1

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

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
$${{{{\sin }^8}x} \over 8} + {{{{\cos }^8}x} \over {27}} = {2 \over {125}}$$
3

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
$$\,{5\pi \over 12}$$
4

IIT-JEE 1999

MCQ (More than One Correct Answer)
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
$${f_5}\left( {{\pi \over {128}}} \right) = 1$$

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