NTA JEE Main 2025 April 8th Shift 2

Let the values of $$\lambda$$ for which the shortest distance between the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$ and $$\frac{x-\lambda}{3} = \frac{y-4}{4} = \frac{z-5}{5}$$ is $$\frac{1}{\sqrt{6}}$$ be $$\lambda_1$$ and $$\lambda_2$$. Then the radius of the circle passing through the points $$(0, 0)$$, $$(\lambda_1, \lambda_2)$$ and $$(\lambda_2, \lambda_1)$$ is :

Let $$\alpha$$ be a solution of $$x^2 + x + 1 = 0$$, and for some a and b in $$\mathbb{R}$$, $$[4 \; a \; b] \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = [0 \; 0 \; 0]$$. If $$\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$$, then $$m + n$$ is equal to :

Let the function $$f(x) = \frac{x}{3} + \frac{3}{x} + 3$$, $$x \ne 0$$ be strictly increasing in $$(-\infty, \alpha_1) \cup (\alpha_2, \infty)$$ and strictly decreasing in $$(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$$. Then $$\sum_{i=1}^{5} \alpha_i^2$$ is equal to :

If A and B are two events such that $$P(A) = 0.7$$, $$P(B) = 0.4$$ and $$P(A \cap \bar{B}) = 0.5$$, where $$\bar{B}$$ denotes the complement of B, then $$P\left(B\mid(A \cup \bar{B})\right)$$ is equal to :

If $$\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty = \frac{\pi^4}{90}$$, $$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty = \alpha$$, $$\frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty = \beta$$, then $$\frac{\alpha}{\beta}$$ is equal to :

The sum of the squares of the roots of $$|x - 2|^2 + |x - 2| - 2 = 0$$ and the squares of the roots of $$x^2 - 2|x - 3| - 5 = 0$$, is

Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $$\alpha$$ with the positive x-axis and the equations of its diagonals are $$(\sqrt{3}+1)x + (\sqrt{3}-1)y = 0$$ and $$(\sqrt{3}-1)x - (\sqrt{3}+1)y + 8\sqrt{3} = 0$$. Then $$a^2$$ is equal to

Let f(x) be a positive function and $$I_1 = \int_{-\frac{1}{2}}^{1} 2xf(2x(1-2x)) \, dx$$ and $$I_2 = \int_{-1}^{2} f(x(1-x)) \, dx$$. Then the value of $$\frac{I_2}{I_1}$$ is equal to :

Let $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$$. Let $$\hat{c}$$ be a unit vector in the plane of the vectors $$\vec{a}$$ and $$\vec{b}$$ and be perpendicular to $$\vec{a}$$. Then such a vector $$\hat{c}$$ is :

Let the ellipse $$3x^2 + py^2 = 4$$ pass through the centre C of the circle $$x^2 + y^2 - 2x - 4y - 11 = 0$$ of radius r. Let $$f_1$$, $$f_2$$ be the focal distances of the point C on the ellipse. Then $$6f_1f_2 - r$$ is equal to