Let $$S = \{x : x \in \mathbb{R}$$ and $$\left(\sqrt{3}+\sqrt{2}\right)^{x^2-4}+\left(\sqrt{3}-\sqrt{2}\right)^{x^2-4}\ =\ 10$$. Then $$nS$$ is equal to
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Let $$S = \{x : x \in \mathbb{R}$$ and $$\left(\sqrt{3}+\sqrt{2}\right)^{x^2-4}+\left(\sqrt{3}-\sqrt{2}\right)^{x^2-4}\ =\ 10$$. Then $$nS$$ is equal to
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If the center and radius of the circle $$\frac{z-2}{z-3} = 2$$ are respectively $$\left(\alpha,\beta\right)$$ and $$\gamma$$, then $$3(\alpha + \beta + \gamma)$$ is equal to
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The sum to 10 terms of the series $$\frac{1}{1+1^2+1^4} + \frac{2}{1+2^2+2^4} + \frac{3}{1+3^2+3^4} + \ldots$$ is:
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The value of $$\frac{1}{1!50!} + \frac{1}{3!48!} + \frac{1}{5!46!} + \ldots + \frac{1}{49!2!} + \frac{1}{51!1!}$$ is
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The combined equation of the two lines $$ax + by + c = 0$$ and $$a'x + b'y + c' = 0$$ can be written as $$ax + by + ca'x + b'y + c' = 0$$. The equation of the angle bisectors of the lines represented by the equation $$2x^2 + xy - 3y^2 = 0$$ is
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If the orthocentre of the triangle, whose vertices are $$\left(1,2\right)$$, $$\left(2,3\right)$$ and $$\left(3,1\right)$$ is $$\alpha, \beta$$, then the quadratic equation whose roots are $$\alpha + 4\beta$$ and $$4\alpha + \beta$$, is
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The negation of the expression $$q \vee ((\sim q) \wedge p)$$ is equivalent to
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The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes of the remaining two observations is
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For a triangle $$ABC$$, the value of $$\cos 2A + \cos 2B + \cos 2C$$ is least. If its inradius is 3 and incentre is $$M$$, then which of the following is NOT correct?
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Let $$R$$ be a relation on $$\mathbb{R}$$, given by $$R = \{a, b : 3a - 3b + \sqrt{7}$$ is an irrational number$$\}$$. Then $$R$$ is
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Let $$S$$ denote the set of all real values of $$\lambda$$ such that the system of equations
$$\lambda x + y + z = 1$$
$$x + \lambda y + z = 1$$
$$x + y + \lambda z = 1$$
is inconsistent, then $$\sum_{\lambda \in S} |\lambda|^2 + |\lambda|$$ is equal to
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Let $$S$$ be the set of all solutions of the equation $$\cos^{-1}(2x) - 2\cos^{-1}\sqrt{1-x^2} = \pi$$, $$x \in [-\frac{1}{2}, \frac{1}{2}]$$. Then $$\sum_{x \in S} 2\sin^{-1}(x^2 - 1)$$ is equal to
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Let $$f(x) = 2x + \tan^{-1}x$$ and $$g(x) = \log_e(\sqrt{1+x^2} + x)$$, $$x \in [0, 3]$$. Then
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Let $$f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1+\sin 2x \end{vmatrix}$$, $$x \in [\frac{\pi}{6}, \frac{\pi}{3}]$$. If $$\alpha$$ and $$\beta$$ respectively are the maximum and the minimum values of $$f$$, then
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$$\lim_{n \to \infty} \frac{1}{1+n} + \frac{1}{2+n} + \frac{1}{3+n} + \ldots + \frac{1}{2n}$$ is equal to:
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The area enclosed by the closed curve $$C$$ given by the differential equation $$\frac{dy}{dx} + \frac{x+a}{y-2} = 0$$, $$y(1) = 0$$ is $$4\pi$$. Let $$P$$ and $$Q$$ be the points of intersection of the curve $$C$$ and the y-axis. If normals at $$P$$ and $$Q$$ on the curve $$C$$ intersect x-axis at points $$R$$ and $$S$$ respectively, then the length of the line segment $$RS$$ is
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If $$y = yx$$ is the solution curve of the differential equation $$\frac{dy}{dx} + y\tan x = x\sec x$$, $$0 \leq x \leq \frac{\pi}{3}$$, $$y(0) = 1$$, then $$y(\frac{\pi}{6})$$ is equal to
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Let the image of the point $$P(2, -1, 3)$$ in the plane $$x + 2y - z = 0$$ be $$Q$$. Then the distance of the plane $$3x + 2y + z + 29 = 0$$ from the point $$Q$$ is
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The shortest distance between the lines $$\frac{x-5}{1} = \frac{y-2}{2} = \frac{z-4}{-3}$$ and $$\frac{x+3}{1} = \frac{y+5}{4} = \frac{z-1}{-5}$$ is
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In a binomial distribution B(n, p), the sum and product of the mean & variance are 5 and 6 respectively, then find $$6(n + p - q)$$ is equal to :-
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The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____.
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Let $$a_1 = 8, a_2, a_3, \ldots, a_n$$ be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170, then the product of its middle two terms is _____.
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The number of 3-digit numbers, that are divisible by either 2 or 3 but not divisible by 7 is _____.
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The remainder when $$19^{200} + 23^{200}$$ is divided by 49, is _____.
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If $$f(x) = x^2 + g'(1)x + g''(2)$$ and $$g(x) = f(1)x^2 + xf'(x) + f''(x)$$, then the value of $$f(4) - g(4)$$ is equal to _____.
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If $$\int_0^1 (x^{21} + x^{14} + x^7)(2x^{14} + 3x^7 + 6)^{1/7} dx = \frac{1}{l}(11)^{m/n}$$ where $$l, m \in \mathbb{N}$$, $$m$$ and $$n$$ are co-prime then $$l + m + n$$ is equal to _____.
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Let $$A$$ be the area bounded by the curve $$y = x|x-3|$$, the x-axis and the ordinates $$x = -1$$ and $$x = 2$$. Then $$12A$$ is equal to _____.
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a differentiable function such that $$f'(x) + f(x) = \int_0^2 f(t) dt$$. If $$f(0) = e^{-2}$$, then $$2f(0) - f(2)$$ is equal to _____.
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Let $$\vec{v} = \alpha\hat{i} + 2\hat{j} - 3\hat{k}$$, $$\vec{w} = 2\alpha\hat{i} + \hat{j} - \hat{k}$$, and $$\vec{u}$$ be a vector such that $$|\vec{u}| = \alpha > 0$$. If the minimum value of the scalar triple product $$[\vec{u}\ \vec{v}\ \vec{w}]$$ is $$-\alpha\sqrt{3401}$$, and $$|\vec{u} \cdot \hat{i}|^2 = \frac{m}{n}$$ where $$m$$ and $$n$$ are coprime natural numbers, then $$m + n$$ is equal to _____.
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$$A(2, 6, 2)$$, $$B(-4, 0, \lambda)$$, $$C(2, 3, -1)$$ and $$D(4, 5, 0)$$, $$|\lambda| \leq 5$$ are the vertices of a quadrilateral $$ABCD$$. If its area is 18 square units, then $$5 - 6\lambda$$ is equal to _____.
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