If the pair of straight lines $xy-x-y+1=0$ and the line $ax+2y-3a=0$ are concurrent,then $a$ is equal to

The least positive integer $n$ for which $(1+i)^n=(1-i)^n$ is

In the circuit shown,the heat produced in $5 \Omega$ resistance due to current $I_2$ is $50 \text{ J/s}$. Then,the heat generated per second in $2 \Omega$ resistance is (in $\text{ J/s}$)

$A$ $100 \ mL$ solution of $0.1 \ N$ $HCl$ was titrated with $0.2 \ N$ $NaOH$ solution. The titration was discontinued after adding $30 \ mL$ of $NaOH$ solution. The remaining titration was completed by adding $0.25 \ N$ $KOH$ solution. The volume of $KOH$ required for completing the titration is $............ \ mL$.

$A$ metallic ring of radius $r$ and cross-sectional area $A$ is fitted into a wooden circular disc of radius $R$ $(R > r)$. If the Young's modulus of the material of the ring is $Y$,the force with which the metal ring expands is

The stress versus strain graphs for wires of two materials $A$ and $B$ are as shown in the figure. If ${Y_A}$ and ${Y_B}$ are the Young's moduli of the materials,then