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If $$\alpha$$ and $$\beta$$ are the roots of the equation $$375x^2 - 25x - 2 = 0$$, then $$\lim_{n \to \infty} \sum_{r=1}^{n} \alpha^r + \lim_{n \to \infty} \sum_{r=1}^{n} \beta^r$$ is equal to:
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The equation $$|z - i| = |z - 1|$$, $$i = \sqrt{-1}$$, represents:
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The Number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is:
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If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is:
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Let S$$_n$$ denote the sum of the first n terms of an A.P. If S$$_4$$ = 16 and S$$_6$$ = -48, then S$$_{10}$$ is equal to:
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The coefficient of $$x^{18}$$ in the product $$(1 + x)(1 - x)^{10}(1 + x + x^2)^9$$ is
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The equation $$y=\sin x\sin\left(x+2\right)-\sin^2(x+1)$$ represents a straight line lying in:
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The number of solutions of the equation $$1 + \sin^4 x = \cos^2 3x$$, $$x \in \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right]$$ is:
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If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is 90°, then the length (in cm) of their common chord is:
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If the normal to the ellipse $$3x^2 + 4y^2 = 12$$ at a point P on it is parallel to the line, $$2x + y = 4$$ and the tangent to the ellipse at P passes through Q(4, 4) then PQ is equal to:
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Let P be the point of intersection of the common tangents to the parabola $$y^2 = 12x$$ and the hyperbola $$8x^2 - y^2 = 8$$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio:
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If the truth value of the statement $$p \to (\sim q \vee r)$$ is false F, then the truth values of the statements p, q, r are respectively
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If the data $$x_1, x_2, \ldots, x_{10}$$ is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2000, then the standard deviation of this data is:
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If $$B = \begin{pmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{pmatrix}$$ is the inverse of a 3$$\times$$3 matrix A, then the sum of all values of $$\alpha$$ for which det(A) + 1 = 0, is:
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If A is a symmetric matrix and B is skew-symmetric matrix such that $$A + B = \begin{pmatrix} 2 & 3 \\ 5 & -1 \end{pmatrix}$$, then AB is equal to:
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The value of $$\sin^{-1}\frac{12}{13} - \sin^{-1}\frac{3}{5}$$ is equal to:
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For $$x \in R$$, Let [x] denotes the greatest integer $$\leq x$$, then the sum of the series $$\left[-\frac{1}{3}\right] + \left[-\frac{1}{3} - \frac{1}{100}\right] + \left[-\frac{1}{3} - \frac{2}{100}\right] + \ldots + \left[-\frac{1}{3} - \frac{99}{100}\right]$$ is
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For $$x \in \left(0, \frac{3}{2}\right)$$, let $$f(x) = \sqrt{x}$$, $$g(x) = \tan x$$ and $$h(x) = \frac{1 - x^2}{1 + x^2}$$. If $$\phi(x) = ((h \circ f) \circ g)(x)$$, then $$\phi\left(\frac{\pi}{3}\right)$$ is equal to:
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If $$e^y + xy = e$$, the ordered pair $$\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)$$ at x = 0 is equal to
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A 2m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is:
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If m is the minimum value of k for which the function $$f(x) = x\sqrt{kx - x^2}$$ is increasing in the interval [0, 3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to:
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The integral $$\int \frac{2x^3 - 1}{x^4 + x} dx$$, is equal to
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Let $$f: R \to R$$ be a continuous and differentiable function such that $$f(2) = 6$$ and $$f'(2) = \frac{1}{48}$$. If $$\int_6^{f(x)}4t^3dt=\left(x-2\right)g(x)$$, then $$\lim_{x \to 2} g(x)$$ is equal to
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If $$\int_0^{\pi/2} \frac{\cot x}{\cot x + \text{cosec} x} dx = m(\pi + n)$$, then mn is equal to
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If the area (in sq. units) of the region $$\{(x, y): y^2 \leq 4x, x + y \leq 1, x \geq 0, y \geq 0\}$$ is $$a\sqrt{2} + b$$, then a - b is
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Consider the differential equation, $$y^2 dx + \left(x - \frac{1}{y}\right) dy = 0$$. If value of y is 1 when x = 1, then the value of x for which y = 2, is
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If the volume of parallelepiped formed by the vectors $$\hat{i} + \lambda\hat{j} + \hat{k}$$, $$\hat{j} + \lambda\hat{k}$$ and $$\lambda\hat{i} + \hat{k}$$ is minimum, then $$\lambda$$ is
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Let $$\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}$$ and $$\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$$ be two vectors. If a vector perpendicular to both the vectors $$\vec{a} + \vec{b}$$ and $$\vec{a} - \vec{b}$$ has the magnitude 12 then one such vector is:
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If the line $$\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-1}{-1}$$ intersects the plane $$2x + 3y - z + 13 = 0$$ at a point P and the plane $$3x + y + 4z = 16$$ at a point Q, then PQ is equal to
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Let a random variable X has a binomial distribution with mean 8 and variance 4. If $$P(X \leq 2) = \frac{k}{2^{16}}$$, then the value of k is equal to
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