If 2 and 6 are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the quadratic equation whose roots are $$\frac{1}{2a+b}$$ and $$\frac{1}{6a+b}$$ is:
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If 2 and 6 are the roots of the equation $$ax^2 + bx + 1 = 0$$, then the quadratic equation whose roots are $$\frac{1}{2a+b}$$ and $$\frac{1}{6a+b}$$ is:
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Let α and β be the sum and the product of all the non-zero solutions of the equation $$(\bar{z})^2 + |z| = 0,\ z \in \mathbb{C}$$. Then $$4(\alpha^2 + \beta^2)$$ is equal to:
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There are 5 points $$P_1, P_2, P_3, P_4, P_5$$ on the side AB, excluding A and B, of a triangle ABC. Similarly there are 6 points $$P_6, P_7,\ldots, P_{11}$$ on the side BC and 7 points $$P_{12}, P_{13},\ldots, P_{18}$$ on the side CA of the triangle. The number of triangles, that can be formed using the points $$P_1, P_2,\ldots, P_{18}$$ as vertices, is:
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Let the first three terms 2, p and q, with q ≠ 2, of a G.P. be respectively the 7th, 8th and 13th terms of an A.P. If the $$5^{th}$$ term of the G.P. is the $$n^{th}$$ term of the A.P., then n is equal to:
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The sum of all rational terms in the expansion of $$\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$$ is equal to:
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The vertices of a triangle are A(−1, 3), B(−2, 2) and C(3, −1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is:
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A square is inscribed in the circle $$x^2 + y^2 - 10x - 6y + 30 = 0$$. One side of this square is parallel to y = x + 3. If $$(x_i, y_i)$$ are the vertices of the square, then $$\sum(x_i^2 + y_i^2)$$ is equal to:
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Let $$\alpha, \beta \in {R}$$. Let the mean and the variance of 6 observations −3, 4, 7, −6, α, β be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:
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Let $$\alpha \in (0,\infty)$$ and $$A = \begin{bmatrix}1 & 2 & \alpha\\ 1 & 0 & 1\\ 0 & 1 & 2\end{bmatrix}$$. If $$\det(\text{adj}(2A-A^T)\cdot\text{adj}(A-2A^T)) = 2^8$$, then $$(\det(A))^2$$ is equal to:
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If the system of equations $$x + (\sqrt{2}\sin\alpha)y + (\sqrt{2}\cos\alpha)z = 0$$, $$x + (\cos\alpha)y + (\sin\alpha)z = 0$$, $$x + (\sin\alpha)y - (\cos\alpha)z = 0$$ has a non-trivial solution, then $$\alpha \in \left(0,\frac{\pi}{2}\right)$$ is equal to:
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If the domain of the function $$\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$$ is $$(\alpha, \beta]$$, then $$3\alpha + 10\beta$$ is equal to:
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Let the sum of the maximum and the minimum values of the function $$f(x) = \frac{2x^2-3x+8}{2x^2+3x+8}$$ be $$\frac{m}{n}$$, where gcd(m, n) = 1. Then m + n is equal to:
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Let $$f : R to R$$ be a function given by $$f(x) = \begin{cases}\frac{1-\cos 2x}{x^2}, & x < 0\\ \alpha, & x = 0\\ \frac{\beta\sqrt{1-\cos x}}{x}, & x > 0\end{cases}$$, where $$\alpha, \beta \in R$$. If f is continuous at x = 0, then $$\alpha^2 + \beta^2$$ is equal to:
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Let $$f(x) = x^5 + 2e^{x/4}$$ for all $$x \in R$$. Consider a function g(x) such that $$(g \circ f)(x) = x$$ for all $$x \in R$$. Then the value of $$8g'(2)$$ is:
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Let $$f(x) = \begin{cases}-2, & -2 \le x \le 0\\ x-2, & 0 < x \le 2\end{cases}$$ and $$h(x) = f(|x|) + |f(x)|$$. Then $$\int_{-2}^{2}h(x)dx$$ is equal to:
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One of the points of intersection of the curves $$y = 1 + 3x - 2x^2$$ and $$y = \frac{1}{x}$$ is $$\left(\frac{1}{2}, 2\right)$$. Let the area of the region enclosed by these curves be $$\frac{1}{24}(l\sqrt{5}+m) - n\log_e(1+\sqrt{5})$$, where $$l, m, n \in N$$. Then $$l + m + n$$ is equal to:
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If the solution $$y = y(x)$$ of the differential equation $$(x^4 + 2x^3 + 3x^2 + 2x + 2)dy - (2x^2 + 2x + 3)dx = 0$$ satisfies $$y(-1) = -\frac{\pi}{4}$$, then y(0) is equal to:
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Let a unit vector which makes an angle of 60° with $$2\hat{i} + 2\hat{j} - \hat{k}$$ and angle 45° with $$\hat{i} - \hat{k}$$ be $$\overrightarrow{C}$$. Then $$\overrightarrow{C} + \left(-\frac{1}{2}\hat{i} + \frac{1}{3\sqrt{2}}\hat{j} - \frac{\sqrt{2}}{3}\hat{k}\right)$$ is:
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Let the point, on the line passing through the points P(1, −2, 3) and Q(5, −4, 7), farther from the origin and at distance of 9 units from the point P, be $$(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ is equal to:
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Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is:
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Let $$a = 1 + \frac{^2C_2}{3!} + \frac{^3C_2}{4!} + \frac{^4C_2}{5!} + \ldots$$, $$b = 1 + \frac{^1C_0 + ^1C_1}{1!} + \frac{^2C_0+^2C_1+^2C_2}{2!} + \frac{^3C_0+^3C_1+^3C_2+^3C_3}{3!} + \ldots$$. Then $$\frac{2b}{a^2}$$ is equal to ______.
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Let the length of the focal chord PQ of the parabola $$y^2 = 12x$$ be 15 units. If the distance of PQ from the origin is p, then $$10p^2$$ is equal to ______.
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Let A be a square matrix of order 2 such that |A| = 2 and the sum of its diagonal elements is −3. If the points (x, y) satisfying $$A^2 + xA + yI = O$$ lie on a hyperbola whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then $$81(e^4 + l^2)$$ is equal to ______.
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If $$\lim_{x\to 1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$$, where gcd(m, n) = 1, then $$8m + 12n$$ is equal to ______.
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In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then $$m + n$$ is equal to ______.
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Let A be a 3×3 matrix of non-negative real elements such that $$A\begin{bmatrix}1\\1\\1\end{bmatrix} = 3\begin{bmatrix}1\\1\\1\end{bmatrix}$$. Then the maximum value of det(A) is ______.
If $$\int_0^{\pi/4}\frac{\sin^2 x}{1+\sin x\cos x}dx = \frac{1}{a}\log_e\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}}$$, where $$a, b \in N$$, then $$a + b$$ is equal to ______.
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Let the solution $$y = y(x)$$ of the differential equation $$\frac{dy}{dx} - y = 1 + 4\sin x$$ satisfy $$y(\pi) = 1$$. Then $$y\left(\frac{\pi}{2}\right) + 10$$ is equal to ______.
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Let ABC be a triangle of area $$15\sqrt{2}$$ and the vectors $$\overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}$$, $$\overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}$$ and $$\overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}$$, d > 0. Then the square of the length of the largest side of the triangle ABC is ______.
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If the shortest distance between the lines $$\frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4}$$ and $$\frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}$$ is $$\frac{38}{3\sqrt{5}}k$$, and $$\int_0^k [x^2]dx = \alpha - \sqrt{\alpha}$$, where [x] denotes the greatest integer function, then $$6\alpha^3$$ is equal to ______.
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