1
JEE Advanced 2022 Paper 2 Online
Numerical
+3
-1
Consider the hyperbola

$$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$

with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S_{1}=\alpha$, with $\alpha<\frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1} P$ at $P_{1}$. Let $\delta$ be the distance of $P$ from the straight line $S P_{1}$, and $\beta=S_{1} P$. Then the greatest integer less than or equal to $\frac{\beta \delta}{9} \sin \frac{\alpha}{2}$ is ________.
2
JEE Advanced 2022 Paper 2 Online
Numerical
+3
-1
Consider the functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$$f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases}$$

If $\alpha$ is the area of the region

$$\left\{(x, y) \in \mathbb{R} \times \mathbb{R}:|x| \leq \frac{3}{4}, 0 \leq y \leq \min \{f(x), g(x)\}\right\},$$

then the value of $9 \alpha$ is
3
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Let $P Q R S$ be a quadrilateral in a plane, where

$Q R=1, \angle P Q R=\angle Q R S=70^{\circ}, \angle P Q S=15^{\circ}$ and $\angle P R S=40^{\circ}$.

If $\angle R P S=\theta^{\circ}, P Q=\alpha$ and $P S=\beta$, then the interval(s) that contain(s) the value of

$4 \alpha \beta \sin \theta^{\circ}$ is/are
A
$(0, \sqrt{2})$
B
$(1,2)$
C
$(\sqrt{2}, 3)$
D
$(2 \sqrt{2}, 3 \sqrt{2})$
4
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
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
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