If the equation $$x^2 + bx + 45 = 0$$, $$b \in R$$ has conjugate complex roots and they satisfy $$|z + 1| = 2\sqrt{10}$$, then
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If the equation $$x^2 + bx + 45 = 0$$, $$b \in R$$ has conjugate complex roots and they satisfy $$|z + 1| = 2\sqrt{10}$$, then
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Let $$f : R \rightarrow R$$ be such that for all $$x \in R$$ $$(2^{1+x} + 2^{1-x})$$, $$f(x)$$ and $$(3^x + 3^{-x})$$ are in A.P., then the minimum value of $$f(x)$$ is
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If a, b and c are the greatest values of $$^{19}C_p$$, $$^{20}C_q$$ and $$^{21}C_r$$ respectively, then:
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Let two points be $$A(1, -1)$$ and $$B(0, 2)$$. If a point P(x', y') be such that the area of $$\triangle PAB = 5$$ sq. units and it lies on the line $$3x + y - 4\lambda = 0$$, then a value of $$\lambda$$ is
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The locus of a point which divides the line segment joining the point $$(0, -1)$$ and a point on the parabola $$x^2 = 4y$$ internally in the ratio 1 : 2 is:
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For $$a > 0$$, let the curves $$C_1 : y^2 = ax$$ and $$C_2 : x^2 = ay$$ intersect at origin O and a point P. Let the line $$x = b$$ ($$0 < b < a$$) intersect the chord OP and the x-axis at points Q and R, respectively. If the line $$x = b$$ bisects the area bounded by the curves, $$C_1$$ and $$C_2$$, and the area of $$\triangle OQR = \frac{1}{2}$$, then 'a' satisfies the equation:
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Let the line $$y = mx$$ and the ellipse $$2x^2 + y^2 = 1$$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $$\left(-\frac{1}{3\sqrt{2}}, 0\right)$$ and $$(0, \beta)$$, then $$\beta$$ is equal to
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$$\lim_{x \to 0} \left(\frac{3x^2+2}{7x^2+2}\right)^{\frac{1}{x^2}}$$ is equal to
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Which one of the following is a tautology?
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The mean and the standard deviation (s.d.) of 10 observations are 20 and 2 respectively. Each of these 10 observations is multiplied by $$p$$ and then reduced by $$q$$, where $$p \neq 0$$ and $$q \neq 0$$. If the new mean and new s.d. become half of their original values, then $$q$$ is equal to
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For which of the following ordered pairs $$(\mu, \delta)$$, the system of linear equations
$$x + 2y + 3z = 1$$
$$3x + 4y + 5z = \mu$$
$$4x + 4y + 4z = \delta$$
is inconsistent?
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The inverse function of $$f(x) = \frac{8^{2x} - 8^{-2x}}{8^{2x} + 8^{-2x}}$$, $$x \in (-1, 1)$$, is
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Let $$f(x) = (\sin(\tan^{-1} x) + \sin(\cot^{-1} x))^2 - 1$$, $$|x| \gt 1$$. If $$\frac{dy}{dx} = \frac{1}{2}\frac{d}{dx}(\sin^{-1}(f(x)))$$ and $$y(\sqrt{3}) = \frac{\pi}{6}$$, then $$y(-\sqrt{3})$$ is equal to:
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If $$c$$ is a point at which Rolle's theorem holds for the function, $$f(x) = \log_e\left(\frac{x^2 + \alpha}{7x}\right)$$ in the interval [3, 4], where $$\alpha \in R$$, then $$f''(c)$$ is equal to
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Let $$f(x) = x\cos^{-1}(-\sin|x|)$$, $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$, then which of the following is true?
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If $$\int \frac{\cos x \, dx}{\sin^3 x (1+\sin^6 x)^{2/3}} = f(x)(1 + \sin^6 x)^{1/\lambda} + c$$, where c is a constant of integration, then $$\lambda f\left(\frac{\pi}{3}\right)$$ is equal to
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Let $$y = y(x)$$ be a solution of the differential equation, $$\sqrt{1 - x^2}\frac{dy}{dx} + \sqrt{1 - y^2} = 0$$, $$|x| < 1$$. If $$y\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2}$$, then $$y\left(\frac{-1}{\sqrt{2}}\right)$$ is equal to
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Let the volume of a parallelepiped whose coterminous edges are given by $$\vec{u} = \hat{i} + \hat{j} + \lambda\hat{k}$$, $$\vec{v} = \hat{i} + \hat{j} + 3\hat{k}$$ and $$\vec{w} = 2\hat{i} + \hat{j} + \hat{k}$$ be 1 cu. unit. If $$\theta$$ be the angle between the edges $$\vec{u}$$ and $$\vec{w}$$, then the value of $$\cos\theta$$ can be
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The shortest distance between the lines $$\frac{x-3}{3} = \frac{y-8}{-1} = \frac{z-3}{1}$$ and $$\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$$ is
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Let A and B be two independent events such that $$P(A) = \frac{1}{3}$$ and $$P(B) = \frac{1}{6}$$. Then, which of the following is true?
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The least positive value of 'a' for which the equation, $$2x^2 + (a - 10)x + \frac{33}{2} = 2a$$ has real roots is
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An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then, the number of ways in which 4 marbles can be drawn so that at the most three of them are red is
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The sum $$\sum_{k=1}^{20} (1 + 2 + 3 + \ldots + k)$$ is
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The number of all $$3 \times 3$$ matrices A, with entries from the set $$\{-1, 0, 1\}$$ such that the sum of the diagonal elements of $$AA^T$$ is 3, is
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Let the normal at a point P on the curve $$y^2 - 3x^2 + y + 10 = 0$$ intersect the y-axis at $$\left(0, \frac{3}{2}\right)$$. If $$m$$ is the slope of the tangent at P to the curve, then $$|m|$$ is equal to
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