In $\triangle ABC$,$\frac{r_1-r}{a}+\frac{r_2-r}{b}+\frac{r_3-r}{c} =$

$A$ galvanometer of resistance $50\,\Omega$ is connected to a battery of $3\,V$ along with a resistance of $2950\,\Omega$ in series. $A$ full-scale deflection of $30$ divisions is obtained in the galvanometer. In order to reduce this deflection to $20$ divisions,the resistance in series should be.......$\Omega$.

The concentration of a $100 \ mL$ solution containing $X \ g$ of $Na_2CO_3$ (molecular weight $= 106$) is $Y \ M$. The values of $X$ and $Y$ are respectively.

Find the value of $k$,if the angle between the straight lines represented by $2x^2 + 5xy + 3y^2 + 6x + 7y + 4 = 0$ is $\tan^{-1}(k)$.

In a metre bridge experiment,the ratio of the left gap resistance to the right gap resistance is $2:3$. The balance point from the left is: (in $~cm$)

$A$ mass $M$,attached to a horizontal spring,executes $S.H.M.$ with amplitude $A_1$. When the mass $M$ passes through its mean position,a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left( \frac{A_1}{A_2} \right)$ is