Algebra underpins almost every Paper 1 question. These four questions test the discriminant, the remainder theorem, laws of indices and geometric series, with the trap a wrong answer usually falls into.
Try each question without a calculator, allowing about 4 minutes, then open the solution.
Question 1. For how many integers k does the equation x2 + kx + 9 = 0 have no real roots?
Answer: C. No real roots means the discriminant is negative: k2 − 36 < 0, so −6 < k < 6. The integers −5 to 5 give 11. Trap: including ±6 gives 13, but then the discriminant is 0 and there is a repeated real root.
Question 2. When x3 − 4x2 + px + 6 is divided by (x − 2) the remainder is 4. What is p?
Answer: C. By the remainder theorem, substitute x = 2: 8 − 16 + 2p + 6 = 4, so 2p = 6 and p = 3.
Question 3. If 4x × 81 − x = 32, what is x?
Answer: A. Write everything as a power of 2: 22x × 23 − 3x = 23 − x and 32 = 25. So 3 − x = 5, giving x = −2.
Question 4. The sum to infinity of the geometric series 6 + 4 + 83 + … is
Answer: C. First term 6, common ratio 23, and |r| < 1 so it converges. Sum = 6 ÷ (1 − 23) = 6 × 3 = 18.
The Admit Maths course has 12 modules mapped to the official specification, worked solutions for every question, and two full timed mock papers. It opens in December 2026, in time for the January sitting.
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