Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}$$. Then $$f$$ is :
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3}$$. Then $$f$$ is :
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Consider the quadratic equation $$(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, n \in \mathbb{R}$$. Let $$\alpha$$ be the minimum value of the product of its roots and $$\beta$$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $$\alpha$$ and the common ratio is $$\dfrac{\alpha}{\beta}$$, is :
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Let $$S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0\}$$. Then $$\displaystyle\sum_{z \in S} z^8$$ is equal to :
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The sum of all possible values of $$\theta \in [0, 2\pi]$$, for which the system of equations :
$$x\cos 3\theta - 8y - 12z = 0$$
$$x\cos 2\theta + 3y + 3z = 0$$
$$x + y + 3z = 0$$
has a non-trivial solution, is equal to :
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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$$ and $$B = [b_{ij}], 1 \le i, j \le 3$$. If $$B = A^{99} - I$$, then the value of $$\dfrac{b_{31} - b_{21}}{b_{32}}$$ is :
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The sum $$1 + \dfrac{1}{2}(1^2 + 2^2) + \dfrac{1}{3}(1^2 + 2^2 + 3^2) + \ldots$$ upto 10 terms is equal to :
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A building has ground floor and 10 more floors. Nine persons enter a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :
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Let the mean and the variance of seven observations 2, 4, $$\alpha$$, 8, $$\beta$$, 12, 14, $$\alpha < \beta$$, be 8 and 16 respectively. Then the quadratic equation whose roots are $$3\alpha + 2$$ and $$2\beta + 1$$ is :
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A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :
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Let C be a circle having centre in the first quadrant and touching the $$x$$-axis at a distance of 3 units from the origin. If the circle C has an intercept of length $$6\sqrt{3}$$ on $$y$$-axis, then the length of the chord of the circle C on the line $$x - y = 3$$ is :
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The eccentricity of an ellipse E with centre at the origin O is $$\dfrac{\sqrt{3}}{2}$$ and its directrices are $$x = \pm \dfrac{4\sqrt{6}}{3}$$. Let $$H: \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of E, and whose length of latus rectum is equal to the length of minor axis of E. Then the distance between the foci of H is :
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Let $$x = 9$$ be a directrix of an ellipse E, whose centre is at the origin and eccentricity is $$\dfrac{1}{3}$$. Let $$P(\alpha, 0)$$, $$\alpha > 0$$, be a focus of E and AB be a chord passing through P. Then the locus of the mid point of AB is :
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If $$\sin(\tan^{-1}(x\sqrt{2})) = \cot(\sin^{-1}\sqrt{1 - x^2})$$, $$x \in (0, 1)$$, then the value of $$x$$ is :
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The shortest distance between the lines $$\dfrac{x - 4}{1} = \dfrac{y - 3}{2} = \dfrac{z - 2}{-3}$$ and $$\dfrac{x + 2}{2} = \dfrac{y - 6}{4} = \dfrac{z - 5}{-5}$$ is :
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Let $$\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}$$ and $$\vec{b} = 6\hat{i} + 3\hat{j} + 3\hat{k}$$. Then the square of the area of the triangle with adjacent sides determined by the vectors $$(2\vec{a} + 3\vec{b})$$ and $$(\vec{a} - \vec{b})$$ is :
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Let $$\displaystyle\lim_{x \to 2} \dfrac{(\tan(x - 2))(rx^2 + (p - 2)x - 2p)}{(x - 2)^2} = 5$$ for some $$r, p \in \mathbb{R}$$. If the set of all possible values of q, such that the roots of the equation $$rx^2 - px + q = 0$$ lie in $$(0, 2)$$, be the interval $$(\alpha, \beta]$$, then $$4(\alpha + \beta)$$ equals :
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Let $$A = \begin{bmatrix} 1 & 3 & -1 \\ 2 & 1 & \alpha \\ 0 & 1 & -1 \end{bmatrix}$$ be a singular matrix. Let $$f(x) = \displaystyle\int_0^x (t^2 + 2t + 3)\,dt$$, $$x \in [1, \alpha]$$. If M and m are respectively the maximum and the minimum values of $$f$$ in $$[1, \alpha]$$, then $$3(M - m)$$ is equal to :
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be such that $$f(xy) = f(x)f(y)$$, for all $$x, y \in \mathbb{R}$$ and $$f(0) \ne 0$$. Let $$g: [1, \infty) \to \mathbb{R}$$ be a differentiable function such that $$x^2 g(x) = \int_1^x (t^2 f(t) - tg(t))\,dt.$$ Then $$g(2)$$ is equal to :
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The area of the region $$\{(x, y) : x^2 - 8x \le y \le -x\}$$ is :
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The value of the integral $$\displaystyle\int_{-1}^{1} \left(\dfrac{x^3 + |x| + 1}{x^2 + 2|x| + 1}\right) dx$$ is equal to :
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Let $$R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e(x + y) \le 2\}$$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is __________.
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If $$(1 - x^3)^{10} = \displaystyle\sum_{r=0}^{10} a_r x^r (1 - x)^{30 - 2r}$$, then $$\dfrac{9a_9}{a_{10}}$$ is equal to __________.
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Let the line $$x - y = 4$$ intersect the circle $$C: (x - 4)^2 + (y + 3)^2 = 9$$ at the points Q and R. If $$P(\alpha, \beta)$$ is a point on C such that $$PQ = PR$$, then $$(6\alpha + 8\beta)^2$$ is equal to __________.
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Let the image of the point $$P(0, -5, 0)$$ in the line $$\dfrac{x - 1}{2} = \dfrac{y}{1} = \dfrac{z + 1}{-2}$$ be the point R and the image of the point $$Q\left(0, \dfrac{-1}{2}, 0\right)$$ in the line $$\dfrac{x - 1}{-1} = \dfrac{y + 9}{4} = \dfrac{z + 1}{1}$$ be the point S. Then the square of the area of the parallelogram PQRS is __________.
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Let $$f(x) = \begin{cases} x^3 + 8, & x < 0 \\ x^2 - 4, & x \ge 0 \end{cases}$$ and $$g(x) = \begin{cases} (x - 8)^{1/3}, & x < 0 \\ (x + 4)^{1/2}, & x \ge 0 \end{cases}$$. Then the number of points, where the function $$g \circ f$$ is discontinuous, is __________.
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