1
JEE Main 2021 (Online) 26th February Morning Shift
+4
-1
The value of $$\sum\limits_{n = 1}^{100} {\int\limits_{n - 1}^n {{e^{x - [x]}}dx} }$$, where [ x ] is the greatest integer $$\le$$ x, is :
A
100e
B
100(e $$-$$ 1)
C
100(1 + e)
D
100(1 $$-$$ e)
2
JEE Main 2021 (Online) 25th February Evening Shift
+4
-1
If $${I_n} = \int\limits_{{\pi \over 4}}^{{\pi \over 2}} {{{\cot }^n}x\,dx}$$, then :
A
$${1 \over {{I_2} + {I_4}}},{1 \over {{I_3} + {I_5}}},{1 \over {{I_4} + {I_6}}}$$ are in A.P.
B
I2 + I4, I3 + I5, I4 + I6 are in A.P.
C
$${1 \over {{I_2} + {I_4}}},{1 \over {{I_3} + {I_5}}},{1 \over {{I_4} + {I_6}}}$$ are in G.P.
D
I2 + I4, (I3 + I5)2, I4 + I6 are in G.P.
3
JEE Main 2021 (Online) 25th February Evening Shift
+4
-1
Out of Syllabus
$$\mathop {\lim }\limits_{n \to \infty } \left[ {{1 \over n} + {n \over {{{(n + 1)}^2}}} + {n \over {{{(n + 2)}^2}}} + ........ + {n \over {{{(2n + 1)}^2}}}} \right]$$ is equal to :
A
$${{1 \over 2}}$$
B
$${{1 \over 3}}$$
C
1
D
$${{1 \over 4}}$$
4
JEE Main 2021 (Online) 25th February Morning Shift
+4
-1
The value of $$\int\limits_{ - 1}^1 {{x^2}{e^{[{x^3}]}}} dx$$, where [ t ] denotes the greatest integer $$\le$$ t, is :
A
$${{e + 1} \over 3}$$
B
$${{e - 1} \over {3e}}$$
C
$${1 \over {3e}}$$
D
$${{e + 1} \over {3e}}$$
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