If the orthocentre of the triangle formed by the lines $$y = x + 1$$, $$y = 4x - 8$$ and $$y = mx + c$$ is at $$(3, -1)$$, then $$m - c$$ is :
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If the orthocentre of the triangle formed by the lines $$y = x + 1$$, $$y = 4x - 8$$ and $$y = mx + c$$ is at $$(3, -1)$$, then $$m - c$$ is :
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Let $$\vec{a}$$ and $$\vec{b}$$ be the vectors of the same magnitude such that $$\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|} = \sqrt{2}+1$$. Then $$\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$$ is :
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Let $$A = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : |\alpha - 1| \le 4 \text{ and } |\beta - 5| \le 6\}$$ and $$B = \{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R} : 16(\alpha - 2)^2 + 9(\beta - 6)^2 \le 144\}$$. Then
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If the range of the function $$f(x) = \frac{5 - x}{x^2 - 3x + 2}$$, $$x \ne 1, 2$$, is $$(-\infty, \alpha] \cup [\beta, \infty)$$, then $$\alpha^2 + \beta^2$$ is equal to :
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A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$n^2 - m^2$$ is equal to :
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Let a random variable X take values 0, 1, 2, 3 with $$P(X = 0) = P(X = 1) = p$$, $$P(X = 2) = P(X = 3) = q$$ and $$E(X^2) = 2E(X)$$. Then the value of $$8p - 1$$ is :
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If the area of the region $$\{(x, y) : 1 + x^2 \le y \le \min\{x + 7, 11 - 3x\}\}$$ is A, then $$3A$$ is equal to
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Let $$f : \mathbf{R} \to \mathbf{R}$$ be a polynomial function of degree four having extreme values at $$x = 4$$ and $$x = 5$$. If $$\lim_{x \to 0} \frac{f(x)}{x^2} = 5$$, then $$f(2)$$ is equal to :
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The number of solutions of the equation $$\cos 2\theta \cos\frac{\theta}{2} + \cos\frac{5\theta}{2} = 2\cos^3\frac{5\theta}{2}$$ in $$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$ is :
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Let $$a_n$$ be the $$n^{\text{th}}$$ term of an A.P. If $$S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$$, $$a_6 = 7$$ and $$S_7 = 7$$, then $$a_{n}$$ is equal to :
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If the locus of $$z \in \mathbb{C}$$, such that $$\text{Re}\left(\frac{z-1}{2z+i}\right) + \text{Re}\left(\frac{\bar{z}-1}{2\bar{z}-i}\right) = 2$$, is a circle of radius $$r$$ and center $$(a, b)$$ then $$\frac{15ab}{r^2}$$ is equal to :
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Let the length of a latus rectum of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be 10. If its eccentricity is the minimum value of the function $$f(t) = t^2 + t + \frac{11}{12}$$, $$t \in \mathbf{R}$$, then $$a^2 + b^2$$ is equal to :
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Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$$, $$y(0) = 1$$. Then $$\int_{-3}^{3} y(x) \, dx$$ is :
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If the equation of the line passing through the point $$\left(0, -\frac{1}{2}, 0\right)$$ and perpendicular to the lines $$\vec{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})$$ and $$\vec{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})$$ is $$\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}$$, then $$a + b + c + d$$ is equal to :
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Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If $$p + q = 126$$, then the eccentricity of the ellipse $$\frac{x^2}{16} + \frac{y^2}{n} = 1$$ is :
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Consider the lines $$L_1 : x - 1 = y - 2 = z$$ and $$L_2 : x - 2 = y = z - 1$$. Let the feet of the perpendiculars from the point $$P(5, 1, -3)$$ on the lines $$L_1$$ and $$L_2$$ be Q and R respectively. If the area of the triangle PQR is A, then $$4A^2$$ is equal to :
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The number of real roots of the equation $$x|x - 2| + 3|x - 3| + 1 = 0$$ is :
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Let $$e_1$$ and $$e_2$$ be the eccentricities of the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{25} = 1$$ and the hyperbola $$\frac{x^2}{16} - \frac{y^2}{b^2} = 1$$, respectively. If $$b < 5$$ and $$e_1 e_2 = 1$$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :
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Let the system of equations $$x + 5y - z = 1$$, $$4x + 3y - 3z = 7$$, $$24x + y + \lambda z = \mu$$, $$\lambda, \mu \in \mathbf{R}$$, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $$7 \le x + y + z \le 77$$, is
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If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :
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If the function $$f(x) = \frac{\tan(\tan x) - \sin(\sin x)}{\tan x - \sin x}$$ is continuous at $$x = 0$$, then $$f(0)$$ is equal to _____.
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If $$\int \left(\frac{1}{x} + \frac{1}{x^3}\right)\left(\sqrt[23]{3x^{-24} + x^{-26}}\right) dx = -\frac{\alpha}{3(\alpha+1)}(3x^\beta + x^\gamma)^{\frac{\alpha+1}{\alpha}} + C$$, $$x > 0$$, $$(\alpha, \beta, \gamma \in \mathbb{Z})$$, where C is the constant of integration, then $$\alpha + \beta + \gamma$$ is equal to _____.
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For $$t > -1$$, let $$\alpha_t$$ and $$\beta_t$$ be the roots of the equation $$\left((t+2)^{1/7} - 1\right)x^2 + \left((t+2)^{1/6} - 1\right)x + \left((t+2)^{1/21} - 1\right) = 0$$. If $$\lim_{t \to -1^+} \alpha_t = a$$ and $$\lim_{t \to -1^+} \beta_t = b$$, then $$72(a + b)^2$$ is equal to _____.
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Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a and 2b, respectively, and one focus and the corresponding directrix of this hyperbola be $$(-5, 0)$$ and $$5x + 9 = 0$$, respectively. If the product of the focal distances of a point $$\left(\alpha, 2\sqrt{5}\right)$$ on the hyperbola is p, then 4p is equal to _____.
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The sum of the series $$2 \times 1 \times {^{20}C_4} - 3 \times 2 \times {^{20}C_5} + 4 \times 3 \times {^{20}C_6} - 5 \times 4 \times {^{20}C_7} + \ldots + 18 \times 17 \times {^{20}C_{20}}$$, is equal to
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