Let the lengths of intercepts on \(x\) -axis and \(y\) -axis made by the circle \(x^{2}+y^{2}+a x+2 a y+c=0\) \((a < 0)\) be \(2 \sqrt{2}\) and \(2 \sqrt{5}\), respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line \(x +2 y =0,\) is euqal to :

For \(\alpha, \beta \in\left(0, \frac{\pi}{2}\right)\), let \(3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)\) and a real number \(k\) be such that \(\tan \alpha=k \tan \beta\). Then the value of \(\mathrm{k}\) is equal to :

A box \('A'\) contanis \(2\) white, \(3\) red and \(2\) black balls. Another box \('B'\) contains \(4\) white, \(2\) red and \(3\) black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box \('B'\) is

Consider a matrix \(A =\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]\), where \(\alpha, \beta, \gamma\) are three distinct natural numbers. If \(\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}\), then the number of such \(3 -\) tuples \((\alpha, \beta, \gamma)\) is \(.....\)

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

If \(1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots\) \(+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k\), where \(m, n, k \in \mathbf{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :

The number of solutions of the equation \(\sin x=\) \(\cos ^{2} x\) in the interval \((0,10)\) is

Let \(S\) be the set of all values of \(a_1\) for which the mean deviation about the mean of \(100\) consecutive positive integers \(a _1, a _2, a _3, \ldots ., a _{100}\) is \(25\). Then \(S\) is

If the system of linear equations \(2 x+y-z=3\) \(x-y-z=\alpha\) \(3 x+3 y+\beta z=3\) has infinitely many solution, then \(\alpha+\beta-\alpha \beta\) is equal to .... .

The minimum value of the sum of the squares of the roots of \(x^{2}+(3-a) x+1=2 a\) is.

Out of all the patients in a hospital \(89\, \%\) are found to be suffering from heart ailment and \(98\, \%\) are suffering from lungs infection. If \(\mathrm{K}\, \%\) of them are suffering from both ailments, then \(\mathrm{K}\) can not belong to the set :

\(\text { If } \int_{0}^{100 \pi} \frac{\sin ^{2} x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}} d x=\frac{\alpha \pi^{3}}{1+4 \pi^{2}}, \alpha \in R\) where \([x]\) is the greatest integer less than or equal to \(x\), then the value of \(\alpha\) is :

Consider \(10\) observation \(\mathrm{x}_1, \mathrm{x}_2, \ldots, \mathrm{x}_{10}\). such that \(\sum_{i=1}^{10}\left(x_i-\alpha\right)=2\) and \(\sum_{i=1}^{10}\left(x_i-\beta\right)^2=40\), where \(\alpha, \beta\) are positive integers. Let the mean and the variance of the observations be \(\frac{6}{5}\) and \(\frac{84}{25}\) respectively. The \(\frac{\beta}{\alpha}\) is equal to :