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Let A be a matrix of order $$3 \times 3$$ and $$|A| = 5$$. If $$|2\text{adj}(3A \text{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}$$, $$\alpha, \beta, \gamma \in \mathbb{N}$$ then $$\alpha + \beta + \gamma$$ is equal to
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Let a line passing through the point $$(4, 1, 0)$$ intersect the line $$L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$ at the point $$A(\alpha, \beta, \gamma)$$ and the line $$L_2 : x - 6 = y = -z + 4$$ at the point $$B(a, b, c)$$. Then $$\begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix}$$ is equal to
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Let $$\alpha$$ and $$\beta$$ be the roots of $$x^2 + \sqrt{3}x - 16 = 0$$, and $$\gamma$$ and $$\delta$$ be the roots of $$x^2 + 3x - 1 = 0$$. If $$P_n = \alpha^n + \beta^n$$ and $$Q_n = \gamma^n + \delta^n$$, then $$\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$$ is equal to
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The sum of all rational terms in the expansion of $$(2 + \sqrt{3})^8$$ is
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Let $$A = \{-3, -2, -1, 0, 1, 2, 3\}$$. Let R be a relation on A defined by $$xRy$$ if and only if $$0 \leq x^2 + 2y \leq 4$$. Let $$l$$ be the number of elements in R and $$m$$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $$l + m$$ is equal to
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A line passing through the point $$P(\sqrt{5}, \sqrt{5})$$ intersects the ellipse $$\frac{x^2}{36} + \frac{y^2}{25} = 1$$ at A and B such that $$(PA) \cdot (PB)$$ is maximum. Then $$5(PA^2 + PB^2)$$ is equal to :
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The sum $$1 + 3 + 11 + 25 + 45 + 71 + \ldots$$ upto 20 terms, is equal to
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If the domain of the function $$f(x) = \log_e\left(\frac{2x - 3}{5 + 4x}\right) + \sin^{-1}\left(\frac{4 + 3x}{2 - x}\right)$$ is $$[\alpha, \beta)$$, then $$\alpha^2 + 4\beta$$ is equal to
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If $$\sum_{r=1}^{9} \left(\frac{r+3}{2^r}\right) \cdot \,^{9}C_r = \alpha\left(\frac{3}{2}\right)^9 - \beta$$, $$\alpha, \beta \in \mathbb{N}$$, then $$(\alpha + \beta)^2$$ is equal to
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The number of solutions of the equation $$2x + 3\tan x = \pi$$, $$x \in [-2\pi, 2\pi] - \left\{\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}\right\}$$ is
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If $$y(x) = \begin{vmatrix} \sin x & \cos x & \sin x + \cos x + 1 \\ 27 & 28 & 27 \\ 1 & 1 & 1 \end{vmatrix}$$, $$x \in \mathbb{R}$$, then $$\frac{d^2y}{dx^2} + y$$ is equal to
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Let g be a differentiable function such that $$\int_0^x g(t)\,dt = x - \int_0^x tg(t)\,dt$$, $$x \geq 0$$ and let $$y = y(x)$$ satisfy the differential equation $$\frac{dy}{dx} - y\tan x = 2(x+1)\sec x \cdot g(x)$$, $$x \in \left[0, \frac{\pi}{2}\right)$$. If $$y(0) = 0$$, then $$y\left(\frac{\pi}{3}\right)$$ is equal to
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A line passes through the origin and makes equal angles with the positive coordinate axes. It intersects the lines $$L_1 : 2x + y + 6 = 0$$ and $$L_2 : 4x + 2y - p = 0$$, $$p \gt 0$$, at the points A and B, respectively. If $$AB = \frac{9}{\sqrt{2}}$$ and the foot of the perpendicular from the point A on the line $$L_2$$ is M, then $$\frac{AM}{BM}$$ is equal to
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Let $$z \in \mathbb{C}$$ be such that $$\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$$. Then the sum of all possible values of $$z^2$$ is
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Let $$f(x) = \int x^3\sqrt{3 - x^2}\,dx$$. If $$5f(\sqrt{2}) = -4$$, then $$f(1)$$ is equal to
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Let $$a_1, a_2, a_3, \ldots$$ be a G.P. of increasing positive numbers. If $$a_3 a_5 = 729$$ and $$a_2 + a_4 = \frac{111}{4}$$, then $$24(a_1 + a_2 + a_3)$$ is equal to
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Let the domain of the function $$f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$$ be $$(a, b)$$. If $$\int_0^{b-a} [x^2]\,dx = p - \sqrt{q} - \sqrt{r}$$, $$p, q, r \in \mathbb{N}$$, $$\gcd(p, q, r) = 1$$, then $$p + q + r$$ is equal to
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The radius of the smallest circle which touches the parabolas $$y = x^2 + 2$$ and $$x = y^2 + 2$$ is
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Let $$f(x) = \begin{cases} (1 + ax)^{1/x}, & x \lt 0 \\ 1 + b, & x = 0 \\ \frac{(x+4)^{1/2} - 2}{(x+c)^{1/3} - 2}, & x \gt 0 \end{cases}$$ be continuous at $$x = 0$$. Then $$e^a bc$$ is equal to
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Line $$L_1$$ passes through the point $$(1, 2, 3)$$ and is parallel to z-axis. Line $$L_2$$ passes through the point $$(\lambda, 5, 6)$$ and is parallel to y-axis. Let for $$\lambda = \lambda_1, \lambda_2$$, $$\lambda_2 \lt \lambda_1$$, the shortest distance between the two lines be 3. Then the square of the distance of the point $$(\lambda_1, \lambda_2, 7)$$ from the line $$L_1$$ is
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All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $$n$$ be denoted by $$W_n$$. Let the probability $$P(W_n)$$ of choosing the word $$W_n$$ satisfy $$P(W_n) = 2P(W_{n-1})$$, $$n \gt 1$$. If $$P(CDBEA) = \frac{2^{\alpha}}{2^{\beta} - 1}$$, $$\alpha, \beta \in \mathbb{N}$$, then $$\alpha + \beta$$ is equal to __________.
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Let the product of the focal distances of the point $$P(4, 2\sqrt{3})$$ on the hyperbola $$H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q. Then $$p^2 + q^2$$ is equal to __________.
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Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} + 2\hat{j} - \hat{k}$$, $$\vec{c} = \lambda\hat{j} + \mu\hat{k}$$ and $$\hat{d}$$ be a unit vector such that $$\vec{a} \times \hat{d} = \vec{b} \times \hat{d}$$ and $$\vec{c} \cdot \hat{d} = 1$$. If $$\vec{c}$$ is perpendicular to $$\vec{a}$$, then $$|3\lambda\hat{d} + \mu\vec{c}|^2$$ is equal to __________.
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If the number of seven-digit numbers, such that the sum of their digits is even, is $$m \cdot n \cdot 10^n$$; $$m, n \in \{1, 2, 3, \ldots, 9\}$$, then $$m + n$$ is equal to __________.
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The area of the region bounded by the curve $$y = \max\{|x|, x|x-2|\}$$, then x-axis and the lines $$x = -2$$ and $$x = 4$$ is equal to __________.
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