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Let $$\alpha$$, $$\beta$$ be the roots of the quadratic equation $$x^2 + \sqrt{6}x + 3 = 0$$. Then $$\frac{\alpha^{23} + \beta^{23} + \alpha^{14} + \beta^{14}}{\alpha^{15} + \beta^{15} + \alpha^{10} + \beta^{10}}$$ is equal to
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Let $$C$$ be the circle in the complex plane with centre $$z_0 = \frac{1}{2}(1 + 3i)$$ and radius $$r = 1$$. Let $$z_1 = 1 + i$$ and the complex number $$z_2$$ be outside circle $$C$$ such that $$|z_1 - z_0||z_2 - z_0| = 1$$. If $$z_0$$, $$z_1$$ and $$z_2$$ are collinear, then the smaller value of $$|z_2|^2$$ is equal to
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The number of five-digit numbers, greater than 40000 and divisible by 5, which can be formed using the digits 0, 1, 3, 5, 7 and 9 without repetition, is equal to
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Let $$\langle a_n \rangle$$ be a sequence such that $$a_1 + a_2 + \ldots + a_n = \frac{n^2 + 3n}{(n+1)(n+2)}$$. If $$28 \sum_{k=1}^{10} \frac{1}{a_k} = p_1 p_2 p_3 \ldots p_m$$, where $$p_1, p_2, \ldots p_m$$ are the first $$m$$ prime numbers, then $$m$$ is equal to
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If $$\frac{1}{n+1}$$ $$^nC_n + \frac{1}{n}$$ $$^nC_{n-1} + \ldots + \frac{1}{2}$$ $$^nC_1 + ^nC_0 = \frac{1023}{10}$$, then $$n$$ is equal to
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The sum of the coefficients of the first 50 terms in the binomial expansion of $$(1 - x)^{100}$$ is equal to
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If the point $$\left(\alpha, \frac{7\sqrt{3}}{3}\right)$$ lies on the curve traced by the mid-points of the line segments of the lines $$x \cos\theta + y \sin\theta = 7$$, $$\theta \in \left(0, \frac{\pi}{2}\right)$$ between the co-ordinates axes, then $$\alpha$$ is equal to
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In a triangle $$ABC$$, if $$\cos A + 2\cos B + \cos C = 2$$ and the lengths of the sides opposite to the angles $$A$$ and $$C$$ are 3 and 7 respectively, then $$\cos A - \cos C$$ is equal to
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Let $$P\left(\frac{2\sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right)$$, Q, R and S be four points on the ellipse $$9x^2 + 4y^2 = 36$$. Let PQ and RS be mutually perpendicular and pass through the origin. If $$\frac{1}{(PQ)^2} + \frac{1}{(RS)^2} = \frac{p}{q}$$, where $$p$$ and $$q$$ are coprime, then $$p + q$$ is equal to
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Among the two statements
$$(S_1) : (p \Rightarrow q) \wedge (p \wedge (\sim q))$$ is a contradiction and
$$(S_2) : (p \wedge q) \vee ((\sim p) \wedge q) \vee (p \wedge (\sim q)) \vee ((\sim p) \wedge (\sim q))$$ is a tautology
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Let $$A = \begin{bmatrix} 1 & \frac{1}{51} \\ 0 & 1 \end{bmatrix}$$. If $$B = \begin{bmatrix} 1 & 2 \\ -1 & -1 \end{bmatrix} A \begin{bmatrix} -1 & -2 \\ 1 & 1 \end{bmatrix}$$, then the sum of all the elements of the matrix $$\sum_{n=1}^{50} B^n$$ is equal to
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Let $$D$$ be the domain of the function $$f(x) = \sin^{-1}\left(\log_{3x}\left(\frac{6 + 2\log_{3}x}{-5x}\right)\right)$$. If the range of the function $$g : D \to \mathbb{R}$$ defined by $$g(x) = x - [x]$$, ($$[x]$$ is the greatest integer function), is $$(\alpha, \beta)$$, then $$\alpha^2 + \frac{5}{\beta}$$ is equal to
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If the total maximum value of the function $$f(x) = \left(\frac{\sqrt{3e}}{2\sin x}\right)^{\sin^2 x}$$, $$x \in \left(0, \frac{\pi}{2}\right)$$, is $$\frac{k}{e}$$, then $$\left(\frac{k}{e}\right)^8 + \frac{k^8}{e^5} + k^8$$ is equal to
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The area of the region enclosed by the curve $$y = x^3$$ and its tangent at the point $$(-1, -1)$$ is
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Let $$y = y(x)$$, $$y > 0$$, be a solution curve of the differential equation $$(1 + x^2)dy = y(x - y)dx$$. If $$y(0) = 1$$ and $$y(2\sqrt{2}) = \beta$$, then
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Let a, b, c be three distinct real numbers, none equal to one. If the vectors $$a\hat{i} + \hat{j} + \hat{k}$$, $$\hat{i} + b\hat{j} + \hat{k}$$ and $$\hat{i} + \hat{j} + c\hat{k}$$ are coplanar, then $$\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}$$ is equal to
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Let $$\lambda \in \mathbb{Z}$$, $$\vec{a} = \lambda \hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = 3\hat{i} - \hat{j} + 2\hat{k}$$. Let $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = 0$$, $$\vec{a} \cdot \vec{c} = -17$$ and $$\vec{b} \cdot \vec{c} = -20$$. Then $$|\vec{c} \times (\lambda\hat{i} + \hat{j} + \hat{k})|^2$$ is equal to
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Let the lines $$L_1 : \frac{x+5}{3} = \frac{y+4}{1} = \frac{z-\alpha}{-2}$$ and $$L_2 : 3x + 2y + z - 2 = 0 = x - 3y + 2z - 13$$ be coplanar. If the point $$P(a, b, c)$$ on $$L_1$$ is nearest to the point $$Q(-4, -3, 2)$$, then $$|a| + |b| + |c|$$ is equal to
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Let the plane $$P : 4x - y + z = 10$$ be rotated by an angle $$\frac{\pi}{2}$$ about its line of intersection with the plane $$x + y - z = 4$$. If $$\alpha$$ is the distance of the point $$(2, 3, -4)$$ from the new position of the plane $$P$$, then $$35\alpha$$ is equal to
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Two dice $$A$$ and $$B$$ are rolled. Let the numbers obtained on $$A$$ and $$B$$ be $$\alpha$$ and $$\beta$$ respectively. If the variance of $$\alpha - \beta$$ is $$\frac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then the sum of the positive divisors of $$p$$ is equal to
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Let the digits $$a$$, $$b$$, $$c$$ be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?
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Two circles in the first quadrant of radii $$r_1$$ and $$r_2$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x + y = 2$$. Then $$r_1^2 + r_2^2 - r_1 r_2$$ is equal to _____.
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Let the positive numbers $$a_1, a_2, a_3, a_4$$ and $$a_5$$ be in a G.P. Let their mean and variance be $$\frac{31}{10}$$ and $$\frac{m}{n}$$ respectively, where $$m$$ and $$n$$ are co-prime. If the mean of their reciprocals is $$\frac{31}{40}$$ and $$a_3 + a_4 + a_5 = 14$$, then $$m + n$$ is equal to _____.
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The number of relations, on the set $$\{1, 2, 3\}$$ containing $$(1, 2)$$ and $$(2, 3)$$ which are reflexive and transitive but not symmetric, is _____.
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Let $$D_k = \begin{vmatrix} 1 & 2k & 2k-1 \\ n & n^2+n+2 & n^2 \\ n & n^2+n & n^2+n+2 \end{vmatrix}$$. If $$\sum_{k=1}^{n} D_k = 96$$, then $$n$$ is equal to _____.
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Let $$[x]$$ be the greatest integer $$\leq x$$. Then the number of points in the interval $$(-2, 1)$$ where the function $$f(x) = |[x]| + \sqrt{x - [x]}$$ is discontinuous, is _____.
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Let $$I(x) = \int \sqrt{\frac{x+7}{x}} \ dx$$ and $$I(9) = 12 + 7\log_e 7$$. If $$I(1) = \alpha + 7\log_e\left(1 + 2\sqrt{2}\right)$$, then $$\alpha^4$$ is equal to _____.
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If $$\int_{-0.15}^{0.15} |100x^2 - 1| \ dx = \frac{k}{3000}$$, then $$k$$ is equal to _____.
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Let the plane $$x + 3y - 2z + 6 = 0$$ meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle $$ABC$$ is $$\left(\alpha, \beta, \frac{6}{7}\right)$$, then $$98(\alpha + \beta)^2$$ is equal to _____.
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A fair $$n$$ ($$n > 1$$) faces die is rolled repeatedly until a number less than $$n$$ appears. If the mean of the number of tosses required is $$\frac{n}{9}$$, then $$n$$ is equal to _____.
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