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Let the values of $$\lambda$$ for which the shortest distance between the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$ and $$\frac{x-\lambda}{3} = \frac{y-4}{4} = \frac{z-5}{5}$$ is $$\frac{1}{\sqrt{6}}$$ be $$\lambda_1$$ and $$\lambda_2$$. Then the radius of the circle passing through the points $$(0, 0)$$, $$(\lambda_1, \lambda_2)$$ and $$(\lambda_2, \lambda_1)$$ is :
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Let $$\alpha$$ be a solution of $$x^2 + x + 1 = 0$$, and for some a and b in $$\mathbb{R}$$, $$[4 \; a \; b] \begin{bmatrix} 1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8 \end{bmatrix} = [0 \; 0 \; 0]$$. If $$\frac{4}{\alpha^4} + \frac{m}{\alpha^a} + \frac{n}{\alpha^b} = 3$$, then $$m + n$$ is equal to :
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Let the function $$f(x) = \frac{x}{3} + \frac{3}{x} + 3$$, $$x \ne 0$$ be strictly increasing in $$(-\infty, \alpha_1) \cup (\alpha_2, \infty)$$ and strictly decreasing in $$(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$$. Then $$\sum_{i=1}^{5} \alpha_i^2$$ is equal to :
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If A and B are two events such that $$P(A) = 0.7$$, $$P(B) = 0.4$$ and $$P(A \cap \bar{B}) = 0.5$$, where $$\bar{B}$$ denotes the complement of B, then $$P\left(B\mid(A \cup \bar{B})\right)$$ is equal to :
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If $$\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty = \frac{\pi^4}{90}$$, $$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty = \alpha$$, $$\frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty = \beta$$, then $$\frac{\alpha}{\beta}$$ is equal to :
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The sum of the squares of the roots of $$|x - 2|^2 + |x - 2| - 2 = 0$$ and the squares of the roots of $$x^2 - 2|x - 3| - 5 = 0$$, is
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Let a be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle $$\alpha$$ with the positive x-axis and the equations of its diagonals are $$(\sqrt{3}+1)x + (\sqrt{3}-1)y = 0$$ and $$(\sqrt{3}-1)x - (\sqrt{3}+1)y + 8\sqrt{3} = 0$$. Then $$a^2$$ is equal to
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Let f(x) be a positive function and $$I_1 = \int_{-\frac{1}{2}}^{1} 2xf(2x(1-2x)) \, dx$$ and $$I_2 = \int_{-1}^{2} f(x(1-x)) \, dx$$. Then the value of $$\frac{I_2}{I_1}$$ is equal to :
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Let $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{j} - \hat{k}$$. Let $$\hat{c}$$ be a unit vector in the plane of the vectors $$\vec{a}$$ and $$\vec{b}$$ and be perpendicular to $$\vec{a}$$. Then such a vector $$\hat{c}$$ is :
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Let the ellipse $$3x^2 + py^2 = 4$$ pass through the centre C of the circle $$x^2 + y^2 - 2x - 4y - 11 = 0$$ of radius r. Let $$f_1$$, $$f_2$$ be the focal distances of the point C on the ellipse. Then $$6f_1f_2 - r$$ is equal to
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The integral $$\int_{-1}^{\frac{3}{2}} \left(\left|\pi^2 x \sin(\pi x)\right|\right) dx$$ is equal to :
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A line passing through the point P(a, $$\theta$$) makes an acute angle $$\alpha$$ with the positive x-axis. Let this line be rotated about the point P through an angle $$\frac{\alpha}{2}$$ in the clock-wise direction. If in the new position, the slope of the line is $$2 - \sqrt{3}$$ and its distance from the origin is $$\frac{1}{\sqrt{2}}$$, then the value of $$3a^2\tan^2\alpha - 2\sqrt{3}$$ is
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There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is
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Let $$A = \left\{\theta \in [0, 2\pi] : 1 + 10\text{Re}\left(\frac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta}\right) = 0\right\}$$. Then $$\sum_{\theta \in A} \theta^2$$ is equal to
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Let $$A = \{0, 1, 2, 3, 4, 5\}$$. Let R be a relation on A defined by $$(x, y) \in R$$ if and only if $$\max\{x, y\} \in \{3, 4\}$$. Then among the statements
$$(S_1)$$ : The number of elements in R is 18, and
$$(S_2)$$ : The relation R is symmetric but neither reflexive nor transitive
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The number of integral terms in the expansion of $$\left(5^{\frac{1}{2}} + 7^{\frac{1}{8}}\right)^{1016}$$ is
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Let $$f(x) = x - 1$$ and $$g(x) = e^x$$ for $$x \in \mathbb{R}$$. If $$\frac{dy}{dx} = \left(e^{-2\sqrt{x}} g(f(f(x))) - \frac{y}{\sqrt{x}}\right)$$, $$y(0) = 0$$, then $$y(1)$$ is :-
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The value of $$\cot^{-1}\left(\frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)}\right) - \cot^{-1}\left(\frac{\sqrt{1 + \tan^2(\frac{1}{2})} + 1}{\tan(\frac{1}{2})}\right)$$ is equal to
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Let $$A = \begin{bmatrix} 2 & 2+p & 2+p+q \\ 4 & 6+2p & 8+3p+2q \\ 6 & 12+3p & 20+6p+3q \end{bmatrix}$$. If $$\det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n$$, $$m, n \in \mathbb{N}$$, then $$m + n$$ is equal to
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Given below are two statements :
Statement I : $$\lim_{x \to 0} \left(\frac{\tan^{-1}x + \log_{e}\sqrt{\frac{1+x}{1-x}} - 2x}{x^5}\right) = \frac{2}{5}$$
Statement II : $$\lim_{x \to 1} \left(x^{\frac{2}{1-x}}\right) = \frac{1}{e^2}$$
In the light of the above statements, choose the correct answer :
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Let the area of the bounded region $$\{(x, y) : 0 \le 9x \le y^2, y \ge 3x - 6\}$$ be A. Then 6A is equal to _____.
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Let the domain of the function $$f(x) = \cos^{-1}\left(\frac{4x+5}{3x-7}\right)$$ be $$[\alpha, \beta]$$ and the domain of $$g(x) = \log_2(2 - 6\log_{27}(2x+5))$$ be $$(\gamma, \delta)$$. Then $$|7(\alpha + \beta) + 4(\gamma + \delta)|$$ is equal to _____.
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Let the area of the triangle formed by the lines $$x + 2 = y - 1 = z$$, $$\frac{x-3}{5} = \frac{y}{-1} = \frac{z-1}{1}$$ and $$\frac{x}{-3} = \frac{y-3}{3} = \frac{z-2}{1}$$ be A. Then $$A^2$$ is equal to _____.
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The product of the last two digits of $$(1919)^{1919}$$ is _____.
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Let r be the radius of the circle, which touches the x-axis at point $$(a, 0)$$, $$a < 0$$ and the parabola $$y^2 = 9x$$ at the point $$(4, 6)$$. Then r is equal to _____.
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