Let $$S = \{x \in [-6, 3] - \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$$ and $$T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$$. Then the number of elements in $$S \cap T$$ is
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Let $$S = \{x \in [-6, 3] - \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$$ and $$T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$$. Then the number of elements in $$S \cap T$$ is
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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + \sqrt{6} = 0$$ and $$\frac{1}{\alpha^2+1}, \frac{1}{\beta^2+1}$$ be the roots of the equation $$x^2 + ax + b = 0$$. Then the roots of the equation $$x^2 - (a+b-2)x + (a+b+2) = 0$$ are:
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Let the tangents at two points A and B on the circle $$x^2 + y^2 - 4x + 3 = 0$$ meet at origin $$O(0,0)$$. Then the area of the triangle OAB is
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Let the hyperbola $$H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ pass through the point $$(2\sqrt{2}, -2\sqrt{2})$$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is $$e$$ times the length of the latus rectum of H, where $$e$$ is the eccentricity of H, then which of the following points lies on the parabola?
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Let
$$p$$: Ramesh listens to music.
$$q$$: Ramesh is out of his village
$$r$$: It is Sunday
$$s$$: It is Saturday
Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as
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A horizontal park is in the shape of a triangle OAB with $$AB = 16$$. A vertical lamp post OP is erected at the point O such that $$\angle PAO = \angle PBO = 15^\circ$$ and $$\angle PCO = 45^\circ$$, where C is the midpoint of AB. Then $$(OP)^2$$ is equal to
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Let A and B be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?
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Let $$f(x) = ax^2 + bx + c$$ be such that $$f(1) = 3, f(-2) = \lambda$$ and $$f(3) = 4$$. If $$f(0) + f(1) + f(-2) + f(3) = 14$$, then $$\lambda$$ is equal to
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The function $$f: \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = \lim_{n \to \infty} \frac{\cos(2\pi x) - x^{2n}\sin(x-1)}{1 + x^{2n+1} - x^{2n}}$$ is continuous for all $$x$$ in
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Let $$x(t) = 2\sqrt{2}\cos t\sqrt{\sin 2t}$$ and $$y(t) = 2\sqrt{2}\sin t\sqrt{\sin 2t}$$, $$t \in (0, \frac{\pi}{2})$$. Then $$\frac{1 + \left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}$$ at $$t = \frac{\pi}{4}$$ is equal to
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The function $$f(x) = xe^{x(1-x)}$$, $$x \in \mathbb{R}$$, is
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The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ in the interval $$[0, \pi]$$ is
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Let $$I_n(x) = \int_0^x \frac{1}{(t^2+5)^n} dt$$, $$n = 1, 2, 3, \ldots$$. Then
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The area enclosed by the curves $$y = \log_e(x+e^2)$$, $$x = \log_e\left(\frac{2}{y}\right)$$, above the line $$x = \log_e 2$$ and $$y = 1$$ is
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Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} + \frac{1}{x^2-1}y = \left(\frac{x-1}{x+1}\right)^{1/2}$$, $$x > 1$$ passing through the point $$\left(2, \sqrt{\frac{1}{3}}\right)$$. Then $$\sqrt{7}y(8)$$ is equal to
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The differential equation of the family of circles passing through the points (0, 2) and (0, -2) is
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Let S be the set of all $$a \in \mathbb{R}$$ for which the angle between the vectors $$\vec{u} = a(\log_e b)\hat{i} - 6\hat{j} + 3\hat{k}$$ and $$\vec{v} = (\log_e b)\hat{i} + 2\hat{j} + 2a(\log_e b)\hat{k}$$, $$(b > 1)$$ is acute. Then S is equal to
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Let the lines $$\frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2}$$ and $$\frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda}$$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?
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A plane P is parallel to two lines whose direction ratios are $$(-2, 1, -3)$$, and $$(-1, 2, -2)$$ and it contains the point $$(2, 2, -2)$$. Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $$\alpha, \beta, \gamma$$. If V is the volume of the tetrahedron OABC, where O is the origin and $$p = \alpha + \beta + \gamma$$, then the ordered pair $$(V, p)$$ is equal to
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Let A and B be two events such that $$P(B|A) = \frac{2}{5}$$, $$P(A|B) = \frac{1}{7}$$ and $$P(A \cap B) = \frac{1}{9}$$. Consider $$(S_1): P(A' \cup B) = \frac{5}{6}$$, $$(S_2): P(A' \cap B') = \frac{1}{18}$$. Then
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Let $$z = a + ib$$, $$b \neq 0$$ be complex numbers satisfying $$z^2 = \bar{z} \cdot 2^{1-|z|}$$. Then the least value of $$n \in \mathbb{N}$$, such that $$z^n = (z+1)^n$$, is equal to _____
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A class contains b boys and g girls. If the number of ways of selecting 3 boys and 2 girls from the class is 168, then $$b + 3g$$ is equal to
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If $$\dfrac{6}{3^{12}} + \dfrac{10}{3^{11}} + \dfrac{20}{3^{10}} + \dfrac{40}{3^9} + \ldots + \dfrac{10240}{3} = 2^n \cdot m$$, where $$m$$ is odd, then $$m \cdot n$$ is equal to _____
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Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}} + \beta x\right)^4$$, $$(1 - 3\beta x)^2$$ and $$\left(1 - \frac{\beta}{2}x\right)^6$$, $$\beta > 0$$ respectively form the first three terms of an A.P. If $$d$$ is the common difference of this A.P., then $$50 - \frac{2d}{\beta^2}$$ is equal to _____
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If $$1 + (2 + {}^{49}C_1 + {}^{49}C_2 + \ldots + {}^{49}C_{49})({}^{50}C_2 + {}^{50}C_4 + \ldots + {}^{50}C_{50})$$ is equal to $$2^n \cdot m$$, where $$m$$ is odd, then $$n + m$$ is equal to _____
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Let $$S = [-\pi, \frac{\pi}{2}) - \{-\frac{\pi}{2}, -\frac{\pi}{4}, -\frac{3\pi}{4}, \frac{\pi}{4}\}$$. Then the number of elements in the set $$A = \{\theta \in S : \tan\theta(1 + \sqrt{5}\tan(2\theta)) = \sqrt{5} - \tan(2\theta)\}$$ is _____
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Two tangent lines $$l_1$$ and $$l_2$$ are drawn from the point (2, 0) to the parabola $$2y^2 = -x$$. If the lines $$l_1$$ and $$l_2$$ are also tangent to the circle $$(x-5)^2 + y^2 = r$$, then $$17r^2$$ is equal to
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Let the tangents at the points P and Q on the ellipse $$\frac{x^2}{2} + \frac{y^2}{4} = 1$$ meet at the point $$R(\sqrt{2}, 2\sqrt{2}-2)$$. If S is the focus of the ellipse on its negative major axis, then $$SP^2 + SQ^2$$ is equal to
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The value of the integral $$\int_0^{\pi/2} \frac{60\sin(6x)}{\sin x} dx$$ is equal to
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A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If $$\sigma^2$$ is the variance of X, then $$100\sigma^2$$ is equal to
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