The mean and the standard deviation \((s.d.)\)  of five observations are \(9\) and \(0,\) respectively. If one of the observations is changed such that the mean of the new set of five observations becomes \(10,\)  then their \(s.d.\)  is?

The solution curve, of the differential equation \(2 \mathrm{y} \frac{\mathrm{dy}}{\mathrm{dx}}+3=5 \frac{\mathrm{dy}}{\mathrm{dx}}\), passing through the point \((0,1)\) is a conic, whose vertex lies on the line :

Let each of the two ellipses \(E _1: \frac{ x ^2}{ a ^2}+\frac{ y ^2}{b^2}=1,( a > b )\) and \(E _2: \frac{ x ^2}{A^2}+\frac{ y ^2}{B^2}=1,(A< B )\) have eccentricity \(\frac{4}{5}\). Let the lengths of the latus recta of \(E_1\) and \(E_2\) be \(\ell_1\) and \(\ell_2\), respectively, such that \(2 \ell_1^2=9 \ell_2\). If the distance between the foci of \(E_1\) is 8 , then the distance between the foci of \(E _2\) is

If \((1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\), then \(\dfrac{9a_9}{a_{10}}\) is equal to __________.

In a group of \(100\) persons \(75\) speak English and \(40\) speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is \(\alpha\) and the number of persons who speak only Hindi is \(\beta\), then the eccentricity of the ellipse \(25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2\) is \(.......\)

Let \(f(x)=\frac{1}{7-\sin 5 x}\) be a function defined on \(R\). Then the range of the function \(f(x)\) is equal to:

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 the mean and variance of six observations \(7,10,11,15, a, b\) are \(10\) and \(\frac{20}{3}\), respectively, then the value of \(|a-b|\) is equal to:

Let \(y=y(x)\) be a solution of the differential equation \((x \cos x) d y+(x y \sin x+y \cos x-1) d x=0\), \(0 < x < \frac{\pi}{2}\). If \(\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}\), then \(\left|\frac{\pi}{6} y^{\prime \prime}\left(\frac{\pi}{6}\right)+2 y^{\prime}\left(\frac{\pi}{6}\right)\right|\) is equal to \(.........\).

If \(\quad A=\left[\begin{array}{cc}\cos \theta & \text { isin } \theta \\ \operatorname{isin} \theta & \cos \theta\end{array}\right], \left(\theta=\frac{\pi}{24}\right)\) and \(A^{5}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) where \(i=\sqrt{-1},\) then which one of the following is not true?

The probabilities of three events \(A , B\) and \(C\) are given by \(P ( A )=0.6, P ( B )=0.4\) and \(P ( C )=0.5\) If \(P ( A \cup B )=0.8, P ( A \cap C )=0.3, P ( A \cap B \cap\) \(C)=0.2, P(B \cap C)=\beta\) and \(P(A \cup B \cup C)=\alpha\) where \(0.85 \leq \alpha \leq 0.95,\) then \(\beta\) lies in the interval

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at \(315^{\text {th }}\) position in this arrangement is :

If \(z =2+3 i\), then \(z ^{5}+(\overline{ z })^{5}\) is equal to.