For $$z \in \mathbb{C}$$, if the minimum value of $$(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$$ is $$5\sqrt{2}$$, then a value of $$p$$ is
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For $$z \in \mathbb{C}$$, if the minimum value of $$(|z - 3\sqrt{2}| + |z - p\sqrt{2}i|)$$ is $$5\sqrt{2}$$, then a value of $$p$$ is
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The sum $$\displaystyle\sum_{n=1}^{21} \dfrac{3}{(4n-1)(4n+3)}$$ is equal to
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The remainder when $$(11)^{1011} + (1011)^{11}$$ is divided by $$9$$ is
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The value of $$2\sin\dfrac{\pi}{22} \sin\dfrac{3\pi}{22} \sin\dfrac{5\pi}{22} \sin\dfrac{7\pi}{22} \sin\dfrac{9\pi}{22}$$ is
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Let the point $$P(\alpha, \beta)$$ be at a unit distance from each of the two lines $$L_1: 3x - 4y + 12 = 0$$, and $$L_2: 8x + 6y + 11 = 0$$. If $$P$$ lies below $$L_1$$ and above $$L_2$$, then $$100(\alpha + \beta)$$ is equal to
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The tangents at the points $$A(1, 3)$$ and $$B(1, -1)$$ on the parabola $$y^2 - 2x - 2y = 1$$ meet at the point $$P$$. Then the area (in $$\text{unit}^2$$) of the triangle $$PAB$$ is:
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If the ellipse $$\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$$ meets the line $$\dfrac{x}{7} + \dfrac{y}{2\sqrt{6}} = 1$$ on the $$x$$-axis and the line $$\dfrac{x}{7} - \dfrac{y}{2\sqrt{6}} = 1$$ on the $$y$$-axis, then the eccentricity of the ellipse is
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Let the foci of the ellipse $$\dfrac{x^2}{16} + \dfrac{y^2}{7} = 1$$ and the hyperbola $$\dfrac{x^2}{144} - \dfrac{y^2}{\alpha} = \dfrac{1}{25}$$ coincide. Then the length of the latus rectum of the hyperbola is:
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$$\displaystyle\lim_{x \to \frac{\pi}{4}} \dfrac{8\sqrt{2} - (\cos x + \sin x)^7}{\sqrt{2} - \sqrt{2}\sin 2x}$$ is equal to
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Consider the following statements:
$$P$$: Ramu is intelligent.
$$Q$$: Ramu is rich.
$$R$$: Ramu is not honest.
The negation of the statement "Ramu is intelligent and honest if and only if Ramu is not rich" can be expressed as:
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If the mean deviation about median for the numbers $$3, 5, 7, 2k, 12, 16, 21, 24$$ arranged in the ascending order, is $$6$$ then the median is
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The number of real values of $$\lambda$$, such that the system of linear equations
$$2x - 3y + 5z = 9$$
$$x + 3y - z = -18$$
$$3x - y + (\lambda^2 - |\lambda|)z = 16$$
has no solutions, is
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The number of bijective functions $$f(\{1, 3, 5, 7, \ldots, 99\}) \to \{2, 4, 6, 8, \ldots, 100\}$$ if $$f(3) > f(5) > f(7) \ldots > f(99)$$ is
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$$\displaystyle\lim_{n \to \infty} \dfrac{1}{2n}\left(\dfrac{1}{\sqrt{1 - \frac{1}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{2}{2n}}} + \dfrac{1}{\sqrt{1 - \frac{3}{2n}}} + \ldots + \dfrac{1}{\sqrt{1 - \frac{2n-1}{2n}}}\right)$$ is equal to
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Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\displaystyle\int_{-3}^{101} ([\sin(\pi x)] + e^{[\cos(2\pi x)]}) dx$$ is equal to
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Let a smooth curve $$y = f(x)$$ be such that the slope of the tangent at any point $$(x, y)$$ on it is directly proportional to $$\left(\dfrac{-y}{x}\right)$$. If the curve passes through the points $$(1, 2)$$ and $$(8, 1)$$, then $$\left|y\left(\dfrac{1}{8}\right)\right|$$ is equal to
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Let $$\vec{a} = \hat{i} - \hat{j} + 2\hat{k}$$ and let $$\vec{b}$$ be a vector such that $$\vec{a} \times \vec{b} = 2\hat{i} - \hat{k}$$ and $$\vec{a} \cdot \vec{b} = 3$$. Then the projection of $$\vec{b}$$ on the vector $$\vec{a} - \vec{b}$$ is:
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A plane $$E$$ is perpendicular to the two planes $$2x - 2y + z = 0$$ and $$x - y + 2z = 4$$, and passes through the point $$P(1, -1, 1)$$. If the distance of the plane $$E$$ from the point $$Q(a, a, 2)$$ is $$3\sqrt{2}$$, then $$(PQ)^2$$ is equal to
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The shortest distance between the lines $$\dfrac{x+7}{-6} = \dfrac{y-6}{7} = z$$ and $$\dfrac{7-x}{2} = y - 2 = z - 6$$ is
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If $$A$$ and $$B$$ are two events such that $$P(A) = \dfrac{1}{3}$$, $$P(B) = \dfrac{1}{5}$$ and $$P(A \cup B) = \dfrac{1}{2}$$, then $$P\left(\dfrac{A}{B'}\right) + P\left(\dfrac{B}{A'}\right)$$ is equal to
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Let $$f(x)$$ be a quadratic polynomial with leading coefficient $$1$$ such that $$f(0) = p, p \neq 0$$, and $$f(1) = \dfrac{1}{3}$$. If the equations $$f(x) = 0$$ and $$fofofof(x) = 0$$ have a common real root, then $$f(-3)$$ is equal to ______.
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If the circles $$x^2 + y^2 + 6x + 8y + 16 = 0$$ and $$x^2 + y^2 + 2(3 - \sqrt{3})x + 2(4 - \sqrt{6})y = k + 6\sqrt{3} + 8\sqrt{6}$$, $$k > 0$$, touch internally at the point $$P(\alpha, \beta)$$, then $$(\alpha + \sqrt{3})^2 + (\beta + \sqrt{6})^2$$ is equal to ______.
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Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Define $$B = \{T \subseteq A :$$ either $$1 \notin T$$ or $$2 \in T\}$$ and $$C = \{T \subseteq A :$$ the sum of all the elements of $$T$$ is a prime number $$\}$$. Then the number of elements in the set $$B \cup C$$ is ______.
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Let $$A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$, $$a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}$$, $$A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix}$$ then $$n + a + b$$ is equal to ______.
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Let $$x = \sin(2\tan^{-1}\alpha)$$ and $$y = \sin\left(\dfrac{1}{2}\tan^{-1}\dfrac{4}{3}\right)$$. If $$S = \{\alpha \in \mathbb{R} : y^2 = 1 - x\}$$, then $$\displaystyle\sum_{\alpha \in S} 16\alpha^3$$ is equal to ______.
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The sum of the maximum and minimum values of the function $$f(x) = |5x - 7| + [x^2 + 2x]$$ in the interval $$\left[\dfrac{5}{4}, 2\right]$$, where $$[t]$$ is the greatest integer $$\le t$$, is ______.
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Let the area enclosed by the $$x$$-axis, and the tangent and normal drawn to the curve $$4x^3 - 3xy^2 + 6x^2 - 5xy - 8y^2 + 9x + 14 = 0$$ at the point $$(-2, 3)$$ be $$A$$. Then $$8A$$ is equal to ______.
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Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f'(0) = 4$$ and $$f(x) + \displaystyle\int_0^x (x-t)f'(t) dt = (e^{2x} + e^{-2x})\cos 2x + \dfrac{2}{a}x$$, then $$(2a+1)^5a^2$$ is equal to ______.
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Let $$a_n = \displaystyle\int_{-1}^{n} \left(1 + \dfrac{x}{2} + \dfrac{x^2}{3} + \ldots + \dfrac{x^{n-1}}{n}\right) dx$$ for every $$n \in \mathbb{N}$$. Then the sum of all the elements of the set $$\{n \in \mathbb{N} : a_n \in (2, 30)\}$$ is ______.
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Let $$y = y(x)$$ be the solution of the differential equation $$\dfrac{dy}{dx} = \dfrac{4y^3 + 2yx^2}{3xy^2 + x^3}$$, $$y(1) = 1$$. If for some $$n \in \mathbb{N}$$, $$y(2) \in [n-1, n)$$, then $$n$$ is equal to ______.
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