JEE Advanced 2026 · Paper 2, Mathematics question 6

Integer roots. If the roots $r,s$ are integers, $r+s=-\frac ba$ and $rs=\frac ca$ are integers. Put $b=ma$, $c=na$; the AP condition $a+c=2b$ gives $n=2m-1$.

Both roots are negative (all coefficients positive). With $p=-r$, $q=-s$:
$$p+q=m,\quad pq=2m-1=2(p+q)-1\implies(p-2)(q-2)=3.$$
With $p,q\ge1$ this forces $\{p,q\}=\{3,5\}$. So $m=8$, $n=15$: $b=8a$, $c=15a$, and the roots are $-3,-5$.

• (A) $c-b=7a$. True.
• (B) $-3$ and $-5$ are odd. True.
• (C) $c=15\Rightarrow a=1,\ b=8$, so $ab=8$. True.
• (D) $b=8\Rightarrow a=1$; the roots are $-3,-5$, not $3$. False.