If $$z = \frac{1}{2} - 2i$$, is such that $$|z + 1| = \alpha z + \beta(1 + i)$$, $$i = \sqrt{-1}$$ and $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to
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If $$z = \frac{1}{2} - 2i$$, is such that $$|z + 1| = \alpha z + \beta(1 + i)$$, $$i = \sqrt{-1}$$ and $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to
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In an A.P., the sixth term $$a_6 = 2$$. If the $$a_1 a_4 a_5$$ is the greatest, then the common difference of the A.P., is equal to
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If in a G.P. of $$64$$ terms, the sum of all the terms is $$7$$ times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to
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If $$\alpha$$, $$-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$$ is the solution of $$4\cos\theta + 5\sin\theta = 1$$, then the value of $$\tan\alpha$$ is
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Let $$(5, \frac{a}{4})$$, be the circumcenter of a triangle with vertices $$A(a, -2)$$, $$B(a, 6)$$ and $$C(\frac{a}{4}, -2)$$. Let $$\alpha$$ denote the circumradius, $$\beta$$ denote the area and $$\gamma$$ denote the perimeter of the triangle. Then $$\alpha + \beta + \gamma$$ is
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In a $$\Delta ABC$$, suppose $$y = x$$ is the equation of the bisector of the angle $$B$$ and the equation of the side $$AC$$ is $$2x - y = 2$$. If $$2AB = BC$$ and the point $$A$$ and $$B$$ are respectively $$(4, 6)$$ and $$(\alpha, \beta)$$, then $$\alpha + 2\beta$$ is equal to
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$$\lim_{x \rightarrow \frac{\pi}{2}} \left(\frac{1}{(x - \frac{\pi}{2})^2} \int_{x^3}^{(\frac{\pi}{2})^3} \cos\left(\frac{1}{t^3}\right) dt\right)$$ is equal to
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Let $$R$$ be a relation on $$Z \times Z$$ defined by $$(a, b)R(c, d)$$ if and only if $$ad - bc$$ is divisible by $$5$$. Then $$R$$ is
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Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{bmatrix}$$ and $$|2A|^3 = 2^{21}$$ where $$\alpha, \beta \in Z$$, Then a value of $$\alpha$$ is
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Let A be a square matrix such that $$AA^T = I$$. Then $$\frac{1}{2}A\left[(A + A^T)^2 + (A - A^T)^2\right]$$ is equal to
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If $$f(x) = \begin{cases} 2 + 2x, & -1 \leq x < 0 \\ 1 - \frac{x}{3}, & 0 \leq x \leq 3 \end{cases}$$; $$g(x) = \begin{cases} -x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1 \end{cases}$$, then range of $$(f \circ g(x))$$ is
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Consider the function $$f : [\frac{1}{2}, 1] \to R$$ defined by $$f(x) = 4\sqrt{2}x^3 - 3\sqrt{2}x - 1$$. Consider the statements
(I) The curve $$y = f(x)$$ intersects the $$x$$-axis exactly at one point
(II) The curve $$y = f(x)$$ intersects the $$x$$-axis at $$x = \cos\frac{\pi}{12}$$
Then
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Suppose $$f(x) = \frac{(2^x + 2^{-x})\tan x \sqrt{\tan^{-1}(x^2 - x + 1)}}{(7x^2 + 3x + 1)^3}$$. Then the value of $$f'(0)$$ is equal to
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If the value of the integral $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{x^2 \cos x}{1 + \pi^x} + \frac{1 + \sin^2 x}{1 + e^{(\sin x)^{2023}}}\right) dx = \frac{\pi}{4}(\pi + a) - 2$$, then the value of $$a$$ is
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For $$x \in (-\frac{\pi}{2}, \frac{\pi}{2})$$, if $$y(x) = \int \frac{\csc x + \sin x}{\csc x \sec x + \tan x \sin^2 x} dx$$ and $$\lim_{x \to (\frac{\pi}{2})^-} y(x) = 0$$ then $$y(\frac{\pi}{4})$$ is equal to
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A function $$y = f(x)$$ satisfies $$f(x)\sin 2x + \sin x - (1 + \cos^2 x)f'(x) = 0$$ with condition $$f(0) = 0$$. Then $$f(\frac{\pi}{2})$$ is equal to
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that $$\vec{b}$$ and $$\vec{c}$$ are non-collinear. If $$\vec{a} + 5\vec{b}$$ is collinear with $$\vec{c}$$, $$\vec{b} + 6\vec{c}$$ is collinear with $$\vec{a}$$ and $$\vec{a} + \alpha\vec{b} + \beta\vec{c} = \vec{0}$$, then $$\alpha + \beta$$ is equal to
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Let $$O$$ be the origin and the position vector of $$A$$ and $$B$$ be $$2\hat{i} + 2\hat{j} + \hat{k}$$ and $$2\hat{i} + 4\hat{j} + 4\hat{k}$$ respectively. If the internal bisector of $$\angle AOB$$ meets the line $$AB$$ at $$C$$, then the length of $$OC$$ is
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Let $$PQR$$ be a triangle with $$R(-1, 4, 2)$$. Suppose $$M(2, 1, 2)$$ is the mid point of $$PQ$$. The distance of the centroid of $$\Delta PQR$$ from the point of intersection of the line $$\frac{x-2}{0} = \frac{y}{2} = \frac{z+3}{-1}$$ and $$\frac{x-1}{1} = \frac{y+3}{-3} = \frac{z+1}{1}$$ is
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A fair die is thrown until $$2$$ appears. Then the probability, that $$2$$ appears in even number of throws, is
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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - x + 2 = 0$$ with $$\text{Im}(\alpha) > \text{Im}(\beta)$$. Then $$\alpha^6 + \alpha^4 + \beta^4 - 5\alpha^2$$ is equal to _______
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All the letters of the word $$GTWENTY$$ are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word $$GTWENTY$$ is _______
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If $$\frac{^{11}C_1}{2} + \frac{^{11}C_2}{3} + \ldots + \frac{^{11}C_9}{10} = \frac{n}{m}$$ with $$\gcd(n, m) = 1$$, then $$n + m$$ is equal to _______
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Equations of two diameters of a circle are $$2x - 3y = 5$$ and $$3x - 4y = 7$$. The line joining the points $$(-\frac{22}{7}, -4)$$ and $$(-\frac{1}{7}, 3)$$ intersects the circle at only one point $$P(\alpha, \beta)$$. Then $$17\beta - \alpha$$ is equal to _______
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If the points of intersection of two distinct conics $$x^2 + y^2 = 4b$$ and $$\frac{x^2}{16} + \frac{y^2}{b^2} = 1$$ lie on the curve $$y^2 = 3x^2$$, then $$3\sqrt{3}$$ times the area of the rectangle formed by the intersection points is _______.
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If the mean and variance of the data $$65, 68, 58, 44, 48, 45, 60, \alpha, \beta, 60$$ where $$\alpha > \beta$$ are $$56$$ and $$66.2$$ respectively, then $$\alpha^2 + \beta^2$$ is equal to _______
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Let $$f(x) = 2^x - x^2$$, $$x \in R$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y = f(x)$$ and $$y = f'(x)$$ intersects the $$x$$-axis, then the value of $$m + n$$ is _______
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The area (in sq. units) of the part of circle $$x^2 + y^2 = 169$$ which is below the line $$5x - y = 13$$ is $$\frac{\pi\alpha}{2\beta} - \frac{65}{2} + \frac{\alpha}{\beta}\sin^{-1}(\frac{12}{13})$$ where $$\alpha, \beta$$ are coprime numbers. Then $$\alpha + \beta$$ is equal to _______
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If the solution curve $$y = y(x)$$ of the differential equation $$(1 + y^2)(1 + \log_e x)dx + xdy = 0$$, $$x > 0$$ passes through the point $$(1, 1)$$ and $$y(e) = \frac{\alpha - \tan(\frac{3}{2})}{\beta + \tan(\frac{3}{2})}$$, then $$\alpha + 2\beta$$ is _______
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A line with direction ratio $$2, 1, 2$$ meets the lines $$x = y + 2 = z$$ and $$x + 2 = 2y = 2z$$ respectively at the point $$P$$ and $$Q$$. If the length of the perpendicular from the point $$(1, 2, 12)$$ to the line $$PQ$$ is $$l$$, then $$l^2$$ is _______
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