(A)          We have a + b = a² + b² and ab = a² b²

$\therefore$          Number of quadratic equations with real roots are 3. Ans                 

(B)          We have

               tan 3π = tan (π – 2θ)

⇒            3π = nπ + (π – 2θ), $\mathrm{n}\in \mathrm{I}$

⇒            θ = (n + 1) $\frac{\mathrm{\pi }}{5},\mathrm{n}\in \mathrm{I}$

            Putting n = – 1, 0, 1, 2, 3, 4

            we get θ = 0,  $\frac{\mathrm{\pi }}{5},\frac{2\mathrm{\pi }}{5},\frac{3\mathrm{\pi }}{5},\frac{4\mathrm{\pi }}{5},\mathrm{\pi }$

$\therefore$       Number of solutionsof the equation in the interval [0, π] = 6.  Ans.

(C)        Using A.M. $\ge$ G.M. (for positive numbers)

So,        $\frac{\frac{2{\mathrm{x}}_1}{{\mathrm{x}}_2}+\frac{128{\mathrm{x}}_3^2}{{\mathrm{x}}_2^2}+\frac{{\mathrm{x}}_2^3}{4{\mathrm{x}}_1{\mathrm{x}}_3^2}}{3}\ge 4$

⇒            Minimum value = 12 Ans.

(D)          Let M (x₁, y₁) be the mid-point of chord.             

               m₁m₂ = – 1 $\Rightarrow \left(\frac{{\mathrm{y}}_1-3}{{\mathrm{x}}_1-1}\right)\left(\frac{{\mathrm{y}}_1}{{\mathrm{x}}_1}\right)=-1$

⇒            Locus of M (x₁, y₁) is x² + y² – x – 3y = 0.