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Let $$f, g : (1, \infty) \to \mathbb{R}$$ be defined as $$f(x) = \dfrac{2x + 3}{5x + 2}$$ and $$g(x) = \dfrac{2 - 3x}{1 - x}$$. If the range of the function $$f \circ g : [2, 4] \to \mathbb{R}$$ is $$[\alpha, \beta]$$, then $$\dfrac{1}{\beta - \alpha}$$ is equal to
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Consider the sets $$A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + y^2 = 25\}$$, $$B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : x^2 + 9y^2 = 144\}$$, $$C = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} : x^2 + y^2 \le 4\}$$, and $$D = A \cap B$$. The total number of one-one functions from the set D to the set C is:
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Let $$A = \{1, 6, 11, 16, \ldots\}$$ and $$B = \{9, 16, 23, 30, \ldots\}$$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $$n(A \cup B)$$ is
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For an integer $$n \ge 2$$, if the arithmetic mean of all coefficients in the binomial expansion of $$(x + y)^{2n-3}$$ is 16, then the distance of the point $$P(2n - 1, n^2 - 4n)$$ from the line $$x + y = 8$$ is:
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The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is:
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Let the shortest distance between the lines $$\dfrac{x - 3}{3} = \dfrac{y - \alpha}{-1} = \dfrac{z - 3}{1}$$ and $$\dfrac{x + 3}{-3} = \dfrac{y + 7}{2} = \dfrac{z - \beta}{4}$$ be $$3\sqrt{30}$$. Then the positive value of $$5\alpha + \beta$$ is
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If $$\displaystyle\lim_{x \to 1} \dfrac{(x - 1)(6 + \lambda \cos(x - 1)) + \mu \sin(1 - x)}{(x - 1)^3} = -1$$, where $$\lambda, \mu \in \mathbb{R}$$, then $$\lambda + \mu$$ is equal to
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Let $$f : [0, \infty) \to \mathbb{R}$$ be differentiable function such that $$f(x) = 1 - 2x + \displaystyle\int_0^x e^{x-t} f(t) \, dt$$ for all $$x \in [0, \infty)$$. Then the area of the region bounded by $$y = f(x)$$ and the coordinate axes is
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Let A and B be two distinct points on the line $$L : \dfrac{x - 6}{3} = \dfrac{y - 7}{2} = \dfrac{z - 7}{-2}$$. Both A and B are at a distance $$2\sqrt{17}$$ from the foot of perpendicular drawn from the point $$(1, 2, 3)$$ on the line L. If O is the origin, then $$\vec{OA} \cdot \vec{OB}$$ is equal to:
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a continuous function satisfying $$f(0) = 1$$ and $$f(2x) - f(x) = x$$ for all $$x \in \mathbb{R}$$. If $$\displaystyle\lim_{n \to \infty} \left\{f(x) - f\left(\dfrac{x}{2^n}\right)\right\} = G(x)$$, then $$\displaystyle\sum_{r=1}^{10} G(r^2)$$ is equal to
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$$1 + 3 + 5^2 + 7 + 9^2 + 11 + 13^2 + \ldots$$ upto 40 terms is equal to
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In the expansion of $$\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^n$$, $$n \in \mathbb{N}$$, if the ratio of $$15^{\text{th}}$$ term from the end to the $$15^{\text{th}}$$ term from the beginning is $$\dfrac{1}{6}$$, then the value of $${}^nC_3$$ is:
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Considering the principal values of the inverse trigonometric functions, $$\sin^{-1}\left(\dfrac{\sqrt{3}}{2} x + \dfrac{1}{2}\sqrt{1 - x^2}\right)$$, $$-\dfrac{1}{2} \lt x \lt \dfrac{1}{\sqrt{2}}$$, is equal to
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Consider two vectors $$\vec{u} = 3\hat{i} - \hat{j}$$ and $$\vec{v} = 2\hat{i} + \hat{j} - \lambda\hat{k}$$, $$\lambda \gt 0$$. The angle between them is given by $$\cos^{-1}\left(\dfrac{\sqrt{5}}{2\sqrt{7}}\right)$$. Let $$\vec{v} = \vec{v}_1 + \vec{v}_2$$, where $$\vec{v}_1$$ is parallel to $$\vec{u}$$ and $$\vec{v}_2$$ is perpendicular to $$\vec{u}$$. Then the value $$|\vec{v}_1|^2 + |\vec{v}_2|^2$$ is equal to
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Let the three sides of a triangle are on the lines $$4x - 7y + 10 = 0$$, $$x + y = 5$$ and $$7x + 4y = 15$$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $$x = 0$$, $$y = 0$$ and $$x + y = 1$$ is
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The value of $$\displaystyle\int_{-1}^{1} \dfrac{\left(1 + \sqrt{|x| - x}\right)e^{-x} + \left(\sqrt{|x| - x}\right)e^{-x}}{e^{x} + e^{-x}} \, dx$$ is equal to
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The length of the latus-rectum of the ellipse, whose foci are $$(2, 5)$$ and $$(2, -3)$$ and eccentricity is $$\dfrac{4}{5}$$, is
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Consider the equation $$x^2 + 4x - n = 0$$, where $$n \in [20, 100]$$ is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to
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A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is
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If $$10\sin^4\theta + 15\cos^4\theta = 6$$, then the value of $$\dfrac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta}$$ is:
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If the area of the region $$\{(x, y) : |x - 5| \le y \le 4\sqrt{x}\}$$ is A, then 3A is equal to ________.
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Let $$A = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 & \cos\theta \end{bmatrix}$$. If for some $$\theta \in (0, \pi)$$, $$A^2 = A^T$$, then the sum of the diagonal elements of the matrix $$(A + I)^3 + (A - I)^3 - 6A$$ is equal to ________.
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Let $$A = \{z \in \mathbb{C} : |z - 2 - i| = 3\}$$, $$B = \{z \in \mathbb{C} : \text{Re}(z - iz) = 2\}$$ and $$S = A \cap B$$. Then $$\displaystyle\sum_{z \in S} |z|^2$$ is equal to ________.
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Let C be the circle $$x^2 + (y - 1)^2 = 2$$, $$E_1$$ and $$E_2$$ be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line $$x + y = 3$$ touch the curves C, $$E_1$$ and $$E_2$$ at $$P(x_1, y_1)$$, $$Q(x_2, y_2)$$ and $$R(x_3, y_3)$$ respectively. Given that P is the mid-point of the line segment QR and $$PQ = \dfrac{2\sqrt{2}}{3}$$, the value of $$9(x_1 y_1 + x_2 y_2 + x_3 y_3)$$ is equal to ________.
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Let m and n be the number of points at which the function $$f(x) = \max\{x, x^3, x^5, \ldots, x^{21}\}$$, $$x \in \mathbb{R}$$, is not differentiable and not continuous, respectively. Then $$m + n$$ is equal to ________.
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