1
JEE Main 2022 (Online) 24th June Evening Shift
+4
-1
Out of Syllabus

$$\mathop {\lim }\limits_{n \to \infty } \left( {{{{n^2}} \over {({n^2} + 1)(n + 1)}} + {{{n^2}} \over {({n^2} + 4)(n + 2)}} + {{{n^2}} \over {({n^2} + 9)(n + 3)}} + \,\,....\,\, + \,\,{{{n^2}} \over {({n^2} + {n^2})(n + n)}}} \right)$$ 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
$${\pi \over 8} + {\log _e}\sqrt 2$$
2
JEE Main 2021 (Online) 1st September Evening Shift
+4
-1
Let f : R $$\to$$ R be a continuous function. Then $$\mathop {\lim }\limits_{x \to {\pi \over 4}} {{{\pi \over 4}\int\limits_2^{{{\sec }^2}x} {f(x)\,dx} } \over {{x^2} - {{{\pi ^2}} \over {16}}}}$$ is equal to :
A
f (2)
B
2f (2)
C
2f $$\left( {\sqrt 2 } \right)$$
D
4f (2)
3
JEE Main 2021 (Online) 1st September Evening Shift
+4
-1
Let $${J_{n,m}} = \int\limits_0^{{1 \over 2}} {{{{x^n}} \over {{x^m} - 1}}dx}$$, $$\forall$$ n > m and n, m $$\in$$ N. Consider a matrix $$A = {[{a_{ij}}]_{3 \times 3}}$$ where $${a_{ij}} = \left\{ {\matrix{ {{j_{6 + i,3}} - {j_{i + 3,3}},} & {i \le j} \cr {0,} & {i > j} \cr } } \right.$$. Then $$\left| {adj{A^{ - 1}}} \right|$$ is :
A
(15)2 $$\times$$ 242
B
(15)2 $$\times$$ 234
C
(105)2 $$\times$$ 238
D
(105)2 $$\times$$ 236
4
JEE Main 2021 (Online) 1st September Evening Shift
+4
-1
The function f(x), that satisfies the condition
$$f(x) = x + \int\limits_0^{\pi /2} {\sin x.\cos y\,f(y)\,dy}$$, is :
A
$$x + {2 \over 3}(\pi - 2)\sin x$$
B
$$x + (\pi + 2)\sin x$$
C
$$x + {\pi \over 2}\sin x$$
D
$$x + (\pi - 2)\sin x$$
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