If $$\alpha$$ and $$\beta$$ are roots of the equation, $$x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$$ for some $$k$$, and $$\alpha^2 + \beta^2 = 66$$, then $$\alpha^3 + \beta^3$$ is equal to:
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If $$\alpha$$ and $$\beta$$ are roots of the equation, $$x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$$ for some $$k$$, and $$\alpha^2 + \beta^2 = 66$$, then $$\alpha^3 + \beta^3$$ is equal to:
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If $$z_1, z_2$$ and $$z_3, z_4$$ are 2 pairs of complex conjugate numbers, then $$\arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right)$$ equals:
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An eight digit number divisible by 9 is to be formed using digits from 0 to 9 without repeating the digits. The number of ways in which this can be done is:
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In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is:
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The sum of the first 20 terms common between the series 3 + 7 + 11 + 15 + ... and 1 + 6 + 11 + 16 + ... is:
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The coefficient of $$x^{50}$$ in the binomial expansion of $$(1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \ldots + x^{1000}$$ is:
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If $$2\cos\theta + \sin\theta = 1$$ ($$\theta \neq \frac{\pi}{2}$$), then $$7\cos\theta + 6\sin\theta$$ is equal to:
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The base of an equilateral triangle is along the line given by $$3x + 4y = 9$$. If a vertex of the triangle is $$(1, 2)$$, then the length of a side of the triangle is:
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The set of all real values of $$\lambda$$ for which exactly two common tangents can be drawn to the circles $$x^2 + y^2 - 4x - 4y + 6 = 0$$ and $$x^2 + y^2 - 10x - 10y + \lambda = 0$$ is the interval:
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Let L$$_1$$ be the length of the common chord of the curves $$x^2 + y^2 = 9$$ and $$y^2 = 8x$$, and L$$_2$$ be the length of the latus rectum of $$y^2 = 8x$$, then:
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A stair-case of length $$l$$ rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the staircase begins to slide on the floor, then the locus of P is:
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Let P($$3\sec\theta, 2\tan\theta$$) and Q($$3\sec\phi, 2\tan\phi$$) where $$\theta + \phi = \frac{\pi}{2}$$, be two distinct points on the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$. Then the ordinate of the point of intersection of the normals at P and Q is:
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If $$\lim_{x \to 2} \frac{\tan(x - 2)\{x^2 + (k+2)x - 2k\}}{x^2 - 4x + 4} = 5$$, then k is equal to:
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The proposition $$\sim (p \vee \sim q) \vee \sim (p \vee q)$$ is logically equivalent to:
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Two ships A and B are sailing straight away from a fixed point O along routes such that $$\angle AOB$$ is always 120°. At a certain instance, OA = 8 km, OB = 6 km and the ship A is sailing at the rate of 20 km/hr while the ship B sailing at the rate of 30 km/hr. Then the distance between A and B is changing at the rate (in km/hr):
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The angle of elevation of the top of a vertical tower from a point P on the horizontal ground was observed to be $$\alpha$$. After moving a distance 2 metres from P towards the foot of the tower, the angle of elevation changes to $$\beta$$. Then the height (in metres) of the tower is:
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Let A(2, 3, 5), B($$-1, 3, 2$$) and C($$\lambda, 5, \mu$$) be the vertices of a $$\triangle$$ABC. If the median through A is equally inclined to the coordinate axes, then:
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Let A be a $$3 \times 3$$ matrix such that
$$A\begin{bmatrix} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}$$
Then A$$^{-1}$$ is:
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Let for i = 1, 2, 3, $$p_i(x)$$ be a polynomial of degree 2 in $$x$$, $$p'_i(x)$$ and $$p''_i(x)$$ be the first and second order derivatives of $$p_i(x)$$ respectively. Let,
$$A(x) = \begin{bmatrix} p_1(x) & p'_1(x) & p''_1(x) \\ p_2(x) & p'_2(x) & p''_2(x) \\ p_3(x) & p'_3(x) & p''_3(x) \end{bmatrix}$$
and $$B(x) = [A(x)]^T A(x)$$. Then determinant of B(x):
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Let f be an odd function defined on the set of real numbers such that for $$x \geq 0$$, $$f(x) = 3\sin x + 4\cos x$$. Then $$f(x)$$ at $$x = -\frac{11\pi}{6}$$ is equal to:
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Let $$f(x) = x|x|$$, $$g(x) = \sin x$$ and $$h(x) = (g \circ f)(x)$$. Then:
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For the curve $$y = 3\sin\theta\cos\theta$$, $$x = e^\theta\sin\theta$$, $$0 \leq \theta \leq \pi$$, the tangent is parallel to x-axis when $$\theta$$ is:
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The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius = $$\sqrt{3}$$ is:
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The integral $$\int x \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) dx$$ ($$x > 0$$) is equal to:
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If for $$n \geq 1$$, $$P_n = \int_1^e (\log x^n) dx$$, then $$P_{10} - 90P_8$$ is equal to:
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If the general solution of the differential equation $$y' = \frac{y}{x} + \Phi\left(\frac{x}{y}\right)$$, for some function $$\Phi$$, is given by $$y\ln|cx| = x$$, where c is an arbitrary constant, then $$\Phi(2)$$ is equal to:
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If $$|\vec{c}|^2 = 60$$ and $$\vec{c} \times (\hat{i} + 2\hat{j} + 5\hat{k}) = \vec{0}$$, then a value of $$\vec{c} \cdot (-7\hat{i} + 2\hat{j} + 3\hat{k})$$ is:
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The plane containing the line $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$$ and parallel to the line $$\frac{x}{1} = \frac{y}{1} = \frac{z}{4}$$ passes through the point:
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A set S contains 7 elements. A non-empty subset A of S and an element x of S are chosen at random. Then the probability that $$x \in A$$ is:
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If X has a binomial distribution, B(n, p) with parameters n and p such that P(X = 2) = P(X = 3), then E(X), the mean of variable X, is:
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