The least integral value $$\alpha$$ of $$x$$ such that $$\frac{x-5}{x^2+5x-14} > 0$$, satisfies :
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The least integral value $$\alpha$$ of $$x$$ such that $$\frac{x-5}{x^2+5x-14} > 0$$, satisfies :
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Let $$a = \text{Im}\left(\frac{1+z^2}{2iz}\right)$$, where z is any non-zero complex number. The set $$A = \{a : |z| = 1$$ and $$z \neq \pm 1\}$$ is equal to:
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The sum of the series : $$(2)^2 + 2(4)^2 + 3(6)^2 + \ldots$$ upto 10 terms is :
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If $$a_1, a_2, a_3, \ldots, a_n, \ldots$$ are in A.P. such that $$a_4 - a_7 + a_{10} = m$$, then the sum of first 13 terms of this A.P., is :
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The sum of the rational terms in the binomial expansion of $$\left(2^{\frac{1}{2}} + 3^{\frac{1}{5}}\right)^{10}$$ is :
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The number of solutions of the equation $$\sin 2x - 2\cos x + 4\sin x = 4$$ in the interval $$[0, 5\pi]$$ is :
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If two lines $$L_1$$ and $$L_2$$ in space, are defined by
$$L_1 = \{x = \sqrt{\lambda}y + (\sqrt{\lambda} - 1), z = (\sqrt{\lambda} - 1)y + \sqrt{\lambda}\}$$ and
$$L_2 = \{x = \sqrt{\mu}y + (1 - \sqrt{\mu}), z = (1 - \sqrt{\mu})y + \sqrt{\mu}\}$$
then $$L_1$$ is perpendicular to $$L_2$$, for all nonnegative reals $$\lambda$$ and $$\mu$$, such that :
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Let $$\theta_1$$ be the angle between two lines $$2x + 3y + c_1 = 0$$ and $$-x + 5y + c_2 = 0$$ and $$\theta_2$$ be the angle between two lines $$2x + 3y + c_1 = 0$$ and $$-x + 5y + c_3 = 0$$, where $$c_1, c_2, c_3$$ are any real numbers :
Statement-1: If $$c_2$$ and $$c_3$$ are proportional, then $$\theta_1 = \theta_2$$.
Statement-2: $$\theta_1 = \theta_2$$ for all $$c_2$$ and $$c_3$$.
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If the circle $$x^2 + y^2 - 6x - 8y + (25 - a^2) = 0$$ touches the axis of x, then a equals.
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The point of intersection of the normals to the parabola $$y^2 = 4x$$ at the ends of its latus rectum is :
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A tangent to the hyperbola $$\frac{x^2}{4} - \frac{y^2}{2} = 1$$ meets x-axis at P and y-axis at Q. Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). Then R lies on :
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For integers $$m$$ and $$n$$, both greater than 1, consider the following three statements : P : m divides n Q : m divides $$n^2$$ R : m is prime, then
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If the median and the range of four numbers $$\{x, y, 2x + y, x - y\}$$, where $$0 < y < x < 2y$$, are 10 and 28 respectively, then the mean of the numbers is :
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If the extremities of the base of an isosceles triangle are the points (2a, 0) and (0, a) and the equation of one of the sides is $$x = 2a$$, then the area of the triangle, in square units, is :
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On the sides AB, BC, CA of a $$\triangle ABC$$, 3, 4, 5 distinct points (excluding vertices A, B, C) are respectively chosen. The number of triangles that can be constructed using these chosen points as vertices are :
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Let $$R = \{(x, y) : x, y \in N$$ and $$x^2 - 4xy + 3y^2 = 0\}$$, where N is the set of all natural numbers. Then the relation R is :
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Let A, other than I or - I, be a $$2 \times 2$$ real matrix such that $$A^2 = I$$, I being the unit matrix. Let Tr(A) be the sum of diagonal elements of A.
Statement-1: Tr(A) = 0
Statement-2: det(A) = -1
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Statement-1: The system of linear equations
$$x + (\sin\alpha)y + (\cos\alpha)z = 0$$
$$x + (\cos\alpha)y + (\sin\alpha)z = 0$$
$$x - (\sin\alpha)y - (\cos\alpha)z = 0$$
has a non-trivial solution for only one value of $$\alpha$$ lying in the interval $$(0, \frac{\pi}{2})$$.
Statement-2: The equation in $$\alpha$$ $$\begin{vmatrix} \cos\alpha & \sin\alpha & \cos\alpha \\ \sin\alpha & \cos\alpha & \sin\alpha \\ \cos\alpha & -\sin\alpha & \cos\alpha \end{vmatrix} = 0$$ has only one solution lying in the interval $$(0, \frac{\pi}{2})$$.
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$$S = \tan^{-1}\left(\frac{1}{n^2+n+1}\right) + \tan^{-1}\left(\frac{1}{n^2+3n+3}\right) + \ldots + \tan^{-1}\left(\frac{1}{1+(n+19)(n+20)}\right)$$, then $$\tan S$$ is equal to :
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Let $$f$$ be a composite function of $$x$$ defined by $$f(u) = \frac{1}{u^2+u-2}$$, $$u(x) = \frac{1}{x-1}$$. Then the number of points $$x$$ where $$f$$ is discontinuous is :
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If $$f(x) = \sin(\sin x)$$ and $$f''(x) + \tan x \cdot f'(x) + g(x) = 0$$, then $$g(x)$$ is :
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If the curves $$\frac{x^2}{\alpha} + \frac{y^2}{4} = 1$$ and $$y^3 = 16x$$ intersect at right angles, then a value of $$\alpha$$ is :
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The cost of running a bus from A to B is Rs. $$(av + b/v)$$ where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/h) of the bus is :
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If a curve passes through the point $$(2, \frac{7}{2})$$ and has slope $$\left(1 - \frac{1}{x^2}\right)$$ at any point $$(x, y)$$ on it, then the ordinate of the point on the curve whose abscissa is -2 is :
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The integral $$\int \frac{x \ dx}{2-x^2+\sqrt{2-x^2}}$$ equals :
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The value of $$\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x} dx$$ is :
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The area under the curve $$y = |\cos x - \sin x|$$, $$0 \leq x \leq \frac{\pi}{2}$$, and above x-axis is :
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If $$\vec{a}$$ and $$\vec{b}$$ are non-collinear vectors, then the value of $$\alpha$$ for which the vectors $$\vec{u} = (\alpha - 2)\vec{a} + \vec{b}$$ and $$\vec{v} = (2 + 3\alpha)\vec{a} - 3\vec{b}$$ are collinear is :
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If the projections of a line segment on the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :
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A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are $$\frac{3}{4}$$, $$\frac{1}{2}$$, $$\frac{5}{8}$$. The probability that the target is hit by A or B but not by C is :
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