NTA JEE Main 27th July 2021 Shift 2

Let $$\alpha = \max_{x \in R}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$$ and $$\beta = \min_{x \in R}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$$. If $$8x^2 + bx + c = 0$$ is a quadratic equation whose roots are $$\alpha^{1/5}$$ and $$\beta^{1/5}$$, then the value of $$c - b$$ is equal to:

Let $$\mathbb{C}$$ be the set of all complex numbers. Let $$S_1 = \{z \in \mathbb{C} : |z - 2| \leq 1\}$$ and $$S_2 = \{z \in \mathbb{C} : z(1 + i) + \bar{z}(1 - i) \geq 4\}$$. Then, the maximum value of $$\left|z - \frac{5}{2}\right|^2$$ for $$z \in S_1 \cap S_2$$ is equal to:

If $$\tan\left(\frac{\pi}{9}\right), x, \tan\left(\frac{7\pi}{18}\right)$$ are in arithmetic progression and $$\tan\left(\frac{\pi}{9}\right), y, \tan\left(\frac{5\pi}{18}\right)$$ are also in arithmetic progression, then $$|x - 2y|$$ is equal to:

A possible value of $$x$$, for which the ninth term in the expansion of $$\left\{3^{\log_3 \sqrt{25^{x-1}+7}} + 3^{\left(-\frac{1}{5}\right)\log_3(5^{x-1}+1)}\right\}^{10}$$ in the increasing powers of $$3^{\left(-\frac{1}{5}\right)\log_3(5^{x-1}+1)}$$ is equal to 180, is:

The point $$P(a, b)$$ undergoes the following three transformations successively:
(a) reflection about the line $$y = x$$.
(b) translation through 2 units along the positive direction of $$x$$-axis.
(c) rotation through angle $$\frac{\pi}{4}$$ about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point $$P$$ are $$\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$$, then the value of $$2a + b$$ is equal to:

Two sides of a parallelogram are along the lines $$4x + 5y = 0$$ and $$7x + 2y = 0$$. If the equation of one of the diagonals of the parallelogram is $$11x + 7y = 9$$, then other diagonal passes through the point:

Consider a circle $$C$$ which touches the $$y$$-axis at $$(0, 6)$$ and cuts off an intercept $$6\sqrt{5}$$ on the $$x$$-axis. Then the radius of the circle $$C$$ is equal to:

The value of $$\lim_{x \to 0}\left(\frac{x}{\sqrt[8]{1 - \sin x} - \sqrt[8]{1 + \sin x}}\right)$$ is equal to:

Which of the following is the negation of the statement "for all $$M \gt 0$$, there exists $$x \in S$$ such that $$x \geq M$$"?

Let the mean and variance of the frequency distribution
$$x$$:       $$x_1 = 2$$       $$x_2 = 6$$       $$x_3 = 8$$        $$x_4 = 9$$
$$f$$:             4                  4                  $$\alpha$$                   $$\beta$$
be 6 and 6.8 respectively. If $$x_3$$ is changed from 8 to 7, then the mean for the new data will be: