Mathematics
The Mathematics section of the SAMPAD Grade 10 entrance exam rewards structured thinking over memorised formulas: expect multi-step arithmetic, algebra, plane geometry and elementary number theory, often dressed up as short puzzles.
Core rules to make automatic
- Algebraic identities: (a+b)² = a²+2ab+b², a³+b³ = (a+b)(a²−ab+b²), and the x + 1/x family - square it to get x² + 1/x², then build x³ + 1/x³ from the two.
- Modular arithmetic: remainders repeat in cycles. Find the cycle length of the base modulo m before raising to a huge power.
- Similarity: for similar figures with ratio k, lengths scale by k, areas by k², volumes by k³.
- Counting: multiplication principle for arrangements, nCr for selections where order does not matter.
Worked example
Find the remainder when 3¹⁰⁰ is divided by 7. Compute powers modulo 7: 3¹≡3, 3²≡2, 3³≡6, 3⁴≡4, 3⁵≡5, 3⁶≡1. The cycle length is 6, so only the exponent mod 6 matters. Since 100 = 6×16 + 4, we get 3¹⁰⁰ ≡ 3⁴ ≡ 4 (mod 7). Answer: 4.
Traps that cost marks
- Scaling area by k instead of k² in similarity questions.
- Percentages: a 20% rise followed by a 20% fall leaves 96%, not 100%.
- Mixing metres and centimetres inside one formula, or dropping units from the final answer.
- Discarding a negative root when the question does not forbid it, or keeping one when a length is required.