Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = ||x + 2| - 2|x||$$. If m is the number of points of local minima and n is the number of points of local maxima of f, then $$m + n$$ is
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = ||x + 2| - 2|x||$$. If m is the number of points of local minima and n is the number of points of local maxima of f, then $$m + n$$ is
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Each of the angles $$\beta$$ and $$\gamma$$ that a given line makes with the positive y- and z-axes, respectively, is half of the angle that this line makes with the positive x-axes. Then the sum of all possible values of the angle $$\beta$$ is
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If the four distinct points $$(4, 6)$$, $$(-1, 5)$$, $$(0, 0)$$ and $$(k, 3k)$$ lie on a circle of radius r, then $$10k + r^2$$ is equal to
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Let the Mean and Variance of five observations $$x_1 = 1, x_2 = 3, x_3 = a, x_4 = 7$$ and $$x_5 = b$$, $$a \gt b$$, be 5 and 10 respectively. Then the Variance of the observations $$n + x_n$$, $$n = 1, 2, \ldots, 5$$ is
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Consider the lines $$x(3\lambda + 1) + y(7\lambda + 2) = 17\lambda + 5$$, $$\lambda$$ being a parameter, all passing through a point P. One of these lines (say L) is farthest from the origin. If the distance of L from the point $$(3, 6)$$ is d, then the value of $$d^2$$ is
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Let $$A = \{-2, -1, 0, 1, 2, 3\}$$. Let R be a relation on A defined by $$xRy$$ if and only if $$y = \max\{x, 1\}$$. Let $$\ell$$ be the number of elements in R. Let m and n be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $$\ell + m + n$$ is equal to
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Let the equation $$x(x + 2)(12 - k) = 2$$ have equal roots. Then the distance of the point $$\left(k, \dfrac{k}{2}\right)$$ from the line $$3x + 4y + 5 = 0$$ is
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Line $$L_1$$ of slope 2 and line $$L_2$$ of slope $$\dfrac{1}{2}$$ intersect at the origin O. In the first quadrant, $$P_1, P_2, \ldots P_{12}$$ are 12 points on line $$L_1$$ and $$Q_1, Q_2, \ldots Q_9$$ are 9 points on line $$L_2$$. Then the total number of triangles, that can be formed having vertices at three of the 22 points O, $$P_1, P_2, \ldots P_{12}$$, $$Q_1, Q_2, \ldots Q_9$$, is:
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The integral $$\displaystyle\int_0^{\pi} \dfrac{8x \, dx}{4\cos^2 x + \sin^2 x}$$ is equal to
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Let f be a function such that $$f(x) + 3f\left(\dfrac{24}{x}\right) = 4x$$, $$x \neq 0$$. Then $$f(3) + f(8)$$ is equal to
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The area of the region $$\{(x, y) : |x - y| \le y \le 4\sqrt{x}\}$$ is
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If the domain of the function $$f(x) = \log_x(1 - \log_4(x^2 - 9x + 18))$$ is $$(\alpha, \beta) \cup (\gamma, \delta)$$, then $$\alpha + \beta + \gamma + \delta$$ is equal to
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If the probability that the random variable X takes the value x is given by $$P(X = x) = k(x + 1)3^{-x}$$, $$x = 0, 1, 2, 3, \ldots$$, where k is a constant, then $$P(X \ge 3)$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$\dfrac{dy}{dx} + 3(\tan^2 x) \, y + 3y = \sec^2 x$$, $$y(0) = \dfrac{1}{3} + e^3$$. Then $$y\left(\dfrac{\pi}{4}\right)$$ is equal to
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If $$z_1, z_2, z_3 \in \mathbb{C}$$ are the vertices of an equilateral triangle, whose centroid is $$z_0$$, then $$\displaystyle\sum_{k=1}^{3} (z_k - z_0)^2$$ is equal to
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The number of solutions of equation $$(4 - \sqrt{3})\sin x - 2\sqrt{3}\cos^2 x = \dfrac{-4}{1 + \sqrt{3}}$$, $$x \in \left[-2\pi, \dfrac{5\pi}{2}\right]$$ is
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Let C be the circle of minimum area enclosing the ellipse $$E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ with eccentricity $$\dfrac{1}{2}$$ and foci $$(\pm 2, 0)$$. Let PQR be a variable triangle, whose vertex P is on the circle C and the side QR of length 2a is parallel to the major axis of E and contains the point of intersection of E with the negative y-axis. Then the maximum area of the triangle PQR is:
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The shortest distance between the curves $$y^2 = 8x$$ and $$x^2 + y^2 + 12y + 35 = 0$$ is:
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The distance of the point $$(7, 10, 11)$$ from the line $$\dfrac{x - 4}{1} = \dfrac{y - 4}{0} = \dfrac{z - 2}{3}$$ along the line $$\dfrac{x - 9}{2} = \dfrac{y - 13}{3} = \dfrac{z - 17}{6}$$ is
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The sum $$1 + \dfrac{1 + 3}{2!} + \dfrac{1 + 3 + 5}{3!} + \dfrac{1 + 3 + 5 + 7}{4!} + \ldots$$ upto $$\infty$$ terms, is equal to
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Let I be the identity matrix of order $$3 \times 3$$ and for the matrix $$A = \begin{bmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{bmatrix}$$, $$|A| = -1$$. Let B be the inverse of the matrix $$\text{adj}(A \cdot \text{adj}(A^2))$$. Then $$|(\lambda B + I)|$$ is equal to ________.
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Let $$(1 + x + x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{20} x^{20}$$. If $$(a_1 + a_3 + a_5 + \ldots + a_{19}) - 11a_2 = 121k$$, then k is equal to ________.
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If $$\displaystyle\lim_{x \to 0} \left(\dfrac{\tan x}{x}\right)^{1/x^2} = p$$, then $$96 \log_e p$$ is equal to ________.
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Let $$\vec{a} = \hat{i} + 2\hat{j} + \hat{k}$$, $$\vec{b} = 3\hat{i} - 3\hat{j} + 3\hat{k}$$, $$\vec{c} = 2\hat{i} - \hat{j} + 2\hat{k}$$ and $$\vec{d}$$ be a vector such that $$\vec{b} \times \vec{d} = \vec{c} \times \vec{d}$$ and $$\vec{a} \cdot \vec{d} = 4$$. Then $$|\vec{a} \times \vec{d}|^2$$ is equal to ________.
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If the equation of the hyperbola with foci $$(4, 2)$$ and $$(8, 2)$$ is $$3x^2 - y^2 - \alpha x + \beta y + \gamma = 0$$, then $$\alpha + \beta + \gamma$$ is equal to ________.
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