Let $$\alpha$$ and $$\beta$$ be the roots of equation $$x^2 - 6x - 2 = 0$$. If $$a_n = \alpha^n - \beta^n$$, $$\forall n \geq 1$$, then the value of $$\frac{a_{10} - 2a_8}{2a_9}$$ is equal to
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Let $$\alpha$$ and $$\beta$$ be the roots of equation $$x^2 - 6x - 2 = 0$$. If $$a_n = \alpha^n - \beta^n$$, $$\forall n \geq 1$$, then the value of $$\frac{a_{10} - 2a_8}{2a_9}$$ is equal to
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A complex number $$z$$ is said to be unimodular if $$|z| = 1$$. Let $$z_1$$ and $$z_2$$ are complex numbers such that $$\frac{z_1 - 2z_2}{2 - z_1\bar{z_2}}$$ is unimodular and $$z_2$$ is not unimodular, then the point $$z_1$$ lies on a
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The number of integers greater than 6000 that can be formed, using the digits 3, 5, 6, 7 and 8, without repetition is
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The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0) is
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Let $$A$$ and $$B$$ be two sets containing four and two elements respectively. Then the number of subsets of the set $$A \times B$$, each having at least three elements is
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The sum of first 9 terms of the series $$\frac{1^3}{1} + \frac{1^3 + 2^3}{1+3} + \frac{1^3 + 2^3 + 3^3}{1+3+5} + \ldots$$ is
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If $$m$$ is the A.M. of two distinct real numbers $$l$$ and $$n$$ $$(l, n > 1)$$ and $$G_1$$, $$G_2$$ and $$G_3$$ are three geometric means between $$l$$ and $$n$$, then $$G_1^4 + 2G_2^4 + G_3^4$$ equals
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The sum of coefficients of integral powers of $$x$$ in the binomial expansion of $$(1 - 2\sqrt{x})^{50}$$ is
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Locus of the image of the point (2, 3) in the line $$(2x - 3y + 4) + k(x - 2y + 3) = 0$$, k $$\in \mathbb{R}$$, is a
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The number of common tangents to the circles $$x^2 + y^2 - 4x - 6y - 12 = 0$$ and $$x^2 + y^2 + 6x + 18y + 26 = 0$$, is
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Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $$x^2 = 8y$$. If the point $$P$$ divides the line segment $$OQ$$ internally in the ratio 1 : 3, then the locus of $$P$$ is
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The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $$\frac{x^2}{9} + \frac{y^2}{5} = 1$$, is
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$$\lim_{x \to 0} \frac{(1 - \cos 2x)(3 + \cos x)}{x \tan 4x} =$$
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The negation of $$\sim s \vee (\sim r \wedge s)$$ is equivalent to
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The mean of a data set comprising of 16 observations is 16. If one of the observation value 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data is
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If the angles of elevation of the top of a tower from three collinear points $$A$$, $$B$$ and $$C$$ on a line leading to the foot of the tower are $$30^\circ$$, $$45^\circ$$ and $$60^\circ$$ respectively, then the ratio $$AB : BC$$, is
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If $$A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$$ is a matrix satisfying the equation $$AA^T = 9I$$, where $$I$$ is $$3 \times 3$$ identity matrix, then the ordered pair $$(a, b)$$ is equal to
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The set of all values of $$\lambda$$ for which the system of linear equations:
$$2x_1 - 2x_2 + x_3 = \lambda x_1$$
$$2x_1 - 3x_2 + 2x_3 = \lambda x_2$$
$$-x_1 + 2x_2 = \lambda x_3$$
has a non-trivial solution,
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Let $$\tan^{-1} y = \tan^{-1} x + \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$$, where $$|x| \lt \frac{1}{\sqrt{3}}$$. Then a value of $$y$$ is
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If the function $$g(x) = \begin{cases} k\sqrt{x+1}, & 0 \leq x \leq 3 \\ mx + 2, & 3 < x \leq 5 \end{cases}$$ is differentiable, then the value of $$k + m$$ is
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The normal to the curve $$x^2 + 2xy - 3y^2 = 0$$, at (1, 1)
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Let $$f(x)$$ be a polynomial of degree four and having its extreme values at $$x = 1$$ and $$x = 2$$. If $$\lim_{x \to 0}\left[1 + \frac{f(x)}{x^2}\right] = 3$$, then $$f(2)$$ is equal to
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The integral $$\int \frac{dx}{x^2(x^4+1)^{3/4}}$$ equals to
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The integral $$\int_2^4 \frac{\log x^2}{\log x^2 + \log(6-x)^2} dx$$ is equal to
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The area (in sq. units) of the region described by $$\{(x,y) : y^2 \leq 2x$$ and $$y \geq 4x - 1\}$$ is
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Let $$y(x)$$ be the solution of the differential equation $$(x \log x)\frac{dy}{dx} + y = 2x \log x$$, $$(x \geq 1)$$. Then $$y(e)$$ is equal to
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Let $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be three non-zero vectors such that no two of them are collinear and $$(\vec{a} \times \vec{b}) \times \vec{c} = \frac{1}{3}|\vec{b}||\vec{c}|\vec{a}$$. If $$\theta$$ is the angle between vectors $$\vec{b}$$ and $$\vec{c}$$, then a value of $$\sin \theta$$ is
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The distance of the point (1, 0, 2) from the point of intersection of the line $$\frac{x-2}{3} = \frac{y+1}{4} = \frac{z-2}{12}$$ and the plane $$x - y + z = 16$$, is
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The equation of the plane containing the line of intersection of $$2x - 5y + z = 3$$; $$x + y + 4z = 5$$, and parallel to the plane, $$x + 3y + 6z = 1$$, is
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If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is
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