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Let $$m$$ and $$n$$ be the numbers of real roots of the quadratic equations $$x^2 - 12x + [x] + 31 = 0$$ and $$x^2 - 5|x + 2| - 4 = 0$$ respectively, where $$[x]$$ denotes the greatest integer $$\leq x$$. Then $$m^2 + mn + n^2$$ is equal to
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Let $$A = \left\{\theta \in (0, 2\pi) : \frac{1 + 2i\sin\theta}{1 - i\sin\theta} \text{ is purely imaginary}\right\}$$. Then the sum of the elements in $$A$$ is
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If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $$(6!)k$$ then $$k$$ is equal to
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Let $$a_n$$ be $$n^{th}$$ term of the series $$5 + 8 + 14 + 23 + 35 + 50 + \ldots$$ and $$S_n = \sum_{k=1}^{n} a_k$$. Then $$S_{30} - a_{40}$$ is equal to
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The absolute difference of the coefficients of $$x^{10}$$ and $$x^7$$ in the expansion of $$\left(2x^2 + \frac{1}{2x}\right)^{11}$$ is equal to
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$$25^{190} - 19^{190} - 8^{190} + 2^{190}$$ is divisible by
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The value of $$36(4\cos^2 9^\circ - 1)(4\cos^2 27^\circ - 1)(4\cos^2 81^\circ - 1)(4\cos^2 243^\circ - 1)$$ is
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Let $$A(0, 1)$$, $$B(1, 1)$$ and $$C(1, 0)$$ be the mid-points of the sides of a triangle with incentre at the point $$D$$. If the focus of the parabola $$y^2 = 4ax$$ passing through $$D$$ is $$\left(\alpha + \beta\sqrt{2}, 0\right)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha}{\beta^2}$$ is equal to
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Let $$O$$ be the origin and $$OP$$ and $$OQ$$ be the tangents to the circle $$x^2 + y^2 - 6x + 4y + 8 = 0$$ at the points $$P$$ and $$Q$$ on it. If the circumcircle of the triangle $$OPQ$$ passes through the point $$\left(\alpha, \frac{1}{2}\right)$$, then a value of $$\alpha$$ is
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If $$\alpha > \beta > 0$$ are the roots of the equation $$ax^2 + bx + 1 = 0$$, and
$$\lim_{x \to \frac{1}{\alpha}} \left(\frac{1 - \cos(x^2 + bx + a)}{2(1 - \alpha x)^2}\right)^{\frac{1}{2}} = \frac{1}{k}\left(\frac{1}{\beta} - \frac{1}{\alpha}\right)$$, then $$k$$ is equal to
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The negation of $$(p \wedge (-q)) \vee (-p)$$ is equivalent to
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Let the mean and variance of 12 observations be $$\frac{9}{2}$$ and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime, then $$m + n$$ is equal to
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Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the relation $$R = \{(x, y) \in A \times A : x + y = 7\}$$ is
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If $$A = \begin{bmatrix} 1 & 5 \\ \lambda & 10 \end{bmatrix}$$, $$A^{-1} = \alpha A + \beta I$$ and $$\alpha + \beta = -2$$, then $$4\alpha^2 + \beta^2 + \lambda^2$$ is equal to:
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Let $$S$$ be the set of all values of $$\theta \in [-\pi, \pi]$$ for which the system of linear equations
$$x + y + \sqrt{3}z = 0$$
$$-x + (\tan\theta)y + \sqrt{7}z = 0$$
$$x + y + (\tan\theta)z = 0$$
has non-trivial solution. Then $$\frac{120}{\pi}\sum_{\theta \in S} \theta$$ is equal to
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If domain of the function $$\log_e\left(\frac{6x^2 + 5x + 1}{2x - 1}\right) + \cos^{-1}\left(\frac{2x^2 - 3x + 4}{3x - 5}\right)$$ is $$(\alpha, \beta) \cup (\gamma, \delta)$$, then $$18(\alpha^2 + \beta^2 + \gamma^2 + \delta^2)$$ is equal to
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The integral $$\int\left(\left(\frac{x}{2}\right)^x + \left(\frac{2}{x}\right)^x\right) \log_2 x \, dx$$ is equal to
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Let the vectors $$\vec{u}_1 = \hat{i} + \hat{j} + a\hat{k}$$, $$\vec{u}_2 = \hat{i} + b\hat{j} + \hat{k}$$, and $$\vec{u}_3 = c\hat{i} + \hat{j} + \hat{k}$$ be coplanar. If the vectors $$\vec{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$$, $$\vec{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$$ and $$\vec{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$$ are also coplanar, then $$6(a + b + c)$$ is equal to
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Let $$P$$ be the plane passing through the line $$\frac{x-1}{1} = \frac{y-2}{-3} = \frac{z+5}{7}$$ and the point $$(2, 4, -3)$$. If the image of the point $$(-1, 3, 4)$$ in the plane $$P$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to
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If the probability that the random variable $$X$$ takes values $$x$$ is given by $$P(X = x) = k(x + 1)3^{-x}$$, $$x = 0, 1, 2, 3, \ldots$$, where $$k$$ is a constant, then $$P(X \geq 2)$$ is equal to
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Let $$0 < z < y < x$$ be three real numbers such that $$\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$$ are in an arithmetic progression and $$x, \sqrt{2}y, z$$ are in a geometric progression. If $$xy + yz + zx = \frac{3}{\sqrt{2}}xyz$$, then $$3(x + y + z)^2$$ is equal to _____.
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The ordinates of the points $$P$$ and $$Q$$ on the parabola with focus $$(3, 0)$$ and directrix $$x = -3$$ are in the ratio 3 : 1. If $$R(\alpha, \beta)$$ is the point of intersection of the tangents to the parabola at $$P$$ and $$Q$$, then $$\frac{\beta^2}{\alpha}$$ is equal to _____.
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Let $$R = \{a, b, c, d, e\}$$ and $$S = \{1, 2, 3, 4\}$$. Total number of onto functions $$f : R \to S$$ such that $$f(a) \neq 1$$, is equal to _____.
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Let k and m be positive real numbers such that the function $$f(x) = \begin{cases} 3x^2 + k\sqrt{x + 1}, & 0 < x < 1 \\ mx^2 + k^2, & x \geq 1 \end{cases}$$ is differentiable for all $$x > 0$$. Then $$\frac{8f'(8)}{f'(\frac{1}{8})}$$ is equal to _____.
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Let $$[t]$$ denote the greatest integer function. If $$\int_0^{2.4} [x^2] dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3} + \delta\sqrt{5}$$, then $$\alpha + \beta + \gamma + \delta$$ is equal to _____.
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Let the area enclosed by the lines $$x + y = 2$$, $$y = 0$$, $$x = 0$$ and the curve $$f(x) = \min\left\{x^2 + \frac{3}{4}, 1 + [x]\right\}$$ where $$[x]$$ denotes the greatest integer $$\leq x$$, be $$A$$. Then the value of $$12A$$ is _____.
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Let the solution curve $$x = x(y)$$, $$0 < y < \frac{\pi}{2}$$, of the differential equation $$(\log_e(\cos y))^2 \cos y \, dx - (1 + 3x \log_e(\cos y)) \sin y \, dy = 0$$ satisfy $$x\left(\frac{\pi}{3}\right) = \frac{1}{2\log_e 2}$$. If $$x\left(\frac{\pi}{6}\right) = \frac{1}{\log_e m - \log_e n}$$, where $$m$$ and $$n$$ are coprime, then $$mn$$ is equal to _____.
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The area of the quadrilateral $$ABCD$$ with vertices $$A(2, 1, 1)$$, $$B(1, 2, 5)$$, $$C(-2, -3, 5)$$ and $$D(1, -6, -7)$$ is equal to _____.
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For $$a, b \in \mathbb{Z}$$ and $$|a - b| \leq 10$$, let the angle between the plane $$P: ax + y - z = b$$ and the line $$L: x - 1 = a - y = z + 1$$ be $$\cos^{-1}\left(\frac{1}{3}\right)$$. If the distance of the point $$(6, -6, 4)$$ from the plane $$P$$ is $$3\sqrt{6}$$, then $$a^4 + b^2$$ is equal to _____.
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Let $$P_1$$ be the plane $$3x - y - 7z = 11$$ and $$P_2$$ be the plane passing through the points $$(2, -1, 0)$$, $$(2, 0, -1)$$, and $$(5, 1, 1)$$. If the foot of the perpendicular drawn from the point $$(7, 4, -1)$$ on the line of intersection of the planes $$P_1$$ and $$P_2$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to _____.
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