Pearson Edexcel International A Level Pure Mathematics 1 Comprehensive Study Guide

Course Overview and Assessment Objectives

• Pearson Edexcel International A Level Pure Mathematics 1 (P1) is a compulsory unit for the International Advanced Subsidiary (IAS) and International Advanced A Level (IAL) in Mathematics and Pure Mathematics.

• The assessment for P1 consists of a 1 hour 30 minute examination worth 75 marks. The paper code is WMA11/01. Assessments are available in January, June, and October.

• P1 accounts for $33 \frac{1}{3}\%$ of the total IAS marks and $16 \frac{2}{3}\%$ of the total IAL marks.

• Assessment Objectives (AO) and minimum weightings in IAL:

  • AO1: Recall, select, and use mathematical facts, concepts, and techniques ($30\%$).

  • AO2: Construct rigorous mathematical arguments and proofs ($30\%$).

  • AO3: Use standard mathematical models to represent real-world situations ($10\%$).

  • AO4: Comprehend translations of realistic contexts into mathematics ($5\%$).

  • AO5: Use technology accurately and efficiently ($5\%$).

• Calculator Requirements: Students must have calculators with the following functions: $+$, $-$, $\times$, $\div$, $\pi$, $x^2$, $\sqrt{x}$, $\frac{1}{x}$, $x^y$, $\ln(x)$, $e^x$, $x!$, $\sin$, $\cos$, and $\tan$ (including inverses), and memory. Radians and degrees must be supported.

• Prohibited Calculator Facilities: Databanks, retrieval of text or formulae, built-in symbolic algebra manipulations (CAS), symbolic differentiation/integration, and language translators.

Chapter 1: Algebraic Expressions

• Index Laws: For powers with the same base, the following laws apply:

  • $a^m \times a^n = a^{m+n}$

  • $a^m \div a^n = a^{m-n}$

  • $(a^m)^n = a^{mn}$

  • $(ab)^n = a^n b^n$

• Expanding Brackets: To find the product of two expressions, every term in the first expression must be multiplied by every term in the second expression. For example, $(x+5)(4x-2y+3) = 4x^2 - 2xy + 3x + 20x - 10y + 15 = 4x^2 - 2xy + 23x - 10y + 15$.

• Factorising: This process is the inverse of expanding brackets.

  • A quadratic expression takes the form $ax^2 + bx + c$. To factorise, find two numbers that multiply to $ac$ and add to $b$.

  • Difference of Two Squares: $x^2 - y^2 = (x + y)(x - y)$.

  • Simple cubic expressions such as $x^3 - 2x^2$ factorise to $x^2(x - 2)$.

• Rational Indices:

  • $a^{\frac{1}{n}} = \sqrt[n]{a}$

  • $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$

  • $a^{-m} = \frac{1}{a^m}$

  • $a^0 = 1$

• Surds: An irrational multiple of $\sqrt{n}$ (where $n$ is not a square number).

  • Rules for manipulation: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ and $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$.

  • Rationalising the Denominator:

  • For $\frac{1}{\sqrt{a}}$, multiply numerator and denominator by $\sqrt{a}$.

  • For $\frac{1}{a+\sqrt{b}}$, multiply by $(a-\sqrt{b})$.

  • For $\frac{1}{a-\sqrt{b}}$, multiply by $(a+\sqrt{b})$.

Chapter 2: Quadratics

• Solving Quadratic Equations ($ax^2 + bx + c = 0$):

  • Factorisation: Set each factor to zero.

  • The Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

  • Completing the Square: $x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2$. For a general quadratic: $ax^2 + bx + c = a(x + \frac{b}{2a})^2 + (c - \frac{b^2}{4a})$.

• Functions: A relationship mapping inputs (Domain) to a single output (Range). The roots of a function $f(x)$ are the values where $f(x) = 0$.

• Quadratic Graphs (Parabolas):

  • If $a > 0$, the curve is U-shaped (minimum). If $a < 0$, it is n-shaped (maximum).

  • The graph crosses the y-axis at $(0, c)$.

  • The graph crosses the x-axis at the roots of $f(x)$.

  • The turning point of $y = a(x + p)^2 + q$ is at $(-p, q)$.

• The Discriminant: The term $b^2 - 4ac$ determines the nature of the roots:

  • $b^2 - 4ac > 0$: Two distinct real roots.

  • $b^2 - 4ac = 0$: One repeated real root.

  • $b^2 - 4ac < 0$: No real roots.

Chapter 3: Equations and Inequalities

• Linear Simultaneous Equations: Solved via elimination or substitution. They represent the point of intersection between two straight lines.

• Linear + Quadratic Simultaneous Equations: Always use substitution. They can have 0, 1, or 2 solutions.

• Graphical Solutions: The x-coordinates of points where $y = f(x)$ and $y = g(x)$ intersect are the solutions to $f(x) = g(x)$.

  • If the discriminant of the resulting quadratic is positive, there are 2 intersections.

  • If zero, the line is a tangent (1 intersection).

  • If negative, there are no intersections.

• Inequalities:

  • Linear: Solve like equations, but flip the sign when multiplying/dividing by a negative number.

  • Quadratic: Find critical values (roots), sketch the graph, and identify the required region (above or below the x-axis).

  • Graphs and Regions: $y > f(x)$ is the region above the curve. Use dotted lines for strict inequalities ($<$, $>$) and solid lines for inclusive inequalities ($\le$, $\ge$).

Chapter 4: Graphs and Transformations

• Cubic Graphs ($y = ax^3 + bx^2 + cx + d$):

  • Can have up to three roots.

  • A repeated root $(x - p)^2$ means the graph touches the axis at $p$.

  • A triple repeated root $(x - p)^3$ means the graph crosses the axis with a point of inflection at $p$.

• Reciprocal Graphs:

  • $y = \frac{k}{x}$: Asymptotes at $x = 0$ and $y = 0$.

  • $y = \frac{k}{x^2}$: Values are always positive if $k > 0$. Asymptotes at $x = 0$ and $y = 0$.

• Points of Intersection: Finding intersections of curves like $y = x^2$ and $y = x^2(x - 3)$ identifies solutions to $x^2 = x^2(x - 3)$.

• Transformations of $y = f(x)$:

  • Translation by vector $\begin{pmatrix} 0 \\ a \end{pmatrix}$: $y = f(x) + a$.

  • Translation by vector $\begin{pmatrix} -a \\ 0 \end{pmatrix}$: $y = f(x + a)$.

  • Vertical stretch (SF $a$): $y = af(x)$.

  • Horizontal stretch (SF $\frac{1}{a}$): $y = f(ax)$.

  • Reflection in x-axis: $y = -f(x)$.

  • Reflection in y-axis: $y = f(-x)$.

Chapter 5: Straight Line Graphs

• Gradient ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Forms of Equation:

  • $y = mx + c$ ($m$ = gradient, $c$ = y-intercept).

  • $ax + by + c = 0$ ($a, b, c$ are integers).

  • $y - y_1 = m(x - x_1)$ (passes through $(x_1, y_1)$).

• Parallel and Perpendicular Lines:

  • Parallel lines: $m_1 = m_2$.

  • Perpendicular lines: $m_1 \times m_2 = -1$. If gradient is $m$, the perpendicular is $-\frac{1}{m}$.

• Distance between $(x_1, y_1)$ and $(x_2, y_2)$: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Chapter 6: Trigonometric Ratios

• Cosine Rule:

  • Missing side: $a^2 = b^2 + c^2 - 2bc \cos(A)$.

  • Missing angle: $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$.

• Sine Rule:

  • Sides: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$.

  • Angles: $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$.

  • Ambiguous Case: $\sin(\theta) = \sin(180^\circ - \theta)$. Two possible triangles may exist given two sides and a non-included acute angle.

• Area of any triangle: $\text{Area} = \frac{1}{2}ab \sin(C)$.

• Trig Graphs (Periodicity):

  • $y = \sin(x)$: Period $360^\circ$, range $[-1, 1]$, roots at multiples of $180^\circ$.

  • $y = \cos(x)$: Period $360^\circ$, range $[-1, 1]$, roots at $90^\circ$, $270^\circ$, etc.

  • $y = \tan(x)$: Period $180^\circ$, asymptotes at $90^\circ, 270^\circ$, etc.

Chapter 7: Radians

• Definition: 1 radian is the angle subtended when arc length equals radius.

• Conversion: $2\pi\,\text{rad} = 360^\circ$, $\pi\,\text{rad} = 180^\circ$.

• Important Angles: $30^\circ = \frac{\pi}{6}$, $45^\circ = \frac{\pi}{4}$, $60^\circ = \frac{\pi}{3}$, $90^\circ = \frac{\pi}{2}$.

• Arc Length ($l$): $l = r\theta$.

• Area of Sector ($A$): $A = \frac{1}{2}r^2\theta$.

• Area of Segment: $A = \frac{1}{2}r^2(\theta - \sin(\theta))$.

Chapter 8: Differentiation

• Definition from First Principles: $f'(x) = \lim_{h \rightarrow 0} \frac{f(x + h) - f(x)}{h}$.

• Derivative of $x^n$: If $y = ax^n$, then $\frac{dy}{dx} = anx^{n-1}$.

• Differentiating Sums: Differentiate each term individually.

• Gradients, Tangents, and Normals:

  • Gradient of tangent at $a = f'(a)$.

  • Equation of tangent: $y - f(a) = f'(a)(x - a)$.

  • Gradient of normal at $a = -\frac{1}{f'(a)}$.

  • Equation of normal: $y - f(a) = -\frac{1}{f'(a)}(x - a)$.

• Second Order Derivative (Rate of change of gradient): $f''(x) = \frac{d^2y}{dx^2}$.

Chapter 9: Integration

• Integration as Inverse Differentiation: If $\frac{dy}{dx} = x^n$, then $y = \frac{x^{n+1}}{n+1} + c$ (for $n \neq -1$).

• Indefinite Integrals: $\int f'(x)\,dx = f(x) + c$.

• Differentiating and Integrating Term-by-Term: Always express terms as $kx^n$ before performing calculus.

• Finding Functions: If a point $(x, y)$ on the curve is known, use it to solve for the constant of integration $c$.