NTA JEE Main 2nd September 2020 Shift 1 - Mathematics

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:

Let $$S$$ be the set of all $$\lambda \in R$$ for which the system of linear equations
$$2x - y + 2z = 2$$
$$x - 2y + \lambda z = -4$$
$$x + \lambda y + z = 4$$
has no solution. Then the set $$S$$:

The domain of the function $$f(x) = \sin^{-1}\left(\frac{|x|+5}{x^2+1}\right)$$ is $$(-\infty, -a] \cup [a, \infty)$$, then $$a$$ is equal to:

If a function $$f(x)$$ defined by $$f(x) = \begin{cases} ae^x + be^{-x}, & -1 \le x < 1 \\ cx^2, & 1 \le x \le 3 \\ ax^2 + 2cx, & 3 < x \le 4 \end{cases}$$ be continuous for some $$a, b, c \in R$$ and $$f'(0) + f'(2) = e$$, then the value of $$a$$ is:

If the tangent to the curve $$y = x + \sin y$$ at a point $$(a, b)$$ is parallel to the line joining $$(0, \frac{3}{2})$$ and $$(\frac{1}{2}, 2)$$, then:

If $$p(x)$$ be a polynomial of degree three that has a local maximum value 8 at $$x = 1$$ and a local minimum value 4 at $$x = 2$$ then $$p(0)$$ is equal to:

Let $$P(h, k)$$ be a point on the curve $$y = x^2 + 7x + 2$$, nearest to the line, $$y = 3x - 3$$. Then the equation of the normal to the curve at $$P$$ is:

Area (in sq. units) of the region outside $$\frac{|x|}{2} + \frac{|y|}{3} = 1$$ and inside the ellipse $$\frac{x^2}{4} + \frac{y^2}{9} = 1$$ is:

Let $$y = y(x)$$ be the solution of the differential equation, $$\frac{2+\sin x}{y+1} \cdot \frac{dy}{dx} = -\cos x$$, $$y > 0$$, $$y(0) = 1$$. If $$y(\pi) = a$$ and $$\frac{dy}{dx}$$ at $$x = \pi$$ is $$b$$, then the ordered pair $$(a, b)$$ is equal to:

The plane passing through the points $$(1, 2, 1)$$, $$(2, 1, 2)$$ and parallel to the line, $$2x = 3y$$, $$z = 1$$ also passes through the point:

Box 1 contains 30 cards numbered 1 to 30 and Box 2 contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box 1 is:

If the letters of the word 'MOTHER' be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word 'MOTHER' is ___________.

The number of integral values of $$k$$ for which the line, $$3x + 4y = k$$ intersects the circle, $$x^2 + y^2 - 2x - 4y + 4 = 0$$ at two distinct points is ___________.

If $$\lim_{x \to 1} \frac{x + x^2 + x^3 + \ldots + x^n - n}{x - 1} = 820$$, $$(n \in N)$$ then the value of $$n$$ is equal to ___________.

The integral $$\int_0^2 ||x - 1| - x| \; dx$$ is equal to ___________.

Let $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be three unit vectors such that $$|\vec{a} - \vec{b}|^2 + |\vec{a} - \vec{c}|^2 = 8$$. Then $$|\vec{a} + 2\vec{b}|^2 + |\vec{a} + 2\vec{c}|^2$$ is equal to ___________.