If, for a positive integer $$n$$, the quadratic equation,
$$x(x+1) + (x+1)(x+2) + \ldots + (x+\overline{n-1})(x+n) = 10n$$
has two consecutive integral solutions, then $$n$$ is equal to:
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If, for a positive integer $$n$$, the quadratic equation,
$$x(x+1) + (x+1)(x+2) + \ldots + (x+\overline{n-1})(x+n) = 10n$$
has two consecutive integral solutions, then $$n$$ is equal to:
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Let $$\omega$$ be a complex number such that $$2\omega + 1 = z$$ where $$z = \sqrt{-3}$$. If
$$\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7} \end{vmatrix} = 3k$$,
then $$k$$ can be equal to:
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A man $$X$$ has 7 friends, 4 of them are ladies and 3 are men. His wife $$Y$$ also has 7 friends, 3 of them are ladies and 4 are men. Assume $$X$$ and $$Y$$ have no common friends. Then the total number of ways in which $$X$$ and $$Y$$ together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of $$X$$ and $$Y$$ are in this party is:
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For any three positive real numbers $$a$$, $$b$$ and $$c$$. If $$9(25a^{2} + b^{2}) + 25(c^{2} - 3ac) = 15b(3a + c)$$. Then
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The value of $$(^{21}C_{1} - ^{10}C_{1}) + (^{21}C_{2} - ^{10}C_{2}) + \dots + (^{21}C_{10} - ^{10}C_{10})$$ is:
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If $$5\tan^{2}x - \cos^{2}x = 2\cos 2x + 9$$, then the value of $$\cos 4x$$ is:
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Let $$k$$ be an integer such that the triangle with vertices $$(k, -3k)$$, $$(5, k)$$ and $$(-k, 2)$$ has area 28 sq. units. Then the orthocenter of this triangle is at the point:
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The radius of a circle, having minimum area, which touches the curve $$y = 4 - x^{2}$$ and the lines, $$y = |x|$$ is:
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The eccentricity of an ellipse whose centre is at the origin is $$\frac{1}{2}$$. If one of its directrices is $$x = -4$$, then the equation of the normal to it at $$\left(1, \frac{3}{2}\right)$$ is:
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A hyperbola passes through the point $$P(\sqrt{2}, \sqrt{3})$$ and has foci at $$( \pm 2, 0)$$. Then the tangent to this hyperbola at $$P$$ also passes through the point
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$$\lim_{x \to \frac{\pi}{2}} \frac{\cot x - \cos x}{(\pi - 2x)^{3}}$$ equals
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The statement $$p \to q \to (\sim p \to q \to q)$$ is
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A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:
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Let a vertical tower $$AB$$ have its end $$A$$ on the level ground. Let $$C$$ be the mid-point of $$AB$$ and $$P$$ be a point on the ground such that $$AP = 2AB$$. If $$\angle BPC = \beta$$, then $$\tan\beta$$ is equal to:
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If $$A = \begin{pmatrix} 2 & -3 \\ -4 & 1 \end{pmatrix}$$, then Adj$$(3A^{2} + 12A)$$ is equal to:
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If $$S$$ is the set of distinct values of $$b$$ for which the following system of linear equations
$$x + y + z = 1$$
$$x + ay + z = 1$$
$$ax + by + z = 0$$
has no solution, then $$S$$ is:
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The function $$f : R \to \left(-\frac{1}{2}, \frac{1}{2}\right)$$ defined as $$f(x) = \frac{x}{1+x^{2}}$$, is:
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Let $$a, b, c \in R$$. If $$f(x) = ax^{2} + bx + c$$ is such that $$a + b + c = 3$$ and $$f(x + y) = f(x) + f(y) + xy$$, $$\forall$$ $$x, y \in R$$, then $$\sum_{n=1}^{10} f(n)$$ is equal to:
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If for $$x \in \left(0, \frac{1}{4}\right)$$, the derivative of $$\tan^{-1}\left(\frac{6x\sqrt{x}}{1-9x^{3}}\right)$$ is $$\sqrt{x} \cdot g(x)$$, then $$g(x)$$ equals:
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Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:
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The normal to the curve $$y(x-2)(x-3) = x + 6$$ at the point where the curve intersects the $$y$$-axis passes through the point:
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Let $$I_{n} = \int \tan^{n}x \, dx$$ ($$n > 1$$). If $$I_{4} + I_{6} = a\tan^{5}x + bx^{5} + c$$, then the ordered pair $$(a, b)$$, is equal to
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The integral $$\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}} \frac{dx}{1 + \cos x}$$ is equal to
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The area (in sq. units) of the region $$\{(x, y) : x \geq 0, x + y \leq 3, x^{2} \leq 4y \text{ and } y \leq 1 + \sqrt{x}\}$$ is
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If $$(2 + \sin x)\frac{dy}{dx} + (y+1)\cos x = 0$$ and $$y(0) = 1$$, then $$y\left(\frac{\pi}{2}\right)$$ is equal to
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Given, $$\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$$ and $$\vec{b} = \hat{i} + \hat{j}$$. Let $$\vec{c}$$ be a vector such that $$|\vec{c} - \vec{a}| = 3$$, $$|\vec{a} \times \vec{b} \times \vec{c}| = 3$$ and the angle between $$\vec{c}$$ and $$\vec{a} \times \vec{b}$$ be 30°. Then $$\vec{a} \cdot \vec{c}$$ is equal to:
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If the image of the point $$P(1, -2, 3)$$ in the plane, $$2x + 3y - 4z + 22 = 0$$ measured parallel to the line, $$\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$$ is $$Q$$, then $$PQ$$ is equal to:
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The distance of the point $$(1, 3, -7)$$ from the plane passing through the point $$(1, -1, -1)$$, having normal perpendicular to both the lines $$\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-4}{3}$$ and $$\frac{x-2}{2} = \frac{y+1}{-1} = \frac{z+7}{-1}$$, is:
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For three events, $$A$$, $$B$$ and $$C$$, $$P$$(Exactly one of $$A$$ or $$B$$ occurs) = $$P$$(Exactly one of $$B$$ or $$C$$ occurs) = $$P$$(Exactly one of $$C$$ or $$A$$ occurs) = $$\frac{1}{4}$$ and $$P$$(All the three events occur simultaneously) = $$\frac{1}{16}$$. Then the probability that at least one of the events occurs, is:
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If two different numbers are taken from the set $$\{0, 1, 2, 3, \ldots, 10\}$$; then the probability that their sum as well as absolute difference are both multiple of 4, is:
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