If $$\alpha$$ and $$\beta$$ be two roots of the equation $$x^2 - 64x + 256 = 0$$. Then the value of $$\left(\frac{\alpha^3}{\beta^5}\right)^{1/8} + \left(\frac{\beta^3}{\alpha^5}\right)^{1/8}$$ is:
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If $$\alpha$$ and $$\beta$$ be two roots of the equation $$x^2 - 64x + 256 = 0$$. Then the value of $$\left(\frac{\alpha^3}{\beta^5}\right)^{1/8} + \left(\frac{\beta^3}{\alpha^5}\right)^{1/8}$$ is:
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The region represented by $$\{z = x + iy \in \mathbb{C} : |z| - \text{Re}(z) \leq 1\}$$ is also given by the inequality:
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Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?
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Let $$a, b, c, d$$ and $$p$$ be non-zero distinct real numbers such that $$(a^2 + b^2 + c^2)p^2 - 2(ab + bc + cd)p + (b^2 + c^2 + d^2) = 0$$. Then:
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If $$\{p\}$$ denotes the fractional part of the number $$p$$, then $$\left\{\frac{3^{200}}{8}\right\}$$ is equal to:
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A ray of light coming from the point $$(2, 2\sqrt{3})$$ is incident at an angle $$30^\circ$$ on the line $$x = 1$$ at the point A. The ray gets reflected on the line $$x = 1$$ and meets $$x$$-axis at the point B. Then, the line AB passes through the point:
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Let $$L_1$$ be a tangent to the parabola $$y^2 = 4(x+1)$$ and $$L_2$$ be a tangent to the parabola $$y^2 = 8(x+2)$$ such that $$L_1$$ and $$L_2$$ intersect at right angles. Then $$L_1$$ and $$L_2$$ meet on the straight line:
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Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $$\frac{x^2}{4} + \frac{y^2}{2} = 1$$ from any of its foci?
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The negation of the Boolean expression $$p \vee (\sim p \wedge q)$$ is equivalent to:
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If $$\sum_{i=1}^{n}(x_i - a) = n$$ and $$\sum_{i=1}^{n}(x_i - a)^2 = na$$, $$(n, a \gt 1)$$, then the standard deviation of $$n$$ observations $$x_1, x_2, \ldots, x_n$$ is:
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Let m and M be respectively the minimum and maximum values of $$\begin{vmatrix} \cos^2 x & 1 + \sin^2 x & \sin 2x \\ 1 + \cos^2 x & \sin^2 x & \sin 2x \\ \cos^2 x & \sin^2 x & 1 + \sin 2x \end{vmatrix}$$. Then the ordered pair (m, M) is equal to:
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The values of $$\lambda$$ and $$\mu$$ for which the system of linear equations $$x + y + z = 2$$, $$x + 2y + 3z = 5$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, are respectively:
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If $$f(x+y) = f(x)f(y)$$ and $$\sum_{x=1}^{\infty} f(x) = 2$$, $$x, y \in N$$, where $$N$$ is the set of all natural numbers, then the value of $$\frac{f(4)}{f(2)}$$ is:
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The position of a moving car at time $$t$$ is given by $$f(t) = at^2 + bt + c$$, $$t > 0$$, where $$a$$, $$b$$ and $$c$$ are real numbers greater than 1. Then the average speed of the car over the time interval $$[t_1, t_2]$$ is attained at the point:
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If $$I_1 = \int_0^1 (1-x^{50})^{100}\,dx$$ and $$I_2 = \int_0^1 (1-x^{50})^{101}\,dx$$ such that $$I_2 = \alpha I_1$$ then $$\alpha$$ equals to:
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$$\lim_{x \to 1}\left(\frac{\int_0^{(x-1)^2} t\cos t^2\,dt}{(x-1)\sin(x-1)}\right)$$
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The area (in sq. units) of the region $$A = \{(x, y) : |x| + |y| \leq 1,\; 2y^2 \geq |x|\}$$
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The general solution of the differential equation $$\sqrt{1 + x^2 + y^2 + x^2y^2} + xy\frac{dy}{dx} = 0$$ (where C is a constant of integration)
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The shortest distance between the lines $$\frac{x-1}{0} = \frac{y+1}{-1} = \frac{z}{1}$$ and $$x + y + z + 1 = 0$$, $$2x - y + z + 3 = 0$$ is:
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Out of 11 consecutive natural numbers if three numbers are selected at random (without repetition), then the probability that they are in A.P. with positive common difference is:
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The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $$45^\circ$$. After walking a distance of $$80$$ meters towards the top, up a slope inclined at angle of $$30^\circ$$ to the horizontal plane the angle of elevation of the top of the hill becomes $$75^\circ$$. Then the height of the hill (in meters) is_____.
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Set $$A$$ has $$m$$ elements and set $$B$$ has $$n$$ elements. If the total number of subsets of $$A$$ is 112 more than the total number of subsets of $$B$$, then the value of $$m \cdot n$$ is___.
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Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} x^5\sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5\cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$. The value of $$\lambda$$ for which $$f''(0)$$ exists, is___.
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Let $$AD$$ and $$BC$$ be two vertical poles at $$A$$ and $$B$$ respectively on a horizontal ground. If $$AD = 8\,\text{m}$$, $$BC = 11\,\text{m}$$, $$AB = 10\,\text{m}$$; then the distance (in meters) of a point M lying in between AB from the point A such that $$MD^2 + MC^2$$ is minimum, is___.
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If $$\vec{a}$$ and $$\vec{b}$$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|$$ is___.
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