NTA JEE Main 31st January 2023 Shift 2

The equation $$e^{4x} + 8e^{3x} + 13e^{2x} - 8e^x + 1 = 0$$, $$x \in R$$ has:

The complex number $$z = \dfrac{i-1}{\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}}$$ is equal to:

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$a_7 = 3$$, the product $$(a_1 a_4)$$ is minimum and the sum of its first $$n$$ terms is zero then $$n! - 4a_{n(n+2)}$$ is equal to

The coefficient of $$x^{-6}$$, in the expansion of $$\left(\dfrac{4x}{5} + \dfrac{5}{2x^2}\right)^9$$, is

If $$^{2n+1}P_{n-1} : ^{2n-1}P_n = 11 : 21$$, then $$n^2 + n + 15$$ is equal to:

The set of all values of $$a^2$$ for which the line $$x + y = 0$$ bisects two distinct chords drawn from a point $$P\left(\dfrac{1+a}{2}, \dfrac{1-a}{2}\right)$$ on the circle $$2x^2 + 2y^2 - (1+a)x - (1-a)y = 0$$, is equal to:

Let H be the hyperbola, whose foci are $$(1 \pm \sqrt{2}, 0)$$ and eccentricity is $$\sqrt{2}$$. Then the length of its latus rectum is:

$$\lim_{x \to \infty} \dfrac{\left(\sqrt{3x+1}+\sqrt{3x-1}\right)^6 + \left(\sqrt{3x+1}-\sqrt{3x-1}\right)^6}{\left(x+\sqrt{x^2-1}\right)^6 + \left(x-\sqrt{x^2-1}\right)^6} \cdot x^3$$

The number of values of $$r \in \{p, q, \sim p, \sim q\}$$ for which $$((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$$ is a tautology, is:

Let the mean and standard deviation of marks of class A of 100 students be respectively 40 and $$\alpha(> 0)$$, and the mean and standard deviation of marks of class B of $$n$$ students be respectively 55 and $$30 - \alpha$$. If the mean and variance of the marks of the combined class of $$100 + n$$ students are respectively 50 and 350, then the sum of variances of classes A and B is