10 free IGCSE Additional Mathematics Algebra questions written in the style of Cambridge Additional Mathematics papers, each with its answer and mark scheme. 4 of them are A* challenge questions, built to catch the mistakes strong students still make.
This topic covers Functions, Quadratic functions, Factors of polynomials, Equations, inequalities & graphs, Simultaneous equations, Logarithmic & exponential functions. Work through the questions below, then generate a full Algebra paper with a fresh set every time.
Generate a full Algebra paper →Find the set of values of m for which the line y = mx − 7 does not meet the curve y = x2 + 18.
| Part | Answer and marking points | Marks |
|---|---|---|
| −10 < m < 10 x2 − mx + 25 = 0 M1 b2 − 4ac < 0 M1 critical values ±10 A1 −10 < m < 10 A1 | 4 |
The cubic equation x3 − 3x2 − x + 3 = 0 has a root x = 3.
Find the other two roots.
| Part | Answer and marking points | Marks |
|---|---|---|
| −1, 1 divide by (x − 3) M1 quadratic factor A1 −1, 1 A1 | 3 |
The line y + 2x = 4 meets the curve xy = 2 at the points A and B.
Find the coordinates of A and B and the length of AB.
| Part | Answer and marking points | Marks |
|---|---|---|
| (1, 2), (1, 2); 0 substitute y = 4 − 2x M1 2x2 − 4x + 2 = 0 A1 solve M1 (1, 2) and (1, 2) A1 distance formula M1 0 A1 | 6 |
Variables x and y are related by y = axn. When lg y is plotted against lg x, a straight line is obtained with gradient 2.5 and intercept 0.477 on the lg y axis.
Find the values of a and n, and hence find y when x = 4.
| Part | Answer and marking points | Marks |
|---|---|---|
| a = 3, n = 2.5; y = 96 lg y = n lg x + lg a M1 n = 2.5 A1 a = 3 A1 y = 96 A1 | 4 |
The number of bacteria, N, in a culture after t hours is given by N = 800e0.05t.
Find the initial number of bacteria.
Find the number of bacteria after 30 hours.
Find the time taken for the number to reach 2400.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | 800 800 B1 | 1 |
| (b) | 3585 3585 B1 | 1 |
| (c) | 22 hours e0.05t = 3 M1 take ln M1 22 A1 | 3 |
Find the values of k for which the line y = kx + 4 is a tangent to the curve y = x2 + 20.
| Part | Answer and marking points | Marks |
|---|---|---|
| k = ±8 x2 − kx + 16 = 0 M1 b2 − 4ac = 0: k2 = 64 M1 k = ±8 A1 | 3 |
f(x) = (x + 3)2 + 5 for x ≥ −3
State the range of f.
Find f−1(x) and state its domain.
Describe how the graphs of y = f(x) and y = f−1(x) are related.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | f(x) ≥ 5 f(x) ≥ 5 B1 | 1 |
| (b) | −3 + √(x − 5), x ≥ 5 swap x and y and rearrange M1 f−1(x) = √(x − 5) − 3 A1 domain x ≥ 5 B1 | 3 |
| (c) | reflection in y = x reflection in the line y = x B1 | 1 |
p(x) = x3 + ax2 + bx − 48
It is given that x + 4 is a factor of p(x) and that when p(x) is divided by x − 1 the remainder is −60.
Find the values of a and b.
Factorise p(x) completely.
Hence solve p(x) = 0.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | a = 3, b = −16 p(−4) = 0 used M1 p(1) = −60 used M1 Solve simultaneously M1 a = 3, b = −16 A1 | 4 |
| (b) | (x + 4)(x + 3)(x − 4) Quadratic factor found M1 Complete factorisation A1 | 2 |
| (c) | −4, −3, 4 −4, −3, 4 B1 | 1 |
Solve log3 x + log3 (x + 8) = 2.
Solve 22x − 10 × 2x + 16 = 0.
| Part | Answer and marking points | Marks |
|---|---|---|
| (a) | x = 1 log3 x(x + 8) = 2 M1 x2 + 8x − 9 = 0 M1 Solve quadratic M1 x = 1 only (reject negative root) A1 | 4 |
| (b) | x = 1, x = 3 Let y = 2x: y2 − 10y + 16 = 0 M1 y = 2, y = 8 A1 Convert back M1 x = 1, x = 3 A1 | 4 |
Solve the simultaneous equations
y = −x − 4
x2 + y2 = 40
| Part | Answer and marking points | Marks |
|---|---|---|
| (2, −6) and (−6, 2) substitute to form a quadratic M1 2x2 + 8x − 24 = 0 A1 solve M1 x = 2, x = −6 A1 y = −6, y = 2 A1 | 5 |
In the 0606 syllabus, Algebra covers Functions, Quadratic functions, Factors of polynomials, Equations, inequalities & graphs, Simultaneous equations, Logarithmic & exponential functions.
This page has 10 Algebra questions with answers. For more, the free generator on mentorriyaz.in builds a new Algebra paper with a full mark scheme each time you click.
Riyaz Shaikh has mentored IGCSE, IB and A Level students for 15+ years, one-on-one, online and in Mumbai. The hardest questions on this site are the ones he uses to find a student's weak points.
Meet Riyaz →Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in