NTA JEE Main 27th July 2022 Shift 1

Let $$S = \{z \in \mathbb{C} : \bar{z}^2 + \bar{z} = 0\}$$. Then $$\sum_{z \in S} (\text{Re}(z) + \text{Im}(z))$$ is equal to______.

Let $$f(x) = 2x^2 - x - 1$$ and $$S = \{n \in \mathbb{Z} : |f(n)| \leq 800\}$$. Then, the value of $$\sum_{n \in S} f(n)$$ is equal to

If the length of the latus rectum of the ellipse $$x^2 + 4y^2 + 2x + 8y - \lambda = 0$$ is $$4$$, and $$l$$ is the length of its major axis, then $$\lambda + l$$ is equal to_______.

An ellipse $$E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ passes through the vertices of the hyperbola $$H : \frac{x^2}{49} - \frac{y^2}{64} = -1$$. Let the major and minor axes of the ellipse $$E$$ coincide with the transverse and conjugate axes of the hyperbola $$H$$. Let the product of the eccentricities of $$E$$ and $$H$$ be $$\frac{1}{2}$$. If $$l$$ is the length of the latus rectum of the ellipse $$E$$, then the value of $$113l$$ is equal to______.

The mean and variance of $$10$$ observations were calculated as $$15$$ and $$15$$ respectively by a student who took by mistake $$25$$ instead of $$15$$ for one observation. Then, the correct standard deviation is______.

Let $$S$$ be the set containing all $$3 \times 3$$ matrices with entries from $$\{-1, 0, 1\}$$. The total number of matrices $$A \in S$$ such that the sum of all the diagonal elements of $$A^T A$$ is $$6$$ is

For $$k \in \mathbb{R}$$, let the solutions of the equation $$\cos\left(\sin^{-1}\left(x \cot\left(\tan^{-1}\left(\cos(\sin^{-1} x)\right)\right)\right)\right) = k$$, $$0 < |x| < \frac{1}{\sqrt{2}}$$ be $$\alpha$$ and $$\beta$$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $$x^2 - bx - 5 = 0$$ are $$\frac{1}{\alpha^2} + \frac{1}{\beta^2}$$ and $$\frac{\alpha}{\beta}$$, then $$\frac{b}{k^2}$$ is equal to

Let $$M$$ and $$N$$ be the number of points on the curve $$y^5 - 9xy + 2x = 0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-axis, respectively. Then the value of $$M + N$$ equals

Let $$y = y(x)$$ be the solution curve of the differential equation $$\sin(2x^2) \log_e(\tan x^2) \, dy + \left(4xy - 4\sqrt{2}x \sin\left(x^2 - \frac{\pi}{4}\right)\right) dx = 0$$, $$0 < x < \sqrt{\frac{\pi}{2}}$$, which passes through the point $$\left(\sqrt{\frac{\pi}{6}}, 1\right)$$. Then $$\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|$$ is equal to

Let the line $$\dfrac{x-3}{7} = \dfrac{y-2}{-1} = \dfrac{z-3}{-4}$$ intersect the plane containing the lines $$\dfrac{x-4}{1} = \dfrac{y+1}{-2} = \dfrac{z}{1}$$ and $$4ax - y + 5z - 7a = 0 = 2x - 5y - z - 3$$, $$a \in \mathbb{R}$$ at the point $$P(\alpha, \beta, \gamma)$$. Then the value of $$\alpha + \beta + \gamma$$  equals ______.