If the image of the point $$P(1, 0, 3)$$ in the line joining the points $$A(4, 7, 1)$$ and $$B(3, 5, 3)$$ is $$Q(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to
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If the image of the point $$P(1, 0, 3)$$ in the line joining the points $$A(4, 7, 1)$$ and $$B(3, 5, 3)$$ is $$Q(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to
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Let $$f : [1, \infty) \to [2, \infty)$$ be a differentiable function. If $$10\int_{1}^{x} f(t)\,dt = 5xf(x) - x^5 - 9$$ for all $$x \geq 1$$, then the value of $$f(3)$$ is :
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The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $$\frac{21}{2}$$. Then the number of terms which are integers in the A.P. is :
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Let $$A = \{1, 2, 3, \ldots, 100\}$$ and R be a relation on A such that $$R = \{(a, b) : a = 2b + 1\}$$. Let $$(a_1, a_2), (a_2, a_3), (a_3, a_4), \ldots, (a_k, a_{k+1})$$ be a sequence of k elements of R such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k, for which such a sequence exists, is equal to :
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If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
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The line $$L_1$$ is parallel to the vector $$\vec{a} = -3\hat{i} + 2\hat{j} + 4\hat{k}$$ and passes through the point $$(7, 6, 2)$$ and the line $$L_2$$ is parallel to the vector $$\vec{b} = 2\hat{i} + \hat{j} + 3\hat{k}$$ and passes through the point $$(5, 3, 4)$$. The shortest distance between the lines $$L_1$$ and $$L_2$$ is :
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Let $$(a, b)$$ be the point of intersection of the curve $$x^2 = 2y$$ and the straight line $$y - 2x - 6 = 0$$ in the second quadrant. Then the integral $$I = \int_{a}^{b} \frac{9x^2}{1 + 5^x}\,dx$$ is equal to :
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If the system of equations
$$2x + \lambda y + 3z = 5$$
$$3x + 2y - z = 7$$
$$4x + 5y + \mu z = 9$$
has infinitely many solutions, then $$(\lambda^2 + \mu^2)$$ is equal to :
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If $$\theta \in \left[-\frac{7\pi}{6}, \frac{4\pi}{3}\right]$$, then the number of solutions of $$\sqrt{3}\csc^2\theta - 2(\sqrt{3} - 1)\csc\theta - 4 = 0$$, is equal to
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Given three identical bags each containing 10 balls, whose colours are as follows :
Bag I : 3 Red, 2 Blue, 5 Green
Bag II : 4 Red, 3 Blue, 3 Green
Bag III : 5 Red, 1 Blue, 4 Green
A person chooses a bag at random and takes out a ball. If the ball is Red, the probability that it is from bag I is p and if the ball is Green, the probability that it is from bag III is q, then the value of $$\left(\frac{1}{p} + \frac{1}{q}\right)$$ is :
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If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $$a + b + ab$$ is equal to :
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If the domain of the function $$f(x) = \frac{1}{\sqrt{10 + 3x - x^2}} + \frac{1}{\sqrt{x + |x|}}$$ is $$(a, b)$$, then $$(1 + a)^2 + b^2$$ is equal to :
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$$4\int_{0}^{1} \frac{1}{\sqrt{3 + x^2} + \sqrt{1 + x^2}} \, dx - 3\log_e(\sqrt{3})$$ is equal to :
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If $$\lim_{x \to 0} \frac{\cos(2x) + a\cos(4x) - b}{x^4}$$ is finite, then $$(a + b)$$ is equal to :
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If $$\sum_{r=0}^{10} \left(\frac{10^{r+1} - 1}{10^r}\right) \cdot \,^{11}C_{r+1} = \frac{\alpha^{11} - 11^{11}}{10^{10}}$$, then $$\alpha$$ is equal to :
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The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can be placed in a box, is :
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Let the point P of the focal chord PQ of the parabola $$y^2 = 16x$$ be $$(1, -4)$$. If the focus of the parabola divides the chord PQ in the ratio $$m : n$$, $$\gcd(m, n) = 1$$, then $$m^2 + n^2$$ is equal to :
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Let $$\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} + 2\hat{j} + 5\hat{k}$$ and a vector $$\vec{c}$$ be such that $$(\vec{a} - \vec{c}) \times \vec{b} = -18\hat{i} - 3\hat{j} + 12\hat{k}$$ and $$\vec{a} \cdot \vec{c} = 3$$. If $$\vec{b} \times \vec{c} = \vec{d}$$, then $$|\vec{a} \cdot \vec{d}|$$ is equal to :
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Let the area of the triangle formed by a straight line $$L : x + by + c = 0$$ with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line L makes an angle of 45° with the positive x-axis, then the value of $$b^2 + c^2$$ is :
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Let A be a $$3 \times 3$$ real matrix such that $$A^2(A - 2I) - 4(A - I) = O$$, where I and O are the identity and null matrices, respectively. If $$A^5 = \alpha A^2 + \beta A + \gamma I$$, where $$\alpha, \beta$$ and $$\gamma$$ are real constants, then $$\alpha + \beta + \gamma$$ is equal to :
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + 2y\sec^2 x = 2\sec^2 x + 3\tan x \cdot \sec^2 x$$ such that $$y(0) = \frac{5}{4}$$. Then $$12\left(y\left(\frac{\pi}{4}\right) - e^{-2}\right)$$ is equal to __________.
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If the sum of the first 10 terms of the series $$\frac{4 \cdot 1}{1 + 4 \cdot 1^4} + \frac{4 \cdot 2}{1 + 4 \cdot 2^4} + \frac{4 \cdot 3}{1 + 4 \cdot 3^4} + \ldots$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to __________.
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If $$y = \cos\left(\frac{\pi}{3} + \cos^{-1}\frac{x}{2}\right)$$, then $$(x - y)^2 + 3y^2$$ is equal to __________.
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Let $$A(4, -2)$$, $$B(1, 1)$$ and $$C(9, -3)$$ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.
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If the set of all $$a \in \mathbb{R} - \{1\}$$, for which the roots of the equation $$(1 - a)x^2 + 2(a - 3)x + 9 = 0$$ are positive is $$(-\infty, -\alpha] \cup [\beta, \gamma)$$, then $$2\alpha + \beta + \gamma$$ is equal to __________.
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