1
JEE Advanced 2024 Paper 1 Online
+3
-1

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
$(\mathrm{P}) \rightarrow(4)$ $(\mathrm{Q}) \rightarrow(2)$ $(\mathrm{R}) \rightarrow(1)$ $(\mathrm{S}) \rightarrow(3)$
2
JEE Advanced 2022 Paper 2 Online
+3
-1
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
$3+\frac{3}{4} \log _{e} 7$
3
JEE Advanced 2018 Paper 2 Offline
+3
-1
Let $${f_1}:R \to R,\,{f_2}:\left( { - {\pi \over 2},{\pi \over 2}} \right) \to R,\,{f_3}:( - 1,{e^{\pi /2}} - 2) \to R$$ and $${f_4}:R \to R$$ be functions defined by

(i) $${f_1}(x) = \sin (\sqrt {1 - {e^{ - {x^2}}}} )$$,

(ii) $${f_2}(x) = \left\{ \matrix{ {{|\sin x|} \over {\tan { - ^1}x}}if\,x \ne 0,\,where \hfill \cr 1\,if\,x = 0 \hfill \cr} \right.$$

the inverse trigonometric function tan$$-$$1x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$,

(iii) $${f_3}(x) = [\sin ({\log _e}(x + 2))]$$, where for $$t \in R,\,[t]$$ denotes the greatest integer less than or equal to t,

(iv) $${f_4}(x) = \left\{ \matrix{ {x^2}\sin \left( {{1 \over x}} \right)\,if\,x \ne 0 \hfill \cr 0\,if\,x = 0 \hfill \cr} \right.$$
LIST-I LIST-II
P. The function $$f_1$$ is 1. NOT continuous at $$x = 0$$
Q. The function $$f_2$$ is 2. continuous at $$x = 0$$ and NOT differentiable at $$x = 0$$
R. The function $$f_3$$ is 3. differentiable at $$x = 0$$ and its derivative is NOT continuous at $$x = 0$$
S. The function $$f_4$$ is 4. differentiable at $$x = 0$$ and its derivative is continuous at $$x = 0$$
A
P $$\to$$ 2 ; Q $$\to$$ 3 ; R $$\to$$ 1 ; S $$\to$$ 4
B
P $$\to$$ 4 ; Q $$\to$$ 1 ; R $$\to$$ 2 ; S $$\to$$ 3
C
P $$\to$$ 4 ; Q $$\to$$ 2 ; R $$\to$$ 1 ; S $$\to$$ 3
D
P $$\to$$ 2 ; Q $$\to$$ 1 ; R $$\to$$ 4 ; S $$\to$$ 3
4
JEE Advanced 2017 Paper 2 Offline
+3
-1
If f : R $$\to$$ R is a twice differentiable function such that f"(x) > 0 for all x$$\in$$R, and $$f\left( {{1 \over 2}} \right) = {1 \over 2}$$, f(1) = 1, then
A
f'(1) $$\le$$ 0
B
f'(1) > 1
C
0 < f'(1) $$\le$$ $${1 \over 2}$$
D
$${1 \over 2}$$ < f'(1) $$\le$$ 1
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