The number of all possible positive integral values of \(\alpha \) for which the roots of the quadratic equation, \(6x^2 - 11x +\alpha =0\) are rational numbers is

If the functions \(f ( x )=\frac{ x ^3}{3}+2 bx +\frac{a x^2}{2}\) and \(g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b\) have a common extreme point, then \(a+2 b+7\) is equal to

The area enclosed between the curves \(y^2+4 x=4 \text { and } y-2 x=2 \text { is}\)

Equation of two diameters of a circle are \(2 x-3 y=5\) and \(3 x-4 y=7\). The line joining the points \(\left(-\frac{22}{7},-4\right)\) and \(\left(-\frac{1}{7}, 3\right)\) intersects the circle at only one point \(P(\alpha, \beta)\). Then \(17 \beta-\alpha\) is equal to

A set \(S\) contains \(7\) elements. A non-empty subset \(A\) of \(S\) and an element \(x\) of \(S\) are chosen at random. Then the probability that \(x \in A\) is

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

The locus of the mid point of the line segment joining the point \((4,3)\) and the points on the ellipse \(x^{2}+2 y^{2}=4\) is an ellipse with eccentricity

Let \(x _1, x _2 \ldots ., x _{100}\) be in an arithmetic progression, with \(x _1=2\) and their mean equal to \(200\) . If \(y_i=i\left(x_i-i\right), 1 \leq i \leq 100\), then the mean of \(y _1, y _2\), \(y _{100}\) is

Let a random variable X take values \(0,1,2,3\) with \(\mathrm{P}(\mathrm{X}=0)=\mathrm{P}(\mathrm{X}=1)=\mathrm{p}, \mathrm{P}(\mathrm{X}=2)=\mathrm{P}(\mathrm{X}=3)\) and \(\mathrm{E}\left(\mathrm{X}^2\right)=2 \mathrm{E}(\mathrm{X})\). Then the value of \(8 \mathrm{p}-1\) is :

Consider function \(f: A \rightarrow B\) and \(g: B \rightarrow C(A, B, C \subseteq R)\) such that \((gof) ^{-1}\) exists, then:

If the sum of the first \(n\) terms of the series \(\sqrt 3  + \sqrt {75}  + \sqrt {243}  + \sqrt {507}  + ......\) is \(435\sqrt 3 \) , then \(n\) equals

Let \(\alpha, \beta\) be the roots of the equation \(x^2 - x + p = 0\) and \(\gamma, \delta\) be the roots the equation \(x^2 - 4x + q = 0\); \(p, q \in \mathbf{Z}\). If \(\alpha, \beta, \gamma, \delta\) are in G.P., then \(|p + q|\) equals :

Let \(\bigcup \limits_{i=1}^{50} X_{i}=\bigcup \limits_{i=1}^{n} Y_{i}=T\) where each \(X_{i}\) contains \(10\) elements and each \(Y_{i}\) contains \(5\) elements. If each element of the set \(T\) is an element of exactly \(20\) of sets \(X_{i}\) 's and exactly \(6\) of sets \(Y_{i}\) 's, then \(n\) is equal to