1
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1
Out of Syllabus

Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then

$$S=\left\{x \in \mathbb{R}: \tan ^{-1}(\sqrt{f(x)})+\sin ^{-1}(\sqrt{f(x)+1})=\frac{\pi}{2}\right\}$$ :

A
is an empty set
B
contains exactly one element
C
contains exactly two elements
D
is an infinite set
2
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
Out of Syllabus
Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to :
A
76
B
81
C
64
D
72
3
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :
A
$\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.
B
$\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in G.P.
C
$d, e, f$ are in A.P.
D
$d, e, f$ are in G.P.
4
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1
Out of Syllabus

If $$\mathrm{P}(\mathrm{h}, \mathrm{k})$$ be a point on the parabola $$x=4 y^{2}$$, which is nearest to the point $$\mathrm{Q}(0,33)$$, then the distance of $$\mathrm{P}$$ from the directrix of the parabola $$\quad y^{2}=4(x+y)$$ is equal to :

A
8
B
2
C
6
D
4
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