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1

JEE Advanced 2013 Paper 1 Offline

MCQ (Single Correct Answer)
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
2

IIT-JEE 2011 Paper 1 Offline

MCQ (Single Correct Answer)

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
$$P = Q$$

Explanation

$$P = \{ \theta :\sin \theta - \cos \theta = \sqrt 2 \cos \theta \} $$

$$ \Rightarrow \cos \theta \left( {\sqrt 2 + 1} \right) = \sin \theta $$

$$ \Rightarrow \tan \theta = \sqrt 2 + 1$$ ..... (i)

$$Q = \{ \theta :\sin \theta + \cos \theta = \sqrt 2 \sin \theta \} $$

$$ \Rightarrow \sin \theta \left( {\sqrt 2 - 1} \right) = \cos \theta $$

$$ \Rightarrow \tan \theta = {1 \over {\sqrt 2 - 1}} \times {{\sqrt 2 + 1} \over {\sqrt 2 + 1}}$$

$$ = \left( {\sqrt 2 + 1} \right)$$ ...... (ii)

$$\therefore$$ $$P = Q$$

3

IIT-JEE 2007

MCQ (Single Correct Answer)
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
4

IIT-JEE 2006

MCQ (Single Correct Answer)
Let $$\theta \in \left( {0,{\pi \over 4}} \right)$$ and $${t_1} = {\left( {\tan \theta } \right)^{\tan \theta }},\,\,\,\,{t_2} = \,\,{\left( {\tan \theta } \right)^{\cot \theta }}$$, $${t_3}\, = \,\,{\left( {\cot \theta } \right)^{\tan \theta }}$$ and $${t_4}\, = \,\,{\left( {\cot \theta } \right)^{\cot \theta }},$$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
$${t_2} > {t_3} > {t_1} > {t_4}$$

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