1
JEE Advanced 2021 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation $${{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1$$. Let $$S = \{ {y_{\alpha ,\beta }}(x):\alpha ,\beta \in R\} $$. Then which of the following functions belong(s) to the set S?
A
$$f(x) = {{{x^2}} \over 2}{e^{ - x}} + \left( {e - {1 \over 2}} \right){e^{ - x}}$$
B
$$f(x) = - {{{x^2}} \over 2}{e^{ - x}} + \left( {e + {1 \over 2}} \right){e^{ - x}}$$
C
$$f(x) = {{{e^x}} \over 2}\left( {x - {1 \over 2}} \right) + \left( {e - {{{e^2}} \over 4}} \right){e^{ - x}}$$
D
$$f(x) = {{{e^x}} \over 2}\left( {{1 \over 2} - x} \right) + \left( {e + {{{e^2}} \over 4}} \right){e^{ - x}}$$
2
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let b be a nonzero real number. Suppose f : R $$ \to $$ R is a differentiable function such that f(0) = 1. If the derivative f' of f satisfies the equation $$f'(x) = {{f(x)} \over {{b^2} + {x^2}}}$$

for all x$$ \in $$R, then which of the following statements is/are TRUE?
A
If b > 0, then f is an increasing function
B
If b < 0, then f is a decreasing function
C
f(x) f($$-$$x) = 1 for all x$$ \in $$R
D
f(x) $$-$$f($$-$$x) = 0 for all x$$ \in $$R
3
JEE Advanced 2020 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-2
Change Language
Which of the following inequalities is/are TRUE?
A
$$\int_0^1 {x\cos xdx\, \ge \,{3 \over 8}} $$
B
$$\int_0^1 {x\sin xdx\, \ge \,{3 \over {10}}} $$
C
$$\int_0^1 {{x^2}\cos xdx\, \ge \,{1 \over 2}} $$
D
$$\int_0^1 {{x^2}\sin xdx\, \ge \,{2 \over 9}} $$
4
JEE Advanced 2018 Paper 1 Offline
MCQ (More than One Correct Answer)
+4
-1
Let f : [0, $$\infty $$) $$ \to $$ R be a continuous function such that

$$f(x) = 1 - 2x + \int_0^x {{e^{x - t}}f(t)dt} $$ for all x $$ \in $$ [0, $$\infty $$). Then, which of the following statement(s) is (are) TRUE?
A
The curve y = f(x) passes through the point (1, 2)
B
The curve y = f(x) passes through the point (2, $$-$$1)
C
The area of the region $$\{ (x,y) \in [0,1] \times R:f(x) \le y \le \sqrt {1 - {x^2}} \} $$ is $${{\pi - 2} \over 4}$$
D
The area of the region $$\{ (x,y) \in [0,1] \times R:f(x) \le y \le \sqrt {1 - {x^2}} \} $$ is $${{\pi - 1} \over 4}$$
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