GCSE-readyInternational GCSE Further Pure Mathematics: common mistakes
The wrong answers students pick most often in our Further Pure Mathematics questions, and why they are wrong. Read these before a test, then open the lesson for anything that surprises you.
Getting started
- Algebra fluency: expanding and factorising
- Not “2(x2 − 4)”. x2 − 4 can still be factorised.
- Not “(2x − 4)(x + 2)”. This is correct in value but 2x − 4 still has a common factor of 2.
- Not “2(x − 2)2”. That expands to 2x2 − 8x + 8.
- Algebra fluency: indices, algebraic fractions and formulae
- Not “2x6”. The 2 must be squared too.
- Not “4x5”. Powers of powers multiply: 3 × 2 = 6.
- Not “−9x − 6”. You cannot cancel terms, only common factors.
1 Logarithmic functions and indices
- Exponential and logarithmic graphs
- Not “(0, 0)”. 30 is 1, not 0.
- Not “(0, 3)”. That is the point at x = 1, not where it crosses the y-axis.
- Not “(1, 0)”. y = 3x is never 0.
- Laws of logarithms
- Not “loga (x + y)”. A sum of logs is not the log of a sum.
- Not “loga x × loga y”. Logs are added, not multiplied.
- Surds and rationalising the denominator
- Not “(2√3 + 1)13”. (2√3)2 is 12, so the bottom is 12 − 1, not 12 + 1.
- Not “(2√3 + 1)5”. (2√3)2 = 4 × 3 = 12, not 6.
2 The quadratic function
- Forming equations from roots
- Not “x2 − 2x − 15 = 0”. The sum is −2, so the x term is +2x.
3 Identities and inequalities
- Algebraic division
- Not “x2 + 3”. That is the quotient.
- Linear and quadratic inequalities
- Not “x < −2 or x > 7”. That is where the graph is above the axis, for > 0.
- Not “x ≥ 3”. x = −5 also works: 25 − 9 ≥ 0.
- Regions and linear programming
- Not “(2, 1)”. y > x fails: 1 is not greater than 2.
- Not “(3, 3)”. x + y = 6, which is more than 4.
7 Scalar and vector quantities
- Vectors: components, magnitude and unit vectors
- Not “3i + 2j”. Subtracting −4j adds 4j.
- Not “−3i + 4j”. That is BA = a − b.
- Vectors in geometry
- Not “23a + 13b”. That is nearer A. P is closer to B because the ratio 2 : 1 is measured from A.
- Not “12(b − a)”. That is AM, not OM.
9 Calculus
- Differentiating powers, sin, cos and eˣ
- Not “5xe5x − 1”. That treats e5x like xn, which it is not.
- Integration and areas
- Not “e2x + 1(2x + 1) + c”. e2x is not a power of x.
- Tangents and normals
- Not “y = −15x + 215”. That is the normal.