If the coefficients of \(x^{7}\) and \(x^{8}\) in the expansion of \(\left(2+\frac{x}{3}\right)^{n}\) are equal, then the value of \(n\) is equal to \(.....\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three units vectors such that \(\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0} .\) If \(\lambda=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}} \) and \(\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}},\) then the ordered pair \((\lambda, {\mathrm{\vec d}})\) is equal to 

A bag contains \(4\) red and \(6\) black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are re­ turned to the bag. Ifnow a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

If \(\alpha \), \(\beta \) and \(\gamma \) are three consecutive terms of a non-constant \(G.P.\) such that the equations \(\alpha x^2 + 2\beta x + \gamma  = 0\) and \(x^2 + x -1 = 0\) have a common root, then \(\alpha(\beta  + \gamma )\) is equal to

If \(\dfrac{\pi}{4} + \displaystyle\sum_{p=1}^{11} \tan^{-1}\left(\dfrac{2^{p-1}}{1 + 2^{2p-1}}\right) = \alpha\), then \(\tan\alpha\) is equal to __________.

Let \(\mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]\) and \(\mathrm{B}=\mathrm{I}+\operatorname{adj}(\mathrm{A})+(\operatorname{adj} \mathrm{A})^2+\ldots+\) \((\operatorname{adj} \mathrm{A})^{10}\). Then, the sum of all the elements of the matrix \(B\) is :

The square of the distance of the point \(P(5, 6, 7)\) from the line \(\dfrac{x-2}{2} = \dfrac{y-5}{3} = \dfrac{z-2}{4}\) is equal to:

An ellipse passes through the foci of the hyperbola, \(9x^2 - 4y^2 = 36\) and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is \(\frac {1}{2}\), then which of the following points does not lie on the ellipse?

If \(\lambda \) be the ratio of the roots of the quadratic equation in \(x, 3m^2x^2 + m(m -4)x + 2 = 0\), then the least value of \(m\) for which \(\lambda  + \frac{1}{\lambda } = 1\), is

If \(|x|<1,|y|<1\) and \(x \neq y,\) then the sum to infinity of the following series \((x+y)+\left(x^{2}+x y+y^{2}\right)+\left(x^{3}+x^{2} y+x y^{2}+y^{3}\right)+\ldots .\)

If \(f(x)\) is a quadratic expression such that \(f(1) + f (2)\, = 0\) , and \(-1\) is a root of \(f(x)\, = 0\), then the other root of \(f(x)\, = 0\) is

\(S = {\tan ^{ - 1}}\left( {\frac{1}{{{n^2} + n + 1}}} \right) + {\tan ^{ - 1}}\left( {\frac{1}{{{n^2} + 3n + 3}}} \right) + ..... + {\tan ^{ - 1}}\left( {\frac{1}{{1 + \left( {n + 19} \right)\left( {n + 20} \right)}}} \right)\) , then \(tan\,S\) is equal to 

The probability that a randomly chosen \(2 \times 2\) matrix with all the entries from the set of first \(10\) primes, is singular, is equal to