1
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

If the maximum distance of normal to the ellipse $$\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1, b < 2$$, from the origin is 1, then the eccentricity of the ellipse is :

A
$$\frac{\sqrt{3}}{4}$$
B
$$\frac{1}{2}$$
C
$$\frac{1}{\sqrt{2}}$$
D
$$\frac{\sqrt{3}}{2}$$
2
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$$. Then, at x = 1,

A
$$2 y^{\prime}+\sqrt{3} \pi^{2} y=0$$
B
$$y^{\prime}+3 \pi^{2} y=0$$
C
$$\sqrt{2} y^{\prime}-3 \pi^{2} y=0$$
D
$$2 y^{\prime}+3 \pi^{2} y=0$$
3
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

$$(\mathrm{S} 1)~(p \Rightarrow q) \vee(p \wedge(\sim q))$$ is a tautology

$$(\mathrm{S} 2)~((\sim p) \Rightarrow(\sim q)) \wedge((\sim p) \vee q)$$ is a contradiction.

Then

A
only (S2) is correct
B
both (S1) and (S2) are correct
C
only (S1) is correct
D
both (S1) and (S2) are wrong
4
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus
Change Language

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
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