GATE CS 2017 Set 2 Q24 — Sets, Relations and Functions

We are given the quadratic equation $x^2 - 13x + 36 = 0$ in base $b$, with roots $x = 5$ and $x = 6$.

For any quadratic equation $x^2 - Sx + P = 0$, where $S$ is the sum of the roots and $P$ is the product of the roots:

1. Sum of roots: $$S = 5 + 6 = 11 \text{ (in decimal)}$$ In base $b$, the coefficient of $x$ is $(13)_b$. Converting this to decimal: $$(13)_b = 1 \cdot b^1 + 3 \cdot b^0 = b + 3$$ Equating the two expressions for the sum of roots: $$b + 3 = 11 \implies b = 8$$
2. Product of roots (Verification): $$P = 5 \times 6 = 30 \text{ (in decimal)}$$ In base $b$, the constant term is $(36)_b$. Converting this to decimal: $$(36)_b = 3 \cdot b^1 + 6 \cdot b^0 = 3b + 6$$ Substituting $b = 8$: $$3(8) + 6 = 24 + 6 = 30$$ This perfectly matches the product of the roots.

Thus, the base of the representation is 8.