1. Initial equilibrium in air

Let each string make angle $\theta$ with the vertical. Then $\alpha = 2\theta.$

For one sphere, in air:

• Weight downward: $mg$
• Tension: $T$
• Electrostatic repulsion horizontally: $F$

Equilibrium gives $T\cos\theta = mg,$ $T\sin\theta = F.$

Hence, $\tan\theta = \frac{F}{mg}. \qquad (1)$

2. After immersion in dielectric liquid

Now two things change:

• The electrostatic force reduces by dielectric constant $K=21$: $F' = \frac{F}{K} = \frac{F}{21}.$

• There is buoyant force upward.

If density of sphere material is $\rho$, sphere volume is $V$, then $m = \rho V.$

Weight of sphere: $mg = \rho V g.$

Buoyant force in liquid of density $\rho_l = 800\,\text{kg m}^{-3}$: $B = \rho_l V g = 800 V g.$

So effective downward force is $mg - B = (\rho - 800)Vg.$

Let tension in liquid be $T'$. Equilibrium gives $T'\cos\theta = mg - B,$ $T'\sin\theta = F'.$

Therefore, $\tan\theta = \frac{F'}{mg-B}. \qquad (2)$

3. Given angle remains the same

Since the angle remains same, $\theta$ remains same. So from (1) and (2), $\frac{F}{mg} = \frac{F'}{mg-B}.$

Substitute $F' = F/21$: $\frac{F}{mg} = \frac{F/21}{mg-B}.$

Cancel $F$: $\frac{1}{mg} = \frac{1}{21(mg-B)}.$

So, $21(mg-B)=mg.$

$21mg - 21B = mg$ $20mg = 21B.$

Now substitute $mg = \rho V g, \qquad B = 800Vg.$

Then $20\rho V g = 21(800Vg).$

Cancel $Vg$: $20\rho = 16800$ $\rho = 840\,\text{kg m}^{-3}.$

So option C is correct.

4. Check each option

Option A: electric force between the spheres remains unchanged

In a dielectric medium, $F' = \frac{F}{21},$ so the electric force does not remain unchanged. It reduces.

So A is false.

Option B: electric force between the spheres reduces

Yes, in dielectric liquid, $F' = \frac{F}{21} < F.$ So B is true.

Option C: mass density of the spheres is $840\,\text{kg m}^{-3}$

Derived above. So C is true.

Option D: tension in the strings holding the spheres remains unchanged

Initially, $T = \frac{mg}{\cos\theta}.$ After immersion, $T' = \frac{mg-B}{\cos\theta}.$ Since $B>0$, $T' < T.$ So tension does not remain unchanged.

Thus D is false.

5. Final derived answer

Correct options are: $\boxed{B,\ C}$

6. Comparison with stored answer

Stored correct answer: $C, A$

My derivation shows that option A is false and option B is true. Hence I disagree with the stored answer.