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Let $$f(x)$$ be a quadratic polynomial such that $$f(-1) + f(2) = 0$$. If one of the roots of $$f(x) = 0$$ is 3, then its other root lies in:
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The imaginary part of $$(3 + 2\sqrt{-54})^{\frac{1}{2}} - (3 - 2\sqrt{-54})^{\frac{1}{2}}$$ can be:
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Let $$n > 2$$ be an integer. Suppose that there are $$n$$ Metro stations in a city located around a circular path. Each pair of the nearest stations is connected by a straight track only. Further, each pair of the nearest station is connected by blue line, whereas all remaining pairs of stations are connected by red line. If number of red lines is 99 times the number of blue lines, then the value of $$n$$ is:
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If the sum of first 11 terms of an A.P. $$a_1, a_2, a_3, \ldots$$ is $$0$$ $$(a_1 \ne 0)$$ then the sum of the A.P. $$a_1, a_3, a_5, \ldots, a_{23}$$ is $$ka_1$$ where $$k$$ is equal to:
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Let $$S$$ be the sum of the first 9 terms of the series: $$\{x + ka\} + \{x^2 + (k+2)a\} + \{x^3 + (k+4)a\} + \{x^4 + (k+6)a\} + \ldots$$ where $$a \ne 0$$ and $$x \ne 1$$. If $$S = \frac{x^{10} - x + 45a(x-1)}{x-1}$$, then $$k$$ is equal to:
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If the equation $$\cos^4\theta + \sin^4\theta + \lambda = 0$$ has real solutions for $$\theta$$ then $$\lambda$$ lies in interval:
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The set of all possible values of $$\theta$$ in the interval $$(0, \pi)$$ for which the points $$(1, 2)$$ and $$(\sin\theta, \cos\theta)$$ lie on the same side of the line $$x + y = 1$$ is:
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The area (in sq. units) of an equilateral triangle inscribed in the parabola $$y^2 = 8x$$, with one of its vertices on the vertex of this parabola is:
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For some $$\theta \in \left(0, \frac{\pi}{2}\right)$$, if the eccentricity of the hyperbola, $$x^2 - y^2\sec^2\theta = 10$$ is $$\sqrt{5}$$ times the eccentricity of the ellipse, $$x^2\sec^2\theta + y^2 = 5$$, then the length of the latus rectum of the ellipse, is:
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$$\lim_{x \to 0} \left(\tan\left(\frac{\pi}{4} + x\right)\right)^{1/x}$$ is equal to:
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Which of the following is a tautology?
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Let $$A = \left\{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\right\}$$ where $$P = \begin{bmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{bmatrix}$$ then the set $$A$$:
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Let $$a, b, c \in R$$ be all non-zero and satisfies $$a^3 + b^3 + c^3 = 2$$. If the matrix $$A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$$ satisfies $$A^TA = I$$, then a value of $$abc$$ can be:
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Let $$f : R \to R$$ be a function which satisfies $$f(x + y) = f(x) + f(y)$$, $$\forall x, y \in R$$. If $$f(1) = 2$$ and $$g(n) = \sum_{k=1}^{(n-1)} f(k)$$, $$n \in N$$ then the value of $$n$$, for which $$g(n) = 20$$, is:
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The equation of the normal to the curve $$y = (1+x)^{2y} + \cos^2(\sin^{-1}x)$$, at $$x = 0$$ is:
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Let $$f : (-1, \infty) \to R$$ be defined by $$f(0) = 1$$ and $$f(x) = \frac{1}{x}\log_e(1 + x)$$, $$x \ne 0$$. Then the function $$f$$:
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Consider a region $$R = \{(x, y) \in R^2 : x^2 \le y \le 2x\}$$. If a line $$y = \alpha$$ divides the area of region $$R$$ into two equal parts, then which of the following is true?
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If a curve $$y = f(x)$$, passing through the point $$(1, 2)$$, is the solution of the differential equation $$2x^2dy = (2xy + y^2)dx$$, then $$f\left(\frac{1}{2}\right)$$ is equal to:
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A plane passing through the point $$(3, 1, 1)$$ contains two lines whose direction ratios are 1, -2, 2 and 2, 3, -1 respectively. If this plane also passes through the point $$(\alpha, -3, 5)$$, then $$\alpha$$ is equal to:
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Let $$E^C$$ denote the complement of an event $$E$$. Let $$E_1$$, $$E_2$$ and $$E_3$$ be any pairwise independent events with $$P(E_1) > 0$$ and $$P(E_1 \cap E_2 \cap E_3) = 0$$ then $$P\left((E_2^C \cap E_3^C)/E_1\right)$$ is equal to:
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For a positive integer $$n$$, $$\left(1 + \frac{1}{x}\right)^n$$ is expanded in increasing powers of $$x$$. If three consecutive coefficients in this expansion are in the ratio, 2 : 5 : 12, then $$n$$ is equal to ___________.
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If the variance of the terms in an increasing A.P. $$b_1, b_2, b_3, \ldots, b_{11}$$ is 90 then the common difference of this A.P. is ___________.
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If $$y = \sum_{k=1}^{6} k\cos^{-1}\left\{\frac{3}{5}\cos kx - \frac{4}{5}\sin kx\right\}$$ then $$\frac{dy}{dx}$$ at $$x = 0$$ is ___________.
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of $$\int_1^2 |2x - [3x]| \; dx$$ is ___________.
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Let the position vectors of points 'A' and 'B' be $$\hat{i} + \hat{j} + \hat{k}$$ and $$2\hat{i} + \hat{j} + 3\hat{k}$$, respectively. A point 'P' divides the line segment AB internally in the ratio $$\lambda : 1$$ $$(\lambda > 0)$$. If O is the origin and $$\vec{OB} \cdot \vec{OP} - 3|\vec{OA} \times \vec{OP}|^2 = 6$$ then $$\lambda$$ is equal to ___________.
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