1
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
$$a = 4,L = {{{e^{4\pi }} + 1} \over {{e^\pi } + 1}}$$
2
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that
$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F\left( x \right) < 0$$ for all $$x \in \left( {{1 \over 2},3} \right).$$ Let $$f\left( x \right) = xF\left( x \right)$$ for all $$x \in R.$$

If $$\int_1^3 {{x^2}F'\left( x \right)dx = - 12}$$ and $$\int_1^3 {{x^3}F''\left( x \right)dx = 40,}$$ then the correct expression(s) is (are)

A
$$9f'\left( 3 \right) + f'\left( 1 \right) - 32 = 0$$
B
$$\int_1^3 {f\left( x \right)dx = 12}$$
C
$$9f'\left( 3 \right) - f'\left( 1 \right) + 32 = 0$$
D
$$\int_1^3 {f\left( x \right)dx = -12}$$
3
JEE Advanced 2015 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$F:R \to R$$ be a thrice differentiable function. Suppose that
$$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F'\left( x \right) < 0$$ for all $$x \in \left( {{1 \over 2},3} \right).$$ Let $$f\left( x \right) = xF\left( x \right)$$ for all $$x \in R.$$

The correct statement(s) is (are)

A
$$f'\left( 1 \right) < 0$$
B
$$f\left( 2 \right) < 0$$
C
$$f'\left( x \right) \ne 0$$ for any $$x \in \left( {1,3} \right)$$
D
$$f'\left( x \right) = 0$$ for some $$x \in \left( {1,3} \right)$$
4
JEE Advanced 2014 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let $$f:\left( {0,\infty } \right) \to R$$ be given by $$f\left( x \right) = \int\limits_{{1 \over x}}^x {{e^{ - \left( {t + {1 \over t}} \right){{dt} \over t}}}} .$$ 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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