The values of 'a' for which one root of the equation $$x^2 - (a+1)x + a^2 + a - 8 = 0$$ exceeds 2 and the other is lesser than 2, are given by :
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The values of 'a' for which one root of the equation $$x^2 - (a+1)x + a^2 + a - 8 = 0$$ exceeds 2 and the other is lesser than 2, are given by :
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If $$Z_1 \neq 0$$ and $$Z_2$$ be two complex numbers such that $$\frac{Z_2}{Z_1}$$ is a purely imaginary number, then $$\left|\frac{2Z_1 + 3Z_2}{2Z_1 - 3Z_2}\right|$$ is equal to:
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A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :
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Let $$a_1, a_2, a_3, \ldots$$ be an A.P, such that $$\frac{a_1 + a_2 + \ldots + a_p}{a_1 + a_2 + a_3 + \ldots + a_q} = \frac{p^3}{q^3}$$; $$p \neq q$$. Then $$\frac{a_6}{a_{21}}$$ is equal to:
The sum of the series: $$1 + \frac{1}{1+2} + \frac{1}{1+2+3} + \ldots$$ upto 10 terms, is:
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The ratio of the coefficient of $$x^{15}$$ to the term independent of $$x$$ in the expansion of $$\left(x^2 + \frac{2}{x}\right)^{15}$$ is:
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A value of $$x$$ for which $$\sin\left(\cot^{-1}(1+x)\right) = \cos\left(\tan^{-1}x\right)$$, is :
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A light ray emerging from the point source placed at P(1, 3) is reflected at a point Q in the axis of x. If the reflected ray passes through the point R(6, 7), then the abscissa of Q is:
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If the three lines $$x - 3y = p$$, $$ax + 2y = q$$ and $$ax + y = r$$ form a right-angled triangle then :
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If each of the lines $$5x + 8y = 13$$ and $$4x - y = 3$$ contains a diameter of the circle $$x^2 + y^2 - 2(a^2 - 7a + 11)x - 2(a^2 - 6a + 6)y + b^3 + 1 = 0$$, then :
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Statement-1: The slope of the tangent at any point P on a parabola, whose axis is the axis of x and vertex is at the origin, is inversely proportional to the ordinate of the point P.
Statement-2: The system of parabolas $$y^2 = 4ax$$ satisfies a differential equation of degree 1 and order 1.
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Equation of the line passing through the points of intersection of the parabola $$x^2 = 8y$$ and the ellipse $$\frac{x^2}{3} + y^2 = 1$$ is :
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If $$a$$ and $$c$$ are positive real numbers and the ellipse $$\frac{x^2}{4c^2} + \frac{y^2}{c^2} = 1$$ has four distinct points in common with the circle $$x^2 + y^2 = 9a^2$$, then
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The value of $$\lim_{x \to 0} \frac{1}{x}\left[\tan^{-1}\left(\frac{x+1}{2x+1}\right) - \frac{\pi}{4}\right]$$ is :
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Statement-1: The statement $$A \rightarrow (B \rightarrow A)$$ is equivalent to $$A \rightarrow (A \vee B)$$.
Statement-2: The statement $$\sim [(A \wedge B) \rightarrow (\sim A \vee B)]$$ is a Tautology.
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The mean of a data set consisting of 20 observations is 40. If one observation 53 was wrongly recorded as 33, then the correct mean will be:
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The matrix $$A^2 + 4A - 5I$$, where $$I$$ is identity matrix and $$A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}$$, equals :
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If $$a, b, c$$ are sides of a scalene triangle, then the value of $$\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix}$$ is :
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Let $$A = \{1, 2, 3, 4\}$$ and $$R : A \rightarrow A$$ be the relation defined by $$R = \{(1,1), (2,3), (3,4), (4,2)\}$$. The correct statement is :
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Let $$f(x) = \frac{x^2 - x}{x^2 + 2x}$$, $$x \neq 0, -2$$. Then $$\frac{d}{dx}\left[f^{-1}(x)\right]$$ (wherever it is defined) is equal to:
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Statement-1: The equation $$x\log x = 2 - x$$ is satisfied by at least one value of $$x$$ lying between 1 and 2.
Statement-2: The function $$f(x) = x\log x$$ is an increasing function in $$[1, 2]$$ and $$g(x)=2-x$$ is a decreasing function in $$[1, 2]$$ and the graphs represented by these functions intersect at a point in $$[1, 2]$$.
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If the surface area of a sphere of radius $$r$$ is increasing uniformly at the rate 8 cm$$^2$$/s, then the rate of change of its volume is:
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If $$\int \frac{dx}{x+x^7} = p(x)$$ then, $$\int \frac{x^6}{x+x^7}dx$$ is equal to:
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If $$x = \int_0^y \frac{dt}{\sqrt{1+t^2}}$$, then $$\frac{d^2y}{dx^2}$$ is equal to :
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The area bounded by the curve $$y = \ln(x)$$ and the lines $$y = 0$$, $$y = \ln(3)$$ and $$x = 0$$ is equal to:
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Let $$\vec{a} = 2\hat{i} - \hat{j} + \hat{k}$$, $$\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{c} = \hat{i} + \hat{j} - 2\hat{k}$$ be three vectors. A vector of the type $$\vec{b} + \lambda\vec{c}$$ for some scalar $$\lambda$$, whose projection on $$\vec{a}$$ is of magnitude $$\sqrt{\frac{2}{3}}$$ is :
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The vector $$(\hat{i} \times \vec{a} \cdot \vec{b})\hat{i} + (\hat{j} \times \vec{a} \cdot \vec{b})\hat{j} + (\hat{k} \times \vec{a} \cdot \vec{b})\hat{k}$$ is equal to:
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A vector $$\vec{n}$$ is inclined to the $$x$$-axis at 45°, to the $$y$$-axis at 60° and at an acute angle to the $$z$$-axis. If $$\vec{n}$$ is a normal to a plane passing through the point $$(\sqrt{2}, -1, 1)$$ then the equation of the plane is :
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If the lines $$\frac{x+1}{2} = \frac{y-1}{1} = \frac{z+1}{3}$$ and $$\frac{x+2}{2} = \frac{y-k}{3} = \frac{z}{4}$$ are coplanar, then the value of $$k$$ is :
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The probability of a man hitting a target is $$\frac{2}{5}$$. He fires at the target $$k$$ times ($$k$$, a given number). Then the minimum $$k$$, so that the probability of hitting the target at least once is more than $$\frac{7}{10}$$, is :
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