The sum of all the real values of $$x$$ satisfying the equation $$2^{(x-1)(x^2+5x-50)} = 1$$ is:
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The sum of all the real values of $$x$$ satisfying the equation $$2^{(x-1)(x^2+5x-50)} = 1$$ is:
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The equation $$Im\left(\frac{iz - 2}{z - i}\right) + 1 = 0$$, $$z \in \mathbb{C}$$, $$z \neq i$$ represents a part of a circle having radius equal to:
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The number of ways in which 5 boys and 3 girls can be seated on a round table if a particular boy $$B_1$$ and a particular girl $$G_1$$ never sit adjacent to each other, is:
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If three positive numbers $$a$$, $$b$$ and $$c$$ are in A.P. such that $$abc = 8$$, then the minimum possible value of $$b$$ is:
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Let $$S_n = \frac{1}{1^3} + \frac{1+2}{1^3+2^3} + \frac{1+2+3}{1^3+2^3+3^3} + \ldots + \frac{1+2+\ldots+n}{1^3+2^3+\ldots+n^3}$$. If 100 $$S_n = n$$, then $$n$$ is equal to:
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The coefficient of $$x^{-5}$$ in the binomial expansion of $$\left(\frac{x+1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} - \frac{x-1}{x - x^{\frac{1}{2}}}\right)^{10}$$ where $$x \neq 0, 1$$ is
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The lengths of two adjacent sides of a cyclic quadrilateral are 2 units and 5 units and the angle between them is 60°. If the area of the quadrilateral is $$4\sqrt{3}$$ sq. units, then the perimeter of the quadrilateral is
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A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is:
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A line drawn through the point $$P(4, 7)$$ cuts the circle $$x^2 + y^2 = 9$$ at the points $$A$$ and $$B$$. Then $$P_A \cdot P_B$$ is equal to.
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If $$y = mx + c$$ is the normal at a point on the parabola $$y^2 = 8x$$ whose focal distance is 8 units, then $$|c|$$ is equal to:
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The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $$(4, -1)$$ and $$(-2, 2)$$ is
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The contrapositive of the statement 'If two numbers are not equal, then their squares are not equal', is
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The sum of 100 observations and the sum of their squares are 400 & 2475, respectively. Later on, three observations 3, 4 & 5 were found to be incorrect. If the incorrect observations are omitted, then the variance of the remaining observations is
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For two $$3 \times 3$$ matrices $$A$$ and $$B$$, let $$A + B = 2B'$$ and $$3A + 2B = I_3$$, where $$B'$$ is the transpose of $$B$$ and $$I_3$$ is $$3 \times 3$$ identity matrix. Then:
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If $$x = a$$, $$y = b$$, $$z = c$$ is a solution of the system of linear equations
$$x + 8y + 7z = 0$$
$$9x + 2y + 3z = 0$$
$$x + y + z = 0$$
Such that the point $$(a, b, c)$$ lies on the plane $$x + 2y + z = 6$$, then $$2a + b + c$$ equals:
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A value of $$x$$ satisfying the equation $$\sin\left[\cot^{-1}(1 + x)\right] = \cos\left[\tan^{-1}x\right]$$, is:
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The function $$f : N \to I$$ defined by $$f(x) = x - 5\left[\frac{x}{5}\right]$$, where $$N$$ is the set of natural numbers and $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is:
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The value of $$k$$ which the function $$f(x) = \begin{cases} \left(\frac{4}{5}\right)^{\frac{\tan 4x}{\tan 5x}}, & 0 < x < \frac{\pi}{2} \\ k + \frac{2}{5}, & x = \frac{\pi}{2} \end{cases}$$ is continuous at $$x = \frac{\pi}{2}$$, is
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If $$2x = y^{\frac{1}{5}} + y^{-\frac{1}{5}}$$ and $$(x^2 - 1)\frac{d^2y}{dx^2} + \lambda x \frac{dy}{dx} + ky = 0$$, then $$\lambda + k$$ is equal to
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The function $$f$$ defined by $$f(x) = x^3 - 3x^2 + 5x + 7$$ is:
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If $$f\left(\frac{3x - 4}{3x + 4}\right) = x + 2$$, $$x \neq -\frac{4}{3}$$, and $$\int f(x)dx = A \log|1 - x| + Bx + C$$, then the ordered pair $$(A, B)$$ is equal to
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If $$\int_1^2 \frac{dx}{(x^2 - 2x + 4)^{\frac{3}{2}}} = \frac{k}{k+5}$$, then $$k$$ is equal to
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If $$\lim_{n \to \infty} \left(\frac{1^a + 2^a + \ldots + n^a}{(n+1)^{a-1}[(na+1) + (na+2) + \ldots + (na+n)]}\right) = \frac{1}{60}$$ for some positive real number $$a$$, then $$a$$ is equal to
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Let $$f$$ be a polynomial function such that $$f(3x) = f'(x) \cdot f''(x)$$, for all $$x \in R$$. Then:
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A tangent to the curve, $$y = f(x)$$ at $$P(x, y)$$ meets x-axis at $$A$$ and y-axis at $$B$$. If $$AP : BP = 1 : 3$$ and $$f(1) = 1$$, then the curve also passes through the point
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If the vector $$\vec{b} = 3\hat{j} + 4\hat{k}$$ is written as the sum of a vector $$\vec{b_1}$$, parallel to $$\vec{a} = \hat{i} + \hat{j}$$ and a vector $$\vec{b_2}$$, perpendicular to $$\vec{a}$$, then $$\vec{b_1} \times \vec{b_2}$$ is equal to:
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If the line, $$\frac{x - 3}{1} = \frac{y + 2}{-1} = \frac{z + \lambda}{-2}$$ lies in the plane, $$2x - 4y + 3z = 2$$, then the shortest distance between this line and the line, $$\frac{x - 1}{12} = \frac{y}{9} = \frac{z}{4}$$ is
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If a variable plane, at a distance of 3 units from the origin, intersects the coordinate axes at $$A$$, $$B$$ & $$C$$, then the locus of the centroid of $$\triangle ABC$$ is
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From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is:
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Let $$E$$ & $$F$$ be two independent events. The probability that $$E$$ & $$F$$ happen is $$\frac{1}{12}$$ and the probability that neither $$E$$ nor $$F$$ happens is $$\frac{1}{2}$$, then a value of $$\frac{P(E)}{P(F)}$$ is:
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