1
JEE Main 2016 (Online) 10th April Morning Slot
+4
-1
The value of the integral

$$\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,$$

where [x] denotes the greatest integer less than or equal to x, is :
A
6
B
3
C
7
D
$${1 \over 3}$$
2
JEE Main 2016 (Online) 10th April Morning Slot
+4
-1
For x $$\in$$ R, x $$\ne$$ 0, if y(x) is a differentiable function such that

x $$\int\limits_1^x y$$ (t) dt = (x + 1) $$\int\limits_1^x ty$$ (t) dt,  then y (x) equals :

(where C is a constant.)
A
$${C \over x}{e^{ - {1 \over x}}}$$
B
$${C \over {{x^2}}}{e^{ - {1 \over x}}}$$
C
$${C \over {{x^3}}}{e^{ - {1 \over x}}}$$
D
$$C{x^3}\,{1 \over {{e^x}}}$$
3
JEE Main 2016 (Online) 9th April Morning Slot
+4
-1
If   $$2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,$$

then $$\int\limits_0^1 {{{\tan }^{ - 1}}} \left( {1 - x + {x^2}} \right)dx$$ is equalto :
A
log4
B
$${\pi \over 2}$$ + log2
C
log2
D
$${\pi \over 2}$$ $$-$$ log4
4
JEE Main 2016 (Offline)
+4
-1
Out of Syllabus
$$\mathop {\lim }\limits_{n \to \infty } {\left( {{{\left( {n + 1} \right)\left( {n + 2} \right)...3n} \over {{n^{2n}}}}} \right)^{{1 \over n}}}$$ is equal to:
A
$${9 \over {{e^2}}}$$
B
$$3\,\log \,3 - 2$$
C
$${{18} \over {{e^4}}}$$
D
$${{27} \over {{e^2}}}$$
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