1
JEE Advanced 2025 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language

Let denote the set of all real numbers. Let f: ℝ → ℝ be defined by

$f(x) = \begin{cases} \dfrac{6x + \sin x}{2x + \sin x}, & \text{if } x \neq 0, \\ \dfrac{7}{3}, & \text{if } x = 0. \end{cases}$

Then which of the following statements is (are) TRUE?

A

The point $x = 0$ is a point of local maxima of $f$

B

The point $x = 0$ is a point of local minima of $f$

C

Number of points of local maxima of $f$ in the interval $[\pi, 6\pi]$ is 3

D

Number of points of local minima of $f$ in the interval $[2\pi, 4\pi]$ is 1

2
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let

$$ \alpha=\sum\limits_{k = 1}^\infty {{{\sin }^{2k}}\left( {{\pi \over 6}} \right)} $$

Let $g:[0,1] \rightarrow \mathbb{R}$ be the function defined by

$$ g(x)=2^{\alpha x}+2^{\alpha(1-x)} . $$

Then, which of the following statements is/are TRUE ?
A
The minimum value of $g(x)$ is $2^{\frac{7}{6}}$
B
The maximum value of $g(x)$ is $1+2^{\frac{1}{3}}$
C
The function $g(x)$ attains its maximum at more than one point
D
The function $g(x)$ attains its minimum at more than one point
3
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Change Language
Let f : R $$ \to $$ R be given by

$$f(x) = (x - 1)(x - 2)(x - 5)$$. Define

$$F(x) = \int\limits_0^x {f(t)dt} $$, x > 0

Then which of the following options is/are correct?
A
F(x) $$ \ne $$ 0 for all x $$ \in $$ (0, 5)
B
F has a local maximum at x = 2
C
F has two local maxima and one local minimum in (0, $$\infty $$)
D
F has a local minimum at x = 1
4
JEE Advanced 2019 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-1
Change Language
Let, $$f(x) = {{\sin \pi x} \over {{x^2}}}$$, x > 0

Let x1 < x2 < x3 < ... < xn < ... be all the points of local maximum of f and y1 < y2 < y3 < ... < yn < ... be all the points of local minimum of f.

Then which of the following options is/are correct?
A
$$|{x_n} - {y_n}|\, > 1$$ for every n
B
$${x_{n + 1}} - {x_n}\, > 2$$ for every n
C
x1 < y1
D
$${x_n} \in \left( {2n,\,2n + {1 \over 2}} \right)$$ for every n
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