NTA JEE Main 2025 April 02 Shift 2

If the image of the point $$P(1, 0, 3)$$ in the line joining the points $$A(4, 7, 1)$$ and $$B(3, 5, 3)$$ is $$Q(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to

Let $$f : [1, \infty) \to [2, \infty)$$ be a differentiable function. If $$10\int_{1}^{x} f(t)\,dt = 5xf(x) - x^5 - 9$$ for all $$x \geq 1$$, then the value of $$f(3)$$ is :

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $$\frac{21}{2}$$. Then the number of terms which are integers in the A.P. is :

Let $$A = \{1, 2, 3, \ldots, 100\}$$ and R be a relation on A such that $$R = \{(a, b) : a = 2b + 1\}$$. Let $$(a_1, a_2), (a_2, a_3), (a_3, a_4), \ldots, (a_k, a_{k+1})$$ be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to :

If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :

The line $$L_1$$ is parallel to the vector $$\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$$ and passes through the point $$(7, 6, 2)$$ and the line $$L_2$$ is parallel to the vector $$\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$$ and passes through the point $$(5, 3, 4)$$. The shortest distance between the lines $$L_1$$ and $$L_2$$ is :

Let $$(a, b)$$ be the point of intersection of the curve $$x^2 = 2y$$ and the straight line $$y - 2x - 6 = 0$$ in the second quadrant. Then the integral $$I = \int_{a}^{b} \frac{9x^2}{1 + 5^x}\,dx$$ is equal to :

If the system of equations
$$2x + \lambda y + 3z = 5$$
$$3x + 2y - z = 7$$
$$4x + 5y + \mu z = 9$$
has infinitely many solutions, then $$(\lambda^2 + \mu^2)$$ is equal to :

If $$\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$$, then the number of solutions of $$\sqrt{3}\csc^2\theta - 2(\sqrt{3} - 1)\csc\theta - 4 = 0$$, is equal to

Given three identical bags each containing 10 balls, whose colours are as follows :

Bag I : 3 Red, 2 Blue, 5 Green
Bag II : 4 Red, 3 Blue, 3 Green
Bag III : 5 Red, 1 Blue, 4 Green

A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q, then the value of $$\left(\frac{1}{p} + \frac{1}{q}\right)$$ is :