NTA JEE Main 28th July 2022 Shift 2

Let $$z = a + ib$$, $$b \neq 0$$ be complex numbers satisfying $$z^2 = \bar{z} \cdot 2^{1-|z|}$$. Then the least value of $$n \in \mathbb{N}$$, such that $$z^n = (z+1)^n$$, is equal to _____

A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $$b + 3g$$ is equal to

If $$\dfrac{6}{3^{12}} + \dfrac{10}{3^{11}} + \dfrac{20}{3^{10}} + \dfrac{40}{3^9} + \ldots + \dfrac{10240}{3} = 2^n \cdot m$$, where $$m$$ is odd, then $$m \cdot n$$ is equal to _____

Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}} + \beta x\right)^4$$, $$(1 - 3\beta x)^2$$ and $$\left(1 - \frac{\beta}{2}x\right)^6$$, $$\beta > 0$$ respectively form the first three terms of an A.P. If $$d$$ is the common difference of this A.P., then $$50 - \frac{2d}{\beta^2}$$ is equal to _____

If $$1 + (2 + {}^{49}C_1 + {}^{49}C_2 + \ldots + {}^{49}C_{49})({}^{50}C_2 + {}^{50}C_4 + \ldots + {}^{50}C_{50})$$ is equal to $$2^n \cdot m$$, where $$m$$ is odd, then $$n + m$$ is equal to _____

Let $$S = [-\pi, \frac{\pi}{2}) - \{-\frac{\pi}{2}, -\frac{\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}\}$$. Then the number of elements in the set $$A = \{\theta \in S : \tan\theta(1 + \sqrt{5}\tan(2\theta)) = \sqrt{5} - \tan(2\theta)\}$$ is _____

Two tangent lines $$l_1$$ and $$l_2$$ are drawn from the point (2, 0) to the parabola $$2y^2 = -x$$. If the lines $$l_1$$ and $$l_2$$ are also tangent to the circle $$(x-5)^2 + y^2 = r$$, then $$17r^2$$ is equal to

Let the tangents at the points P and Q on the ellipse $$\frac{x^2}{2} + \frac{y^2}{4} = 1$$ meet at the point $$R(\sqrt{2}, 2\sqrt{2}-2)$$. If S is the focus of the ellipse on its negative major axis, then $$SP^2 + SQ^2$$ is equal to

The value of the integral $$\int_0^{\pi/2} \frac{60\sin(6x)}{\sin x} dx$$ is equal to

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If $$\sigma^2$$ is the variance of X, then $$100\sigma^2$$ is equal to