NTA JEE Main 2025 April 7th Shift 2

If the orthocentre of the triangle formed by the lines $$y = x + 1$$, $$y = 4x - 8$$ and $$y = mx + c$$ is at $$(3, -1)$$, then $$m - c$$ is :

Let $$\vec{a}$$ and $$\vec{b}$$ be the vectors of the same magnitude such that $$\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|} = \sqrt{2}+1$$. Then $$\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$$ is :

Let $$A = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}$$ and $$B = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}$$. Then

If the range of the function $$f(x) = \frac{5 - x}{x^2 - 3x + 2}$$, $$x \ne 1, 2$$, is $$(-\infty, \alpha] \cup [\beta, \infty)$$, then $$\alpha^2 + \beta^2$$ is equal to :

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$n^2 - m^2$$ is equal to :

Let a random variable X take values 0, 1, 2, 3 with $$P(X = 0) = P(X = 1) = p$$, $$P(X = 2) = P(X = 3) = q$$ and $$E(X^2) = 2E(X)$$. Then the value of $$8p - 1$$ is :

If the area of the region $$\{(x, y) : 1 + x^2 \le y \le \min\{x + 7, 11 - 3x\}\}$$ is A, then $$3A$$ is equal to

Let $$f : \mathbf{R} \to \mathbf{R}$$ be a polynomial function of degree four having extreme values at $$x = 4$$ and $$x = 5$$. If $$\lim_{x \to 0} \frac{f(x)}{x^2} = 5$$, then $$f(2)$$ is equal to :

The number of solutions of the equation $$\cos 2\theta \cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$$ in $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ is :

Let $$a_n$$ be the $$n^{\text{th}}$$ term of an A.P. If $$S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$$, $$a_6 = 7$$ and $$S_7 = 7$$, then $$a_{n}$$ is equal to :