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Let p, q and r be real numbers ($$p \neq q, r \neq 0$$), such that the roots of the equation $$\frac{1}{x+p} + \frac{1}{x+q} = \frac{1}{r}$$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to:
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If an angle A of a $$\triangle ABC$$ satisfies $$5\cos A + 3 = 0$$, then the roots of the quadratic equation $$9x^2 + 27x + 20 = 0$$ are:
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The least positive integer n for which $$\left(\frac{1 + i\sqrt{3}}{1 - i\sqrt{3}}\right)^n = 1$$ is:
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The number of numbers between 2,000 and 5,000 that can be formed with the digits 0, 1, 2, 3, 4 (repetition of digits is not allowed) and are multiple of 3 is:
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Let $$\frac{1}{x_1}, \frac{1}{x_2}, \ldots, \frac{1}{x_n}$$ ($$x_i \neq 0$$ for i = 1, 2, ..., n) be in A.P. such that $$x_1 = 4$$ and $$x_{21} = 20$$. If n is the least positive integer for which $$x_n \gt 50$$, then $$\sum_{i=1}^{n}\left(\frac{1}{x_i}\right)$$ is equal to:
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The sum of the first 20 terms of the series $$1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \ldots$$ is:
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The coefficient of $$x^2$$ in the expansion of the product $$(2 - x^2)\left\{(1 + 2x + 3x^2)^6 + (1 - 4x^2)^6\right\}$$ is:
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The locus of the point of intersection of the lines $$\sqrt{2}x - y + 4\sqrt{2}k = 0$$ and $$\sqrt{2}kx + ky - 4\sqrt{2} = 0$$ (k is any non-zero real parameter) is:
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If a circle C, whose radius is 3, touches externally the circle $$x^2 + y^2 + 2x - 4y - 4 = 0$$ at the point (2, 2), then the length of the intercept cut by this circle C on the x-axis is equal to:
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Let P be a point on the parabola $$x^2 = 4y$$. If the distance of P from the center of the circle $$x^2 + y^2 + 6x + 8 = 0$$ is minimum, then the equation of the tangent to the parabola at P is:
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If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is $$\frac{3}{2}$$ units, then its eccentricity is:
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$$\lim_{x \to 0} \frac{(27+x)^{\frac{1}{3}} - 3}{9 - (27+x)^{\frac{2}{3}}}$$ equals:
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If $$p \to (\sim p \vee \sim q)$$ is false, then the truth values of p and q are, respectively:
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The mean and the standard deviation (S.D.) of five observations are 9 and 0, respectively. If one of the observation is increased such that the mean of the new set of five observations becomes 10, then their S.D. is:
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A man on the top of a vertical tower observes a car moving at a uniform speed towards the tower on a horizontal road. If it takes 18 min for the angle of depression of the car to change from 30$$^\circ$$ to 45$$^\circ$$, then the time taken (in min) by the car to reach the foot of the tower is:
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Let N denote the set of all natural numbers. Define two binary relations on N as $$R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$$ and $$R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$$. Then:
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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$$ and $$B = A^{20}$$. Then the sum of the elements of the first column of B is:
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The number of values of k for which the system of linear equations $$(k+2)x + 10y = k$$ and $$kx + (k+3)y = k - 1$$ has no solution is:
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If the function f defined as $$f(x) = \frac{1}{x} - \frac{k-1}{e^{2x} - 1}$$, $$x \neq 0$$ is continuous at $$x = 0$$, then ordered pair (k, f(0)) is equal to:
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If $$x = \sqrt{2^{\text{cosec}^{-1}t}}$$ and $$y = \sqrt{2^{\text{sec}^{-1}t}}$$, ($$|t| \geq 1$$), then $$\frac{dy}{dx}$$ is equal to:
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Let M and m be respectively the absolute maximum and the absolute minimum values of the function, $$f(x) = 2x^3 - 9x^2 + 12x + 5$$ in the interval [0, 3]. Then M - m is equal to:
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If $$\int \frac{\tan x}{1 + \tan x + \tan^2 x} dx = x - \frac{K}{\sqrt{A}} \tan^{-1}\left(\frac{K\tan x + 1}{\sqrt{A}}\right) + C$$, (C is a constant of integration), then the ordered pair (K, A) is equal to:
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If $$f(x) = \int_0^x t(\sin x - \sin t)dt$$, then:
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If the area of the region bounded by the curves, $$y = x^2$$, $$y = \frac{1}{x}$$ and the lines $$y = 0$$ and $$x = t$$ (t > 1) is 1 sq. unit, then t is equal to:
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The differential equation representing the family of ellipses having foci either on the x-axis or on the y-axis, center at the origin and passing through the point (0, 3) is:
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Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{c} = \hat{j} - \hat{k}$$ and a vector $$\vec{b}$$ be such that $$\vec{a} \times \vec{b} = \vec{c}$$ and $$\vec{a} \cdot \vec{b} = 3$$. Then $$|\vec{b}|$$ equals:
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The sum of the intercepts on the coordinate axes of the plane passing through the point (-2, -2, 2) and containing the line joining the points (1, -1, 2) and (1, 1, 1) is:
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If the angle between the lines $$\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$$ and $$\frac{5-x}{-2} = \frac{7y-14}{P} = \frac{z-3}{4}$$ is $$\cos^{-1}\left(\frac{2}{3}\right)$$, then P is equal to:
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Two different families A and B are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is $$\frac{1}{12}$$, then the number of children in each family is:
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Let A, B and C be three events, which are pair-wise independent and $$\bar{E}$$ denotes the complement of an event E. If $$P(A \cap B \cap C) = 0$$ and $$P(C) > 0$$, then $$P\left[(\bar{A} \cap \bar{B}) | C\right]$$ is equal to:
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