1
JEE Advanced 2021 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
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 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 $${{\pi - 1} \over 4}$$
3
JEE Advanced 2017 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
If the line x = $$\alpha$$ divides the area of region R = {(x, y) $$\in$$R2 : x3 $$\le$$ y $$\le$$ x, 0 $$\le$$ x $$\le$$ 1} into two equal parts, then
A
2$$\alpha$$4 $$-$$ 4$$\alpha$$2 + 1 =0
B
$$\alpha$$4 + 4$$\alpha$$2 $$-$$ 1 =0
C
$${1 \over 2} < \alpha < 1$$
D
0 < $$\alpha$$ $$\le$$ $${1 \over 2}$$
4
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
$$\int_1^3 {f\left( x \right)dx = -12}$$
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