IGCSE Mathematics: Algebra and Graphs (Extended) — Practice Questions (Cambridge 0580)

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — Cambridge International holds copyright in its own papers. Use these alongside the official past papers available from your board.

Each question practises a skill from the Extended (Supplement) content of the syllabus. After each answer there is a common mistake to watch out for.

---

Questions

1. (Extended) Simplify fully (x² − 9) / (x² + x − 6). [3]

2. (Extended) Write 3/(x + 2) − 2/(x − 1) as a single fraction in its simplest form. [3]

3. (Extended) Solve the equation 2/(x + 1) + 5/x = 3. Show all your working and give your answers correct to 2 decimal places. [4]

4. (Extended) f(x) = 4x − 3 and g(x) = x² + 2.

(a) Find f⁻¹(x).

(b) Find gf(x), giving your answer in the form ax² + bx + c. [4]

5. (Extended) y is inversely proportional to the square of x. When x = 2, y = 12. Find y when x = 5. [3]

6. (Extended) A curve has equation y = x³ − 6x² + 9x + 1. Find the coordinates of its two turning points and determine which one is a maximum. [5]

---

Answers

1. (Extended) Numerator: x² − 9 = (x − 3)(x + 3) [1]. Denominator: x² + x − 6 = (x + 3)(x − 2) [1]. Cancel (x + 3): (x − 3)/(x − 2) [1].

Common mistake: Cancelling the x² terms or the individual numbers. Only whole common factors (brackets) can be cancelled, so factorise first.

2. (Extended) Common denominator (x + 2)(x − 1) [1]. Numerator: 3(x − 1) − 2(x + 2) = 3x − 3 − 2x − 4 = x − 7 [1]. Answer: (x − 7)/((x + 2)(x − 1)) [1].

Common mistake: Writing −2(x + 2) as −2x + 4. The minus sign multiplies every term in the bracket, giving −2x − 4.

3. (Extended) Multiply through by x(x + 1): 2x + 5(x + 1) = 3x(x + 1) [1]. So 7x + 5 = 3x² + 3x, giving 3x² − 4x − 5 = 0 [1]. Quadratic formula: x = (4 ± √(16 + 60)) / 6 = (4 ± √76) / 6 [1]. x = 2.12 or x = −0.79 [1].

Common mistake: Getting the sign of c wrong inside the formula: b² − 4ac = (−4)² − 4 × 3 × (−5) = 16 + 60, not 16 − 60.

4. (Extended) (a) Let y = 4x − 3, so x = (y + 3)/4 [1]. f⁻¹(x) = (x + 3)/4 [1].

(b) gf(x) = g(4x − 3) = (4x − 3)² + 2 [1] = 16x² − 24x + 9 + 2 = 16x² − 24x + 11 [1].

Common mistake: Working out fg(x) instead of gf(x). In gf(x), apply f first, then put the result into g.

5. (Extended) y = k/x² [1]. 12 = k/2², so k = 48 [1]. When x = 5, y = 48/25 = 1.92 [1].

Common mistake: Using y = k/x (plain inverse proportion) or forgetting to square x when substituting.

6. (Extended) dy/dx = 3x² − 12x + 9 [1]. Set 3x² − 12x + 9 = 0, so 3(x − 1)(x − 3) = 0 and x = 1 or x = 3 [1]. When x = 1, y = 1 − 6 + 9 + 1 = 5; when x = 3, y = 27 − 54 + 27 + 1 = 1, so the turning points are (1, 5) and (3, 1) [1]. d²y/dx² = 6x − 12 [1]. At x = 1, d²y/dx² = −6 < 0, so (1, 5) is the maximum (and (3, 1), where d²y/dx² = 6 > 0, is the minimum) [1].

Common mistake: Finding the x-values but not substituting back into the original equation to find the y-coordinates.

---

Where marks are usually lost

• Cancelling terms rather than factors in algebraic fractions.
• Sign errors when a subtracted fraction has a bracket in its numerator.
• Mixing up the order in composite functions (gf means f first).
• Using the wrong type of proportion (square, root, inverse) from the wording.
• Not justifying which turning point is a maximum and which is a minimum.