| Date | None Specimen | Marks available | 4 | Reference code | SPNone.1.hl.TZ0.3 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Determine | Question number | 3 | Adapted from | N/A |
Question
The positive integer \(N\) is represented by \(4064\) in base \(b\) and \(2612\) in base \(b + 1\) .
Determine the value of \(b\) .
Find the representation of \(N\)
(i) in base \(10\);
(ii) in base \(12\).
Markscheme
the equation satisfied by \(b\) is
\(4{b^3} + 6b + 4 = 2{(b + 1)^3} + 6{(b + 1)^2} + (b + 1) + 2\) M1A1
\(2{b^3} - 12{b^2} - 13b - 7 = 0\) (A1)
\(b = 7\) A1
[4 marks]
(i) \(N = 4 \times {7^3} + 6 \times 7 + 4 = 1418\)
or \(N = 2 \times {8^3} + 6 \times {8^2} + 1 \times 8 + 2 = 1418\) (M1)A1
(ii) \(12\left| \!{\underline {\,
{1418} \,}} \right. \) (M1)
\(12\left| \!{\underline {\,
{118} \,}} \right. \) remainder \(2\) (A1)
\(12\left| \!{\underline {\,
9 \,}} \right. \) remainder A (where A \( = 10\) ) (A1)
\(1418 = {(9{\rm{A}}2)_{12}}\) A1
Note: Accept alternative correct methods.
[6 marks]
