Mean of \(5\) observations is \(7.\) If four of these observations are \(6, 7, 8, 10\) and one is missing then the variance of all the five observations is

 Let \( f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}\) be defined as : \(f(\mathrm{x})=|\mathrm{x}-1|\) and  \(g(x)=\left\{\begin{array}{cc}\mathrm{e}^{\mathrm{x}}, & \mathrm{x} \geq 0 \\ \mathrm{x}+1, & \mathrm{x} \leq 0\end{array}\right.\). Then the function \(f(\mathrm{~g}(\mathrm{x}))\) is

Let \(A=\{1,2,3,4,5\}\) and \(B=\{1,2,3,4,5,6\}\). Then the number of functions \(f: A \rightarrow B\) satisfying \(f(1)+f(2)=f(4)-1\) is equal to

Let \(\mathrm{C}\) be a circle with radius \(\sqrt{10}\) units and centre at the origin. Let the line \(x+y=2\) intersects the circle \(\mathrm{C}\) at the points \(\mathrm{P}\) and \(\mathrm{Q}\). Let \(\mathrm{MN}\) be a chord of \(C\) of length \(2\) unit and slope \(-1\) . Then, a distance (in units) between the chord \(PQ\) and the chord \(MN\) is :

The intercepts on \(x-\) axis made by tangents to the curve \(y = \mathop \smallint \limits_0^x \left| t \right|dt,x \in R\) which is parallel to the line \(y = 2x\) are equal to ::

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

The plane containing the line \(\frac{{x - 1}}{1} = \frac{{y - 2}}{2} = \frac{{z - 3}}{3}\) and parallel to the line \(\frac{x}{1} = \frac{y}{1} = \frac{z}{4}\) passes through the point

Let \(a_1, a_2, \ldots . a_{10}\) be \(10\) observations such that \(\sum_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50\) and \(\sum_{\forall \mathrm{k}<\mathrm{j}} \mathrm{a}_{\mathrm{k}} \cdot \mathrm{a}_{\mathrm{j}}=1100\). Then the standard deviation of \(a_1, a_2, \ldots, a_{10}\) is equal to :

Let the median and the mean deviation about the median of \(7\) observation \(170,125,230,190,210\), \(a, b\) be 1\(70\) and \(\frac{205}{7}\) respectively. Then the mean deviation about the mean of these \(7\) observations is :

Let \(f(x)=(x+3)^2(x-2)^3, x \in[-4,4]\). If \(M\) and \(m\) are the maximum and minimum values of \(f\), respectively in \([-4,4]\), then the value of \(M-m\) is :

If \(\mathrm{n}\) is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then \(\mathrm{n}\) is equal to :

Let \(f(x)=x^{2025}-x^{2000}, x\in[0,1]\)and the minimum value of the function \(f(x)\) in the interval [0, 1] be \((80)^{80}(n)^{-81}\). Then n is equal to

Let \(\alpha, \beta\) be the roots of the equation \(x^2+2 \sqrt{2} x-1=0\). The quadratic equation, whose roots are \(\alpha^4+\beta^4\) and \(\frac{1}{10}\left(\alpha^6+\beta^6\right)\), is :