1
JEE Advanced 2024 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2

Let $S$ be the set of all $(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}$ such that

$$\lim\limits_{x \rightarrow \infty} \frac{\sin \left(x^2\right)\left(\log _e x\right)^\alpha \sin \left(\frac{1}{x^2}\right)}{x^{\alpha \beta}\left(\log _e(1+x)\right)^\beta}=0 .$$

Then which of the following is (are) correct?

A
$(-1,3) \in S$
B
$(-1,1) \in S$
C
$(1,-1) \in S$
D
$(1,-2) \in S$
2
JEE Advanced 2023 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
A
The function $f$ is discontinuous exactly at one point in $(0,1)$
B
There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
C
The function $f$ is NOT differentiable at more than three points in $(0,1)$
D
The minimum value of the function $f$ is $-\frac{1}{512}$
3
JEE Advanced 2021 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Let f : R $$\to$$ R be defined by $$f(x) = {{{x^2} - 3x - 6} \over {{x^2} + 2x + 4}}$$

Then which of the following statements is (are) TRUE?
A
f is decreasing in the interval ($$-$$2, $$-$$1)
B
f is increasing in the interval (1, 2)
C
f is onto
D
Range of f is $$\left[ { - {3 \over 2},2} \right]$$
4
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let f : R $$\to$$ R and g : R $$\to$$ R be functions
satisfying f(x + y) = f(x) + f(y) + f(x)f(y)
and f(x) = xg(x) for all x, y$$\in$$R.
If $$\mathop {\lim }\limits_{x \to 0} g(x) = 1$$, then which of the following statements is/are TRUE?
A
f is differentiable at every x$$\in$$R
B
If g(0) = 1, then g is differentiable at every x$$\in$$R
C
The derivative f'(1) is equal to 1
D
The derivative f'(0) is equal to 1
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