Let $$a, b \in \mathbb{C}$$. Let $$\alpha, \beta$$ be the roots of $$x^2 + ax + b = 0$$. If $$\beta - \alpha = \sqrt{11}$$ and $$\beta^2 - \alpha^2 = 3i\sqrt{11}$$, then $$(\beta^3 - \alpha^3)^2$$ is equal to :
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Let $$a, b \in \mathbb{C}$$. Let $$\alpha, \beta$$ be the roots of $$x^2 + ax + b = 0$$. If $$\beta - \alpha = \sqrt{11}$$ and $$\beta^2 - \alpha^2 = 3i\sqrt{11}$$, then $$(\beta^3 - \alpha^3)^2$$ is equal to :
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Let the sum of first $$n$$ terms of an A.P. is $$3n^2 + 5n$$. The sum of squares of the first 10 terms is :
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Let $$A$$ is a $$3 \times 3$$ matrix such that $$A^T \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}$$, $$A^T \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$$, $$A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 4 \end{bmatrix}$$, $$A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}$$. If $$\det(A) = 1$$, then $$\det(\text{adj}(A^2 + A))$$ is equal to :
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Consider the system of equations in $$x, y, z$$:
$$x + 2y + tz = 0$$,
$$6x + y + 5tz = 0$$,
$$3x + t^2 y + f(t)z = 0$$,
where $$f: \mathbb{R} \to \mathbb{R}$$ is differentiable function. If this system has infinitely many solutions for all $$t \in \mathbb{R}$$, then $$f$$ is :
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$$\displaystyle\sum_{n=1}^{10} \left(\frac{528}{n(n+1)(n+2)}\right)$$ is equal to :
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Let $$\tan A$$ and $$\tan B$$, where $$A, B \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, be the roots of the quadratic equation $$x^2 - 2x - 5 = 0$$. Then $$20\sin^2\left(\frac{A+B}{2}\right)$$ is equal to :
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A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is :
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The mean deviation about the mean for the data
is equal to :
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Let a focus of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be $$S(4, 0)$$ and its eccentricity be $$\frac{4}{5}$$. If $$P(3, \alpha)$$ lies on $$E$$ and $$O$$ is the origin, then the area of $$\triangle POS$$ is equal to:
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Let $$P$$ moving point on the circle $$x^2 + y^2 - 6x - 8y + 21 = 0$$. Then,the maximum distance of $$P$$ from the vertex of the parabola $$x^2 + 6x + y + 13 = 0$$ is :
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In an equilateral triangle $$PQR$$,let the vertex $$P = (3, 5)$$ and the side $$QR$$ be along the line $$x + y = 4$$. If the orthocentre of the triangle $$PQR$$ is $$(\alpha, \beta)$$, then $$9(\alpha + \beta)$$ is equal to :
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The sum of all integral values of $$p$$ such that the equation $$3\sin^2 x + 12\cos x - 3 = p, x \in R,$$ has at least one solution is :
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The square of the distance of the point $$P(5, 6, 7)$$ from the line $$\frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4}$$ is equal to:
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Let $$\vec{a} = \sqrt{7}\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{j} + 2\hat{k}$$. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0}$$ and $$\vec{r} \cdot \vec{a} = 0$$, then $$|3\vec{r}|^2$$ is equal to :
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The square of the distance of the point of intersection of the lines $$\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})$$, $$a \neq 0$$ and $$\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})$$ from the origin is :
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The area of the region $$R = \{(x, y) : xy \leq 27, \; 1 \leq y \leq x^2\}$$ is equal to:
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The product of all values of $$\alpha$$, for which $$\displaystyle\lim_{x \to 0} \left(\frac{1 - \cos(\alpha x) \cos((\alpha+1)x) \cos((\alpha+2)x)}{\sin^2((\alpha+1)x)}\right) = 2$$ is :
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The value of the integral $$\displaystyle\int_0^{\infty} \frac{\log_e (x)}{x^2 + 4} \, dx$$ is:
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Let $$f: \mathbb{R} \to \mathbb{R}$$ be a differentiable function such that $$f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3}$$ for all $$x, y \in R$$ and $$f'(0) = 3$$. Then the minimum value of function $$g(x) = 3 + e^x f(x)$$ is :
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The value of the integral $$\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left(\frac{4 - \operatorname{cosec}^2 x}{\cos^4 x} \right) dx$$ is equal to :
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Let $$A = \{1, 2, 3, 4, 5, 6\}$$. The number of one-one functions $$f: A \to A$$ such that $$f(1) \geq 3$$, $$f(3) \leq 4$$, and $$f(2) + f(3) = 5$$ is :
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Two players $$A$$ and $$B$$ play a series of badminton games. The first player, who wins 5 games first, wins the series. Assuming that no game ends in a draw, the number of ways in which player $$A$$ wins the series is :_____.
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If the sum of the coefficients of $$x^7$$ and $$x^{14}$$ in the expansion of $$\left(\frac{1}{x^3} - x^4\right)^n$$, $$x \neq 0$$, is zero, then the value of n is _______ :
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If $$\frac{\pi}{4} + \displaystyle\sum_{p=1}^{11} \tan^{-1}\left(\frac{2^{p-1}}{1 + 2^{2p-1}}\right) = \alpha$$, then $$\tan \alpha$$ is equal to :
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Let $$y = y(x)$$ be the solution of the differential equation $$x\sin\left(\frac{y}{x}\right)dy = \left(y\sin\left(\frac{y}{x}\right) - x\right)dx$$, $$y(1) = \frac{\pi}{2}$$ and let $$\alpha = \cos\left(\frac{y(e^{12})}{e^{12}}\right)$$. The number of integral values of $$p$$ for which the equation $$x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$$ represents a circle of radius $$r \leq 6$$ is :
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