NTA JEE Main 26th June 2022 Shift 2

If $$p$$ and $$q$$ are real number such that $$p + q = 3, p^4 + q^4 = 369$$, then the value of $$\left(\frac{1}{p} + \frac{1}{q}\right)^{-2}$$ is equal to ______

If $$z^2 + z + 1 = 0, z \in C$$, then $$\left|\sum_{n=1}^{15}\left(z^n + (-1)^n \frac{1}{z^n}\right)^2\right|$$ is equal to ______

The total number of 3-digit numbers, whose greatest common divisor with 36 is 2, is ______

If $$a_1(> 0), a_2, a_3, a_4, a_5$$ are in a G.P., $$a_2 + a_4 = 2a_3 + 1$$ and $$3a_2 + a_3 = 2a_4$$, then $$a_2 + a_4 + 2a_5$$ is equal to ______

If $$^{40}C_0 + ^{41}C_1 + ^{42}C_2 + \cdots + ^{60}C_{20} = \frac{m}{n} \times ^{60}C_{20}$$ where $$m$$ & $$n$$ are co-prime, then $$m + n$$ is equal to ______

Let a line $$L_1$$ be tangent to the hyperbola $$\frac{x^2}{16} - \frac{y^2}{4} = 1$$ and let $$L_2$$ be the line passing through the origin and perpendicular to $$L_1$$. If the locus of the point of intersection of $$L_1$$ and $$L_2$$ is $$(x^2 + y^2)^2 = \alpha x^2 + \beta y^2$$, then $$\alpha + \beta$$ is equal to ______

Let $$X = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$, $$Y = \alpha I + \beta X + \gamma X^2$$ and $$Z = \alpha^2 I - \alpha\beta X + (\beta^2 - \alpha\gamma)X^2, \alpha, \beta, \gamma \in \mathbb{R}$$.
If $$Y^{-1} = \begin{bmatrix} \frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5} \end{bmatrix}$$, then $$(\alpha - \beta + \gamma)^2$$ is equal to ______

Let $$f : \mathbb{R} \to \mathbb{R}$$ satisfy $$f(x + y) = 2^x f(y) + 4^y f(x), \forall x, y \in \mathbb{R}$$. If $$f(2) = 3$$, then $$14 \cdot \frac{f'(4)}{f'(2)}$$ is equal to ______

The integral $$\frac{24}{\pi}\int_0^{\sqrt{2}} \frac{(2-x^2)dx}{(2+x^2)\sqrt{4+x^4}}$$ is equal to ______

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is $$p$$, then $$96p$$ is equal to ______