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Let a circle $$C$$ in complex plane pass through the points $$z_1 = 3 + 4i, z_2 = 4 + 3i$$ and $$z_3 = 5i$$. If $$z \neq z_1$$ is a point on $$C$$ such that the line through $$z$$ and $$z_1$$ is perpendicular to the line through $$z_2$$ and $$z_3$$, then $$\arg z$$ is equal to
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If $$\frac{1}{2 \cdot 3^{10}} + \frac{1}{2^2 \cdot 3^9} + \cdots + \frac{1}{2^{10} \cdot 3} = \frac{K}{2^{10} \cdot 3^{10}}$$, then the remainder when $$K$$ is divided by $$6$$ is
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Let a circle $$C$$ touch the lines $$L_1: 4x - 3y + K_1 = 0$$ and $$L_2: 4x - 3y + K_2 = 0$$, $$K_1, K_2 \in R$$. If a line passing through the centre of the circle $$C$$ intersects $$L_1$$ at $$(-1, 2)$$ and $$L_2$$ at $$(3, -6)$$, then the equation of the circle $$C$$ is
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If $$y = m_1 x + c_1$$ and $$y = m_2 x + c_2$$, $$m_1 \neq m_2$$ are two common tangents of circle $$x^2 + y^2 = 2$$ and parabola $$y^2 = x$$, then the value of $$8|m_1 m_2|$$ is equal to
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Let $$x = 2t, y = \frac{t^2}{3}$$ be a conic. Let $$S$$ be the focus and $$B$$ be the point on the axis of the conic such that $$SA \perp BA$$, where $$A$$ is any point on the conic. If $$k$$ is the ordinate of the centroid of the $$\triangle SAB$$, then $$\lim_{t \to 1} k$$ is equal to
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Let $$f(x)$$ be a polynomial function such that $$f(x) + f'(x) + f''(x) = x^5 + 64$$. Then, the value of $$\lim_{x \to 1} \frac{f(x)}{x - 1}$$ is equal to
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Consider the following two propositions :
$$P_1:\sim(p\to\sim q)$$
$$P_2:(p\land\sim q)\land(\sim p\lor q)$$
If the proposition $$p\to(\sim p\lor q) $$ is evaluated as FALSE, then
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Let $$a, b$$ and $$c$$ be the length of sides of a triangle $$ABC$$ such that $$\frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9}$$. If $$r$$ and $$R$$ are the radius of incircle and radius of circumcircle of the triangle $$ABC$$, respectively, then the value of $$\frac{R}{r}$$ is equal to
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Let $$A = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix}$$. If $$M$$ and $$N$$ are two matrices given by $$M = \sum_{k=1}^{10} A^{2k}$$ and $$N = \sum_{k=1}^{10} A^{2k-1}$$ then $$MN^2$$ is
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Let $$A$$ be a $$3\times 3$$ real matrix such that
$$A\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\1\\0\end{pmatrix},\qquad A\begin{pmatrix}1\\0\\1\end{pmatrix}=\begin{pmatrix}-1\\0\\1\end{pmatrix},\qquad A\begin{pmatrix}0\\0\\1\end{pmatrix}=\begin{pmatrix}1\\1\\2\end{pmatrix}.$$
If $$X=(x_1,x_2,x_3)^T$$ and $$I$$ is an identity matrix of order $$3$$, then the system
$$(A-2I)X=\begin{pmatrix}4\\1\\1\end{pmatrix}$$
has ____________.
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Let $$f : N \to R$$ be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers $$x$$ and $$y$$. If $$f(1) = 2$$, then the value of $$\alpha$$ for which $$\sum_{k=1}^{10} f(\alpha + k) = \frac{512}{3}2^{20} - 1$$ holds, is
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Let $$f : R \to R$$ be defined as $$f(x) = x^3 + x - 5$$. If $$g(x)$$ is a function such that $$f(g(x)) = x, \forall x \in R$$, then $$g'(63)$$ is equal to
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Let $$f : R \to R$$ and $$g : R \to R$$ be two functions defined by $$f(x) = \log_e(x^2 + 1) - e^{-x} + 1$$ and $$g(x) = \frac{1 - 2e^{2x}}{e^x}$$. Then, for which of the following range of $$\alpha$$, the inequality $$f\left(g\left(\frac{\left(\alpha-1\right)^2}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)$$ holds?
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Let $$g : (0, \infty) \to R$$ be a differentiable function such that $$\int \frac{x\cos x - \sin x}{e^x + 1} + \frac{g(x)e^x + 1 - xe^x}{(e^x + 1)^2} dx = \frac{xg(x)}{e^x + 1} + C$$, for all $$x > 0$$, where $$C$$ is an arbitrary constant. Then
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Evaluate
$$I=\int_0^{\pi}\frac{e^{\cos x}\sin x}{(1+\cos^2x)e^{\cos x}+e^{-\cos x}}\,dx.$$
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Let $$y = y(x)$$ be the solution of the differential equation $$(x + 1)y' - y = e^{3x}(x + 1)^2$$, with $$y(0) = \frac{1}{3}$$. Then, the point $$x = -\frac{4}{3}$$ for the curve $$y = y(x)$$ is
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If the solution curve $$y = y(x)$$ of the differential equation $$y^2 dx + (x^2 - xy + y^2)dy = 0$$, which passes through the point $$(1, 1)$$ and intersects the line $$y = \sqrt{3}x$$ at the point $$(\alpha, \sqrt{3}\alpha)$$, then value of $$\log_e \sqrt{3}\alpha$$ is equal to
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Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}, a_i > 0, i = 1, 2, 3$$ be a vector which makes equal angles with the coordinate axes $$OX, OY$$ and $$OZ$$. Also, let the projection of $$\vec{a}$$ on the vector $$3\hat{i} + 4\hat{j}$$ be $$7$$. Let $$\vec{b}$$ be a vector obtained by rotating $$\vec{a}$$ with $$90°$$. If $$\vec{a}, \vec{b}$$ and x-axis are coplanar, then projection of a vector $$\vec{b}$$ on $$3\hat{i} + 4\hat{j}$$ is equal to
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Let $$Q$$ be the mirror image of the point $$P(1, 0, 1)$$ with respect to the plane $$S : x + y + z = 5$$. If a line $$L$$ passing through $$(1, -1, -1)$$, parallel to the line $$PQ$$ meets the plane $$S$$ at $$R$$, then $$QR^2$$ is equal to
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Let $$E_1$$ and $$E_2$$ be two events such that the conditional probabilities $$P(E_1 | E_2) = \frac{1}{2}$$, $$P(E_2 | E_1) = \frac{3}{4}$$ and $$P(E_1 \cap E_2) = \frac{1}{8}$$. Then
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For a natural number $$n$$, let $$\alpha_n = 19^n - 12^n$$. Then, the value of $$\frac{31\alpha_9 - \alpha_{10}}{57\alpha_8}$$ is ______.
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The number of 3-digit odd numbers, whose sum of digits is a multiple of $$7$$, is ______.
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The greatest integer less than or equal to the sum of first $$100$$ terms of the sequence $$\frac{1}{3}, \frac{5}{9}, \frac{19}{27}, \frac{65}{81}, \ldots$$ is equal to ______.
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Let $$C_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1 + x)^{10}$$. If for $$\alpha, \beta \in R$$,
$$C_{1} + 3 \cdot 2C_{2} + 5 \cdot 3C_{3} + \ldots $$ upto 10 terms $$= \frac{\alpha \times 2^{11}}{2^{\beta} - 1}(C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \ldots $$ upto 10 terms ) then the value of $$\alpha + \beta $$ is equal to ______.
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The number of values of $$x$$ in the interval $$\left[\frac{\pi}{4}, \frac{7\pi}{4}\right]$$ for which $$14\cosec^2 x - 2\sin^2 x = 21 - 4\cos^2 x$$ holds, is ______.
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Let the abscissae of the two points $$P$$ and $$Q$$ be the roots of $$2x^2 - rx + p = 0$$ and the ordinates of $$P$$ and $$Q$$ be the roots of $$x^2 - sx - q = 0$$. If the equation of the circle described on $$PQ$$ as diameter is $$2(x^2 + y^2) - 11x - 14y - 22 = 0$$, then $$2r + s - 2q + p$$ is equal to ______.
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Let $$A$$ be a $$3 \times 3$$ matrix having entries from the set $$\{-1, 0, 1\}$$. The number of all such matrices $$A$$ having sum of all the entries equal to $$5$$, is ______.
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Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)(2+x^{25})\right)^{\frac{1}{50}}.$$ If the function $$g(x)=f(f(f(x)))+f(f(x)),$$ then the greatest integer less than or equal to $$g(1)$$ is ____________.
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Let $$\theta$$ be the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$, where $$|\vec{a}| = 4, |\vec{b}| = 3$$ and $$\theta \in \left[\frac{\pi}{4}, \frac{\pi}{3}\right]$$. Then $$|\vec{a} - \vec{b} \times \vec{a} + \vec{b}|^2 + 4|\vec{a} \cdot \vec{b}|^2$$ is equal to ______.
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Let the lines $$L_1 : \vec{r} = \lambda(\hat{i} + 2\hat{j} + 3\hat{k}), \lambda \in R$$ and $$L_2 : \vec{r} = \hat{i} + 3\hat{j} + \hat{k} + \mu(\hat{i} + \hat{j} + 5\hat{k}); \mu \in R$$, intersect at the point $$S$$. If a plane $$ax + by - z + d = 0$$ passes through $$S$$ and is parallel to the lines $$L_1$$ and $$L_2$$, then the value of $$a + b + d$$ is equal to ______.
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