NTA JEE Main 24th January 2023 Shift 2

The number of real solutions of the equation $$3\left(x^2 + \frac{1}{x^2}\right) - 2\left(x + \frac{1}{x}\right) + 5 = 0$$, is

The value of $$\left(\frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}}\right)^3$$ is

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is

If $$\frac{1^3 + 2^3 + 3^3 + \ldots \text{upto n terms}}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots \text{upto n terms}} = \frac{9}{5}$$ then the value of $$n$$ is

If $$\binom{30}{1}^2 + 2\binom{30}{2}^2 + 3\binom{30}{3}^2 + \ldots + 30\binom{30}{30}^2 = \frac{\alpha \cdot 60!}{(30!)^2}$$, then $$\alpha$$ is equal to

The locus of the middle points of the chords of the circle $$C_1: (x-4)^2 + (y-5)^2 = 4$$ which subtend an angle $$\theta_i$$ at the centre of the circle $$C_i$$, is a circle of radius $$r_i$$. If $$\theta_1 = \frac{\pi}{3}$$, $$\theta_3 = \frac{2\pi}{3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $$\theta_2$$ is equal to

The equations of sides $$AB$$ and $$AC$$ of a triangle $$ABC$$ are $$(\lambda + 1)x + \lambda y = 4$$ and $$\lambda x + (1 - \lambda)y + \lambda = 0$$ respectively. Its vertex $$A$$ is on the $$y$$-axis and its orthocentre is $$(1, 2)$$. The length of the tangent from the point $$C$$ to the part of the parabola $$y^2 = 6x$$ in the first quadrant is

The set of values of $$a$$ for which $$\lim_{x \to a} ([x-5] - [2x+2]) = 0$$, where $$[\zeta]$$ denotes the greatest integer less than or equal to $$\zeta$$ is equal to

Let $$p$$ and $$q$$ be two statements. Then $$\sim(p \wedge (p \to \sim q))$$ is equivalent to

Let the six numbers $$a_1, a_2, \ldots, a_6$$ be in A.P. and $$a_1 + a_3 = 10$$. If the mean of these six numbers is $$\frac{19}{2}$$ and their variance is $$\sigma^2$$, then $$8\sigma^2$$ is equal to