NTA JEE Main 2025 April 4th Shift 2

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 :