The sum of the roots of the equation, $$x^2 + |2x - 3| - 4 = 0$$, is:
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The sum of the roots of the equation, $$x^2 + |2x - 3| - 4 = 0$$, is:
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Let $$z \neq -i$$ be any complex number such that $$\frac{z-1}{z+1}$$ is a purely imaginary number. Then $$z + \frac{1}{z}$$ is:
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8-digit numbers are formed using the digits 1, 1, 2, 2, 2, 3, 4, 4. The number of such numbers in which the odd digits do no occupy odd places, is:
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Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $$\frac{1}{a}$$ and $$\frac{1}{b}$$. If $$\frac{1}{M}$$ : G is 4 : 5, then a : b can be:
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The least positive integer n such that $$1 - \frac{2}{3} - \frac{2}{3^2} - \ldots - \frac{2}{3^{n-1}} \lt \frac{1}{100}$$, is:
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If $$1 + x^4 + x^5 = \sum_{i=0}^{5} a_i(1+x)^i$$, for all $$x$$ in R, then $$a_2$$ is:
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If $$\left(2 + \frac{x}{3}\right)^{55}$$ is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:
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If a line intercepted between the coordinate axes is trisected at a point A(4, 3), which is nearer to x-axis, then its equation is:
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If the three distinct lines $$x + 2ay + a = 0$$, $$x + 3by + b = 0$$ and $$x + 4ay + a = 0$$ are concurrent, then the point $$(a, b)$$ lies on a:
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For the two circles $$x^2 + y^2 = 16$$ and $$x^2 + y^2 - 2y = 0$$, there is/are:
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Two tangents are drawn from a point $$(-2, -1)$$ to the curve, $$y^2 = 4x$$. If $$\alpha$$ is the angle between them, then $$|\tan\alpha|$$ is equal to:
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The minimum area of a triangle formed by any tangent to the ellipse $$\frac{x^2}{16} + \frac{y^2}{81} = 1$$ and the co-ordinate axes is:
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Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $$p \Rightarrow (q \vee r)$$ is:
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Let $$\bar{X}$$ and M.D. be the mean and the mean deviation about $$\bar{X}$$ of n observations $$x_i$$, i = 1, 2, n. If each of the observations is increased by 5, then the new mean and the mean deviation about the new mean, respectively, are:
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A relation on the set A = {x : |x| < 3, x $$\in$$ Z}, where Z is the set of integers is defined by R = {(x, y) : y = |x|, x $$\neq$$ $$-1$$}. Then the number of elements in the power set of R is:
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If $$A = \begin{bmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{bmatrix}$$ and $$B = \begin{bmatrix} y \\ x \\ 1 \end{bmatrix}$$ be such that AB = $$\begin{bmatrix} 6 \\ 8 \end{bmatrix}$$, then:
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If $$\begin{vmatrix} a^2 & b^2 & c^2\\ (a+\lambda)^2 & (b+\lambda)^2 & (c+\lambda)^2 \\ (a-\lambda)^2 & (b-\lambda)^2 & (c-\lambda)^2 \end{vmatrix}$$ = $$k\lambda \begin{vmatrix} a^2 & b^2 & c^2 \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix}$$, $$\lambda \neq 0$$, then k is equal to:
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If $$f(\theta) = \begin{vmatrix} 1 & \cos\theta & 1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix}$$ and A and B are respectively the maximum and the minimum values of $$f(\theta)$$, then (A, B) is equal to:
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Statement I: The equation $$(\sin^{-1}x)^3 + (\cos^{-1}x)^3 - a\pi^3 = 0$$ has a solution for all $$a \geq \frac{1}{32}$$.
Statement II: For any x $$\in$$ R, $$\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$$ and $$0 \leq \left(\sin^{-1}x - \frac{\pi}{4}\right)^2 \leq \frac{9\pi^2}{16}$$.
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If $$f(x) = x^2 - x + 5$$, $$x > \frac{1}{2}$$, and g(x) is its inverse function, then g'(7) equals:
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Let f, g : R → R be two functions defined by $$f(x) = \begin{cases} x\sin\left(\frac{1}{x}\right), & x \neq 0 \\ 0, & x = 0 \end{cases}$$ and $$g(x) = xf(x)$$. Statement I: f is a continuous function at x = 0. Statement II: g is a differentiable function at x = 0.
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Let $$f$$ and $$g$$ be two differentiable functions on R such that $$f'(x) > 0$$ and $$g'(x) < 0$$ for all $$x \in R$$. Then for all x:
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The integral $$\int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} dx$$ is equal to:
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If [ ] denotes the greatest integer function, then the integral $$\int_0^{\pi} [\cos x] dx$$ is equal to:
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If for a continuous function f(x), $$\int_{-\pi}^{t} (f(x) + x) dx = \pi^2 - t^2$$, for all $$t \geq -\pi$$, then $$f\left(-\frac{\pi}{3}\right)$$ is equal to:
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The general solution of the differential equation, $$\sin 2x\left(\frac{dy}{dx} - \sqrt{\tan x}\right) - y = 0$$, is:
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If $$\hat{x}$$, $$\hat{y}$$ and $$\hat{z}$$ are three unit vectors in threedimensional space, then the minimum value of $$|\hat{x} + \hat{y}|^2 + |\hat{y} + \hat{z}|^2 + |\hat{z} + \hat{x}|^2$$
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A symmetrical form of the line of intersection of the planes $$x = ay + b$$ and $$z = cy + d$$ is:
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If the distance between planes, $$4x - 2y - 4z + 1 = 0$$ and $$4x - 2y - 4z + d = 0$$ is 7, then d is:
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A number x is chosen at random from the set {1, 2, 3, 4, ..., 100}. Define the event: A = the chosen number x satisfies $$\frac{(x-10)(x-50)}{(x-30)} \geq 0$$. Then P(A) is:
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