1

### JEE Advanced 2013 Paper 1 Offline

MCQ (More than One Correct Answer)
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 $$\left( {n,\,n + 1} \right)$$
2

### IIT-JEE 2012 Paper 1 Offline

MCQ (More than One Correct Answer)
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
$${{3\pi } \over 2} < \varphi < 2\pi$$
3

### IIT-JEE 2009

MCQ (More than One Correct Answer)
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
$${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {2 \over {125}}$$
4

### IIT-JEE 2009

MCQ (More than One Correct Answer)
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
$$\,{5\pi \over 12}$$

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Class 12