1
JEE Main 2022 (Online) 26th July Morning Shift
+4
-1
Out of Syllabus

If $$a = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {{{2n} \over {{n^2} + {k^2}}}}$$ and $$f(x) = \sqrt {{{1 - \cos x} \over {1 + \cos x}}}$$, $$x \in (0,1)$$, then :

A
$$2\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
B
$$f\left( {{a \over 2}} \right)f'\left( {{a \over 2}} \right) = \sqrt 2$$
C
$$\sqrt 2 f\left( {{a \over 2}} \right) = f'\left( {{a \over 2}} \right)$$
D
$$f\left( {{a \over 2}} \right) = \sqrt 2 f'\left( {{a \over 2}} \right)$$
2
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1
Out of Syllabus

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

A
$$\frac{1}{2}$$
B
1
C
2
D
$$-$$2
3
JEE Main 2022 (Online) 25th July Evening Shift
+4
-1

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_{-3}^{101}\left([\sin (\pi x)]+e^{[\cos (2 \pi x)]}\right) d x$$ is equal to

A
$$\frac{52(1-e)}{e}$$
B
$$\frac{52}{e}$$
C
$$\frac{52(2+e)}{e}$$
D
$$\frac{104}{e}$$
4
JEE Main 2022 (Online) 25th July Morning Shift
+4
-1

For any real number $$x$$, let $$[x]$$ denote the largest integer less than equal to $$x$$. Let $$f$$ be a real valued function defined on the interval $$[-10,10]$$ by $$f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$$ Then the value of $$\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$$ is :

A
4
B
2
C
1
D
0
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