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Let $$S_1 = \{z \in C : |z| \leq 5\}$$, $$S_2 = \left\{z \in C : \text{Im}\left(\frac{z + 1 - \sqrt{3}i}{1 - \sqrt{3}i}\right) \geq 0\right\}$$ and $$S_3 = \{z \in C : \text{Re}(z) \geq 0\}$$. Then the area of the region $$S_1 \cap S_2 \cap S_3$$ is :
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60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the $$50^{th}$$ word is :
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For $$x \geq 0$$, the least value of $$K$$, for which $$4^{1+x} + 4^{1-x}, \frac{K}{2}, 16^x + 16^{-x}$$ are three consecutive terms of an A.P., is equal to :
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If the constant term in the expansion of $$\left(\frac{\sqrt[5]{3}}{x} + \frac{2x}{\sqrt[3]{5}}\right)^{12}$$, $$x \neq 0$$, is $$\alpha \times 2^8 \times \sqrt[5]{3}$$, then $$25\alpha$$ is equal to :
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Let $$A(-1, 1)$$ and $$B(2, 3)$$ be two points and $$P$$ be a variable point above the line $$AB$$ such that the area of $$\triangle PAB$$ is 10. If the locus of $$P$$ is $$ax + by = 15$$, then $$5a + 2b$$ is :
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Let $$ABCD$$ and $$AEFG$$ be squares of side 4 and 2 units, respectively. The point $$E$$ is on the line segment $$AB$$ and the point $$F$$ is on the diagonal $$AC$$. Then the radius $$r$$ of the circle passing through the point $$F$$ and touching the line segments $$BC$$ and $$CD$$ satisfies:
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Let the circle $$C_1 : x^2 + y^2 - 2(x + y) + 1 = 0$$ and $$C_2$$ be a circle having centre at $$(-1, 0)$$ and radius 2. If the line of the common chord of $$C_1$$ and $$C_2$$ intersects the $$y$$-axis at the point $$P$$, then the square of the distance of $$P$$ from the centre of $$C_1$$ is :
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Let the set $$S = \{2, 4, 8, 16, \ldots, 512\}$$ be partitioned into 3 sets $$A, B, C$$ with equal number of elements such that $$A \cup B \cup C = S$$ and $$A \cap B = B \cap C = A \cap C = \phi$$. The maximum number of such possible partitions of $$S$$ is equal to:
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Let $$\alpha\beta \neq 0$$ and $$A = \begin{bmatrix} \beta & \alpha & 3 \\ \alpha & \alpha & \beta \\ -\beta & \alpha & 2\alpha \end{bmatrix}$$. If $$B = \begin{bmatrix} 3\alpha & -9 & 3\alpha \\ -\alpha & 7 & -2\alpha \\ -2\alpha & 5 & -2\beta \end{bmatrix}$$ is the matrix of cofactors of the elements of $$A$$, then $$\det(AB)$$ is equal to :
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The values of $$m, n$$, for which the system of equations $$x + y + z = 4$$, $$2x + 5y + 5z = 17$$, $$x + 2y + mz = n$$ has infinitely many solutions, satisfy the equation:
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Let $$f, g : \mathbb{R} \rightarrow \mathbb{R}$$ be defined as : $$f(x) = |x - 1|$$ and $$g(x) = \begin{cases} e^x, & x \geq 0 \\ x + 1, & x \leq 0 \end{cases}$$. Then the function $$f(g(x))$$ is
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Let $$f : [-1, 2] \rightarrow \mathbb{R}$$ be given by $$f(x) = 2x^2 + x + [x^2] - [x]$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. The number of points, where $$f$$ is not continuous, is :
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If $$y(\theta) = \frac{2\cos\theta + \cos 2\theta}{\cos 3\theta + 4\cos 2\theta + 5\cos\theta + 2}$$, then at $$\theta = \frac{\pi}{2}$$, $$y'' + y' + y$$ is equal to :
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Let $$\beta(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} dx$$, $$m, n > 0$$. If $$\int_0^1 (1 - x^{10})^{20} dx = a \times \beta(b, c)$$, then $$100(a + b + c)$$ equals
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The area enclosed between the curves $$y = x|x|$$ and $$y = x - |x|$$ is :
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The differential equation of the family of circles passing through the origin and having centre at the line $$y = x$$ is :
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Consider three vectors $$\vec{a}, \vec{b}, \vec{c}$$. Let $$|\vec{a}| = 2, |\vec{b}| = 3$$ and $$\vec{a} = \vec{b} \times \vec{c}$$. If $$\alpha \in \left[0, \frac{\pi}{3}\right]$$ is the angle between the vectors $$\vec{b}$$ and $$\vec{c}$$, then the minimum value of $$27|\vec{c} - \vec{a}|^2$$ is equal to:
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Let $$\vec{a} = 2\hat{i} + 5\hat{j} - \hat{k}$$, $$\vec{b} = 2\hat{i} - 2\hat{j} + 2\hat{k}$$ and $$\vec{c}$$ be three vectors such that $$(\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i})$$. If $$\vec{a} \cdot \vec{c} = -29$$, then $$\vec{c} \cdot (-2\hat{i} + \hat{j} + \hat{k})$$ is equal to:
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Let $$(\alpha, \beta, \gamma)$$ be the image of the point $$(8, 5, 7)$$ in the line $$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-2}{5}$$. Then $$\alpha + \beta + \gamma$$ is equal to :
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The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are from the set $$\{1, 2, 3, 4, 5, 6\}$$. If the probability of this equation having one real root bigger than the other is $$p$$, then $$216p$$ equals :
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The number of real solutions of the equation $$x|x + 5| + 2|x + 7| - 2 = 0$$ is _________
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If $$1 + \frac{\sqrt{3} - \sqrt{2}}{2\sqrt{3}} + \frac{5 - 2\sqrt{6}}{18} + \frac{9\sqrt{3} - 11\sqrt{2}}{36\sqrt{3}} + \frac{49 - 20\sqrt{6}}{180} + \ldots$$ upto $$\infty = 2 + \left(\sqrt{\frac{b}{a}} + 1\right)\log_e\left(\frac{a}{b}\right)$$, where $$a$$ and $$b$$ are integers with $$\gcd(a, b) = 1$$, then $$11a + 18b$$ is equal to ______
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The number of solutions of $$\sin^2 x + (2 + 2x - x^2)\sin x - 3(x - 1)^2 = 0$$, where $$-\pi \leq x \leq \pi$$, is ________
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Let a line perpendicular to the line $$2x - y = 10$$ touch the parabola $$y^2 = 4(x - 9)$$ at the point $$P$$. The distance of the point $$P$$ from the centre of the circle $$x^2 + y^2 - 14x - 8y + 56 = 0$$ is __________
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Let $$a > 0$$ be a root of the equation $$2x^2 + x - 2 = 0$$. If $$\lim_{x \to \frac{1}{a}} \frac{16(1 - \cos(2 + x - 2x^2))}{(1 - ax)^2} = \alpha + \beta\sqrt{17}$$, where $$\alpha, \beta \in Z$$, then $$\alpha + \beta$$ is equal to ______
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Let the mean and the standard deviation of the probability distribution $$\begin{array}{|c|c|c|c|c|} \hline X & \alpha & 1 & 0 & -3 \\ \hline P(X) & \frac{1}{3} & K & \frac{1}{6} & \frac{1}{4} \\ \hline \end{array}$$ be $$\mu$$ and $$\sigma$$, respectively. If $$\sigma - \mu = 2$$, then $$\sigma + \mu$$ is equal to ________
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Let the maximum and minimum values of $$\left(\sqrt{8x - x^2 - 12} - 4\right)^2 + (x - 7)^2$$, $$x \in \mathbb{R}$$ be $$M$$ and $$m$$, respectively. Then $$M^2 - m^2$$ is equal to _________
If $$f(t) = \int_0^{\pi} \frac{2x \, dx}{1 - \cos^2 t \sin^2 x}$$, $$0 < t < \pi$$, then the value of $$\int_0^{\frac{\pi}{2}} \frac{\pi^2 dt}{f(t)}$$ equals _________
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} + \frac{2x}{(1+x^2)^2} y = xe^{\frac{1}{(1+x^2)}}$$; $$y(0) = 0$$. Then the area enclosed by the curve $$f(x) = y(x)e^{-\frac{1}{(1+x^2)}}$$ and the line $$y - x = 4$$ is __________
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Let the point $$(-1, \alpha, \beta)$$ lie on the line of the shortest distance between the lines $$\frac{x+2}{-3} = \frac{y-2}{4} = \frac{z-5}{2}$$ and $$\frac{x+2}{-1} = \frac{y+6}{2} = \frac{z-1}{0}$$. Then $$(\alpha - \beta)^2$$ is equal to __________
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