$A={(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:\ |\alpha-1|\le 4\ \text{and}\ |\beta-5|\le 6}$ $B={(\alpha,\beta)\in\mathbb{R}\times\mathbb{R}:\ 16(\alpha-2)^2+9(\beta-6)^2\le 144}.$ Then

1

$\text{Consider the lines } L_{1}: , x-1=y-2=z \quad \text{and} \quad L_{2}: , x-2=y=z-1.$ $\text{Let the feet of the p…

Topic: JEE Main 2025 (7 April Evening Shift)

2

The number of solutions of the equation $ \cos 2\theta \cos \frac{\theta}{2} + \cos \frac{5\theta}{2} = 2\cos^3 \frac{…

Topic: JEE Main 2025 (7 April Evening Shift)

3

If the area of the region ${(x,y):, 1+x^2 \le y \le \min{x+7,; 11-3x}}$ is $A$, then $3A$ is equal to:

Topic: JEE Main 2025 (7 April Evening Shift)

4

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is $21$ and the sum of its eighth, tenth a…

Topic: JEE Main 2025 (7 April Evening Shift)

5

If the equation of the line passing through the point $ \left( 0, -\frac{1}{2}, 0 \right) $ and perpendicular to the li…

Topic: JEE Main 2025 (7 April Evening Shift)

6

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\dfrac{x^2}{b^2}+\dfrac{y^2}{25}=1$ and the hyperbola $\dfrac…

Topic: JEE Main 2025 (7 April Evening Shift)

7

Let the length of a latus rectum of an ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ be $10$. If its eccentrici…

Topic: JEE Main 2025 (7 April Evening Shift)

8

The number of real roots of the equation $x|x-2|+3|x-3|+1=0$ is:

Topic: JEE Main 2025 (7 April Evening Shift)

9

If the range of the function $f(x)=\dfrac{5-x}{x^2-3x+2}$, $x\ne1,2$, is $(-\infty,\alpha]\cup[\beta,\infty)$, then $\a…

Topic: JEE Main 2025 (7 April Evening Shift)

10

A bag contains $19$ unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and a head …

Topic: JEE Main 2025 (7 April Evening Shift)

1

Let $\vec a$ and $\vec b$ be vectors of the same magnitude such that $\displaystyle \frac{\lvert\vec a+\vec b\rvert+\lv…

Topic: JEE Main 2025 (7 April Evening Shift)

2

Let the system of equations x + 5y - z = 1 4x + 3y - 3z = 7 24x + y + λz = μ λ, μ ∈ ℝ, have infinitely many solutions. …

Topic: JEE Main 2025 (7 April Evening Shift)

3

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular $n$-gon $P$, and $q$ be …

Topic: JEE Main 2025 (7 April Evening Shift)

4

Let a random variable $X$ take values $0,1,2,3$ with $P(X=0)=P(X=1)=p$, $P(X=2)=P(X=3)$ and $E(X^2)=2E(X)$. Then the va…

Topic: JEE Main 2025 (7 April Evening Shift)

5

If the orthocenter of the triangle formed by the lines y = x + 1, y = 4x - 8 and y = mx + c is at (3, -1), then m - c i…

Topic: JEE Main 2025 (7 April Evening Shift)