Can we give dc source to transformer? if not, why? What's the difference between useful flux and mutual flux? why secondary of CT is short circuited?

But if high value resistor is connected to the primary of transformer, then can we give dc?

Some answers are not totally exact.

A voltage on a winding, whatever the waveform is, will creat a current which is related to the flux flowing inside the ferromagnetic material.

However, the magnetic energy storage is limited in this material (as the water volume in a cup is limited). Consequently, you need a way the decrease this energy either through another winding or by reversing the waveform.

On a basic transformer with sinusoidal waveform, the flux increases and decreases alternatively.

On the transformer of a DC/DC power supply, you can have DC voltage current in primary winding beacause a transistor stops energy storage on primary side and involves the secondary winding to empty magnetic energy toward a diode and a capacitor (see flyback architecture for more informations).

Thanks

no we can't where transformer depends on alternative flux and DC current doesn't give us this flux where frequency is zero

we shall short the secondary of current transformer to protect it from burn, where if we keep it open the voltage value on the secondasry side will be very high

No we can't give any DC supply to the transformer..

It is very simple, you will have no output. This is because a transformer works on the principle of elctromagnetic induction, which requires a voltage source that changes with time (alternating source). Thus since a dc supply is unidirectuional, the transformer cannot work.