NTA JEE Main 6th April 2023 Shift 2

Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X = \{z \in C : Re(az^2 + bz) = a$$ and $$Re(bz^2 + az) = b\}$$ is equal to

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

If $$gcd(m, n) = 1$$ and $$1^2 - 2^2 + 3^2 - 4^2 + \ldots + (2021)^2 - (2022)^2 + (2023)^2 = 1012m^2n$$ then $$m^2 - n^2$$ is equal to

If the coefficients of $$x^7$$ in $$\left(ax^2 + \dfrac{1}{2bx}\right)^{11}$$ and $$x^{-7}$$ in $$\left(ax - \dfrac{1}{3bx^2}\right)^{11}$$ are equal, then

Among the statements:
(S1): $$2023^{2022} - 1999^{2022}$$ is divisible by 8.
(S2): $$13(13)^n - 11n - 13$$ is divisible by 144 for infinitely many $$n \in \mathbb{N}$$

If the tangents at the points $$P$$ and $$Q$$ on the circle $$x^2 + y^2 - 2x + y = 5$$ meet at the point $$R\left(\dfrac{9}{4}, 2\right)$$, then the area of the triangle $$PQR$$ is

$$\lim_{n \to \infty} \left\{\left(2^{1/2} - 2^{1/4}\right)\left(2^{1/2} - 2^{1/8}\right) \cdots \left(2^{1/2} - 2^{1/(2n+1)}\right)\right\}$$ is equal to

Among the statements
(S1): $$(p \Rightarrow q) \lor ((\sim p) \wedge q)$$ is a tautology
(S2): $$(q \Rightarrow p) \Rightarrow ((\sim p) \wedge q)$$ is a contradiction

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons who speak only English is $$\alpha$$ and the number of persons who speaks only Hindi is $$\beta$$, then the eccentricity of the ellipse $$25(\beta^2 x^2 + \alpha^2 y^2) = \alpha^2\beta^2$$ is

Let $$P$$ be a square matrix such that $$P^2 = I - P$$. For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$P^\alpha + P^\beta = \gamma I - 29P$$ and $$P^\alpha - P^\beta = \delta I - 13P$$, then $$\alpha + \beta + \gamma - \delta$$ is equal to