Let \(f:(1,3) \rightarrow \mathrm{R}\) be a function defined by \(f(\mathrm{x})=\frac{\mathrm{x}[\mathrm{x}]}{1+\mathrm{x}^{2}},\) where \([\mathrm{x}]\) denotes the greatest integer \(\leq \mathrm{x} .\) Then the range of \(f\) is

Let the mean of 6 observation \(1,2,4,5, x\) and \(y\) be \(5\) and their variance be \(10\) . Then their mean deviation about the mean is equal to \(........\).

\(\cos ^{-1}(\cos (-5))+\sin ^{-1}(\sin (6))-\tan ^{-1}(\tan (12))\) is equal to : (The inverse trigonometric functions take the principal values)

Let \(\lambda \neq 0\) be a real number. Let \(\alpha, \beta\) be the roots of the equation \(14 x^2-31 x+3 \lambda=0\) and \(\alpha, \gamma\) be the roots of the equation \(35 x^2-53 x+4 \lambda=0\). Then \(\frac{3 \alpha}{\beta}\) and \(\frac{4 \alpha}{\gamma}\) are the roots of the equation :

Let \(S_n\) be the sum to n-terms of an arithmetic progression \(3,7,11, \ldots\) If \(40<\left(\frac{6}{\mathrm{n}(\mathrm{n}+1)} \sum_{\mathrm{k}=1}^{\mathrm{n}} \mathrm{S}_{\mathrm{k}}\right)<42\), then \(\mathrm{n}\) equals

Let \(3,7,11,15, \ldots, 403\) and \(2,5,8,11, \ldots, 404\) be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ...........

An online exam is attempted by \(50\) candidates out of which \(20\) are boys. The average marks obtained by boys is \(12\) with a variance \(2 .\) The variance of marks obtained by \(30\) girls is also \(2 .\) The average marks of all \(50\) candidates is \(15 .\) If \(\mu\) is the average marks of girls and \(\sigma^{2}\) is the variance of marks of \(50\) candidates, then \(\mu+\sigma^{2}\) is equal to ...... .

Let the line \(\ell: x =\frac{1- y }{-2}=\frac{ z -3}{\lambda}, \lambda \in R\) meet the plane \(P : x +2 y +3 z =4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(\ell\) and the plane \(P\) is \(\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)\), then \(\alpha+2 \beta+6 \gamma\) is equal to

Let \(n \geq 5\) be an integer. If \(9^{n}-8 n-1=64 \alpha\) and \(6^{ n }-5 n -1=25 \beta\), then \(\alpha-\beta\) is equal to

Let \(0 < z < y < x\) be three real numbers such that \(\frac{1}{ x }, \frac{1}{ y }, \frac{1}{ z }\) are in an arithmetic progression and \(x\), \(\sqrt{2} y, z\) are in a geometric progression. If \(x y+y z\) \(+z x=\frac{3}{\sqrt{2}} x y z\), then \(3(x+y+z)^2\) is equal to \(............\).

The coefficient of \(x^5\) in the expansion of \(\left(2 x^3-\frac{1}{3 x^2}\right)^5\) is

If \(\left(\frac{1+i}{1-i}\right)^{\frac{m}{2}}=\left(\frac{1+i}{i-1}\right)^{\frac{n}{3}}=1,(m, n \in N)\) then the greatest common divisor of the least values of \(m\) and \(n\) is

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