NTA JEE Main 2025 April 3rd Shift 2

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = ||x + 2| - 2|x||$$. If m is the number of points of local minima and n is the number of points of local maxima of f, then $$m + n$$ is

Each of the angles $$\beta$$ and $$\gamma$$ that a given line makes with the positive y- and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle $$\beta$$ is

If the four distinct points $$(4, 6)$$, $$(-1, 5)$$, $$(0, 0)$$ and $$(k, 3k)$$ lie on a circle of radius r, then $$10k + r^2$$ is equal to

Let the Mean and Variance of five observations $$x_1 = 1, x_2 = 3, x_3 = a, x_4 = 7$$ and $$x_5 = b$$, $$a \gt b$$, be 5 and 10 respectively. Then the Variance of the observations $$n + x_n$$, $$n = 1, 2, \ldots, 5$$ is

Consider the lines $$x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5$$, $$\lambda$$ being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the distance of L from the point $$(3, 6)$$ is d, then the value of $$d^2$$ is

Let $$A = \{-2, -1, 0, 1, 2, 3\}$$. Let R be a relation on A defined by $$xRy$$ if and only if $$y = \max\{x, 1\}$$. Let $$\ell$$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $$\ell + m + n$$ is equal to

Let the equation $$x(x + 2)(12 - k) = 2$$ have equal roots. Then the distance of the point $$\left(k, \dfrac{k}{2}\right)$$ from the line $$3x + 4y + 5 = 0$$ is

Line $$L_1$$ of slope 2 and line $$L_2$$ of slope $$\dfrac{1}{2}$$ intersect at the origin O. In the first quadrant, $$P_1, P_2, \ldots P_{12}$$ are 12 points on line $$L_1$$ and $$Q_1, Q_2, \ldots Q_9$$ are 9 points on line $$L_2$$. Then the total number of triangles, that can be formed having vertices at three of the 22 points O, $$P_1, P_2, \ldots P_{12}$$, $$Q_1, Q_2, \ldots Q_9$$, is:

The integral $$\displaystyle\int_0^{\pi} \dfrac{8x \, dx}{4\cos^2 x + \sin^2 x}$$ is equal to

Let f be a function such that $$f(x) + 3f\left(\dfrac{24}{x}\right) = 4x$$, $$x \neq 0$$. Then $$f(3) + f(8)$$ is equal to