The number of points of intersection $$|z - (4+3i)| = 2$$ and $$|z| + |z-4| = 6, z \in C$$ is
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The number of points of intersection $$|z - (4+3i)| = 2$$ and $$|z| + |z-4| = 6, z \in C$$ is
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Let for some real numbers $$\alpha$$ and $$\beta$$, $$a = \alpha - i\beta$$. If the system of equations $$4ix + (1+i)y = 0$$ and $$8\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right)x + \bar{a}y = 0$$ has more than one solution then $$\frac{\alpha}{\beta}$$ is equal to
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Let $$S = 2 + \frac{6}{7} + \frac{12}{7^2} + \frac{20}{7^3} + \frac{30}{7^4} + \ldots$$ then $$4S$$ is equal to
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If $$a_1, a_2, a_3 \ldots$$ and $$b_1, b_2, b_3 \ldots$$ are A.P. and $$a_1 = 2, a_{10} = 3, a_1b_1 = 1 = a_{10}b_{10}$$ then $$a_4b_4$$ is equal to
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$$\alpha = \sin 36°$$ is a root of which of the following equation
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The set of values of $$k$$ for which the circle $$C : 4x^2 + 4y^2 - 12x + 8y + k = 0$$ lies inside the fourth quadrant and the point $$(1, -\frac{1}{3})$$ lies on or inside the circle $$C$$ is
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If the equation of the parabola, whose vertex is at $$(5, 4)$$ and the directrix is $$3x + y - 29 = 0$$, is $$x^2 + ay^2 + bxy + cx + dy + k = 0$$, then $$a + b + c + d + k$$ is equal to
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Which of the following statement is a tautology?
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The mean and variance of the data $$4, 5, 6, 6, 7, 8, x, y$$ where $$x < y$$ are $$6$$ and $$\frac{9}{4}$$ respectively. Then $$x^4 + y^2$$ is equal to
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Let $$A$$ and $$B$$ be two $$3 \times 3$$ matrices such that $$AB = I$$ and $$|A| = \frac{1}{8}$$ then $$|adj(B \cdot adj(2A))|$$ is equal to
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Let $$f(x) = \begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^2 & ax & a \end{vmatrix}, a \in R$$. Then the sum of the squares of all the values of $$a$$ for
$$2f'(10) - f'(5) + 100 = 0$$ is
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The value of $$\cot\left(\sum_{n=1}^{50} \tan^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$$ is
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If $$m$$ and $$n$$ respectively are the number of local maximum and local minimum points of the function $$f(x) = \int_0^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} dt$$, then the ordered pair $$(m, n)$$ is equal to
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Let $$f$$ be a differentiable function in $$\left(0, \frac{\pi}{2}\right)$$. If $$\int_{\cos x}^{1} t^2 f(t) dt = \sin^3 x + \cos x$$, then $$\frac{1}{\sqrt{3}} f'\left(\frac{1}{\sqrt{3}}\right)$$ is equal to
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The value of the integral $$\int_0^1 \frac{1}{7^{[\frac{1}{x}]}} dx$$, where $$[\cdot]$$ denotes the greatest integer function, is equal to
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If the solution curve of the differential equation $$((\tan^{-1}y) - x)dy = (1 + y^2)dx$$ passes through the point $$(1, 0)$$ then the abscissa of the point on the curve whose ordinate is $$\tan(1)$$ is
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Let $$\vec{a}$$ and $$\vec{b}$$ be the vectors along the diagonal of a parallelogram having area $$2\sqrt{2}$$. Let the angle between $$\vec{a}$$ and $$\vec{b}$$ be acute. $$|\vec{a}| = 1$$ and $$|\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}|$$. If $$\vec{c} = 2\sqrt{2}(\vec{a} \times \vec{b}) - 2\vec{b}$$, then an angle between $$\vec{b}$$ and $$\vec{c}$$ is
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Let the foot of the perpendicular from the point $$(1, 2, 4)$$ on the line $$\frac{x+2}{4} = \frac{y-1}{2} = \frac{z+1}{3}$$ be $$P$$. Then the distance of $$P$$ from the plane $$3x + 4y + 12z + 23 = 0$$ is
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The shortest distance between the lines $$\frac{x-3}{2} = \frac{y-2}{3} = \frac{z-1}{-1}$$ and $$\frac{x+3}{2} = \frac{y-6}{1} = \frac{z-5}{3}$$ is
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If a point $$A(x, y)$$ lies in the region bounded by the y-axis, straight lines $$2y + x = 6$$ and $$5x - 6y = 30$$, then the probability that $$y < 1$$ is
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Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - 4\lambda x + 5 = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$x^2 - (3\sqrt{2} + 2\sqrt{3})x + 7 + 3\lambda\sqrt{3} = 0$$. If $$\beta + \gamma = 3\sqrt{2}$$, then $$(\alpha + 2\beta + \gamma)^2$$ is equal to ______
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If the sum of the coefficients of all the positive powers of $$x$$, in the binomial expansion of $$\left(x^n + \frac{2}{x^5}\right)^7$$ is $$939$$, then the sum of all the possible integral values of $$n$$ is ______
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Let a circle $$C$$ of radius $$5$$ lie below the $$x$$-axis. The line $$L_1 = 4x + 3y + 2$$ passes through the centre $$P$$ of the circle $$C$$ and intersects the line $$L_2 : 3x - 4y - 11 = 0$$ at $$Q$$. The line $$L_2$$ touches $$C$$ at the point $$Q$$. Then the distance of $$P$$ from the line $$5x - 12y + 51 = 0$$ is ______
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Let $$[t]$$ denote the greatest integer $$\leq t$$ and $$\{t\}$$ denote the fractional part of $$t$$. Then integral value of $$\alpha$$ for which the left hand limit of the function $$f(x) = [1+x] + \frac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$$ at $$x = 0$$ is equal to $$\alpha - \frac{4}{3}$$ is ______
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Let $$A$$ be a matrix of order $$2 \times 2$$, whose entries are from the set $$\{0, 1, 2, 3, 4, 5\}$$. If the sum of all the entries of $$A$$ is a prime number $$p, 2 < p < 8$$, then the number of such matrices $$A$$ is ______
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Let $$S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$. Define $$f : S \to S$$ as $$f(n) = \begin{cases} 2n, & \text{if } n = 1,2,3,4,5 \\ 2n-11 & \text{if } n = 6,7,8,9,10 \end{cases}$$
Let $$g : S \geq S$$ be a function such that $$fog(n) = \begin{cases} n+1, & \text{if } n \text{ is odd} \\ n-1, & \text{if } n \text{ is even} \end{cases}$$, then
$$g(10)(g(1) + g(2) + g(3) + g(4) + g(5))$$ is equal to ______
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If $$y(x) = (x^x)^x, x > 0$$ then $$\frac{d^2x}{dy^2} + 20$$ at $$x = 1$$ is equal to ______
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If the area of the region $$\left\{(x,y) : x^{\frac{2}{3}} + y^{\frac{2}{3}} \leq 1, x + y \geq 0, y \geq 0\right\}$$ is $$A$$, then $$\frac{256A}{\pi}$$ is ______
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Let $$y = y(x)$$ be the solution of the differential equation
$$(1 - x^2)dy = \left(xy + (x^3 + 2)\sqrt{1-x^2}\right)dx, -1 < x < 1$$
and $$y(0) = 0$$. If $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^2} y(x)dx = k$$ then $$k^{-1}$$ is equal to ______
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Let $$S = \{E, E_2 \ldots E_8\}$$ be a sample space of a random experiment such that $$P(E_n) = \frac{n}{36}$$ for every $$n = 1, 2 \ldots 8$$. Then the number of elements in the set $$\{A \subset S : P(A) \geq \frac{4}{5}\}$$ is ______
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