Let the sum of two positive integers be $24$. If the probability that their product is not less than $\dfrac{3}{4}$ times their greatest possible product is $\dfrac{m}{n}$, where $\gcd(m,n)=1$, then $n-m$ equals

1

Let $y=y(x)$ be the solution of the differential equation $(1+y^{2})e^{\tan x},dx+\cos^{2}x,(1+e^{2\tan x}),dy=0$, $y(0…

Topic: JEE Main 2024 (8 April Morning Shift)

2

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:

Topic: JEE Main 2024 (8 April Morning Shift)

3

Let $H:\dfrac{-x^2}{a^2}+\dfrac{y^2}{b^2}=1$ be the hyperbola whose eccentricity is $\sqrt{3}$ and the length of the la…

Topic: JEE Main 2024 (8 April Morning Shift)

4

Let $f(x)$ be a positive function such that the area bounded by $y=f(x)$, $y=0$ from $x=0$ to $x=a>0$ is $e^{-a}+4a^{2}…

Topic: JEE Main 2024 (8 April Morning Shift)

5

Let the circles $C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2$ and $C_2:(x-8)^2+\left(y-\dfrac{15}{2}\right)^2=r_2^2$ touch each …

Topic: JEE Main 2024 (8 April Morning Shift)

6

The set of all $\alpha$ for which the vectors $\vec a=\alpha t,\hat i+6,\hat j-3,\hat k$ and $\vec b=t,\hat i-2,\hat j-…

Topic: JEE Main 2024 (8 April Morning Shift)

7

Let $f(x)=4\cos^{3}x+3\sqrt{3}\cos^{2}x-10$. The number of points of local maxima of $f$ in the interval $(0,2\pi)$ is:

Topic: JEE Main 2024 (8 April Morning Shift)

8

The value of $k\in\mathbb{N}$ for which the integral $I_n=\displaystyle\int_{0}^{1}(1-x^{k})^{n},dx,\ n\in\mathbb{N}$, …

Topic: JEE Main 2024 (8 April Morning Shift)

9

Let $A=\left[\begin{array}{lll}2 & a & 0 \\ 1 & 3 & 1 \\ 0 & 5 & b\end{array}\right]$. If $A^3=4 A^2-A-21 I$, where $I$…

Topic: JEE Main 2024 (8 April Morning Shift)

1

For the function $f(x)=\cos x - x + 1,; x\in\mathbb{R}$, consider the statements (S1) $f(x)=0$ for only one value of $x…

Topic: JEE Main 2024 (8 April Morning Shift)

2

Let $P(x,y,z)$ be a point in the first octant whose projection on the $xy$–plane is $Q$. Let $OP=\gamma$; the angle bet…

Topic: JEE Main 2024 (8 April Morning Shift)

3

Let $z$ be a complex number such that $\lvert z+2\rvert=1$ and $\operatorname{Im}!\left(\dfrac{z+1}{z+2}\right)=\dfrac{…

Topic: JEE Main 2024 (8 April Morning Shift)

4

The sum of all the solutions of the equation $(8)^{2x}-16\cdot(8)^x+48=0$ is:

Topic: JEE Main 2024 (8 April Morning Shift)

5

The equations of two sides $AB$ and $AC$ of a triangle $ABC$ are $4x+y=14$ and $3x-2y=5$, respectively. The point $\lef…

Topic: JEE Main 2024 (8 April Morning Shift)

6

If $\sin x=-\frac{3}{5}$, where $\pi< x

Topic: JEE Main 2024 (8 April Morning Shift)

7

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A …

Topic: JEE Main 2024 (8 April Morning Shift)

8

If the set $R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$ has $m$ elements and $\sum_\limits{n=1}^m\left(1-i^{n !}\right…

Topic: JEE Main 2024 (8 April Morning Shift)

9

$L_1:;\vec r=(2+\lambda),\hat i+(1-3\lambda),\hat j+(3+4\lambda),\hat k,;\lambda\in\mathbb R$ $L_2:;\vec r=2(1+\mu),\ha…

Topic: JEE Main 2024 (8 April Morning Shift)

10

Let $I(x)=\displaystyle\int \frac{6}{\sin^{2}x,(1-\cot x)^{2}},dx$. If $I(0)=3$, then $I!\left(\tfrac{\pi}{12}\right)$ …

Topic: JEE Main 2024 (8 April Morning Shift)