NTA JEE Main 5th September 2020 Shift 1 - Mathematics

The product of the roots of the equation $$9x^2 - 18|x| + 5 = 0$$ is:

If the four complex numbers $$z$$, $$\overline{z}$$, $$\overline{z} - 2\,\text{Re}(\overline{z})$$ and $$z - 2\,\text{Re}(z)$$ represent the vertices of a square of side 4 units in the Argand plane, then $$|z|$$ is equal to:

If $$2^{10} + 2^9 \cdot 3^1 + 2^8 \cdot 3^2 + \ldots + 2 \cdot 3^9 + 3^{10} = S - 2^{11}$$, then $$S$$ is equal to:

If $$3^{2\sin 2\alpha - 1}$$, 14 and $$3^{4 - 2\sin 2\alpha}$$ are the first three terms of an A.P. for some $$\alpha$$, then the sixth term of this A.P. is:

If the common tangent to the parabolas, $$y^2 = 4x$$ and $$x^2 = 4y$$ also touches the circle, $$x^2 + y^2 = c^2$$, then $$c$$ is equal to:

If the co-ordinates of two points $$A$$ and $$B$$ are $$\left(\sqrt{7}, 0\right)$$ and $$\left(-\sqrt{7}, 0\right)$$ respectively and $$P$$ is any point on the conic, $$9x^2 + 16y^2 = 144$$, then $$PA + PB$$ is equal to:

If the point $$P$$ on the curve, $$4x^2 + 5y^2 = 20$$ is farthest from the point $$Q(0, -4)$$, then $$PQ^2$$ is equal to:

If $$\alpha$$ is the positive root of the equation, $$p(x) = x^2 - x - 2 = 0$$, then $$\lim_{x \to \alpha^+} \frac{\sqrt{1 - \cos(p(x))}}{x + \alpha - 4}$$ is equal to:

The negation of the Boolean expression $$x \leftrightarrow \sim y$$ is equivalent to:

The mean and variance of 7 observations are 8 and 16, respectively. If five of the observations are 2, 4, 10, 12, 14 then the absolute difference of the remaining two observations is:

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If $$x$$ denotes the percentage of them, who like both coffee and tea, then $$x$$ cannot be:

If the minimum and the maximum values of the function $$f : \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \to R$$, defined by$$f(\theta) = \begin{vmatrix} -\sin^2\theta & -1 - \sin^2\theta & 1 \\ -\cos^2\theta & -1 - \cos^2\theta & 1 \\ 12 & 10 & -2 \end{vmatrix}$$ are $$m$$ and $$M$$ respectively, then the ordered pair $$(m, M)$$ is equal to:

Let $$\lambda \in \mathbb{R}$$. The system of linear equations
$$2x_1 - 4x_2 + \lambda x_3 = 1$$
$$x_{1} - 6x_{2} + x_{3} = 2$$
$$\lambda x_1 - 10x_2 + 4x_3 = 3$$
is inconsistent for:

If $$S$$ is the sum of the first 10 terms of the series, $$\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) + \tan^{-1}\left(\frac{1}{21}\right) + \ldots$$, then $$\tan(S)$$ is equal to:

If the function $$f(x) = \begin{cases} k_1(x - \pi)^2 - 1, & x \leq \pi \\ k_2 \cos x, & x > \pi \end{cases}$$ is twice differentiable, then the ordered pair $$(k_1, k_2)$$ is equal to:

If $$\int (e^{2x} + 2e^x - e^{-x} - 1)e^{(e^x + e^{-x})}\,dx = g(x)e^{(e^x + e^{-x})} + c$$, where $$c$$ is a constant of integration, then $$g(0)$$ is:

The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{1 + e^{\sin x}}\,dx$$ is:

If $$y = y(x)$$ is the solution of the differential equation $$\frac{5 + e^x}{2 + y} \cdot \frac{dy}{dx} + e^x = 0$$ satisfying $$y(0) = 1$$, then value of $$y(\log_e 13)$$ is:

If the volume of a parallelepiped, whose coterminous edges are given by the vectors $$\vec{a} = \hat{i} + \hat{j} + n\hat{k}$$, $$\vec{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$$ and $$\vec{c} = \hat{i} + n\hat{j} + 3\hat{k}$$ $$(n \geq 0)$$ is 158 cubic units, then:

If $$(a, b, c)$$ is the image of the point $$(1, 2, -3)$$ in the line, $$\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$$, then $$a + b + c$$ is equal to:

The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word 'SYLLABUS' such that two letters are distinct and two letters are alike, is

The natural number $$m$$, for which the coefficient of $$x$$ in the binomial expansion of $$\left(x^m + \frac{1}{x^2}\right)^{22}$$ is 1540, is

If the line, $$2x - y + 3 = 0$$ is at a distance $$\frac{1}{\sqrt{5}}$$ and $$\frac{2}{\sqrt{5}}$$ from the lines $$4x - 2y + \alpha = 0$$ and $$6x - 3y + \beta = 0$$ respectively, then the sum of all possible values of $$\alpha$$ and $$\beta$$ is __________.

Let $$f(x) = x \cdot \left[\frac{x}{2}\right]$$, for $$-10 < x < 10$$, where $$[t]$$ denotes the greatest integer function. Then the number of points of discontinuity of $$f(x)$$ is equal to

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five, is