IIT-JEE 2000 ScreeningQuestion 1
For the equation $3x^{2}+px+3=0$, $p>0$, if one of the roots is the square of the other, then $p$ is equal to
Quadratic Equation · Relation Between Roots and Coefficients
Solution
Concept. With roots $\alpha,\alpha^2$ the product is $\alpha^3=1$; $\alpha=1$ gives $p<0$, so $\alpha$ is a non-real cube root of unity. Formulas used. (1) $\omega^3=1$, $1+\omega+\omega^2=0$Step 1 - Product. $\alpha\cdot\alpha^2=\alpha^3=\frac33=1$. Step 2 - $\alpha=1$. Sum $=2=-\frac p3$ gives $p=-6<0$. Rejected. Step 3 - $\alpha=\omega$. By (1), sum $=\omega+\omega^2=-1=-\frac p3$, so $p=3$. Answer: (C)