NTA JEE Main 29th January 2023 Shift 2

Let $$\alpha = 8 - 14i$$, $$A = \left\{z \in \mathbb{C} : \frac{\alpha\bar{z} - \bar{\alpha}z}{z^2 - (\bar{z})^2 - 112i} = 1\right\}$$ and $$B = \{z \in \mathbb{C} : |z + 3i| = 4\}$$. Then, $$\sum_{z \in A \cap B} (Re\ z - Im\ z)$$ is equal to ______.

The total number of 4-digit numbers whose greatest common divisor with $$54$$ is $$2$$, is

Let $$a_1 = b_1 = 1$$ and $$a_n = a_{n-1} + (n-1)$$, $$b_n = b_{n-1} + a_{n-1}$$, $$\forall n \geq 2$$. If $$S = \sum_{n=1}^{10} \left(\frac{b_n}{2^n}\right)$$ and $$T = \sum_{n=1}^{8} \frac{n}{2^{n-1}}$$ then $$2^7(2S - T)$$ is equal to

Let $$\{a_k\}$$ and $$\{b_k\}$$, $$k \in \mathbb{N}$$, be two G.P.s with common ratio $$r_1$$ and $$r_2$$ respectively such that $$a_1 = b_1 = 4$$ and $$r_1 < r_2$$. Let $$c_k = a_k + b_k$$, $$k \in \mathbb{N}$$. If $$c_2 = 5$$ and $$c_3 = \frac{13}{4}$$ then $$\sum_{k=1}^{\infty} c_k - (12a_6 + 8b_4)$$ is equal to

A circle with centre $$(2, 3)$$ and radius $$4$$ intersects the line $$x + y = 3$$ at the points $$P$$ and $$Q$$. If the tangents at $$P$$ and $$Q$$ intersect at the point $$S(\alpha, \beta)$$, then $$4\alpha - 7\beta$$ is equal to

A triangle is formed by the tangents at the point $$(2, 2)$$ on the curves $$y^2 = 2x$$ and $$x^2 + y^2 = 4x$$, and the line $$x + y + 2 = 0$$. If $$r$$ is the radius of its circumcircle, then $$r^2$$ is equal to

Let $$X = \{11, 12, 13, \ldots, 40, 41\}$$ and $$Y = \{61, 62, 63, \ldots, 90, 91\}$$ be the two sets of observations. If $$\bar{x}$$ and $$\bar{y}$$ are their respective means and $$\sigma^2$$ is the variance of all the observations in $$X \cup Y$$, then $$|\bar{x} + \bar{y} - \sigma^2|$$ is equal to

Let A be a symmetric matrix such that $$|A| = 2$$ and $$\begin{bmatrix} 2 & 1 \\ 3 & \frac{3}{2} \end{bmatrix} A = \begin{bmatrix} 1 & 2 \\ \alpha & \beta \end{bmatrix}$$. If the sum of the diagonal elements of A is $$s$$, then $$\frac{\beta s}{\alpha^2}$$ is equal to ______.

Consider a function $$f : \mathbb{N} \to \mathbb{R}$$, satisfying $$f(1) + 2f(2) + 3f(3) + \ldots + xf(x) = x(x+1)f(x)$$; $$x \geq 2$$ with $$f(1) = 1$$. Then $$\frac{1}{f(2022)} + \frac{1}{f(2028)}$$ is equal to

If the equation of the normal to the curve $$y = \frac{x-a}{(x+b)(x-2)}$$ at the point $$(1, -3)$$ is $$x - 4y = 13$$ then the value of $$a + b$$ is equal to ______.