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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - x + p = 0$$ and $$\gamma, \delta$$ be the roots of the equation$$x^2 - 4x + q = 0$$, where $$p, q \in \mathbb{Z}$$. If $$\alpha, \beta, \gamma, \delta$$ are in G.P., then $$|p + q|$$ equals :
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Let $$z_1, z_2 \in \mathbb{C}$$ be the distinct solutions of the equation $$z^2 + 4z - (1 + 12i) = 0$$. Then $$|z_1|^2 + |z_2|^2$$ is equal to :
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Let $$f : \mathbb{N} \to \mathbb{Z}$$ be defined by $$f(n) = \det\begin{bmatrix} n & -1 & -5\\-2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{bmatrix}$$, $$k \in \mathbb{N}$$ and $$\displaystyle\sum_{n=1}^{k} f(n) = 98$$, then $$k$$ is equal to :
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Let $$M$$ be a $$3 \times 3$$ matrix such that $$M\begin{bmatrix}1\\0\\0\end{bmatrix} = \begin{bmatrix}1\\2\\3\end{bmatrix}$$, $$M\begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}0\\1\\0\end{bmatrix}$$, $$M\begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}-1\\1\\1\end{bmatrix}$$. If $$M\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}1\\7\\11\end{bmatrix}$$, then $$x + y + z$$ is equal to :
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The sum of the first 10 terms of the series $$\frac{1}{1 + 1^4 \cdot 4} + \frac{2}{1 + 2^4 \cdot 4} + \frac{3}{1 + 3^4 \cdot 4} + \cdots$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$. Then $$m + n$$ is equal to :
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Let $$A_1, A_2, \ldots, A_{39}$$ be 39 arithmetic means between the numbers 59 and 159. The mean of $$A_{25}, A_{28}, A_{31} and A_{36}$$ is equal to :
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The coefficient of $$x^2$$ in the expansion of $$\left(2x^2 + \frac{1}{x}\right)^{10}$$, $$x \neq 0$$, is :
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The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :
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A box contains 5 blue, 6 yellow, and 4 red balls. The number of ways, of drawing 8 balls containing atleast two balls of each colour, is :
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A variable $$X$$ takes values $$0, 0, 2, 6, 12, 20, \ldots, n(n-1)$$ with frequencies $${^{n} C_{0}}, {^{n} C_{1}}, {^{n} C_{2}}, \ldots, {^{n} C_{n}}$$ respectively. If the mean of the data is 60, then the median is :
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Let the point $$P$$ be the vertex of the parabola $$y = x^2 - 6x + 12$$. If a line passing through the point $$P$$ intersects the circle $$x^2 + y^2 - 2x - 4y + 3 = 0$$ at the points $$R$$ and $$S$$.then the maximum value of $$(PR + PS)^2$$ is :
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Let the directrix of the parabola $$P: y^2 = 8x$$ cuts the x-axis at the point $$A$$.Let $$B(\alpha, \beta)$$, $$\alpha > 1$$, be a point on $$P$$ such that the slope of $$AB$$ is $$3/5$$. If $$BC$$ is a focal chord of chord of $$P$$. then six times the area off $$(\triangle ABC)$$ is :
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Let the eccentricity $$e$$ of a hyperbola satisfy the equation $$6e^2 - 11e + 3 = 0$$. Its foci of the hyperbola are $$(3, 5)$$ and $$(3, -4)$$.then the length of its latus rectum is :
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Let a $$\triangle PQR$$,be such that $$P$$ and $$Q$$ lie on the line $$\frac{x+3}{8} = \frac{y-4}{2} = \frac{z+1}{2}$$ and are at a distance of 6 units from $$R(1, 2, 3)$$. If $$(\alpha, \beta, \gamma)$$ is the centroid of $$\triangle PQR$$, then $$\alpha + \beta + \gamma$$ is equal to :
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Let the distance of the point $$(a, 2, 5)$$ from the image of the point $$(1, 2, 7)$$ in the line $$\frac{x}{1} = \frac{y-1}{1} = \frac{z-2}{2}$$ is 4,then the sum of all possible values of $$a$$ is equal to:
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Let $$O$$ be the origin, $$\overrightarrow{OP} = \vec{a}$$ and $$\overrightarrow{OQ} = \vec{b}$$.If $$R$$ is the point on $$\overrightarrow{OP}$$ such that $$\overrightarrow{OP} = 5\overrightarrow{OR}$$, and $$M$$ is the point such that $$\overrightarrow{OQ} = 5\overrightarrow{RM}$$. Then $$\overrightarrow{PM}$$ is equal to :
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Let $$f(x) = \displaystyle\lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^3}$$. Then the number of solutions of the equation $$f(x) = \sin x$$, $$x \in \mathbb{R}$$, is :
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Let $$(2^{1-a} + 2^{1+a})$$, $$f(a)$$, $$(3^a + 3^{-a})$$ be in A.P. and $$\alpha$$ be the minimum value of $$f(a)$$, Then the value of the integral $$\displaystyle\int_{\log_e(\alpha - 1)}^{\log_e(\alpha)} \frac{dx}{e^{2x} - e^{-2x}}$$ is equal to :
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Let $$f : [1, \infty) \to \mathbb{R}$$ be a differentiable defined as $$f(x) = \displaystyle\int_1^x f(t)\,dt + (1 - x)(\log_e x - 1) + e$$. Then the value of $$f(f(1))$$ is :
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Let $$f(x)$$ and $$g(x)$$ be twice differentiable functions satisfying $$f''(x) = g''(x)$$ for all $$x$$, $$f'(1) = 2g'(1) = 4$$, and $$g(2) = 3f(2) = 9$$. Then $$f(25) - g(25)$$ is equal to :
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Let $$A = \{1, 4, 7\}$$ and $$B = \{2, 3, 8\}$$. Then the number of elements, in the relation $$R = \{((a_1, b_1), (a_2, b_2)) \in (A \times B) \times (A \times B) : a_1 + b_2 \text{ divides } a_2 + b_1\}$$. is :
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From the point $$(-1, -1)$$, two rays are sent making angle of $$45°$$ with the line $$x + y = 0$$. The rays get reflected from the mirror $$x + 2y = 1$$. If the equations of the reflected rays are $$ax + by = 9$$ and $$cx + dy = 7$$,$$a,b,c,d \in \mathbb{Z}$$ thenthe value of $$ad + bc$$ is :
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Let $$S = \left\{\theta \in [-\pi, \pi] : \cos\theta \cos\left(\frac{5\theta}{2}\right) = \cos 7\theta \cos\left(\frac{7\theta}{2}\right)\right\}$$. Then $$n(S)$$ is equal to :
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function such that $$f(x) + 3f\left(\frac{\pi}{2} - x\right) = \sin x, x \in R$$.Let the maximum value of $$f$$ on $$\mathbb{R}$$ be $$\alpha$$. The area of the region bounded by the curves $$g(x) = x^2$$ and $$h(x) = \beta x^3$$, $$\beta > 0$$, is $$\alpha^2$$. Then $$30\beta^3$$ is equal to :
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Let $$y = y(x)$$ be the solution of $$(\tan x)^{1/2}\,dy = (\sec^3 x - (\tan x)^{3/2}\,y)\,dx$$, $$0 < x < \frac{\pi}{2}$$. If $$y\left(\frac{\pi}{4}\right) = \frac{6\sqrt{2}}{5}$$, and $$y\left(\frac{\pi}{3}\right) = \frac{4}{5}\alpha$$, then $$\alpha^4$$ is equal to :
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