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1

IIT-JEE 2006

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
$${t_2} > {t_3} > {t_1} > {t_4}$$
2

IIT-JEE 2006 Screening

The values of $$\theta \in \left( {0,2\pi } \right)$$ for which $$2\,{\sin ^2}\theta - 5\,\sin \theta + 2 > 0,$$ are
A
$$\left( {0,{\pi \over 6}} \right)\, \cup \,\left( {{{5\pi } \over 6},2\pi } \right)$$
B
$$\,\left( {{\pi \over 8},{{5\pi } \over 6}} \right)$$
C
$$\left( {0,{\pi \over 8}} \right)\, \cup \,\left( {{\pi \over 6},{{5\pi } \over 6}} \right)$$ v
D
$$\,\left( {{{41\pi } \over {48}},\,\pi } \right)$$
3

IIT-JEE 2005 Screening

$$\cos \left( {\alpha - \beta } \right) = 1$$ 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
4

IIT-JEE 2004 Screening

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

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