NTA JEE Main 2025 April 02 Shift 2

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $$a + b + ab$$ is equal to :

If the domain of the function $$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$$ is $$(a, b)$$, then $$(1 + a)^2 + b^2$$ is equal to :

$$4\int_{0}^{1} \left(\frac{1}{\sqrt{3 + x^2}} + \frac{1}{\sqrt{1 + x^2}}\right)dx - 3\log_e(\sqrt{3})$$ is equal to :

If $$\lim_{x \to 0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$$ is finite, then $$(a + b)$$ is equal to :

If $$\sum_{r=0}^{10} \left(\frac{10^{r+1} - 1}{10^r}\right) \cdot \,^{11}C_{r+1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$$, then $$\alpha$$ is equal to :

The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :

Let the point P of the focal chord PQ of the parabola $$y^2 = 16x$$ be $$(1, -4)$$. If the focus of the parabola divides the chord PQ in the ratio $$m : n$$, $$\gcd(m, n) = 1$$, then $$m^2 + n^2$$ is equal to :

Let $$\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$$ and a vector $$\vec{c}$$ be such that $$(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$$ and $$\vec{a} \cdot \vec{c} = 3$$. If $$\vec{b} \times \vec{c} = \vec{d}$$, then $$|\vec{a} \cdot \vec{d}|$$ is equal to :

Let the area of the triangle formed by a straight line $$L : x + by + c = 0$$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of 45° with the positive x-axis, then the value of $$b^2 + c^2$$ is :

Let A be a $$3 \times 3$$ real matrix such that $$A^2(A - 2I) - 4(A - I) = O$$, where I and O are the identity and null matrices, respectively. If $$A^5 = \alpha A^2 + \beta A + \gamma I$$, where $$\alpha, \beta$$ and $$\gamma$$ are real constants, then $$\alpha + \beta + \gamma$$ is equal to :