$(30$ pts$)$ Consider the bridge circuit shown below where the output voltage Vo is measured by a multimeter. A platinum resistance thermometer whose resistance value at $20C$ is $500$ and sensitivity is $5\frac{}{}C$ is inserted to the bridge from Ru position. $R1=500,R2=515,R3=480,$ the variable resistance $R5=100$ and $Vs=5V.$

i$.$ Determine the ratio $R\frac{a}{R}b$ such that at $25C$ the bridge is null$-$type.

ii$.$ Determine the output voltage, if the internal resistance of multimeter is $Rm=1k.$