A and B throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. A wins if he throws 6 before B throws 7 and B wins if he throws 7 before \(A\) throws 6 . If \(A\) begins, the probability of his winning is

The triangle with vertices at \((-2,2),(2,-2)\) and \((1,1)\) is

In $∆ABC,A=75°,B=45°$ then the value of $b+c\sqrt{2}=$

The vector equation of any plane passing through the line of intersection of the planes \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_1=\mathrm{q}_1\) and \(\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{m}}_2=\mathrm{q}_2\) is given by \(\overrightarrow{\mathrm{r}}\left(\overrightarrow{\mathrm{m}}_1+\lambda \overrightarrow{\mathrm{m}}_2\right)=\mathrm{q}_1+\lambda \mathrm{q}_2\) for \(\lambda \in \mathbb{R}\). The vector equation of a plane passing through the point \(2 \hat{i}-3 \hat{j}+\hat{k}\) and the line of intersection of the planes r. \((\hat{i}-2 \hat{j}+3 \hat{k})=5\) and \(\vec{r} \cdot(3 \hat{i}+\hat{j}-2 \hat{k})=7\)

If a square matrix \(A\) is such that \(\left(A^T-\frac{1}{2} I\right)\left(A-\frac{1}{2} I\right)\) \(=\left(A^T+\frac{1}{2} I\right)\left(A+\frac{1}{2} I\right)=I\), where \(I\) is a unit matrix, then \(A\) is

The line passing through the point \(2 \bar{a}+\bar{b}\) and parallel to the vector \(\bar{b}-\bar{c}\) and the plane passing through the point \(\bar{a}\) and parallel to the vectors \(\bar{b}+\bar{c}\) and \(\bar{a}+2 \bar{b}-\bar{c}\) intersect at \(P\). The position vector of \(\mathrm{P}\) is

The product of the degree and order of the differential equation \(\left(\frac{d^2 y}{d x^2}\right)^2-\left(\frac{d y}{d x}\right)^3=y^3\) is

The image of the point \((3,8)\) in the line \(x+3 y=7\) is

In a cell, a copper electrode was used as a cathode. What is the elecrode potential (in V ) of the copper electrode dipped in \(0.1 \mathrm{M} \mathrm{Cu}^{2+}\) solution at 298 K ?
\(\left(\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^{\ominus}=0.34 \mathrm{~V} ; \frac{2.303 \mathrm{RT}}{\mathrm{~F}}=0.06 \mathrm{~V}\right)\)

The coefficient of \(x^5\) in the expansion of \(\left(1+x^2\right)^5(1+x)^4\), is :

The statement related to law of definite proportions is

The formula for calculating 'spin only' magnetic moment is

Force is the mutual interaction between the bodies is according to