1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2 Consider the equation

$$\int_{1}^{e} \frac{\left(\log _{\mathrm{e}} x\right)^{1 / 2}}{x\left(a-\left(\log _{\mathrm{e}} x\right)^{3 / 2}\right)^{2}} d x=1, \quad a \in(-\infty, 0) \cup(1, \infty)$$

Which of the following statements is/are TRUE?

A
No $$a$$ satisfies the above equation
B
An integer $$a$$ satisfies the above equation
C
An irrational number $$a$$ satisfies the above equation
D
More than one $$a$$ satisfy the above equation
2
JEE Advanced 2021 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2 Let $$f:\left[ { - {\pi \over 2},{\pi \over 2}} \right] \to R$$ be a continuous function such that $$f(0) = 1$$ and $$\int_0^{{\pi \over 3}} {f(t)dt = 0}$$. Then which of the following statements is(are) TRUE?
A
The equation $$f(x) - 3\cos 3x = 0$$ has at least one solution in $$\left( {0,{\pi \over 3}} \right)$$
B
The equation $$f(x) - 3\sin 3x = - {6 \over \pi }$$ has at least one solution in $$\left( {0,{\pi \over 3}} \right)$$
C
$$\mathop {\lim }\limits_{x \to 0} {{x\int_0^x {f(t)dt} } \over {1 - {e^{{x^2}}}}} = - 1$$
D
$$\mathop {\lim }\limits_{x \to 0} {{\sin x\int_0^x {f(t)dt} } \over {{x^2}}} = - 1$$
3
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
$$f(x) = {{{e^x}} \over 2}\left( {{1 \over 2} - x} \right) + \left( {e + {{{e^2}} \over 4}} \right){e^{ - x}}$$
4
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2 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
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