Let $$[\cdot]$$ denote the greatest integer function. If the domain of $$f(x) = \cos^{-1}\left(\frac{4x + 2[x]}{3}\right)$$ is $$[\alpha, \beta]$$, then $$12(\alpha + \beta)$$ is equal to :
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Let $$[\cdot]$$ denote the greatest integer function. If the domain of $$f(x) = \cos^{-1}\left(\frac{4x + 2[x]}{3}\right)$$ is $$[\alpha, \beta]$$, then $$12(\alpha + \beta)$$ is equal to :
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If the set of all solutions of $$|x^2 + x - 9| = |x| + |x^2 - 9|$$ is $$[\alpha, \beta] \cup [\gamma, \infty)$$, then $$(\alpha^2 + \beta^2 + \gamma^2)$$ is equal to :
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Let $$z$$ be complex such that $$|z + 2| = |z - 2|$$ and $$\arg\left(\frac{z+3}{z-i}\right) = \frac{\pi}{4}$$. Then $$|z|^2$$ is :
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The number of functions $$f: \{1,2,3,4\} \to \{a,b,c\}$$, which are not onto, is :
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Let $$S = \left\{A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a,b,c,d \in \{0,1,2,3,4\} \text{ and } A^2 - 4A + 3I = 0\right\}$$ be a set of $$2 \times 2$$ matrices. Then the number of matrices in $$S$$, for which the sum of the diagonal elements is equal to 4, is :
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Let $$A = \begin{bmatrix} 1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5 \end{bmatrix}$$. Then the sum of all elements of the matrix $$\text{adj}\left(\text{adj}\left(2(\text{adj}\,A)^{-1}\right)\right)$$ is equal to :
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The first term of an A.P. of 30 non-negative terms is $$\frac{10}{3}$$. If the sum of the A.P. is the cube of its last term, then its common difference is :
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The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is :
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Let the smallest value of $$k \in \mathbb{N}$$, for which the coefficient of $$x^3$$ in $$(1+x)^3 + (1+x)^4 + \ldots + (1+x)^{99} + (1+kx)^{100}$$, $$x \neq 0$$, is $$\left(43n + \frac{101}{4}\right)\binom{100}{3}$$ for some $$n \in \mathbb{N}$$, be $$p$$. Then the value of $$p + n$$ is :
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Suppose that the mean and median of the non-negative numbers $$21, 8, 17, a, 51, 103, b, 13, 67$$, $$(a > b)$$, are 40 and 21, respectively. If the mean deviation about the median is 26, then $$2a$$ is equal to :
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Let the line $$L_1: x + 3 = 0$$ intersect the lines $$L_2: x - y = 0$$ and $$L_3: 3x + y = 0$$ at the points $$A$$ and $$B$$, respectively. Let the bisector of the obtuse angle between the lines $$L_2$$ and $$L_3$$ intersect the line $$L_1$$ at the point $$C$$. Then $$BC^2 : AC^2$$ is equal to :
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Let the vertex $$A$$ of a triangle $$ABC$$ be $$(1, 2)$$, and the mid-point of the side $$AB$$ be $$(5, -1)$$. If the centroid of this triangle is $$(3, 4)$$ and its circumcenter is $$(\alpha, \beta)$$, then $$21(\alpha + \beta)$$ is equal to :
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Suppose that two chords, drawn from the point $$(1, 2)$$ on the circle $$x^2 + y^2 + x - 3y = 0$$ are bisected by the $$y$$-axis. If the other ends of these chords are $$R$$ and $$S$$, and the mid point of the line segment $$RS$$ is $$(\alpha, \beta)$$, then $$6(\alpha + \beta)$$ is equal to :
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A line with direction ratios $$1, -1, 2$$ intersects the lines $$\frac{x}{2} = \frac{y}{3} = \frac{z+1}{3}$$ and $$\frac{x+1}{-1} = \frac{y-2}{1} = \frac{z}{4}$$ at the points $$P$$ and $$Q$$, respectively. If the length of the line segment $$PQ$$ is $$\alpha$$, then $$225\alpha^2$$ is equal to :
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The square of the distance of the point $$(-2, -8, 6)$$ from the line $$\frac{x-1}{1} = \frac{y-1}{2} = \frac{z}{-1}$$ along the line $$\frac{x+5}{1} = \frac{y+5}{-1} = \frac{z}{2}$$ is equal to :
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If $$y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right)$$, then $$\frac{dy}{dx}$$ at $$x = \frac{\sqrt{3}}{2}$$ is equal to :
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Let $$f$$ be a real polynomial of degree $$n$$ such that $$f(x) = f'(x) \cdot f''(x)$$, for all $$x \in \mathbb{R}$$. If $$f(0) = 0$$, then $$36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)$$ is equal to :
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The area of the region $$\{(x,y): y \leq \pi - |x|, \; y \leq |x \sin x|, \; y \geq 0\}$$ is :
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Let $$\int_{-2}^{2} (|\sin x| + [x \sin x])\,dx = 2(3 - \cos 2) + \beta$$, where $$[\cdot]$$ is the greatest integer function. Then $$\beta \sin\left(\frac{\beta}{2}\right)$$ equals :
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = (1 + x + x^2)(1 - y + y^2)$$, $$y(0) = \frac{1}{2}$$. Then $$(2y(1) - 1)$$ is equal to :
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A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $$p$$, then $$96p$$ is equal to _____.
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Consider the parabola $$P: y^2 = 4kx$$ and the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$. Let the line segment joining the points of intersection of $$P$$ and $$E$$, be their latus rectums. If the eccentricity of $$E$$ is $$e$$, then $$e^2 + 2\sqrt{2}$$ is equal to _____.
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If $$A = \frac{\sin 3°}{\cos 9°} + \frac{\sin 9°}{\cos 27°} + \frac{\sin 27°}{\cos 81°}$$ and $$B = \tan 81° - \tan 3°$$, then $$\frac{B}{A}$$ is equal to _____.
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Let $$\vec{a_k} = (\tan\theta_k)\hat{i} + \hat{j}$$ and $$\vec{b_k} = \hat{i} - (\cot\theta_k)\hat{j}$$, where $$\theta_k = \frac{2^{k-1}\pi}{2^n + 1}$$, for some $$n \in \mathbb{N}$$, $$n > 5$$. Then the value of $$\frac{\sum_{k=1}^{n}|\vec{a_k}|^2}{\sum_{k=1}^{n}|\vec{b_k}|^2}$$ is _____.
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The number of points, at which the function $$f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\cos\left|x^2 - \frac{1}{4}\right|$$, $$x \in (-\pi, \pi)$$, is not differentiable, is _____.
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