NTA JEE Main 31st August 2021 Shift 2

The sum of the roots of the equation, $$x + 1 - 2\log_2 3 + 2^x + 2\log_4 10 - 2^{-x} = 0$$, is:

The number of solutions of the equation $$32^{\tan^2 x} + 32^{\sec^2 x} = 81$$, $$0 \leq x \leq \frac{\pi}{4}$$ is:

If $$z$$ is a complex number such that $$\frac{z-i}{z-1}$$ is purely imaginary, then the minimum value of $$|z - (3 + 3i)|$$ is:

Let $$a_1, a_2, a_3, \ldots$$ be an A.P. If $$\frac{a_1 + a_2 + \ldots + a_{10}}{a_1 + a_2 + \ldots + a_p} = \frac{100}{p^2}$$, $$p \neq 10$$, then $$\frac{a_{11}}{a_{10}}$$ is equal to:

Let $$A$$ be the set of all points $$\alpha, \beta$$ such that the area of triangle formed by the points $$(5, 6)$$, $$(3, 2)$$ and $$(\alpha, \beta)$$ is 12 square units. Then the least possible length of a line segment joining the origin to a point in $$A$$, is:

The locus of mid-points of the line segments joining -3, -5 and the points on the ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ is:

If $$\alpha = \lim_{x \to \pi/4} \frac{\tan^3 x - \tan x}{\cos x + \frac{\pi}{4}}$$ and $$\beta = \lim_{x \to 0} \cos x^{\cot x}$$ are the roots of the equation, $$ax^2 + bx - 4 = 0$$, then the ordered pair $$a, b$$ is:

Negation of the statement $$(p \vee r) \Rightarrow (q \vee r)$$ is:

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is:

If $$\alpha + \beta + \gamma = 2\pi$$, then the system of equations
$$x + \cos\gamma y + \cos\beta z = 0$$
$$\cos\gamma x + y + \cos\alpha z = 0$$
$$\cos\beta x + \cos\alpha y + z = 0$$
has: