GATE 2024 EC Q.29: solution and answer

GATE 2024 · ECQ.29 · 1 mark · NAT

In a number system of base $r$ , the equation $x^{2} - 12 x + 37 = 0$ has $x = 8$ as one of its solutions. The value of $r$ is ________.

See the question in the paper

Tap to open full size

Answer and solution

ANSWER

The base is $r = 11$ .

Official answer: 11Source: Official answer key, GATE 2024 EC (IISc Bengaluru)

Convert the coefficients from base $r$ to decimal using positional weights. $(12)_{r} = 1 \cdot r + 2 = r + 2$ and $(37)_{r} = 3 \cdot r + 7 = 3 r + 7$ . The numeral $8$ is a single digit, so its value is $8$ in any base where it is allowed.
Check which bases are valid. The digits $8$ and $7$ both appear, and a digit must be smaller than the base, so $r > 8$ , i.e. $r \geq 9$ .
Substitute the root $x = 8$ into the equation, now written in decimal: $8^{2} - 8 (r + 2) + (3 r + 7) = 0$
Solve for $r$ : $64 - 8 r - 16 + 3 r + 7 = 0 \Rightarrow 55 - 5 r = 0 \Rightarrow r = 11$
Check with the sum and product of the roots (Vieta). For $r = 11$ the equation in decimal is $x^{2} - 13 x + 40 = 0$ , which factors as $(x - 8) (x - 5) = 0$ . The roots are $8$ and $5$ . Their sum is $13 = (12)_{11}$ and their product is $40 = (37)_{11}$ , so both coefficients agree. Also $r = 11 > 8$ , so every digit is valid. Hence $r = 11$ .

KEY IDEA

Every multi-digit numeral in the equation is written in base $r$ , so convert each one to decimal with positional weights: $(12)_{r} = r + 2$ and $(37)_{r} = 3 r + 7$ . Then substitute the known root and solve the resulting linear equation in $r$ . The digits that appear must all be smaller than $r$ , so $r \geq 9$ . You can confirm the answer with Vieta: sum of roots $= (12)_{r}$ and product $= (37)_{r}$ .

WATCH OUT FOR

Reading $12$ and $37$ as decimal numbers. Then $x = 8$ gives $64 - 96 + 37 = 5 \neq = 0$ , and some students conclude the question has no answer or start guessing. Another slip is forgetting the digit constraint: the answer has to satisfy $r > 8$ because the digits $8$ and $7$ appear.

From the GATE 2024 EC paper. Every question of this paper · Download the full paper · All 5 Number systems questions, with notes

GATE 2024 EC Q.29: solution and answer

Answer and solution

More Number systems questions