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Let $$\alpha = \max_{x \in R}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$$ and $$\beta = \min_{x \in R}\{8^{2\sin 3x} \cdot 4^{4\cos 3x}\}$$. If $$8x^2 + bx + c = 0$$ is a quadratic equation whose roots are $$\alpha^{1/5}$$ and $$\beta^{1/5}$$, then the value of $$c - b$$ is equal to:
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Let $$\mathbb{C}$$ be the set of all complex numbers. Let $$S_1 = \{z \in \mathbb{C} : |z - 2| \leq 1\}$$ and $$S_2 = \{z \in \mathbb{C} : z(1 + i) + \bar{z}(1 - i) \geq 4\}$$. Then, the maximum value of $$\left|z - \frac{5}{2}\right|^2$$ for $$z \in S_1 \cap S_2$$ is equal to:
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If $$\tan\left(\frac{\pi}{9}\right), x, \tan\left(\frac{7\pi}{18}\right)$$ are in arithmetic progression and $$\tan\left(\frac{\pi}{9}\right), y, \tan\left(\frac{5\pi}{18}\right)$$ are also in arithmetic progression, then $$|x - 2y|$$ is equal to:
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A possible value of $$x$$, for which the ninth term in the expansion of $$\left\{3^{\log_3 \sqrt{25^{x-1}+7}} + 3^{\left(-\frac{1}{5}\right)\log_3(5^{x-1}+1)}\right\}^{10}$$ in the increasing powers of $$3^{\left(-\frac{1}{5}\right)\log_3(5^{x-1}+1)}$$ is equal to 180, is:
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The point $$P(a, b)$$ undergoes the following three transformations successively:
(a) reflection about the line $$y = x$$.
(b) translation through 2 units along the positive direction of $$x$$-axis.
(c) rotation through angle $$\frac{\pi}{4}$$ about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point $$P$$ are $$\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$$, then the value of $$2a + b$$ is equal to:
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Two sides of a parallelogram are along the lines $$4x + 5y = 0$$ and $$7x + 2y = 0$$. If the equation of one of the diagonals of the parallelogram is $$11x + 7y = 9$$, then other diagonal passes through the point:
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Consider a circle $$C$$ which touches the $$y$$-axis at $$(0, 6)$$ and cuts off an intercept $$6\sqrt{5}$$ on the $$x$$-axis. Then the radius of the circle $$C$$ is equal to:
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The value of $$\lim_{x \to 0}\left(\frac{x}{\sqrt[8]{1 - \sin x} - \sqrt[8]{1 + \sin x}}\right)$$ is equal to:
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Which of the following is the negation of the statement "for all $$M \gt 0$$, there exists $$x \in S$$ such that $$x \geq M$$"?
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Let the mean and variance of the frequency distribution
$$x$$: $$x_1 = 2$$ $$x_2 = 6$$ $$x_3 = 8$$ $$x_4 = 9$$
$$f$$: 4 4 $$\alpha$$ $$\beta$$
be 6 and 6.8 respectively. If $$x_3$$ is changed from 8 to 7, then the mean for the new data will be:
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Let $$N$$ be the set of natural numbers and a relation $$R$$ on $$N$$ be defined by $$R = \{(x, y) \in N \times N : x^3 - 3x^2y - xy^2 + 3y^3 = 0\}$$. Then the relation $$R$$ is:
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Let $$A$$ and $$B$$ be two $$3 \times 3$$ real matrices such that $$(A^2 - B^2)$$ is invertible matrix. If $$A^5 = B^5$$ and $$A^3B^2 = A^2B^3$$, then the value of the determinant of the matrix $$A^3 + B^3$$ is equal to:
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Let $$f : R \rightarrow R$$ be defined as $$f(x + y) + f(x - y) = 2f(x)f(y)$$, $$f\left(\frac{1}{2}\right) = -1$$. Then the value of $$\sum_{k=1}^{20} \frac{1}{\sin(k)\sin(k + f(k))}$$ is equal to:
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Let $$f : [0, \infty) \rightarrow [0, 3]$$ be a function defined by $$f(x) = \begin{cases} \max\{\sin t : 0 \leq t \leq \pi\}, & x \in [0, \pi] \\ 2 + \cos x, & x > \pi \end{cases}$$. Then which of the following is true?
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Let $$f : (a, b) \rightarrow R$$ be twice differentiable function such that $$f(x) = \int_a^x g(t) \, dt$$ for a differentiable function $$g(x)$$. If $$f(x) = 0$$ has exactly five distinct roots in $$(a, b)$$, then $$g(x)g'(x) = 0$$ has at least:
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The area of the region bounded by $$y - x = 2$$ and $$x^2 = y$$ is equal to:
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Let $$y = y(x)$$ be the solution of the differential equation $$(x - x^3)dy = (y + yx^2 - 3x^4)dx$$, $$x \gt 2$$. If $$y(3) = 3$$, then $$y(4)$$ is equal to:
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Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three vectors such that $$\vec{a} = \vec{b} \times (\vec{b} \times \vec{c})$$. If magnitudes of the vectors $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ are $$\sqrt{2}, 1$$ and 2 respectively and the angle between $$\vec{b}$$ and $$\vec{c}$$ is $$\theta$$ $$(0 < \theta < \frac{\pi}{2})$$, then the value of $$1 + \tan \theta$$ is equal to:
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For real numbers $$\alpha$$ and $$\beta \neq 0$$, if the point of intersection of the straight lines $$\frac{x - \alpha}{1} = \frac{y - 1}{2} = \frac{z - 1}{3}$$ and $$\frac{x - 4}{\beta} = \frac{y - 6}{3} = \frac{z - 7}{3}$$ lies on the plane $$x + 2y - z = 8$$, then $$\alpha - \beta$$ is equal to:
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A student appeared in an examination consisting of 8 true-false type questions. The student guesses the answers with equal probability. The smallest value of $$n$$, so that the probability of guessing at least $$n$$ correct answers is less than $$\frac{1}{2}$$, is:
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The number of real roots of the equation $$e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0$$ is equal to _________
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If the real part of the complex number $$z = \frac{3 + 2i\cos\theta}{1 - 3i\cos\theta}$$, $$\theta \in \left(0, \frac{\pi}{2}\right)$$ is zero, then the value of $$\sin^2 3\theta + \cos^2 \theta$$ is equal to _________
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Let $$n$$ be a non-negative integer. Then the number of divisors of the form $$4n + 1$$ of the number $$(10)^{10} \cdot (11)^{11} \cdot (13)^{13}$$ is equal to _________.
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Let $$E$$ be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at $$(3, -4)$$, one focus at $$(4, -4)$$ and one vertex at $$(5, -4)$$. If $$mx - y = 4$$, $$m \gt 0$$ is a tangent to the ellipse $$E$$, then the value of $$5m^2$$ is equal to _________.
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Let $$A = \{n \in N \mid n^2 \leq n + 10000\}$$, $$B = \{3k + 1 \mid k \in N\}$$ and $$C = \{2k \mid k \in N\}$$, then the sum of all the elements of the set $$A \cap (B - C)$$ is equal to _________.
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If $$A = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$$ and $$M = A + A^2 + A^3 + \ldots + A^{20}$$, then the sum of all the elements of the matrix $$M$$ is equal to _________.
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If $$\int_0^\pi (\sin^3 x) e^{-\sin^2 x} dx = \alpha - \frac{\beta}{e} \int_0^1 \sqrt{t} \, e^t dt$$, then $$\alpha + \beta$$ is equal to _________
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Let $$y = y(x)$$ be the solution of the differential equation $$dy = e^{\alpha x + y} dx$$; $$\alpha \in N$$. If $$y(\log_e 2) = \log_e 2$$ and $$y(0) = \log_e\left(\frac{1}{2}\right)$$, then the value of $$\alpha$$ is equal to _________.
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Let $$\vec{a} = \hat{i} - \alpha\hat{j} + \beta\hat{k}$$, $$\vec{b} = 3\hat{i} + \beta\hat{j} - \alpha\hat{k}$$ and $$\vec{c} = -\alpha\hat{i} - 2\hat{j} + \hat{k}$$, where $$\alpha$$ and $$\beta$$ are integers. If $$\vec{a} \cdot \vec{b} = -1$$ and $$\vec{b} \cdot \vec{c} = 10$$, then $$(\vec{a} \times \vec{b}) \cdot \vec{c}$$ is equal to _________.
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The distance of the point $$P(3, 4, 4)$$ from the point of intersection of the line joining the points $$Q(3, -4, -5)$$ and $$R(2, -3, 1)$$ and the plane $$2x + y + z = 7$$, is equal to _________.
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