Let the domain of the function $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ be $(a, b)$. If $\int_0^{b - a} [x^2] , dx = p - \sqrt{q - \sqrt{r}}, ; p, q, r \in \mathbb{N}, ; \gcd(p, q, r) = 1$, where $[,]$ is the greatest integer function, then $p + q + r$ is equal to

1

Let $f(x) = \int x^3 \sqrt{3 - x^2} , dx.$ If $5f(\sqrt{2}) = -4$, then $f(1)$ is equal to

Topic: JEE Main 2025 (3 April Morning Shift)

2

The radius of the smallest circle which touches the parabolas $y=x^2+2$ and $x=y^2+2$ is

Topic: JEE Main 2025 (3 April Morning Shift)

3

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by $x \mathrm{R} y$ if and only if $0 \leq x^2+…

Topic: JEE Main 2025 (3 April Morning Shift)

4

$ \text { The number of solutions of the equation } 2 x+3 \tan x=\pi, x \in[-2 \pi, 2 \pi]-\left\{ \pm \frac{\pi}{2}, \…

Topic: JEE Main 2025 (3 April Morning Shift)

5

The sum $1 + 3 + 11 + 25 + 45 + 71 + \ldots$ up to $20$ terms is equal to:

Topic: JEE Main 2025 (3 April Morning Shift)

6

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \dfrac{111}{4…

Topic: JEE Main 2025 (3 April Morning Shift)

7

If the domain of the function $f(x) = \log_e\left(\dfrac{2x - 3}{5 + 4x}\right) + \sin^{-1}\left(\dfrac{4 + 3x}{2 - x}\…

Topic: JEE Main 2025 (3 April Morning Shift)

1

Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x - 16 = 0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3…

Topic: JEE Main 2025 (3 April Morning Shift)

2

Let a line passing through the point $(4,1,0)$ intersect the line $\mathrm{L}_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}…

Topic: JEE Main 2025 (3 April Morning Shift)

3

Let $\quad f(x)= \begin{cases}(1+a x)^{1 / x} & , x0\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to:

Topic: JEE Main 2025 (3 April Morning Shift)

4

A line passing through the point $P(\sqrt{5}, \sqrt{5})$ intersects the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{25} = 1$…

Topic: JEE Main 2025 (3 April Morning Shift)

5

Line $L_1$ passes through the point $(1, 2, 3)$ and is parallel to the $z$-axis. Line $L_2$ passes through the point $(…

Topic: JEE Main 2025 (3 April Morning Shift)

6

Let $g$ be a differentiable function such that $\displaystyle \int_0^x g(t),dt = x - \int_0^x t g(t),dt,; x \ge 0$ and …

Topic: JEE Main 2025 (3 April Morning Shift)

7

The sum of all rational terms in the expansion of $(2 + \sqrt{3})^8$ is:

Topic: JEE Main 2025 (3 April Morning Shift)

8

If $\displaystyle \sum_{r=1}^{9} \left(\dfrac{r + 3}{2^r}\right) \cdot {^9C_r} = \alpha \left(\dfrac{3}{2}\right)^9 - \…

Topic: JEE Main 2025 (3 April Morning Shift)

9

Let $z \in \mathbb{C}$ be such that $\dfrac{z^2+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^2$ is:

Topic: JEE Main 2025 (3 April Morning Shift)

10

$ \text { If } y(x)=\left|\begin{array}{ccc} \sin x & \cos x & \sin x+\cos x+1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{array}…

Topic: JEE Main 2025 (3 April Morning Shift)