Question

Assertion The $V$ -i graph for a conductor at temperatures $T_{1}$ and $T_{2}$ is shown in figure, then $\left(T_{2}-T_{1}\right)$ is proportional to $\cot 2 \theta$.
Reason Resistance of conductor decrease with rise in temperature.

• A. If both Assertion and Reason are true and the Reason is correct explanation of the Assertion
• B. If both Assertion and Reason are true but Reason is not correct explanation of the Assertion
• C. If Assertion is true but Reason is false
• D. If Assertion is false hut the Reason is true

Answer 1

Answer: (B)

From the relation for $V$ -i graph
$R=\tan \theta=R_{0}(1+\alpha T)$
where, $\theta$ is angle made by $V$ -i graph with $i$ axis.
So, $ R_{1} =\tan \theta=R_{0}\left(1+\alpha T_{1}\right)$ ......(i)
$ R_{2} =\tan \left(90^{\circ}-\theta\right)=\cot \theta $
$=R_{0}\left(1+\alpha T_{2}\right) $ .......(ii)
So, $\cot \theta-\tan \theta=R_{0} \alpha\left(T_{2}-T_{1}\right)$
or $\frac{\cos \theta}{\sin \theta}-\frac{\sin \theta}{\cos \theta}=R_{0} \alpha\left(T_{2}-T_{1}\right)$
$\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\sin \theta \cos \theta}=\frac{2 \cos 2 \theta}{\sin 2 \theta}=R_{0} \alpha\left(T_{2}-T_{1}\right)$
Hence, $T_{2}-T_{1} \propto \cot 2 \theta$