If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^4 + x^3 + x^2 + x + 1 = 0$$, then $$\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$$ is equal to
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If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^4 + x^3 + x^2 + x + 1 = 0$$, then $$\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$$ is equal to
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For $$n \in \mathbb{N}$$, let $$S_n = \{z \in \mathbb{C} : |z - 3 + 2i| = \dfrac{n}{4}\}$$ and $$T_n = \{z \in \mathbb{C} : |z - 2 + 3i| = \dfrac{1}{n}\}$$. Then the number of elements in the set $$\{n \in \mathbb{N} : S_n \cap T_n = \phi\}$$ is
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The number of solutions of $$\cos x = |\sin x|$$, such that $$-4\pi \le x \le 4\pi$$ is
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A line, with the slope greater than one, passes through the point $$A(4, 3)$$ and intersects the line $$x - y - 2 = 0$$ at the point $$B$$. If the length of the line segment $$AB$$ is $$\dfrac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line
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Let the locus of the centre $$(\alpha, \beta)$$, $$\beta > 0$$, of the circle which touches the circle $$x^2 + (y - 1)^2 = 1$$ externally and also touches the $$x$$-axis be $$L$$. Then the area bounded by $$L$$ and the line $$y = 4$$ is
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If $$\lim_{n \to \infty} \left(\sqrt{n^2 - n - 1} + n\alpha + \beta\right) = 0$$ then $$8(\alpha + \beta)$$ is equal to
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Which of the following statements is a tautology?
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A tower $$PQ$$ stands on a horizontal ground with base $$Q$$ on the ground. The point $$R$$ divides the tower in two parts such that $$QR = 15 \text{ m}$$. If from a point $$A$$ on the ground the angle of elevation of $$R$$ is $$60°$$ and the part $$PR$$ of the tower subtends an angle of $$15°$$ at $$A$$, then the height of the tower is
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The number of $$\theta \in [0, 4\pi]$$ for which the system of linear equations
$$3(\sin 3\theta) x - y + z = 2$$
$$3(\cos 2\theta) x + 4y + 3z = 3$$
$$6x + 7y + 7z = 9$$
has no solution is
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The total number of functions, $$f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$$ such that $$f(1) + f(2) = f(3)$$, is equal to
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If the absolute maximum value of the function $$f(x) = (x^2 - 2x + 7)e^{(4x^3 - 12x^2 - 180x + 31)}$$ in the interval $$[-3, 0]$$ is $$f(\alpha)$$, then
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The curve $$y(x) = ax^3 + bx^2 + cx + 5$$ touches the $$x$$-axis at the point $$P(-2, 0)$$ and cuts the $$y$$-axis at the point $$Q$$ where $$y'$$ is equal to $$3$$. Then the local maximum value of $$y(x)$$ is
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For any real number $$x$$, let $$[x]$$ denote the largest integer less than or equal to $$x$$. Let $$f$$ be a real-valued function defined on the interval $$[-10, 10]$$ by
$$f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$$
Then, the value of $$\dfrac{\pi^2}{10} \displaystyle\int_{-10}^{10} f(x) \cos \pi x \, dx$$ is
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The area of the region given by $$A = \{(x, y) : x^2 \le y \le \min\{x + 2, 4 - 3x\}\}$$ is
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The slope of the tangent to a curve $$C: y = y(x)$$ at any point $$[x, y)$$ on it is $$\dfrac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}$$. If $$C$$ passes through the points $$\left(0, \dfrac{1}{2} + \dfrac{\pi}{2\sqrt{2}}\right)$$ and $$\left(\alpha, \dfrac{1}{2}e^{2\alpha}\right)$$ then $$e^{\alpha}$$ is equal to
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The general solution of the differential equation $$(x - y^2)dx + y(5x + y^2)dy = 0$$ is
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Let $$ABC$$ be a triangle such that $$\vec{BC} = \vec{a}$$, $$\vec{CA} = \vec{b}$$, $$\vec{AB} = \vec{c}$$, $$|\vec{a}| = 6\sqrt{2}$$, $$|\vec{b}| = 2\sqrt{3}$$ and $$\vec{b} \cdot \vec{c} = 12$$. Consider the statements:
$$S_1: |\vec{a} \times (\vec{b} + \vec{c})| \times |\vec{b} - \vec{c}| = 6(2\sqrt{2} - 1)$$
$$S_2: \angle ABC = \cos^{-1}\sqrt{\dfrac{2}{3}}$$
Then
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Let $$P$$ be the plane containing the straight line $$\dfrac{x - 3}{9} = \dfrac{y + 4}{-1} = \dfrac{z - 7}{-5}$$ and perpendicular to the plane containing the straight lines $$\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z}{5}$$ and $$\dfrac{x}{3} = \dfrac{y}{7} = \dfrac{z}{8}$$. If $$d$$ is the distance of $$P$$ from the point $$(2, -5, 11)$$, then $$d^2$$ is equal to
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If the sum and the product of mean and variance of a binomial distribution are $$24$$ and $$128$$ respectively, then the probability of one or two successes is
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If the numbers appeared on the two throws of a fair six faced die are $$\alpha$$ and $$\beta$$, then the probability that $$x^2 + \alpha x + \beta > 0$$, for all $$x \in \mathbb{R}$$, is
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Let $$a, b$$ be two non-zero real numbers. If $$p$$ and $$r$$ are the roots of the equation $$x^2 - 8ax + 2a = 0$$ and $$q$$ and $$s$$ are the roots of the equation $$x^2 + 12bx + 6b = 0$$, such that $$\dfrac{1}{p}, \dfrac{1}{q}, \dfrac{1}{r}, \dfrac{1}{s}$$ are in A.P., then $$a^{-1} - b^{-1}$$ is equal to ______.
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The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is ______.
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Let $$a_1 = b_1 = 1$$, $$a_n = a_{n-1} + 2$$ and $$b_n = a_n + b_{n-1}$$ for every natural number $$n \ge 2$$. Then $$\displaystyle\sum_{n=1}^{15} a_n \cdot b_n$$ is equal to ______.
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If the maximum value of the term independent of $$t$$ in the expansion of $$\left(t^2 x^{1/5} + \dfrac{(1-x)^{1/10}}{t}\right)^{15}$$, $$x \ge 0$$, is $$K$$, then $$8K$$ is equal to ______.
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The sum of diameters of the circles that touch (i) the parabola $$75x^2 = 64(5y - 3)$$ at the point $$\left(\dfrac{8}{5}, \dfrac{6}{5}\right)$$ and (ii) the $$y$$-axis, is equal to ______.
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Let the equation of two diameters of a circle $$x^2 + y^2 - 2x + 2fy + 1 = 0$$ be $$2px - y = 1$$ and $$2x + py = 4p$$. Then the slope $$m \in (0, \infty)$$ of the tangent to the hyperbola $$3x^2 - y^2 = 3$$ passing through the centre of the circle is equal to ______.
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Let $$A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$$ and $$B = A - I$$. If $$\omega = \dfrac{\sqrt{3}i - 1}{2}$$, then the number of elements in the set $$\{n \in \{1, 2, \ldots, 100\} : A^n + (\omega B)^n = A + B\}$$ is equal to ______.
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Let $$f(x) = \begin{cases} \{4x^2 - 8x + 5\}, & \text{if } 8x^2 - 6x + 1 \ge 0 \\ [4x^2 - 8x + 5], & \text{if } 8x^2 - 6x + 1 < 0 \end{cases}$$, where $$[\alpha]$$ denotes the greatest integer less than or equal to $$\alpha$$ . Then the number of points in $$\mathbb{R}$$ where $$f$$ is not differentiable is ______.
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If $$\displaystyle\lim_{n \to \infty} \dfrac{(n+1)^{k-1}}{n^{k+1}} \left[(nk+1) + (nk+2) + \ldots + (nk+n)\right] = 33 \cdot \lim_{n \to \infty} \dfrac{1}{n^{k+1}} \left(1^k + 2^k + 3^k + \ldots + n^k\right)$$, then the integral value of $$k$$ is equal to ______.
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The line of shortest distance between the lines $$\dfrac{x-2}{0} = \dfrac{y-1}{1} = \dfrac{z}{1}$$ and $$\dfrac{x-3}{2} = \dfrac{y-5}{2} = \dfrac{z-1}{1}$$ makes an angle of $$\sin^{-1}\sqrt{\dfrac{2}{27}}$$ with the plane $$P: ax - y - z = 0$$, $$a > 0$$. If the image of the point $$(1, 1, -5)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta - \gamma$$ is equal to ______.
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