Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1,2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\dfrac{57}{13}, -\dfrac{40}{13}\right)$, then $|\alpha \lambda|$ is equal to

1

For a statistical data $x_1, x_2, \ldots, x_{10}$ of $10$ values, a student obtained the mean as $5.5$ and $\sum_{i=1}^…

Topic: JEE Main 2025 (24 January Morning Shift)

2

Let $S_n = \dfrac{1}{2} + \dfrac{1}{6} + \dfrac{1}{12} + \dfrac{1}{20} + \dots$ up to $n$ terms. If the sum of the firs…

Topic: JEE Main 2025 (24 January Morning Shift)

3

The area of the region ${(x, y) : x^2 + 4x + 2 \le y \le |x + 2|}$ is equal to

Topic: JEE Main 2025 (24 January Morning Shift)

4

Consider the region $R = {(x, y) : x \le y \le 9 - \dfrac{11}{3}x^2, , x \ge 0}$. The area of the largest rectangle of …

Topic: JEE Main 2025 (24 January Morning Shift)

5

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$, $\vec{b} = 3\hat{i} + \hat{j} - \hat{k}$ and $\vec{c}$ be three vectors …

Topic: JEE Main 2025 (24 January Morning Shift)

6

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of $5$ before $B$ throws a sum of $8$, and $B…

Topic: JEE Main 2025 (24 January Morning Shift)

7

Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\dfrac{x - 1}{2} = \dfrac{y + 1}{3} = \d…

Topic: JEE Main 2025 (24 January Morning Shift)

8

Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie…

Topic: JEE Main 2025 (24 January Morning Shift)

9

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1 + x^2}\right)dx + (1 + x^2)dy = 0$,…

Topic: JEE Main 2025 (24 January Morning Shift)

10

Let the product of the focal distances of the point $\left( \sqrt{3}, \dfrac{1}{2} \right)$ on the ellipse $\dfrac{x^2}…

Topic: JEE Main 2025 (24 January Morning Shift)

1

If the system of equations $2x - y + z = 4$, $5x + \lambda y + 3z = 12$, $100x - 47y + \mu z = 212$ has infinitely many…

Topic: JEE Main 2025 (24 January Morning Shift)

2

Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ …

Topic: JEE Main 2025 (24 January Morning Shift)