NTA JEE Main 2nd September 2020 Shift 1

Let $$\alpha$$ and $$\beta$$ be the roots of the equation, $$5x^2 + 6x - 2 = 0$$. If $$S_n = \alpha^n + \beta^n$$, $$n = 1, 2, 3, \ldots$$, then:

The value of $$\left(\frac{1+\sin\frac{2\pi}{9}+i\cos\frac{2\pi}{9}}{1+\sin\frac{2\pi}{9}-i\cos\frac{2\pi}{9}}\right)^3$$ is:

The sum of the first three terms of G.P. is $$S$$ and their product is 27. Then all such $$S$$ lie in:

If $$|x| \lt 1$$, $$|y| \lt 1$$ and $$x \ne 1$$, then the sum to infinity of the following series $$(x + y) + (x^2 + xy + y^2) + (x^3 + x^2y + xy^2 + y^3) + \ldots$$ is:

Let $$\alpha \gt 0, \beta \gt 0$$ be such that $$\alpha^3 + \beta^2 = 4$$. If the maximum value of the term independent of $$x$$ in the binomial expansion of $$\left(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}}\right)^{10}$$ is $$10k$$, then $$k$$ is equal to:

A line parallel to the straight line $$2x - y = 0$$ is tangent to the hyperbola $$\frac{x^2}{4} - \frac{y^2}{2} = 1$$ at the point $$(x_1, y_1)$$. Then $$x_1^2 + 5y_1^2$$ is equal to:

The contrapositive of the statement "If I reach the station in time, then I will catch the train" is:

Let $$X = \{x \in N : 1 \le x \le 17\}$$ and $$Y = \{ax + b : x \in X \text{ and } a, b \in R, a > 0\}$$. If mean and variance of elements of Y are 17 and 216 respectively then $$a + b$$ is equal to:

If $$R = \{(x, y) : x, y \in Z, x^2 + 3y^2 \le 8\}$$ is a relation on the set of integers $$Z$$, then the domain of $$R^{-1}$$ is:

Let $$A$$ be a $$2 \times 2$$ real matrix with entries from $$\{0, 1\}$$ and $$|A| \ne 0$$. Consider the following two statements:
$$(P)$$ If $$A \ne I_2$$, then $$|A| = -1$$
$$(Q)$$ If $$|A| = 1$$, then $$tr(A) = 2$$
Where $$I_2$$ denotes $$2 \times 2$$ identity matrix and $$tr(A)$$ denotes the sum of the diagonal entries of $$A$$. Then: