1
JEE Advanced 2016 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let
$$f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {{{{n^n}\left( {x + n} \right)\left( {x + {n \over 2}} \right)...\left( {x + {n \over n}} \right)} \over {n!\left( {{x^2} + {n^2}} \right)\left( {{x^2} + {{{n^2}} \over 4}} \right)....\left( {{x^2} + {{{n^2}} \over {{n^2}}}} \right)}}} \right)^{{x \over n}}},$$ for

all $$x>0.$$ Then
A
$$f\left( {{1 \over 2}} \right) \ge f\left( 1 \right)$$
B
$$f\left( {{1 \over 3}} \right) \le f\left( {{2 \over 3}} \right)$$
C
$$\,f'\left( 2 \right) \le 0$$
D
$$\,{{f'\left( 3 \right)} \over {f\left( 3 \right)}} \ge {{f'\left( 2 \right)} \over {f\left( 2 \right)}}$$
2
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$f\left( x \right) = 7{\tan ^8}x + 7{\tan ^6}x - 3{\tan ^4}x - 3{\tan ^2}x$$ for all $$x \in \left( { - {\pi \over 2},{\pi \over 2}} \right).$$
Then the correct expression(s) is (are)
A
$$\int\limits_0^{\pi /4} {xf\left( x \right)dx = {1 \over {12}}}$$
B
$$\int\limits_0^{\pi /4} {f\left( x \right)dx = 0}$$
C
$$\int\limits_0^{\pi /4} {xf\left( x \right)dx = {1 \over {6}}}$$
D
$$\int\limits_0^{\pi /4} {f\left( x \right)dx = 1}$$
3
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
The option(s) with the values of a and $$L$$ that satisfy the following equation is (are) $${{\int\limits_0^{4\pi } {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} } \over {\int\limits_0^\pi {{e^t}\left( {{{\sin }^6}at + {{\cos }^4}at} \right)dt} }} = L?$$\$
A
$$a = 2,L = {{{e^{4\pi }} - 1} \over {{e^\pi } - 1}}$$
B
$$a = 2,L = {{{e^{4\pi }} + 1} \over {{e^\pi } + 1}}$$
C
$$a = 4,L = {{{e^{4\pi }} - 1} \over {{e^\pi } - 1}}$$
D
$$a = 4,L = {{{e^{4\pi }} + 1} \over {{e^\pi } + 1}}$$
4
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+3
-0
Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right)$$= $$\int\limits_{{1 \over x}}^x {{{{e^{ - \left( {t + {1 \over t}} \right)}}} \over t}} dt$$. Then
A
$$f(x)$$ is monotonically increasing on $$\left[ {1,\infty } \right)$$
B
$$f(x)$$ is monotonically decreasing on $$(0,1)$$
C
$$f(x)$$ $$+ f\left( {{1 \over x}} \right) = 0$$, for all $$x \in \left( {0,\infty } \right)$$
D
$$f\left( {{2^x}} \right)$$ is an odd function of $$x$$ on $$R$$
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