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Let a complex number be $$w = 1 - \sqrt{3}i$$. Let another complex number $$z$$ be such that $$|zw| = 1$$ and $$\arg(z) - \arg(w) = \frac{\pi}{2}$$. Then the area of the triangle (in sq. units) with vertices origin, $$z$$ and $$w$$ is equal to
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Let $$S_1$$ be the sum of first $$2n$$ terms of an arithmetic progression. Let $$S_2$$ be the sum of first $$4n$$ terms of the same arithmetic progression. If $$(S_2 - S_1)$$ is 1000, then the sum of the first $$6n$$ terms of the arithmetic progression is equal to:
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If $$15 \sin^4 \alpha + 10 \cos^4 \alpha = 6$$, for some $$\alpha \in R$$, then the value of $$27 \sec^6 \alpha + 8\operatorname{cosec}^6 \alpha$$ is equal to :
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Let the centroid of an equilateral triangle $$ABC$$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $$x + y = 3$$. If $$R$$ and $$r$$ be the radius of circumcircle and incircle respectively of $$\triangle ABC$$, then $$(R + r)$$ is equal to :
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Let $$S_1 : x^2 + y^2 = 9$$ and $$S_2 : (x-2)^2 + y^2 = 1$$. Then the locus of center of a variable circle $$S$$ which touches $$S_1$$ internally and $$S_2$$ externally always passes through the points :
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Let a tangent be drawn to the ellipse $$\frac{x^2}{27} + y^2 = 1$$ at $$(3\sqrt{3}\cos\theta, \sin\theta)$$ where $$\theta \in \left(0, \frac{\pi}{2}\right)$$. Then the value of $$\theta$$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
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Consider a hyperbola $$H : x^2 - 2y^2 = 4$$. Let the tangent at a point $$P(4, \sqrt{6})$$ meet the x-axis at $$Q$$ and latus rectum at $$R(x_1, y_1)$$, $$x_1 > 0$$. If $$F$$ is a focus of $$H$$ which is nearer to the point $$P$$, then the area of $$\triangle QFR$$ (in sq. units) is equal to
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If $$P$$ and $$Q$$ are two statements, then which of the following compound statement is a tautology?
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Let in a series of $$2n$$ observations, half of them are equal to $$a$$ and remaining half are equal to $$-a$$. Also by adding a constant $$b$$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $$a^2 + b^2$$ is equal to :
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A pole stands vertically inside a triangular park $$ABC$$. Let the angle of elevation of the top of the pole from each corner of the park be $$\frac{\pi}{3}$$. If the radius of the circumcircle of $$\triangle ABC$$ is 2, then the height of the pole is equal to :
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Define a relation $$R$$ over a class of $$n \times n$$ real matrices $$A$$ and $$B$$ as "$$ARB$$" iff there exists a non-singular matrix $$P$$ such that $$PAP^{-1} = B$$. Then which of the following is true?
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Let the system of linear equations
$$4x + \lambda y + 2z = 0$$
$$2x - y + z = 0$$
$$\mu x + 2y + 3z = 0$$, $$\lambda, \mu \in R$$
has a non-trivial solution. Then which of the following is true?
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Let $$f : R - \{3\} \to R - \{1\}$$ be defined by $$f(x) = \frac{x-2}{x-3}$$. Let $$g : R \to R$$ be given as $$g(x) = 2x - 3$$. Then, the sum of all the values of $$x$$ for which $$f^{-1}(x) + g^{-1}(x) = \frac{13}{2}$$ is equal to
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Let $$f : R \to R$$ be a function defined as
$$$f(x) = \begin{cases} \frac{\sin(a+1)x + \sin 2x}{2x}, & \text{if } x < 0 \\ b, & \text{if } x = 0 \\ \frac{\sqrt{x+bx^3} - \sqrt{x}}{bx^{5/2}}, & \text{if } x > 0 \end{cases}$$$
If $$f$$ is continuous at $$x = 0$$, then the value of $$a + b$$ is equal to :
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Let $$g(x) = \int_0^x f(t)dt$$, where $$f$$ is continuous function in $$[0, 3]$$ such that $$\frac{1}{3} \le f(t) \le 1$$ for all $$t \in [0, 1]$$ and $$0 \le f(t) \le \frac{1}{2}$$ for all $$t \in (1, 3]$$.
The largest possible interval in which $$g(3)$$ lies is :
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The area (in sq. unit) bounded by the curve $$4y^2 = x^2(4-x)(x-2)$$ is equal to
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Let $$y = y(x)$$ be the solution of the differential equation $$\frac{dy}{dx} = (y+1)\left((y+1)e^{x^2/2} - x\right)$$, $$0 < x < 2.1$$, with $$y(2) = 0$$. Then the value of $$\frac{dy}{dx}$$ at $$x = 1$$ is equal to
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In a triangle $$ABC$$, if $$|\overrightarrow{BC}| = 8$$, $$|\overrightarrow{CA}| = 7$$, $$|\overrightarrow{AB}| = 10$$, then the projection of the vector $$\overrightarrow{AB}$$ on $$\overrightarrow{AC}$$ is equal to :
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Let $$\vec{a}$$ and $$\vec{b}$$ be two non-zero vectors perpendicular to each other and $$|\vec{a}| = |\vec{b}|$$, If $$|\vec{a} \times \vec{b}| = |\vec{a}|$$, then the angle between the vectors $$\left(\vec{a} + \vec{b} + (\vec{a} \times \vec{b})\right)$$ and $$\vec{a}$$ is equal to :
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Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then the probability of getting exactly 3 successes is equal to :
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If $$f(x)$$ and $$g(x)$$ are two polynomials such that the polynomial $$P(x) = f(x^3) + xg(x^3)$$ is divisible by $$x^2 + x + 1$$, then $$P(1)$$ is equal to ___.
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If $$\sum_{r=1}^{10} r!(r^3 + 6r^2 + 2r + 5) = \alpha(11!)$$, then the value of $$\alpha$$ is equal to ___.
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The term independent of $$x$$ in the expansion of $$\left[\frac{x+1}{x^{2/3} - x^{1/3} + 1} - \frac{x-1}{x - x^{1/2}}\right]^{10}$$, $$x \neq 1$$, is equal to ___.
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Let $${}^nC_r$$ denote the binomial coefficient of $$x^r$$ in the expansion of $$(1+x)^n$$. If $$\sum_{k=0}^{10} (2^2 + 3k) {}^{10}C_k = \alpha \cdot 3^{10} + \beta \cdot 2^{10}$$, $$\alpha, \beta \in R$$, then $$\alpha + \beta$$ is equal to ___.
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Let $$I$$ be an identity matrix of order $$2 \times 2$$ and $$P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$$. Then the value of $$n \in N$$ for which $$P^n = 5I - 8P$$ is equal to ___.
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Let $$f : R \to R$$ satisfy the equation $$f(x+y) = f(x) \cdot f(y)$$ for all $$x, y \in R$$ and $$f(x) \neq 0$$ for any $$x \in R$$. If the function $$f$$ is differentiable at $$x = 0$$ and $$f'(0) = 3$$, then $$\lim_{h \to 0} \frac{1}{h}(f(h) - 1)$$ is equal to ___.
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Let $$P(x)$$ be a real polynomial of degree 3 which vanishes at $$x = -3$$. Let $$P(x)$$ have local minima at $$x = -1$$, local maxima at $$x = 1$$ and $$\int_{-1}^{1} P(x)dx = 18$$, then the sum of all the coefficients of the polynomial $$P(x)$$ is equal to ___.
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Let $$y = y(x)$$ be the solution of the differential equation $$xdy - ydx = \sqrt{(x^2 - y^2)}dx$$, $$x \ge 1$$, with $$y(1) = 0$$. If the area bounded by the line $$x = 1$$, $$x = e^\pi$$, $$y = 0$$ and $$y = y(x)$$ is $$\alpha e^{2\pi} + \beta$$, then the value of $$10(\alpha + \beta)$$ is equal to ___.
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Let the mirror image of the point $$(1, 3, a)$$ with respect to the plane $$\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) - b = 0$$ be $$(-3, 5, 2)$$. Then the value of $$|a + b|$$ is equal to ___.
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Let $$P$$ be a plane containing the line $$\frac{x-1}{3} = \frac{y+6}{4} = \frac{z+5}{2}$$ and parallel to the line $$\frac{x-3}{4} = \frac{y-2}{-3} = \frac{z+5}{7}$$. If the point $$(1, -1, \alpha)$$ lies on the plane $$P$$, then the value of $$|5\alpha|$$ is equal to ___.
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