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Let $$\alpha$$ and $$\beta$$ be the roots of the quadratic equation $$x^2 \sin\theta - x(\sin\theta \cos\theta + 1) + \cos\theta = 0$$ $$(0 < \theta < 45°)$$, and $$\alpha < \beta$$. Then $$\sum_{n=0}^{\infty}\left(\alpha^n + \frac{(-1)^n}{\beta^n}\right)$$ is equal to:
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Let z be a complex number such that $$|z| + z = 3 + i$$ (where $$i = \sqrt{-1}$$). Then $$|z|$$ is equal to:
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If 19th term of a non-zero A.P. is zero, then its (49th term) : (29th term) is:
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Let $$S_n = 1 + q + q^2 + \ldots + q^n$$ and $$T_n = 1 + \left(\frac{q+1}{2}\right) + \left(\frac{q+1}{2}\right)^2 + \ldots + \left(\frac{q+1}{2}\right)^n$$ where q is a real number and $$q \neq 1$$. If $${}^{101}C_1 + {}^{101}C_2 \cdot S_1 + \ldots + {}^{101}C_{101} \cdot S_{100} = \alpha T_{100}$$, then $$\alpha$$ is equal to:
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Let $$(x + 10)^{50} + (x - 10)^{50} = a_0 + a_1 x + a_2 x^2 + \ldots + a_{50} x^{50}$$, for all $$x \in R$$; then $$\frac{a_2}{a_0}$$ is equal to:
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If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1,2), (3,4) and (2,5), then the equation of the diagonal AD is:
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A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. Then the locus of the centre of this circle, is:
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If the area of the triangle whose one vertex is at the vertex of the parabola, $$y^2 + 4(x - a^2) = 0$$ and the other two vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is:
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Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?
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If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:
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$$\lim_{x \to 0} \frac{x \cot(4x)}{\sin^2 x \cot^2(2x)}$$ is equal to:
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Contrapositive of the statement "If two numbers are not equal, then their squares are not equal" is:
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Given $$\frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13}$$ for a $$\Delta ABC$$ with usual notation. If $$\frac{\cos A}{a} = \frac{\cos B}{\beta} = \frac{\cos C}{\gamma}$$, then the ordered triad $$(\alpha, \beta, \gamma)$$ has a value:
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If $$\begin{vmatrix} a-b-c & 2a & 2a \\ 2b & b-c-a & 2b \\ 2c & 2c & c-a-b \end{vmatrix} = (a + b + c)(x + a + b + c)^2$$, $$x \neq 0$$ and $$a + b + c \neq 0$$, then x is equal to
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Let A and B be two invertible matrices of order $$3 \times 3$$. If $$\det(ABA^T) = 8$$ and $$\det(AB^{-1}) = 8$$, then $$\det(BA^{-1}B^T)$$ is equal to
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All $$x$$ satisfying the inequality $$(\cot^{-1} x)^2 - 7(\cot^{-1} x) + 10 > 0$$, lie in the interval:
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Let a function $$f : (0, \infty) \to (0, \infty)$$ be defined by $$f(x) = \left|1 - \frac{1}{x}\right|$$. Then f is:
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The number of functions f from $$\{1, 2, 3, \ldots, 20\}$$ onto $$\{1, 2, 3, \ldots, 20\}$$ such that $$f(k)$$ is a multiple of 3, whenever k is a multiple of 4 is:
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Let K be the set of all real values of x where the function $$f(x) = \sin|x| - |x| + 2(x - \pi)\cos|x|$$ is not differentiable. Then the set K is equal to:
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Let $$f(x) = \frac{x}{\sqrt{a^2 + x^2}} - \frac{d - x}{\sqrt{b^2 + (d-x)^2}}$$, $$x \in \mathbb{R}$$ where a, b and d are non-zero real constants. Then:
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Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression $$\frac{x^m y^n}{(1+x^{2m})(1+y^{2n})}$$ is:
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If $$\int \frac{x+1}{\sqrt{2x-1}} dx = f(x)\sqrt{2x-1} + C$$, where C is a constant of integration, then $$f(x)$$ is equal to:
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The integral $$\int_{\pi/6}^{\pi/4} \frac{dx}{\sin 2x(\tan^5 x + \cot^5 x)}$$ equals:
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The area (in sq. units) in the first quadrant bounded by the parabola, $$y = x^2 + 1$$, the tangent to it at the point (2, 5) and the coordinate axes is:
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The solution of the differential equation, $$\frac{dy}{dx} = (x - y)^2$$, when $$y(1) = 1$$, is:
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Let $$\sqrt{3}\hat{i} + \hat{j}$$, $$\hat{i} + \sqrt{3}\hat{j}$$ and $$\beta\hat{i} + (1 - \beta)\hat{j}$$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is $$\frac{\sqrt{3}}{\sqrt{2}}$$, then the sum of all possible values of $$\beta$$ is:
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Two lines $$\frac{x-3}{1} = \frac{y+1}{3} = \frac{z-6}{-1}$$ and $$\frac{x+5}{7} = \frac{y-2}{-6} = \frac{z-3}{4}$$ intersect at the point R. The reflection of R in the xy-plane has coordinates:
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If the point $$(2, \alpha, \beta)$$ lies on the plane which passes through the points (3,4,2) and (7,0,6) and is perpendicular to the plane $$2x - 5y = 15$$, then $$2\alpha - 3\beta$$ is equal to:
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Let $$S = \{1, 2, \ldots, 20\}$$. A subset B of S is said to be "nice" if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is:
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A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then $$\left(\frac{\text{mean of X}}{\text{standard deviation of X}}\right)$$ is equal to:
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