NTA JEE Main 30th January 2023 Shift 1

Let $$z = 1 + i$$ and $$z_1 = \frac{1 + i\bar{z}}{\bar{z}(1-z) + \frac{1}{z}}$$. Then $$\frac{12}{\pi} \arg z_1$$ is equal to

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to

$$\sum_{n=0}^{\infty} \frac{n^3((2n)!) + (2n-1)(n!)}{(n!)(2n)!} = ae + \frac{b}{e} + c$$ where $$a, b, c \in \mathbb{Z}$$ and $$e = \sum_{n=0}^{\infty} \frac{1}{n!}$$. Then $$a^2 - b + c$$ is equal to ______.

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted, $$a$$ and $$b$$ are respectively mean and variance of remaining 6 observation, then $$a + 3b - 5$$ is equal to ______.

Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the number of one-one functions $$f: S \to P(S)$$, where $$P(S)$$ denote the power set of $$S$$, such that $$f(n) \subset f(m)$$ where $$n < m$$ is

Let $$f^1(x) = \frac{3x+2}{2x+3}$$, $$x \in R - \{-\frac{3}{2}\}$$. For $$n \geq 2$$, define $$f^n x = f^1 \circ f^{n-1}(x)$$. If $$f^5 x = \frac{ax+b}{bx+a}$$, $$\gcd(a,b) = 1$$, then $$a + b$$ is equal to ______.

$$\lim_{x \to 0} \frac{48}{x^4} \int_0^x \frac{t^3}{t^6+1} dt$$ is equal to

Let $$\alpha$$ be the area of the larger region bounded by the curve $$y^2 = 8x$$ and the lines $$y = x$$ and $$x = 2$$, which lies in the first quadrant. Then the value of $$3\alpha$$ is equal to

If the equation of the plane passing through the point $$(1, 1, 2)$$ and perpendicular to the line $$x - 3y + 2z - 1 = 0 = 4x - y + z$$ is $$Ax + By + Cz = 1$$, then $$140(C - B + A)$$ is equal to

If $$\lambda_1 < \lambda_2$$ are two values of $$\lambda$$ such that the angle between the planes $$P_1: \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7$$ and $$P_2: \vec{r} \cdot (\lambda\hat{i} + \hat{j} - 3\hat{k}) = 9$$ is $$\sin^{-1}\frac{2\sqrt{6}}{5}$$, then the square of the length of perpendicular from the point $$(38\lambda_1, 10\lambda_2, 2)$$ to the plane $$P_1$$ is