Hard Matric Maths Questions with Step-by-Step Solutions

Challenge yourself with the hardest NSC Matric Maths questions — every one comes with a clear step-by-step solution.

Aiming for a distinction? These are the toughest Matric Maths questions — calculus optimisation, advanced trig identities, financial maths and 3D geometry — fully worked. Hard questions are usually ordinary methods combined in an unfamiliar way, like the final parts of a Paper 1 or Paper 2 question. Before reading each solution, write down which methods you think it needs: that habit is what lets you start a question you have never seen before.

Practice questions with solutions

Optimise: max area of rectangle inscribed in y = 4 − x²

  1. A(x) = 2x(4 − x²)
  2. A'(x) = 8 − 6x²
  3. x = √(4/3)

Answer: x = 2/√3, giving a maximum area of 32√3/9 ≈ 6.16 square units

Prove sin(2x) = 2 sin x cos x

  1. Use compound angle: sin(x + x)
  2. = sin x cos x + cos x sin x

Answer: = 2 sin x cos x

Find the general solution of cos2x = sinx

  1. Use cos2x = 1 − 2sin²x
  2. 1 − 2sin²x = sinx, so 2sin²x + sinx − 1 = 0
  3. (2sinx − 1)(sinx + 1) = 0
  4. sinx = ½ or sinx = −1

Answer: x = 30° + k·360°, x = 150° + k·360° or x = 270° + k·360°, k ∈ ℤ

For which values of p does x² − px + 4 = 0 have real roots?

  1. Real roots need Δ ≥ 0: p² − 16 ≥ 0
  2. (p − 4)(p + 4) ≥ 0

Answer: p ≤ −4 or p ≥ 4

How many monthly payments of R1 500 repay a R50 000 loan at 15% p.a. compounded monthly?

  1. 50 000 = 1 500[1 − (1,0125)⁻ⁿ] / 0,0125
  2. (1,0125)⁻ⁿ = 1 − (50 000 × 0,0125)/1 500 = 0,58333…
  3. n = −log 0,58333 / log 1,0125

Answer: n ≈ 43,4, so 44 payments, the last one smaller

Show that the sum of the first n odd numbers is n².

  1. 1; 3; 5; … is arithmetic with a = 1 and d = 2
  2. Sₙ = n/2 [2(1) + (n − 1)(2)] = n/2 × 2n

Answer: Sₙ = n²

If sin α = 3/5 and α is obtuse, find sin 2α without a calculator.

  1. α is in quadrant 2, so cos α is negative: cos α = −4/5
  2. sin 2α = 2 sin α cos α = 2(3/5)(−4/5)

Answer: sin 2α = −24/25

How many 6-digit even numbers can be made from the digits 1 to 6 without repetition?

  1. The last digit must be even: 3 choices (2, 4 or 6)
  2. The other five digits can be arranged in 5! = 120 ways
  3. 3 × 120

Answer: 360

Tips

  • If stuck, work backwards from the answer.
  • Always state your final answer in the required form.

Snap&Learn gives South African Matric learners instant, CAPS-aligned, step-by-step Mathematics solutions. Snap or upload a question and the AI shows every method mark the way the NSC memo awards them. Your first three AI solutions are free.