NTA JEE Main 2025 April 4th Shift 2 - Mathematics

Let $$a > 0$$. If the function $$f(x) = 6x^3 - 45ax^2 + 108a^2x + 1$$ attains its local maximum and minimum values at the points $$x_1$$ and $$x_2$$ respectively such that $$x_1 x_2 = 54$$, then $$a + x_1 + x_2$$ is equal to :

Let f be a differentiable function on $$\mathbf{R}$$ such that $$f(2) = 1$$, $$f'(2) = 4$$. Let $$\lim_{x \to 0} (f(2+x))^{3/x} = e^\alpha$$. Then the number of times the curve $$y = 4x^3 - 4x^2 - 4(\alpha - 7)x - \alpha$$ meets the x-axis is :

The sum of the infinite series $$\cot^{-1}\left(\frac{7}{4}\right) + \cot^{-1}\left(\frac{19}{4}\right) + \cot^{-1}\left(\frac{39}{4}\right) + \cot^{-1}\left(\frac{67}{4}\right) + \ldots$$ is :

Let $$A = \{-3, -2, -1, 0, 1, 2, 3\}$$ and $$R$$ be a relation on $$A$$ defined by $$xRy$$ if and only if $$2x - y \in \{0, 1\}$$. Let $$l$$ 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, respectively. Then $$l + m + n$$ is equal to :

Let the product of $$\omega_1 = (8 + i)\sin\theta + (7 + 4i)\cos\theta$$ and $$\omega_2 = (1 + 8i)\sin\theta + (4 + 7i)\cos\theta$$ be $$\alpha + i\beta$$, $$i = \sqrt{-1}$$. Let p and q be the maximum and the minimum values of $$\alpha + \beta$$ respectively.

Let the values of p, for which the shortest distance between the lines $$\frac{x+1}{3} = \frac{y}{4} = \frac{z}{5}$$ and $$\vec{r} = (p\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$$ is $$\frac{1}{\sqrt{6}}$$, be a, b (a < b). Then the length of the latus rectum of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ is :

The axis of a parabola is the line $$y = x$$ and its vertex and focus are in the first quadrant at distances $$\sqrt{2}$$ and $$2\sqrt{2}$$ units from the origin, respectively. If the point (1, k) lies on the parabola, then a possible value of k is :

Let the domains of the functions $$f(x) = \log_4 \log_3 \log_7 (8 - \log_2(x^2 + 4x + 5))$$ and $$g(x) = \sin^{-1}\left(\frac{7x+10}{x-2}\right)$$ be $$(\alpha, \beta)$$ and $$[\gamma, \delta]$$, respectively. Then $$\alpha^2 + \beta^2 + \gamma^2 + \delta^2$$ is equal to :

A line passing through the point A(-2, 0), touches the parabola P : $$y^2 = x - 2$$ at the point B in the first quadrant. The area of the region bounded by the line AB, parabola P and the x-axis, is :

Let the sum of the focal distances of the point $$P(4, 3)$$ on the hyperbola H : $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be $$8\sqrt{\frac{5}{3}}$$. If for $$H$$, the length of the latus rectum is $$l$$ and the product of the focal distances of the point P is m, then $$9l^2 + 6m$$ is equal to :

Let the matrix $$A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$ satisfy $$A^n = A^{n-2} + A^2 - I$$ for $$n \geq 3$$. Then the sum of all the elements of $$A^{50}$$ is :

If the sum of the first 20 terms of the series $$\frac{4 \cdot 1}{4 + 3 \cdot 1^2 + 1^4} + \frac{4 \cdot 2}{4 + 3 \cdot 2^2 + 2^4} + \frac{4 \cdot 3}{4 + 3 \cdot 3^2 + 3^4} + \frac{4 \cdot 4}{4 + 3 \cdot 4^2 + 4^4} + \ldots$$ is $$\frac{m}{n}$$, where m and n are coprime, then $$m + n$$ is equal to :

If $$1^2 \cdot ({^{15} C_{1}}) + 2^2 \cdot ({^{15} C_{2}}) + 3^2 \cdot ({^{15} C_{3}}) + \ldots + 15^2 \cdot ({^{15} C_{15}}) = 2^m \cdot 3^n \cdot 5^k$$, where $$m, n, k \in \mathbb{N}$$, then $$m + n + k$$ is equal to :

Let for two distinct values of p the lines $$y = x + p$$ touch the ellipse E : $$\frac{x^2}{4^2} + \frac{y^2}{3^2} = 1$$ at the points A and B. Let the line $$y = x$$ intersect E at the points C and D. Then the area of the quadrilateral ABCD is equal to

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of A.P.'s in A and B respectively such that $$D = d + 3, d > 0$$. If $$\frac{p+q}{p-q} = \frac{19}{5}$$, then $$p - q$$ is equal to

If a curve $$y = y(x)$$ passes through the point $$\left(1, \frac{\pi}{2}\right)$$ and satisfies the differential equation $$(7x^4 \cot y - e^x \csc y) \frac{dx}{dy} = x^5, x \geq 1$$, then at $$x = 2$$, the value of $$\cos y$$ is :

The centre of a circle C is at the centre of the ellipse E : $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$. Let C pass through the foci $$F_1$$ and $$F_2$$ of E such that the circle C and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle $$PF_1F_2$$ is 30 and the length of the major axis of E is 17, then the distance between the foci of E is :

Let $$f(x) + 2f\left(\frac{1}{x}\right) = x^2 + 5$$ and $$2g(x) - 3g\left(\frac{1}{2}\right) = x$$, $$x > 0$$. If $$\alpha = \int_1^2 f(x)\,dx$$, and $$\beta = \int_1^2 g(x)\,dx$$, then the value of $$9\alpha + \beta$$ is :

Let A be the point of intersection of the lines $$L_1 : \frac{x-7}{1} = \frac{y-5}{0} = \frac{z-3}{-1}$$ and $$L_2 : \frac{x-1}{3} = \frac{y+3}{4} = \frac{z+7}{5}$$. Let B and C be the points on the lines $$L_1$$ and $$L_2$$ respectively such that $$AB = AC = \sqrt{15}$$. Then the square of the area of the triangle ABC is :

Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and $$\sqrt{2}$$ respectively. Then the mean deviation about the mode of these observations is :

If $$\alpha$$ is a root of the equation $$x^2 + x + 1 = 0$$ and $$\sum_{k=1}^{n} \left(\alpha^k + \frac{1}{\alpha^k}\right)^2 = 20$$, then n is equal to ______.

If $$\int \frac{(\sqrt{1+x^2}+x)^{10}}{(\sqrt{1+x^2}-x)^9} dx = \frac{1}{m}\left((\sqrt{1+x^2}+x)^n\left(n\sqrt{1+x^2}-x\right)\right) + C$$ where C is the constant of integration and $$m, n \in \mathbb{N}$$, then $$m + n$$ is equal to

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $$\frac{11}{50}$$, the n is equal to

Let m and n, (m < n) be two 2-digit numbers. Then the total numbers of pairs (m, n), such that gcd(m, n) = 6, is ______.

Let the three sides of a triangle ABC be given by the vectors $$2\hat{i} - \hat{j} + \hat{k}$$, $$\hat{i} - 3\hat{j} - 5\hat{k}$$ and $$3\hat{i} - 4\hat{j} - 4\hat{k}$$. Let G be the centroid of the triangle ABC. Then $$6\left(|\vec{AG}|^2 + |\vec{BG}|^2 + |\vec{CG}|^2\right)$$ is equal to ______.