JEE Main 2026 (Online) 5th April Evening ShiftQuestion 1
Let $\alpha,\beta$ be the roots of the equation $x^{2}-x+p=0$ and $\gamma,\delta$ be the roots of the equation $x^{2}-4x+q=0$; $p,q\in\mathbf{Z}$. If $\alpha,\beta,\gamma,\delta$ are in G.P., then $|p+q|$ equals :
Quadratic Equation · Relation Between Roots and Coefficients
Solution
Concept. Write the four roots as $\alpha,\alpha r,\alpha r^2,\alpha r^3$ and use the two sums. Formulas used. (1) For $x^2-Sx+P=0$: sum of roots $S$, product $P$Step 1 - Two sums. By (1), $$\alpha(1+r)=1,\qquad\alpha r^2(1+r)=4.$$ Dividing, $r^2=4$, so $r=\pm2$. Step 2 - $r=2$. $\alpha=\frac13$ and $p=\alpha\cdot\alpha r=\frac29\notin\mathbb Z$. Rejected. Step 3 - $r=-2$. $\alpha=-1$, so the roots are $-1,\ 2,\ -4,\ 8$. Then $$p=(-1)(2)=-2,\qquad q=(-4)(8)=-32.$$ Step 4 - Value. $|p+q|=34$. Answer: (C)