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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Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits 1, 2, 3, 4, 5 with repetition, is _____.
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Let $$\alpha$$ denote the greatest integer $$\leq \alpha$$. Then $$\sqrt{1} + \sqrt{2} + \sqrt{3} + \ldots + \sqrt{120}$$ is equal to _____.
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Let $$f(x) = \sum_{k=1}^{10} k \cdot x^k$$, $$x \in \mathbb{R}$$, if $$2f(2) + f'(2) = 119 \cdot 2^n + 1$$ then $$n$$ is equal to _____.
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The remainder, when $$7^{103}$$ is divided by $$17$$, is _____.
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The foci of a hyperbola are $$(\pm 2, 0)$$ and its eccentricity is $$\frac{3}{2}$$. A tangent, perpendicular to the line $$2x + 3y = 6$$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the x- and y-axes are $$a$$ and $$b$$ respectively, then $$|6a| + |5b|$$ is equal to _____.
The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is _____.
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Let $$A = \{-4, -3, -2, 0, 1, 3, 4\}$$ and $$R = \{(a, b) \in A \times A : b = |a|$$ or $$b^2 = a + 1\}$$ be a relation on $$A$$. Then the minimum number of elements, that must be added to the relation $$R$$ so that it becomes reflexive and symmetric, is _____.
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For $$x \in (-1, 1]$$, the number of solutions of the equation $$\sin^{-1}x = 2\tan^{-1}x$$ is equal to _____.
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Let $$f_n = \int_0^{\pi/2} \sum_{k=1}^{n} \sin^{k-1}x \sum_{k=1}^{n} (2k-1)\sin^{k-1}x \cos x \ dx$$, $$n \in \mathbb{N}$$. Then $$f_{21} - f_{20}$$ is equal to _____.
If $$y = y(x)$$ is the solution of the differential equation $$\frac{dy}{dx} + \frac{4x}{x^2-1}y = \frac{x+2}{(x^2-1)^{5/2}}$$, $$x \gt 1$$ such that $$y(2) = \frac{2}{9}\log_e 2 + \sqrt{3}$$ and $$y\sqrt{2} = \alpha\log_e(\sqrt{\alpha} + \beta) + \beta - \sqrt{\gamma}$$, $$\alpha, \beta, \gamma \in \mathbb{N}$$, then $$\alpha\beta\gamma$$ is equal to _____.
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