1
JEE Main 2022 (Online) 25th June Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $${b_n} = \int_0^{{\pi \over 2}} {{{{{\cos }^2}nx} \over {\sin x}}dx,\,n \in N} $$, then

A
$${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$$ are in A.P. with common difference $$-$$2
B
$${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$$ are in an A.P. with common difference 2
C
$${b_3} - {b_2},\,{b_4} - {b_3},\,{b_5} - {b_4}$$ are in a G.P.
D
$${1 \over {{b_3} - {b_2}}},{1 \over {{b_4} - {b_3}}},{1 \over {{b_5} - {b_4}}}$$ are in an A.P. with common difference $$-$$2
2
JEE Main 2022 (Online) 25th June Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of $$\int\limits_0^\pi {{{{e^{\cos x}}\sin x} \over {(1 + {{\cos }^2}x)({e^{\cos x}} + {e^{ - \cos x}})}}dx} $$ is equal to:

A
$${{{\pi ^2}} \over 4}$$
B
$${{{\pi ^2}} \over 2}$$
C
$${\pi \over 4}$$
D
$${\pi \over 2}$$
3
JEE Main 2022 (Online) 24th June Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The value of the integral

$$\int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {(1 + {e^x})({{\sin }^6}x + {{\cos }^6}x)}}} $$ is equal to

A
2$$\pi$$
B
0
C
$$\pi$$
D
$${\pi \over 2}$$
4
JEE Main 2022 (Online) 24th June Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\, + \,\,{{{n^2}} \over {({n^2} + {n^2})(n + n)}}} \right)$$ is equal to :

A
$${\pi \over 8} + {1 \over 4}{\log _e}2$$
B
$${\pi \over 4} + {1 \over 8}{\log _e}2$$
C
$${\pi \over 4} - {1 \over 8}{\log _e}2$$
D
$${\pi \over 8} + {\log _e}\sqrt 2 $$
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