NTA JEE Main 6th September 2020 Shift 1

If $$\alpha$$ and $$\beta$$ be two roots of the equation $$x^2 - 64x + 256 = 0$$. Then the value of $$\left(\frac{\alpha^3}{\beta^5}\right)^{1/8} + \left(\frac{\beta^3}{\alpha^5}\right)^{1/8}$$ is:

The region represented by $$\{z = x + iy \in \mathbb{C} : |z| - \text{Re}(z) \leq 1\}$$ is also given by the inequality:

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

Let $$a, b, c, d$$ and $$p$$ be non-zero distinct real numbers such that $$(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$$. Then:

If $$\{p\}$$ denotes the fractional part of the number $$p$$, then $$\left\{\frac{3^{200}}{8}\right\}$$ is equal to:

A ray of light coming from the point $$(2, 2\sqrt{3})$$ is incident at an angle $$30^\circ$$ on the line $$x = 1$$ at the point A. The ray gets reflected on the line $$x = 1$$ and meets $$x$$-axis at the point B. Then, the line AB passes through the point:

Let $$L_1$$ be a tangent to the parabola $$y^2 = 4(x+1)$$ and $$L_2$$ be a tangent to the parabola $$y^2 = 8(x+2)$$ such that $$L_1$$ and $$L_2$$ intersect at right angles. Then $$L_1$$ and $$L_2$$ meet on the straight line:

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $$\frac{x^2}{4} + \frac{y^2}{2} = 1$$ from any of its foci?

The negation of the Boolean expression $$p \vee (\sim p \wedge q)$$ is equivalent to:

If $$\sum_{i=1}^{n}(x_i - a) = n$$ and $$\sum_{i=1}^{n}(x_i - a)^2 = na$$, $$(n, a \gt 1)$$, then the standard deviation of $$n$$ observations $$x_1, x_2, \ldots, x_n$$ is: