Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\ \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $a>b$, of eccentricity $\dfrac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$ is:

1

If $a=\sin^{-1}(\sin 5)$ and $b=\cos^{-1}(\cos 5)$, then $a^{2}+b^{2}$ is equal to:

Topic: JEE Main 2024 (31 January Evening Shift)

2

The number of ways in which 21 identical apples can be distributed among three children such that each child gets at le…

Topic: JEE Main 2024 (31 January Evening Shift)

3

Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line \[ \frac{x-1}{2}=\frac{y-2}{3}=\frac…

Topic: JEE Main 2024 (31 January Evening Shift)

4

Let $A$ be a $3\times3$ real matrix such that \[ A\!\begin{pmatrix}1\\0\\1\end{pmatrix} =2\!\begin{pmatrix}1\\0\\1\end{…

Topic: JEE Main 2024 (31 January Evening Shift)

5

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probabil…

Topic: JEE Main 2024 (31 January Evening Shift)

1

The number of solutions of the equation $e^{\sin x}-2e^{-\sin x}=2$ is:

Topic: JEE Main 2024 (31 January Evening Shift)

2

The shortest distance, between lines $L_1$ and $L_2$, where $L_1: \frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$…

Topic: JEE Main 2024 (31 January Evening Shift)

3

The area of the region enclosed by the parabolas $y=4x-x^{2}$ and $3y=(x-4)^{2}$ is equal to:

Topic: JEE Main 2024 (31 January Evening Shift)

4

Let $f,g:(0,\infty)\to\mathbb{R}$ be defined by $f(x)=\int_{-x}^{x}\big(|t|-t^{2}\big)e^{-t^{2}}\,dt,\qquad g(x)=\int_{…

Topic: JEE Main 2024 (31 January Evening Shift)

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Let $f:\mathbb{R}\to(0,\infty)$ be a strictly increasing function such that $\displaystyle \lim_{x\to\infty}\frac{f(7x)…

Topic: JEE Main 2024 (31 January Evening Shift)

6

The temperature $T(t)$ of a body at time $t=0$ is $160^\circ\!F$ and it decreases continuously as per the differential …

Topic: JEE Main 2024 (31 January Evening Shift)

7

If the function $f:(-\infty,-1]\to(a,b]$ defined by $f(x)=e^{x^{3}-3x+1}$ is one–one and onto, then the distance of the…

Topic: JEE Main 2024 (31 January Evening Shift)

8

Let the $2^{\text{nd}}, 8^{\text{th}}$ and $44^{\text{th}}$ terms of a non-constant A.P. be respectively the $1^{\text{…

Topic: JEE Main 2024 (31 January Evening Shift)

9

Let $z_1$ and $z_2$ be two complex numbers such that $z_1+z_2=5$ and $z_1^{3}+z_2^{3}=20+15i$. Then, $\,\big|z_1^{4}+z_…

Topic: JEE Main 2024 (31 January Evening Shift)

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If for some $m,n$, $\binom{6}{m}+2\binom{6}{m+1}+\binom{6}{m+2}>8\binom{6}{3}$ and $\,^{\,n-1}\!P_{3}:\,^{\,n}\!P_{4}=1…

Topic: JEE Main 2024 (31 January Evening Shift)