Question

For an $LCR$ series $AC$ circuit match the following two columns.

Answer 1

In L-C-R
$Z=\sqrt{(R^2+(X_L-X_C{)}^2)}$
and $I=V/Z$
$\cos \varphi =R/Z=1/\sqrt{(1+((X_L-X_C)/R{)}^2)}$
$Power=V_{rms}{I}_{rms}\cos \varphi$

For(A) if R increases then Z increases therefore, current decreases.
Now Power can be written as $P=V_{rms}^{2}R/Z^2=V_{rms}^2/(R+((X_L-X_C){)}^2/R)$
Therefore,Power will increases when R increases but $(X_L-X_C{)}^2/R>R$
and power will decrease after $R>(X_L-X_C{)}^2/R$
Therefore,2 and 4

For (B)when capacitance is increased $X_C$ decreases Since, $X_C=1/\omega C$
Current will increases when $X_C>X_L$ and C is increased and current will decreases when $X_C<X_L$ and C is increased
current may increase or decrease.
Now Power is given by$P=V_{rms}^{2}R/Z^2=V_{rms}^2/(R+((X_L-X_C){)}^2/R)$
Therefore, power will increase when $X_C>X_L$ and C is increased
power will decrease when $X_C<X_L$ and C is increased.
Therefore, 3 &4

For (C)when L is increased $X_L$ increases
Therefore, current will increase when $X_L<X_C$ and L is increased
current will decrease when $X_L>X_C$ and L is increased
Similarly for power also.
Therefore, 3&4

Now for frequency also $(X_L-X_C)=\omega L-1/\omega C$
therefore, current will increase when X_L<X_C and frequency is increased.
current will decrease when X_L>X_C and frequency is increased.
Similarly for power will also increase and decrease
Therefore, 3&4