Grade 12 Maths Paper 1 — Exam Walkthrough

Instructions: Work with a calculator. 1 mark ≈ 1 minute. Answers at the bottom — mark yourself honestly.

Question 1 — Algebra & Equations (15 marks)

1.1 Solve for x: 2x² − 5x − 3 = 0 (4) 1.2 Solve simultaneously: y = 2x + 1 and x² + y² = 25 (6) 1.3 Given f(x) = (x−2)/(x+3), for which x is f undefined? (1)

Question 2 — Functions (14 marks)

2.1 Given f(x) = −2(x−1)² + 8, write down the turning point and the axis of symmetry. (3) 2.2 Sketch g(x) = 2^x − 3, showing intercepts. (5) 2.3 Determine the inverse of h(x) = 2x − 4 and sketch both. (6)

Question 3 — Sequences & Series (16 marks)

3.1 In an arithmetic sequence, T₁ = 5 and T₉ = 29. Find the 20th term. (4) 3.2 Prove that the sum of the first n terms of a geometric series with 0 < r < 1 is Sn = a(1−rⁿ)/(1−r). (6) 3.3 A geometric series has S∞ = 40 and S₂ = 10. Find r. (6)

Question 4 — Calculus (20 marks)

4.1 Differentiate from first principles: f(x) = x² − 3x. (5) 4.2 Determine dy/dx if y = (3x² − 2)(x + 5). (4) 4.3 The height of a ball is h(t) = −5t² + 20t. When is it at maximum height? (4) 4.4 Sketch f(x) = x³ − 3x, marking stationary points and concavity. (7)

---

Answers

1.1 x = 3 or x = −½ · 1.2 x = −2, y = −3 or x = 3, y = 7 · 1.3 x = −3 2.1 (1; 8), x = 1 · 2.2 y-intercept (0; −3), asymptote y = −3 · 2.3 h⁻¹(x) = (x+4)/2 3.1 d = 3, T₂₀ = 62 · 3.2 standard proof · 3.3 r = ½ 4.1 2x − 3 · 4.2 9x² + 30x − 2 · 4.3 t = 2 s · 4.4 stationary points at x = ±1, concave down for x > 0