NTA JEE Main 29th June 2022 Shift 1

Let $$S = \{z \in C : |z - 2| \leq 1, z(1+i) + \bar{z}(1-i) \leq 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at $$z_1 \in S$$ and $$z_2 \in S$$. If $$5(|z_1|^2 + |z_2|^2) = \alpha + \beta\sqrt{5}$$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha + \beta$$ is equal to ______.

Let $$b_1b_2b_3b_4$$ be a 4-element permutation with $$b_i \in \{1, 2, 3, \ldots, 100\}$$ for $$1 \leq i \leq 4$$ and $$b_i \neq b_j$$ for $$i \neq j$$, such that either $$b_1, b_2, b_3$$ are consecutive integers or $$b_2, b_3, b_4$$ are consecutive integers. Then the number of such permutations $$b_1b_2b_3b_4$$ is equal to ______.

The number of elements in the set $$S = \{\theta \in [-4\pi, 4\pi] : 3\cos^2 2\theta + 6\cos 2\theta - 10\cos^2\theta + 5 = 0\}$$ is ______.

The number of solutions of the equation $$2\theta - \cos^2\theta + \sqrt{2} = 0$$ in $$R$$ is equal to ______.

Let $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$, $$a > 0$$, $$b > 0$$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $$4(2\sqrt{2} + \sqrt{14})$$. If the eccentricity $$H$$ is $$\frac{\sqrt{11}}{2}$$, then value of $$a^2 + b^2$$ is equal to ______.

$$50\tan\left(3\tan^{-1}\left(\frac{1}{2}\right) + 2\cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right) + 4\sqrt{2}\tan\left(\frac{1}{2}\tan^{-1}(2\sqrt{2})\right)$$ is equal to ______.

Let $$c, k \in R$$. If $$f(x) = (c+1)x^2 + (1-c^2)x + 2k$$ and $$f(x+y) = f(x) + f(y) - xy$$, for all $$x, y \in R$$, then the value of $$|2(f(1) + f(2) + f(3) + \ldots + f(20))|$$ is equal to ______.

Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{\sqrt{2}y}{2\cos^4x - \cos 2x} = xe^{\tan^{-1}(\sqrt{2}\cot 2x)}$$, $$0 < x < \frac{\pi}{2}$$ with $$y\left(\frac{\pi}{4}\right) = \frac{\pi^2}{32}$$. If $$y\left(\frac{\pi}{3}\right) = \frac{\pi^2}{18}e^{-\tan^{-1}(\alpha)}$$, then the value of $$3\alpha^2$$ is equal to ______.

Let $$d$$ be the distance between the foot of perpendiculars of the points $$P(1, 2, -1)$$ and $$Q(2, -1, 3)$$ on the plane $$-x + y + z = 1$$. Then $$d^2$$ is equal to ______.

Let $$P_1 : \vec{r} \cdot (2\hat{i} + \hat{j} - 3\hat{k}) = 4$$ be a plane. Let $$P_2$$ be another plane which passes through the points $$(2, -3, 2)$$, $$(2, -2, -3)$$ and $$(1, -4, 2)$$. If the direction ratios of the line of intersection of $$P_1$$ and $$P_2$$ be $$16, \alpha, \beta$$, then the value of $$\alpha + \beta$$ is equal to ______.