Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + 2 = 0$$. Then $$\alpha^{14} + \beta^{14}$$ is equal to
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Let $$\alpha$$, $$\beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + 2 = 0$$. Then $$\alpha^{14} + \beta^{14}$$ is equal to
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Let $$S = \{z \in \mathbb{C} : \bar{z} = i(z^2 + \text{Re}(\bar{z}))\}$$. Then $$\sum_{z \in S} |z|^2$$ is equal to
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All words, with or without meaning, are made using all the letters of the word $$MONDAY$$. These words are written as in a dictionary with serial numbers. The serial number of the word $$MONDAY$$ is
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Let $$a_1, a_2, a_3, \ldots$$ be a G.P. of increasing positive numbers. Let the sum of its 6$$^{th}$$ and 8$$^{th}$$ terms be 2 and the product of its 3$$^{rd}$$ and 5$$^{th}$$ terms be $$\frac{1}{9}$$. Then $$6a_2 + a_4a_4 + a_6$$ is equal to
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The coefficient of $$x^5$$ in the expansion of $$\left(2x^3 - \frac{1}{3x^2}\right)^5$$ is
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Let $$(\alpha, \beta)$$ be the centroid of the triangle formed by the lines $$15x - y = 82$$, $$6x - 5y = -4$$ and $$9x + 4y = 17$$. Then $$\alpha + 2\beta$$ and $$2\alpha - \beta$$ are the roots of the equation
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Let the centre of a circle $$C$$ be $$\alpha, \beta$$ and its radius $$r < 8$$. Let $$3x + 4y = 24$$ and $$3x - 4y = 32$$ be two tangents and $$4x + 3y = 1$$ be a normal to $$C$$. Then $$(\alpha - \beta + r)$$ is equal to
If $$\lim_{x \to 0} \frac{e^{ax} - \cos(bx) - \frac{cxe^{-cx}}{2}}{1 - \cos(2x)} = 17$$, then $$5a^2 + b^2$$ is equal to
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The statement $$(p \wedge (\sim q)) \vee ((\sim p) \wedge q) \vee ((\sim p) \wedge (\sim q))$$ is equivalent to ____
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Let for $$A = \begin{bmatrix} 1 & 2 & 3 \\ \alpha & 3 & 1 \\ 1 & 1 & 2 \end{bmatrix}$$, $$|A| = 2$$. If $$|2 \ \text{adj}(2 \ \text{adj}(2A))| = 32^n$$, then $$3n + \alpha$$ is equal to
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If the system of equations
$$2x + y - z = 5$$
$$2x - 5y + \lambda z = \mu$$
$$x + 2y - 5z = 7$$
has infinitely many solutions, then $$(\lambda + \mu)^2 + (\lambda - \mu)^2$$ is equal to
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The range of $$f(x) = 4\sin^{-1}\left(\frac{x^2}{x^2+1}\right)$$ is
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The value of $$\frac{e^{-\frac{\pi}{4}} + \int_0^{\frac{\pi}{4}} e^{-x}\tan^{50}x \ dx}{\int_0^{\frac{\pi}{4}} e^{-x}(\tan^{49}x + \tan^{51}x) \ dx}$$
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The area of the region $$\{x, y : x^2 \leq y \leq x^2 - 4, y \geq 1\}$$ is
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Let $$|\vec{a}| = 2$$, $$|\vec{b}| = 3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})|^2$$ is equal to
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Let for a triangle $$ABC$$
$$\vec{AB} = -2\hat{i} + \hat{j} + 3\hat{k}$$
$$\vec{CB} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}$$
$$\vec{CA} = 4\hat{i} + 3\hat{j} + \delta\hat{k}$$
If $$\delta > 0$$ and the area of the triangle $$ABC$$ is $$5\sqrt{6}$$ then $$\vec{CB} \cdot \vec{CA}$$ is equal to
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The plane, passing through the points $$(0, -1, 2)$$ and $$(-1, 2, 1)$$ and parallel to the line passing through $$(5, 1, -7)$$ and $$(1, -1, -1)$$, also passes through the point
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The line, that is coplanar to the line $$\frac{x+3}{-3} = \frac{y-1}{1} = \frac{z-5}{5}$$, is
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Let N be the foot of perpendicular from the point $$P(1, -2, 3)$$ on the line passing through the points $$(4, 5, 8)$$ and $$(1, -7, 5)$$. Then the distance of N from the plane $$2x - 2y + z + 5 = 0$$ is
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The random variable $$X$$ follows binomial distribution $$B(n, p)$$, for which the difference of the mean and the variance is 1.
If $$2P(X = 2) = 3P(X = 1)$$, then $$n^2P(X > 1)$$ is equal to
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