Consider the following frequency distribution :

Class:	\(0-6\)	\(6-12\)	\(12-18\)	\(18-24\)	\(24-30\)
Frequency :	\(a\)	\(b\)	\(12\)	\(9\)	\(5\)

If mean \(=\frac{309}{22}\) and median \(=14\), than value \((a-b)^{2}\) is equal to \(.....\)

Let \(x\) denote the total number of one-one functions from a set \(A\) with \(3\) elements to a set \(B\) with \(5\) elements and \(y\) denote the total number of one-one functions from the set \(A\) to the set \(A \times B\). Then ...... .

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 ...........

Let \(f(x)\) be a polynomial of degree \(3\) such that \(f(-1)=10, f(1)=-6, f(\mathrm{x})\) has a critical point at \(\mathrm{x}=-1\) and \(f^{\prime}(\mathrm{x})\) has a critical point at \(\mathrm{x}=1\) Then \(f(x)\) has a local minima at \(x=\)

If \(a, b\) and \(c\) are the greatest value of \(^{19} \mathrm{C}_{\mathrm{p}},^{20} \mathrm{C}_{\mathrm{q}}\) and \(^{21 }\mathrm{C}_{\mathrm{r}}\) respectively, then

Let \(f(x)=\left|(x-1)\left(x^{2}-2 x-3\right)\right|+x-3, x \in R\). If \(m\) and \(M\) are respectively the number of points of local minimum and local maximum of \(f\) in the interval \((0,4)\), then \(m + M\) is equal to

A company has two plants \(\mathrm{A}\) and \(\mathrm{B}\) to manufacture motorcycles. \(60 \%\) motorcycles are manufactured at plant \(\mathrm{A}\) and the remaining are manufactured at plant B. \(80 \%\) of the motorcycles manufactured at plant \(\mathrm{A}\) are rated of the standard quality, while \(90 \%\) of the motorcycles manufactured at plant \(B\) are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If \(p\) is the probability that it was manufactured at plant \(\mathrm{B}\), then \(126 \mathrm{p}\) is

Let \(\vec a,\vec b\;\) and \(\;\vec c\) be three unit vectors such that  \(\vec a \times \left( {\vec b \times \vec c} \right) = \frac{{\sqrt 3 }}{2}\left( {\vec b + \vec c} \right)\) . If  \(\vec b\) is not parallel to \(\vec c\) then angle between \(\vec a\;\) and \(\;\vec b\) is :

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

If \(\int_{0}^{\pi}\left(\sin ^{3} x\right) e^{-\sin ^{2} x} d x=\alpha-\frac{\beta}{e} \int_{0}^{1} \sqrt{t} e^{t} d t\), then \(\alpha+\beta\) is equal to \(....\)

Let \(p\) and \(p+2\) be prime numbers and let \(\Delta=\left|\begin{array}{ccc}p ! & (p+1) ! & (p+2) ! \\ (p+1) ! & (p+2) ! & (p+3) ! \\ (p+2) ! & (p+3) ! & (p+4) !\end{array}\right|\) Then the sum of the maximum values of \(\alpha\) and \(\beta\), such that \(p ^{\alpha}\) and \(( p +2)^{\beta}\) divide \(\Delta\), is \(........\)

The probabilities that players \(A\) and \(B\) of a team are selected for the captaincy for a tournament are \(0.6\) and \(0.4\), respectively. If \(A\) is selected the captain, the probability that the team wins the tournament is \(0.8\) and if \(B\) is selected the captain, the probability that the team wins the tournament is \(0.7\). Then the probability, that the team wins the tournament, is :

Let \(\alpha\) and \(\beta\) be the roots of \(x^2+\sqrt{3 x}-16=0\), and \(\gamma\) and \(\delta\) be the roots of \(x^2+3 x-1=0\). If \(P_n=\alpha^n+\beta^n\) and \(Q_n=\gamma^n+\delta^n\), then \(\frac{\mathrm{P}_{25}+\sqrt{3 \mathrm{P}_{24}}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}\) is equal to