24$

Question 51: Palette 51/60

JEE Advanced 1987 MCQ

The number of all possible triplets $\left( {{a_1},\,{a_2},\,{a_3}} \right)$ such that ${a_1} + {a_2}\,\,\cos \left( {2x} \right) + {a_3}{\sin ^2}\left( x \right) = 0\,$ for all $x$ is

A

zero

B

one

C

three

D

infinite

Question 52: Palette 52/60

JEE Advanced 1986 MCQ

The expression $2\left[ {{{\sin }^6}\left( {{\pi \over 2} + \alpha } \right) + {{\sin }^6}\left( {5\pi - \alpha } \right)} \right]$ is equal to

A

0

B

1

C

3

D

$\sin \,4\,\alpha + \cos \,6\,\alpha \,\,\,\,$

Question 53: Palette 53/60

JEE Advanced 1984 MCQ

$\left( {1 + \cos {\pi \over 8}} \right)\left( {1 + \cos {{3\pi } \over 8}} \right)\left( {1 + \cos {{5\pi } \over 8}} \right)\left( {1 + \cos {{7\pi } \over 8}} \right)$ is equal to

A

${1 \over 2}$

B

$\cos {\pi \over 8}$

C

${1 \over 8}$

D

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

Question 54: Palette 54/60

JEE Advanced 1984 MCQ

There exists a value of $\theta $ between 0 and $2\pi $ that satisfies the equation $\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$

A

TRUE

B

FALSE

Question 55: Palette 55/60

JEE Advanced 1983 MCQ

If $\tan \,A = \left( {1 - \cos B} \right)/\sin B,$ then $tan2A = tan\,B$.

A

TRUE

B

FALSE

Question 56: Palette 56/60

JEE Advanced 1981 MCQ

The general solution of the trigonometric equation sin x+cos x=1 is given by:

A

$2n\pi ;\,n = 0,\, \pm 1,\, \pm 2....$

B

$x = 2n\pi + \pi /2;\,n = 0,\, \pm 1,\, \pm 2....$

C

$x = n\pi + {\left( { - 1} \right)^n}\,\,\,\,\,\,\,{\pi \over 4} - {\pi \over 4}$ ; $n = 0,\, \pm 1,\, \pm 2..$

D

none of these

Question 57: Palette 57/60

JEE Advanced 1980 MCQ

The equation $\,2{\cos ^2}{x \over 2}{\sin ^2}x = {x^2} + {x^{ - 2}};\,0 < x \le {\pi \over 2}$ has

A

no real solution

B

one real solution

C

more than one solution

D

none of these

Question 58: Palette 58/60

JEE Advanced 1980 MCQ

Given $A = {\sin ^2}\theta + {\cos ^4}\theta $ then for all real values of $\theta $

A

$1 \le A \le 2$

B

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

C

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

D

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

Question 59: Palette 59/60

JEE Advanced 1979 MCQ

If $\alpha + \beta + \gamma = 2\pi ,$ then

A

$tan{\alpha \over 2} + \tan {\beta \over 2} + \tan {\gamma \over 2} = \tan {\alpha \over 2}\tan {\beta \over 2}\tan {\gamma \over 2}$

B

$\tan {\alpha \over 2}\tan {\beta \over 2} + \tan {\beta \over 2}\tan {\gamma \over 2} + \tan {\gamma \over 2}\tan {\alpha \over 2} = 1$

C

$tan{\alpha \over 2} + \tan {\beta \over 2} + \tan {\gamma \over 2} = - \tan {\alpha \over 2}\tan {\beta \over 2}\tan {\gamma \over 2}$

D

None of these.

Question 60: Palette 60/60

JEE Advanced 1979 MCQ

If $\tan \theta = - {4 \over 3},then\sin \theta \,is\,$

A

$ - {4 \over 5}\,but\,not\,{4 \over 5}$

B

$ - {4 \over 5}\,or\,{4 \over 5}$

C

${4 \over 5}\,\,but\,not\, - {4 \over 5}$

D

None of these.