A bag contains 6 blue and 6 green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one blue and one green ball is :

Let \(\vec{u}=\hat{i}-\hat{j}-2 \hat{k}, \vec{v}=2 \hat{i}+\hat{j}-\hat{k}, \vec{v} \cdot \vec{w}=2\) and \(\vec{v} \times \vec{w}=\vec{u}+\lambda \vec{v}\), then \(\vec{u} \cdot \vec{w}\) is equal to

Let the vertices \(Q\) and \(R\) of the triangle \(P Q R\) lie on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}, Q R=5\) and the coordinates of the point \(P\) be \((0,2,3)\). If the area of the triangle \(P Q R\) is \(\frac{m}{n}\) then :

\(\mathop {\lim }\limits_{y \to 0} \frac{{\sqrt {1 + \sqrt {1 + {y^4}} }  - \sqrt 2 }}{{{y^4}}} = \)

If \(x,y,z\)  are in \(A.P.\) and  \({\tan ^{ - 1}}x,{\tan ^{ - 1}}y\) and \({\tan ^{ - 1}}z\) are also in other \(A.P.\) then  . . . 

Let the equation of the plane, that passes through the point \((1,4,-3)\) and contains the line of intersection of the planes \(3 x-2 y+4 z-7=0\) and \(x+5 y-2 z+9=0\), be \(\alpha x+\beta y+\gamma z+3=0\), then \(\alpha+\beta+\gamma\) is equal to :

If \(\frac{1}{{\sqrt \alpha  }}\) and \(\frac{1}{{\sqrt \beta  }}\) are the roots of the equation , \(ax^2 + bx + 1 = 0\) \(\left( {a \ne 0,a,b \in R} \right)\), then the equation, \(x ( x + b^3 ) + (a^3 - 3abx ) = 0\) has roots

Let the values of p , for which the shortest distance between the lines \(\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}\) and \(\overrightarrow{\mathrm{r}}=(\mathrm{p} \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})\) is \(\frac{1}{\sqrt{6}}\), be \(\mathrm{a}, \mathrm{b}\), \((a \lt b)\). Then the length of the latus rectum of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is :-

If \(P = \left[ {\begin{array}{*{20}{c}}1&\alpha &3\\1&3&3\\2&4&4\end{array}} \right]\) is the adjoint of  \(3×3 \) matrix \(A\)  and \( |A|=4\)  then \(\alpha \) is equal to 

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

Let \(H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N\). Let \(k\) be the smallest even value of \(n\) such that the eccentricity of \(H _{ k }\) is a rational number. If \(l\) is length of the latus return of \(H _{ k }\), then \(21 l\) is equal to \(.......\).

Let \(A = \left\{ {\left( {x,y} \right):{y^2} \le 4x,y - 2x \ge  - 4} \right\}\) .The area of the region \(A\) is

Let \(\vec a = 3\hat i + 2\hat j + x\hat k\) and \(\vec b = \hat i - \hat j + \hat k\), for some real \(x\). Then \(\left| {\vec a \times \vec b} \right| = r\) is possible if