If $$\alpha, \beta$$ are the roots of the equation, $$x^2 - x - 1 = 0$$ and $$S_n = 2023\alpha^n + 2024\beta^n$$, then
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If $$\alpha, \beta$$ are the roots of the equation, $$x^2 - x - 1 = 0$$ and $$S_n = 2023\alpha^n + 2024\beta^n$$, then
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Let $$\alpha = \dfrac{(4!)!}{(4!)^{3!}}$$ and $$\beta = \dfrac{(5!)!}{(5!)^{4!}}$$. Then :
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The 20th term from the end of the progression $$20, 19\frac{1}{4}, 18\frac{1}{2}, 17\frac{3}{4}, \ldots, -129\frac{1}{4}$$ is :
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If $$2\tan^2\theta - 5\sec\theta = 1$$ has exactly 7 solutions in the interval $$0, \frac{n\pi}{2}$$, for the least value of $$n \in \mathbb{N}$$ then $$\sum_{k=1}^{n} \frac{k}{2^k}$$ is equal to :
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Let $$A$$ and $$B$$ be two finite sets with $$m$$ and $$n$$ elements respectively. The total number of subsets of the set $$A$$ is 56 more than the total number of subsets of $$B$$. Then the distance of the point $$P(m, n)$$ from the point $$Q(-2, -3)$$ is
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Let R be the interior region between the lines $$3x - y + 1 = 0$$ and $$x + 2y - 5 = 0$$ containing the origin. The set of all values of $$a$$, for which the points $$(a^2, a + 1)$$ lie in R, is :
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Let $$e_1$$ be the eccentricity of the hyperbola $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ and $$e_2$$ be the eccentricity of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a > b$$, which passes through the foci of the hyperbola. If $$e_1 e_2 = 1$$, then the length of the chord of the ellipse parallel to the x-axis and passing through $$(0, 2)$$ is :
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If $$\lim_{x \to 0} \frac{3 + \alpha \sin x + \beta \cos x + \log_e(1 - x)}{3\tan^2 x} = \frac{1}{3}$$, then $$2\alpha - \beta$$ is equal to :
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The values of $$\alpha$$, for which $$\begin{vmatrix} 1 & \frac{3}{2} & \alpha + \frac{3}{2} \\ 1 & \frac{1}{3} & \alpha + \frac{1}{3} \\ 2\alpha + 3 & 3\alpha + 1 & 0 \end{vmatrix} = 0$$, lie in the interval
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Considering only the principal values of inverse trigonometric functions, the number of positive real values of $$x$$ satisfying $$\tan^{-1}(x) + \tan^{-1}(2x) = \frac{\pi}{4}$$ is :
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Let $$f: \mathbb{R} - \frac{-1}{2}\to \mathbb{R}$$ and $$g: \mathbb{R} - \frac{-5}{2} \to \mathbb{R}$$ be defined as $$f(x) = \frac{2x + 3}{2x + 1}$$ and $$g(x) = \frac{|x| + 1}{2x + 5}$$. Then the domain of the function fog is:
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Consider the function $$f: (0, 2) \to \mathbb{R}$$ defined by $$f(x) = \frac{x}{2} + \frac{2}{x}$$ and the function $$g(x)$$ defined by $$g(x) = \begin{cases} \min\{f(t)\},\ 0 < t \leq x & \text{and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases}$$. Then
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Let $$g(x) = 3f\left(\frac{x}{3}\right) + f(3 - x)$$ and $$f''(x) > 0$$ for all $$x \in (0, 3)$$. If g is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8\alpha$$ is
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The integral $$\int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1)\tan^{-1}\left(x^3 + \frac{1}{x^3}\right)} dx$$ is equal to :
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For $$0 < a < 1$$, the value of the integral $$\int_0^{\pi} \frac{dx}{1 - 2a\cos x + a^2}$$ is :
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If $$y = y(x)$$ is the solution curve of the differential equation $$(x^2 - 4)dy - (y^2 - 3y)dx = 0$$, $$x > 2$$, $$y(4) = \frac{3}{2}$$ and the slope of the curve is never zero, then the value of $$y(10)$$ equals :
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The position vectors of the vertices A, B and C of a triangle are $$2\hat{i} - 3\hat{j} + 3\hat{k}$$, $$2\hat{i} + 2\hat{j} + 3\hat{k}$$ and $$-\hat{i} + \hat{j} + 3\hat{k}$$ respectively. Let $$l$$ denotes the length of the angle bisector AD of $$\angle BAC$$ where D is on the line segment BC, then $$2l^2$$ equals :
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Let the position vectors of the vertices A, B and C of a triangle be $$2\hat{i} + 2\hat{j} + \hat{k}$$, $$\hat{i} + 2\hat{j} + 2\hat{k}$$ and $$2\hat{i} + \hat{j} + 2\hat{k}$$ respectively. Let $$l_1, l_2$$ and $$l_3$$ be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides AB, BC and CA respectively, then $$l_1^2 + l_2^2 + l_3^2$$ equals :
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Let the image of the point $$(1, 0, 7)$$ in the line $$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$$ be the point $$(\alpha, \beta, \gamma)$$. Then which one of the following points lies on the line passing through $$(\alpha, \beta, \gamma)$$ and making angles $$\frac{2\pi}{3}$$ and $$\frac{3\pi}{4}$$ with y-axis and z-axis respectively and an acute angle with x-axis?
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An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :
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Let the complex numbers $$\alpha$$ and $$\frac{1}{\bar{\alpha}}$$ lie on the circles $$|z - z_0|^2 = 4$$ and $$|z - z_0|^2 = 16$$ respectively, where $$z_0 = 1 + i$$. Then, the value of $$100|\alpha|^2$$ is _____.
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The coefficient of $$x^{2012}$$ in the expansion of $$(1 - x)^{2008}(1 + x + x^2)^{2007}$$ is equal to _____.
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If the sum of squares of all real values of $$\alpha$$, for which the lines $$2x - y + 3 = 0$$, $$6x + 3y + 1 = 0$$ and $$\alpha x + 2y - 2 = 0$$ do not form a triangle is p, then the greatest integer less than or equal to p is _____.
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Consider a circle $$(x - \alpha)^2 + (y - \beta)^2 = 50$$, where $$\alpha, \beta > 0$$. If the circle touches the line $$y + x = 0$$ at the point P, whose distance from the origin is $$4\sqrt{2}$$, then $$(\alpha + \beta)^2$$ is equal to _____.
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The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12. If $$\mu$$ and $$\sigma^2$$ denote the mean and variance of the correct observations respectively, then $$15(\mu + \mu^2 + \sigma^2)$$ is equal to _____.
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Let A be a $$2 \times 2$$ real matrix and I be the identity matrix of order 2. If the roots of the equation $$|A - xI| = 0$$ be -1 and 3, then the sum of the diagonal elements of the matrix $$A^2$$ is _____.
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Let $$f(x) = \int_0^x g(t)\log_e\frac{1-t}{1+t}dt$$, where g is a continuous odd function. If $$\int_{-\pi/2}^{\pi/2} \left(f(x) + \frac{x^2\cos x}{1 + e^x}\right) dx = \left(\frac{\pi}{\alpha}\right)^2 - \alpha$$, then $$\alpha$$ is equal to _____.
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If the area of the region $$\{(x, y) : 0 \leq y \leq \min(2x, 6x - x^2)\}$$ is A, then 12A is equal to _____.
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If the solution curve, of the differential equation $$\frac{dy}{dx} = \frac{x + y - 2}{x - y}$$ passing through the point $$(2, 1)$$ is $$\tan^{-1}\frac{y-1}{x-1} - \frac{1}{\beta}\log_e\left(\alpha + \left(\frac{y-1}{x-1}\right)^2\right) = \log_e(x-1)$$, then $$5\beta + \alpha$$ is equal to
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The lines $$\frac{x-2}{2} = \frac{y}{-2} = \frac{z-7}{16}$$ and $$\frac{x+3}{4} = \frac{y+2}{3} = \frac{z+2}{1}$$ intersect at the point P. If the distance of P from the line $$\frac{x+1}{2} = \frac{y-1}{3} = \frac{z-1}{1}$$ is $$l$$, then $$14l^2$$ is equal to _____.
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