NTA JEE Main 8th April 2023 Shift 2

Let $$0 < z < y < x$$ be three real numbers such that $$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$$ are in an arithmetic progression and $$x, \sqrt{2}y, z$$ are in a geometric progression. If $$xy + yz + zx = \frac{3}{\sqrt{2}}xyz$$, then $$3(x + y + z)^2$$ is equal to _____.

The ordinates of the points $$P$$ and $$Q$$ on the parabola with focus $$(3, 0)$$ and directrix $$x = -3$$ are in the ratio 3 : 1. If $$R(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$P$$ and $$Q$$, then $$\frac{\beta^2}{\alpha}$$ is equal to _____.

Let $$R = \{a, b, c, d, e\}$$ and $$S = \{1, 2, 3, 4\}$$. Total number of onto functions $$f : R \to S$$ such that $$f(a) \neq 1$$, is equal to _____.

Let k and m be positive real numbers such that the function $$f(x) = \begin{cases} 3x^2 + k\sqrt{x + 1}, & 0 < x < 1 \\ mx^2 + k^2, & x \geq 1 \end{cases}$$ is differentiable for all $$x > 0$$. Then $$\frac{8f'(8)}{f'(\frac{1}{8})}$$ is equal to _____.

Let $$[t]$$ denote the greatest integer function. If $$\int_0^{2.4} [x^2] dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5}$$, then $$\alpha + \beta + \gamma + \delta$$ is equal to _____.

Let the area enclosed by the lines $$x + y = 2$$, $$y = 0$$, $$x = 0$$ and the curve $$f(x) = \min\left\{x^2 + \frac{3}{4}, 1 + [x]\right\}$$ where $$[x]$$ denotes the greatest integer $$\leq x$$, be $$A$$. Then the value of $$12A$$ is _____.

Let the solution curve $$x = x(y)$$, $$0 < y < \frac{\pi}{2}$$, of the differential equation $$(\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0$$ satisfy $$x\left(\frac{\pi}{3}\right) = \frac{1}{2\log_e 2}$$. If $$x\left(\frac{\pi}{6}\right) = \frac{1}{\log_e m - \log_e n}$$, where $$m$$ and $$n$$ are coprime, then $$mn$$ is equal to _____.

The area of the quadrilateral $$ABCD$$ with vertices $$A(2, 1, 1)$$, $$B(1, 2, 5)$$, $$C(-2, -3, 5)$$ and $$D(1, -6, -7)$$ is equal to _____.

For $$a, b \in \mathbb{Z}$$ and $$|a - b| \leq 10$$, let the angle between the plane $$P: ax + y - z = b$$ and the line $$L: x - 1 = a - y = z + 1$$ be $$\cos^{-1}\left(\frac{1}{3}\right)$$. If the distance of the point $$(6, -6, 4)$$ from the plane $$P$$ is $$3\sqrt{6}$$, then $$a^4 + b^2$$ is equal to _____.

Let $$P_1$$ be the plane $$3x - y - 7z = 11$$ and $$P_2$$ be the plane passing through the points $$(2, -1, 0)$$, $$(2, 0, -1)$$, and $$(5, 1, 1)$$. If the foot of the perpendicular drawn from the point $$(7, 4, -1)$$ on the line of intersection of the planes $$P_1$$ and $$P_2$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to _____.