The sum of all real values of $$x$$ satisfying the equation $$\left(x^2 - 5x + 5\right)^{x^2 + 4x - 60} = 1$$ is
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The sum of all real values of $$x$$ satisfying the equation $$\left(x^2 - 5x + 5\right)^{x^2 + 4x - 60} = 1$$ is
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A value of $$\theta$$ for which $$\frac{2 + 3i\sin\theta}{1 - 2i\sin\theta}$$ is purely imaginary, is
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If all the words (with or without meaning) having five letters, formed using the letters of the word $$SMALL$$ and arranged as in a dictionary; then the position of the word $$SMALL$$ is
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If the $$2^{nd}$$, $$5^{th}$$ and $$9^{th}$$ terms of a non-constant arithmetic progression are in geometric progression, then the common ratio of this geometric progression is
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If the sum of the first ten terms of the series $$\left(1\frac{3}{5}\right)^2 + \left(2\frac{2}{5}\right)^2 + \left(3\frac{1}{5}\right)^2 + 4^2 + \left(4\frac{4}{5}\right)^2 + \ldots$$, is $$\frac{16}{5}m$$, then $$m$$ is equal to
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If the number of terms in the expansion of $$\left(1 - \frac{2}{x} + \frac{4}{y^2}\right)^n$$, $$x, y \neq 0$$, is 28, then the sum of the coefficients of all the terms in this expansion is
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If $$0 \leq x < 2\pi$$, then the number of real values of $$x$$, which satisfy the equation $$\cos x + \cos 2x + \cos 3x + \cos 4x = 0$$, is
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Two sides of a rhombus are along the lines, $$x - y + 1 = 0$$ and $$7x - y - 5 = 0$$. If its diagonals intersect at $$(-1, -2)$$, then which one of the following is a vertex of this rhombus?
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The centres of those circles which touch the circle, $$x^2 + y^2 - 8x - 8y - 4 = 0$$, externally and also touch the x-axis, lie on
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If one of the diameters of the circle, given by the equation, $$x^2 + y^2 - 4x + 6y - 12 = 0$$, is a chord of a circle $$S$$, whose centre is at $$(-3, 2)$$, then the radius of $$S$$ is
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Let $$P$$ be the point on the parabola, $$y^2 = 8x$$ which is at a minimum distance from the center $$C$$ of the circle, $$x^2 + (y+6)^2 = 1$$. Then the equation of the circle, passing through $$C$$ and having its center at $$P$$ is
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The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is
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$$\lim_{n \to \infty} \left(\frac{(n+1)(n+2)\ldots 3n}{n^{2n}}\right)^{1/n}$$ is equal to
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Let $$P = \lim_{x \to 0^+} \left(1 + \tan^2\sqrt{x}\right)^{1/2x}$$, then $$\log P$$ is equal to
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The Boolean Expression $$(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$$ is equivalent to
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If the standard deviation of the numbers 2, 3, $$a$$ and 11 is 3.5, then which of the following is true?
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A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point $$A$$ on the path, he observes that the angle of elevation of the top of the pillar is $$30°$$. After walking for 10 minutes from $$A$$ in the same direction, at a point $$B$$, he observes that the angle of elevation of the top of the pillar is $$60°$$. Then the time taken (in minutes) by him, from $$B$$ to reach the pillar, is
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If $$A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$$ and $$A \cdot adj A = A A^T$$, then $$5a + b$$ is equal to
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The system of linear equations
$$x + \lambda y - z = 0$$
$$\lambda x - y - z = 0$$
$$x + y - \lambda z = 0$$
has a non-trivial solution for
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If $$f(x) + 2f\left(\frac{1}{x}\right) = 3x$$, $$x \neq 0$$, and $$S = \{x \in R : f(x) = f(-x)\}$$, then $$S$$
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For $$x \in R$$, $$f(x) = |\log 2 - \sin x|$$ and $$g(x) = f(f(x))$$, then
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Consider $$f(x) = \tan^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right)$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. A normal to $$y = f(x)$$ at $$x = \frac{\pi}{6}$$ also passes through the point
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A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = $$x$$ units and a circle of radius = $$r$$ units. If the sum of the areas of the square and the circle so formed is minimum, then
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The integral $$\int \frac{2x^{12} + 5x^9}{(x^5 + x^3 + 1)^3} dx$$, is equal to
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The area (in sq. units) of the region $$\{(x, y) : y^2 \geq 2x$$ and $$x^2 + y^2 \leq 4x$$, $$x \geq 0$$, $$y \geq 0\}$$ is
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If a curve $$y = f(x)$$ passes through the point $$(1, -1)$$ and satisfies the differential equation, $$y(1 + xy)dx = x\,dy$$, then $$f\left(-\frac{1}{2}\right)$$ is equal to
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Let $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be three unit vectors such that $$\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b} + \vec{c})$$. If $$\vec{b}$$ is not parallel to $$\vec{c}$$, then the angle between $$\vec{a}$$ and $$\vec{b}$$ is
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If the line, $$\frac{x-3}{2} = \frac{y+2}{-1} = \frac{z+4}{3}$$ lies in the plane, $$lx + my - z = 9$$, then $$l^2 + m^2$$ is equal to
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The distance of the point $$(1, -5, 9)$$ from the plane $$x - y + z = 5$$ measured along the line $$x = y = z$$ is
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Let two fair six-faced dice $$A$$ and $$B$$ be thrown simultaneously. If $$E_1$$ is the event that die $$A$$ shows up four, $$E_2$$ is the event that die $$B$$ shows up two and $$E_3$$ is the event that the sum of numbers on both dice is odd, then which of the following statements is not true?
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