1
JEE Advanced 2024 Paper 2 Online
+3
-1
Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.$ and $\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal to
A
$\frac{17}{2}$
B
$\frac{17}{3}$
C
$\frac{17}{4}$
D
$\frac{17}{5}$
2
JEE Advanced 2023 Paper 1 Online
+3
-1
Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=\sqrt{n}$ if $x \in\left[\frac{1}{n+1}, \frac{1}{n}\right)$ where $n \in \mathbb{N}$. Let $g:(0,1) \rightarrow \mathbb{R}$ be a function such that $\int\limits_{x^2}^x \sqrt{\frac{1-t}{t}} d t < g(x) < 2 \sqrt{x}$ for all $x \in(0,1)$. Then $\lim\limits_{x \rightarrow 0} f(x) g(x)$
A
does NOT exist
B
is equal to 1
C
is equal to 2
D
is equal to 3
3
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $${\psi _1}:[0,\infty ) \to R$$, $${\psi _2}:[0,\infty ) \to R$$, f : (0, $$\infty$$) $$\to$$ R and g : [0, $$\infty$$) $$\to$$ R be functions such that f(0) = g(0) = 0,

$${\psi _1}(x) = {e^{ - x}} + x,x \ge 0$$,

$${\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0$$,

$$f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x > 0}$$ and

$$g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x > 0}$$.
Which of the following statements is TRUE?
A
$$f(\sqrt {\ln 3} ) + g(\sqrt {\ln 3} ) = {1 \over 3}$$
B
For every x > 1, there exists an $$\alpha$$ $$\in$$ (1, x) such that $${\psi _1}(x) = 1 + \alpha x$$
C
For every x > 0, there exists a $$\beta$$ $$\in$$ (0, x) such that $${\psi _2}(x) = 2x({\psi _1}(\beta ) - 1)$$
D
f is an increasing function on the interval $$\left[ {0,{3 \over 2}} \right]$$
4
JEE Advanced 2021 Paper 2 Online
+3
-1
Let $${\psi _1}:[0,\infty ) \to R$$, $${\psi _2}:[0,\infty ) \to R$$, f : (0, $$\infty$$) $$\to$$ R and g : [0, $$\infty$$) $$\to$$ R be functions such that f(0) = g(0) = 0,

$${\psi _1}(x) = {e^{ - x}} + x,x \ge 0$$,

$${\psi _2}(x) = {x^2} - 2x - 2{e^{ - x}} + 2,x \ge 0$$,

$$f(x) = \int_{ - x}^x {(|t| - {t^2}){e^{ - {t^2}}}dt,x > 0}$$ and

$$g(x) = \int_0^{{x^2}} {\sqrt t {e^{ - t}}dt,x > 0}$$.
Which of the following statements is TRUE?
A
$${\psi _1}(x) \le 1$$, for all x > 0
B
$${\psi _2}(x) \le 0$$, for all x > 0
C
$$f(x) \ge 1 - {e^{ - {x^2}}} - {2 \over 3}{x^3} + {2 \over 5}{x^5}$$, for all $$x \in \left( {0,{1 \over 2}} \right)$$
D
$$g(x) \le {2 \over 3}{x^3} - {2 \over 5}{x^5} + {1 \over 7}{x^7}$$, for all $$x \in \left( {0,{1 \over 2}} \right)$$
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