Knowledge organisers

Pure Mathematics 1 (P1): Edexcel IAL Maths knowledge organiser

Everything to know about pure mathematics 1 (p1) on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Discriminant: b² − 4ac: positive gives two real roots, zero one repeated root, negative none.
Surd: An irrational root such as √2, kept exact rather than written as a decimal.
Tangent and normal: The tangent has the curve's gradient at the point; the normal is perpendicular to it.
Radian: π radians = 180°; arc length s = rθ and sector area ½r²θ need θ in radians.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Quadratic formula	\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Discriminant \(b^2-4ac\): \(>0\) two real roots, \(=0\) equal roots, \(<0\) no real roots
Completing the square	\(x^2+bx+c=\left(x+\tfrac b2\right)^2+c-\tfrac{b^2}4\)
Indices; surds	\(a^{-n}=\tfrac1{a^n},\) \(a^{\frac mn}=\sqrt[n]{a^m};\) \(\frac{1}{\sqrt a}=\frac{\sqrt a}a\)
Lines	\(y-y_1=m(x-x_1),\) \(m=\frac{y_2-y_1}{x_2-x_1},\) \(m_1m_2=-1\)
Transformations: \(f(x)+a\) up \(a\); \(f(x+a)\) left \(a\); \(af(x)\) vertical ×\(a\); \(f(ax)\) horizontal ×\(\tfrac1a\)
MensurationIn the formula booklet	\(\text{sphere }A=4\pi r^2,\) \(\text{cone curved }A=\pi rl\)
Sine rule	\(\frac a{\sin A}=\frac b{\sin B}\)
Cosine ruleIn the formula booklet	\(a^2=b^2+c^2-2bc\cos A\)
Area; arc; sector (radians)	\(\tfrac12ab\sin C;\) \(s=r\theta;\) \(\tfrac12r^2\theta\)
Differentiate / integrate \(x^n\)	\(\tfrac{d}{dx}x^n=nx^{n-1},\) \(\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+c\ (n\ne-1)\)
Tangent; normal	\(y-y_1=m(x-x_1),\) \(m_\text{n}=-\tfrac1m\)

Worked example

Find the equation of the tangent to y = x³ − 4x at the point where x = 1.

dy/dx = 3x² − 4 = −1 at x = 1
y = 1 − 4 = −3, so the point is (1, −3)
y + 3 = −1(x − 1)

Answer: y = −x − 2

Common mistakes

Calculator warning ignored: surds and fractions done on the calculator
Indices: not writing terms as powers of the same base
(a+b)² written as a²+b²
A^(−n) treated as −aⁿ

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Apply the laws of indices to simplify expressions such as (3x²)³ ÷ 9x, then expand brackets and collect like terms in longer algebraic expressions.
Model real situations such as the path of a ball or the area of a pen with quadratic functions, and interpret roots and turning points in context.
Sketch quartic graphs and reciprocal graphs such as y = a/x and y = a/x², marking intercepts and horizontal and vertical asymptotes.
Use the sine rule to find missing sides and angles, and recognise the ambiguous case where an angle could be acute or obtuse.
Solve multi-step problems with arcs, sectors and segments, such as finding the perimeter or area of shapes made from circles and triangles.
Use a point on a curve and its gradient function to find the constant of integration and the equation of the curve.

The printable sheet

Pure Mathematics 1 (P1) knowledge organiser for Edexcel IAL Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Pure Mathematics 1 (P1) knowledge organiser (Edexcel IAL Maths), A4. Download the PDF.

Revise it next

Other Edexcel IAL Maths topics: Pure Mathematics 2 (P2) · Pure Mathematics 3 (P3) · Pure Mathematics 4 (P4) · Statistics 1 (S1) · Mechanics 1 (M1) · All Edexcel IAL Maths organisers