1
JEE Advanced 2024 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language
Let $k \in \mathbb{R}$. If $\lim \limits_{x \rightarrow 0+}(\sin (\sin k x)+\cos x+x)^{\frac{2}{x}}=e^6$, then the value of $k$ is
A
1
B
2
C
3
D
4
2
JEE Advanced 2024 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by

$$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$

Then which of the following statements is TRUE?

A
$f(x)=0$ has infinitely many solutions in the interval $\left[\frac{1}{10^{10}}, \infty\right)$.
B
$f(x)=0$ has no solutions in the interval $\left[\frac{1}{\pi}, \infty\right)$.
C
The set of solutions of $f(x)=0$ in the interval $\left(0, \frac{1}{10^{10}}\right)$ is finite.
D
$f(x)=0$ has more than 25 solutions in the interval $\left(\frac{1}{\pi^2}, \frac{1}{\pi}\right)$.
3
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by

$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text { otherwise } .\end{cases}\right. $$

Let $a, b, c, d \in \mathbb{R}$. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ by

$$ h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R} . $$

Match each entry in List-I to the correct entry in List-II.

List-I List-II
(P) If $a = 0$, $b = 1$, $c = 0$, and $d = 0$, then (1) $h$ is one-one.
(Q) If $a = 1$, $b = 0$, $c = 0$, and $d = 0$, then (2) $h$ is onto.
(R) If $a = 0$, $b = 0$, $c = 1$, and $d = 0$, then (3) $h$ is differentiable on $\mathbb{R}$.
(S) If $a = 0$, $b = 0$, $c = 0$, and $d = 1$, then (4) the range of $h$ is $[0, 1]$.
(5) the range of $h$ is $\{0, 1\}$.

The correct option is
A
$(\mathrm{P}) \rightarrow(4)$ $(\mathrm{Q}) \rightarrow(3)$ $(\mathrm{R}) \rightarrow(1)$ (S) $\rightarrow$ (2)
B
$(\mathrm{P}) \rightarrow(5)$ $(\mathrm{Q}) \rightarrow(2)$ $(\mathrm{R}) \rightarrow(4)$ (S) $\rightarrow(3)$
C
$(\mathrm{P}) \rightarrow(5)$ $(\mathrm{Q}) \rightarrow(3)$ $(\mathrm{R}) \rightarrow(2)$ $(\mathrm{S}) \rightarrow(4)$
D
$(\mathrm{P}) \rightarrow(4)$ $(\mathrm{Q}) \rightarrow(2)$ $(\mathrm{R}) \rightarrow(1)$ $(\mathrm{S}) \rightarrow(3)$
4
JEE Advanced 2022 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language
For positive integer $n$, define

$$ f(n)=n+\frac{16+5 n-3 n^{2}}{4 n+3 n^{2}}+\frac{32+n-3 n^{2}}{8 n+3 n^{2}}+\frac{48-3 n-3 n^{2}}{12 n+3 n^{2}}+\cdots+\frac{25 n-7 n^{2}}{7 n^{2}} . $$

Then, the value of $$\mathop {\lim }\limits_{n \to \infty } f\left( n \right)$$ is equal to :
A
$3+\frac{4}{3} \log _{e} 7$
B
$4-\frac{3}{4} \log _{e}\left(\frac{7}{3}\right)$
C
$4-\frac{4}{3} \log _{e}\left(\frac{7}{3}\right)$
D
$3+\frac{3}{4} \log _{e} 7$
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