Cambridge International AS & A Level Mathematics: Pure Mathematics 1 Coursebook Notes

Preface and Series Introduction (Pages i–vii)

Series Introduction and Philosophy

Mathematics at AS & A Level is described not just as a facilitating subject for university, but as a discipline that encourages precise, logical thinking and creativity. It is likened to art: a mathematician must master tools (algebra, calculus) to express ideas in novel ways.

Mathematical Problem vs. Exercise:

• Exercise: A question where the method of solution is immediately known.

• Problem: A question where the answer is not immediately obvious, requiring the learner to use different tools and approaches over minutes, hours, or even days.

Key Concepts in the Syllabus:

1. Communication: Exploring and proofing ideas verbally and in writing.

2. Mathematical Modelling: Capturing real-world aspects (e.g., weather, financial markets, population change) through equations to build a model of reality.

Coursebook Features

• Explore Activities: Non-routine problems for classroom group discussion.

• Flagged Questions: Labelled with P (Proof), M (Modelling), or PS (Problem Solving).

• Underground Mathematics: References to web-based resources for developing holistic mathematical thinking.

Chapter 1: Quadratics (Pages 1–32)

Learning Objectives

• Completing the square for $ax^2 + bx + c$.

• Finding and using the discriminant.

• Solving quadratic equations and inequalities.

• Solving simultaneous equations (one linear, one quadratic).

• Solving equations that are quadratic in form (e.g., quartics).

• Understanding the relationship between quadratic graphs and algebraic solutions.

Why Study Quadratics?

Quadratic functions are of the form $y = ax^2 + bx + c$ where $a \neq 0$. Notable historical connection: Galileo discovered that the trajectory of a projectile and the vertical motion of an object follow quadratic models.

1.1 Solving Quadratic Equations by Factorisation

Factorisation utilizes the property that if $pq = 0$, then $p = 0$ or $q = 0$.

Worked Example 1.1: Solve: $9x^2 - 39x - 30 = 0$

1. Divide by common factor 3: $3x^2 - 13x - 10 = 0$.

2. Factorise: $(3x + 2)(x - 5) = 0$.

3. Result: $x = -\frac{2}{3}$ or $x = 5$.

Worked Example 1.4: A rectangle has sides $x\,\text{cm}$ and $(6x - 7)\,\text{cm}$ with Area $= 90\,\text{cm}^2$.

• Equation: $x(6x - 7) = 90 \rightarrow 6x^2 - 7x - 90 = 0$.

• Factorise: $(2x - 9)(3x + 10) = 0$.

• Solutions: $x = 4.5$ or $x = -\frac{10}{3}$.

• Result: Since length must be positive, $x = 4.5\,\text{cm}$. The sides are $4.5\,\text{cm}$ and $20\,\text{cm}$.

1.2 Completing the Square

This process rewrites a quadratic using only one occurrence of the variable. Key Point 1.1:

• $x^2 + 2dx = (x + d)^2 - d^2$

• $x^2 - 2dx = (x - d)^2 - d^2$

Worked Example 1.5: Express $2x^2 - 12x + 3$ in the form $p(x + q)^2 + r$

• $2[x^2 - 6x] + 3$

• $2[(x - 3)^2 - 9] + 3$

• $2(x - 3)^2 - 15$. Result: $p=2, q=-3, r=-15$.

1.3 The Quadratic Formula

For $ax^2 + bx + c = 0$: Key Point 1.2: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

1.4 Solving Simultaneous Equations (One Linear, One Quadratic)

The points of intersection of a line and a curve represent the solutions of their simultaneous equations.

1.5 Solving More Complex Quadratic Equations

Use substitution to transform higher-degree equations into quadratics. Worked Example 1.11: Solve $x - 4\sqrt{x} - 12 = 0$.

• Let $y = \sqrt{x}$. Then $y^2 - 4y - 12 = 0$.

• $(y - 6)(y + 2) = 0 \rightarrow y = 6$ (since $\sqrt{x}$ cannot be negative).

• Result: $x = 36$.

1.6 Maximum and Minimum Values

The graph of a quadratic is a parabola.

• If a > 0, the curve has a minimum point (U-shaped).

• If a < 0, the curve has a maximum point (n-shaped).

• The turning point is called the vertex. A parabola is symmetrical about the vertical line passing through the vertex.

Key Point 1.3: If $f(x) = a(x + h)^2 + k$:

• Line of symmetry: $x = -h$

• Vertex: $(-h, k)$

1.8 The Number of Roots (The Discriminant)

Key Point 1.5: The discriminant is defined as $\Delta = b^2 - 4ac$.

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

• If $b^2 - 4ac = 0$: Two equal real roots (one repeated root).

• If b^2 - 4ac < 0: No real roots (curve is entirely above or below the $x$-axis).

1.9 Intersection of a Line and a Curve

When solving $y = mx + c$ and $y = ax^2 + bx + c$ simultaneously:

• Discriminant > 0: Two distinct points of intersection.

• Discriminant $= 0$: One point of intersection (the line is a tangent to the curve).

• Discriminant < 0: No intersection.

Chapter 2: Functions (Pages 33–69)

2.1 Definition of a Function

A function (or mapping) associates members of a domain (inputs) with members of a codomain (outputs/range).

• One-one function: One unique output for every input, and vice versa.

• Many-one function: Many inputs can result in the same output (e.g., $f(x) = x^2$).

• Constraint: A relation where one input gives two outputs is not a function.

2.2 Composite Functions

Key Point 2.1: $fg(x)$ means $g$ acts on $x$ first, then $f$ acts on the result. $fg$ exists only if the range of $g$ is within the domain of $f$.

2.3 Inverse Functions

An inverse function $f^{-1}(x)$ "undoes" the operation of $f(x)$.

• Key Point 2.4: $f^{-1}(x)$ exists if and only if $f(x)$ is a one-one mapping.

• Key Point 2.5: The graph of $f^{-1}$ is the reflection of $f$ in the line $y = x$.

• Self-inverse function: If $f(x) = f^{-1}(x)$. Example: $f(x) = \frac{1}{x}$.

2.5 Transformations of Functions

For a graph $y = f(x)$, the following transformations apply:

Vertical Transformations (affect $y$-coordinates):

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

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

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

Horizontal Transformations (affect $x$-coordinates):

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

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

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

Order of Combined Transformations:

• Vertical: Follow standard arithmetic order (multiply/stretch before add/translate).

• Horizontal: Follow the opposite order of standard arithmetic.

Chapter 3: Coordinate Geometry (Pages 70–98)

3.1 Length and Midpoint

Key Point 3.1: For points $P(x_1, y_1)$ and $Q(x_2, y_2)$:

• Midpoint $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

• Length $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

3.2 Parallel and Perpendicular Lines

• Parallel: Gradients are equal ($m_1 = m_2$).

• Perpendicular: $m_1 \times m_2 = -1$.

3.3 Equations of Straight Lines

Key Point 3.5: Equation of a line through $(x_1, y_1)$ with gradient $m$ is $y - y_1 = m(x - x_1)$.

3.4 Equation of a Circle

Key Point 3.6: A circle with centre $(a, b)$ and radius $r$ is $(x - a)^2 + (y - b)^2 = r^2$. Key Point 3.7: General form: $x^2 + y^2 + 2gx + 2fy + c = 0$, where centre is $(-g, -f)$ and radius is $\sqrt{g^2 + f^2 - c}$.

Circle Properties:

• Angle in a semicircle is a right angle.

• The perpendicular from the centre to a chord bisects the chord.

• Tangent is perpendicular to the radius at the point of contact.

Chapter 4: Circular Measure (Pages 99–115)

4.1 Radians

Definition: A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

• Conversion: $\pi\,\text{radians} = 180^\circ$.

• To change degrees to radians: multiply by $\frac{\pi}{180}$.

• To change radians to degrees: multiply by $\frac{180}{\pi}$.

4.2 Arc Length and Sector Area

For angle $\theta$ in radians:

• Arc Length: $s = r\theta$

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

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

Chapter 5: Trigonometry (Pages 116–154)

5.1 Exact Values

• $30^\circ \left(\frac{\pi}{6}\right)$: $\sin = \frac{1}{2}$, $\cos = \frac{\sqrt{3}}{2}$, $\tan = \frac{1}{\sqrt{3}}$.

• $45^\circ \left(\frac{\pi}{4}\right)$: $\sin = \frac{1}{\sqrt{2}}$, $\cos = \frac{1}{\sqrt{2}}$, $\tan = 1$.

• $60^\circ \left(\frac{\pi}{3}\right)$: $\sin = \frac{\sqrt{3}}{2}$, $\cos = \frac{1}{2}$, $\tan = \sqrt{3}$.

5.3 Trigonometric Ratios of General Angles

Using the unit circle and sign quadrants (CAST diagram):

• Quadrant 1 (0–90): All positive.

• Quadrant 2 (90–180): Sine positive.

• Quadrant 3 (180–270): Tangent positive.

• Quadrant 4 (270–360): Cosine positive.

5.4 Graphs and Periods

• $y = \sin(x)$: Period $360^\circ$, Amplitude $1$.

• $y = \cos(x)$: Period $360^\circ$, Amplitude $1$.

• $y = \tan(x)$: Period $180^\circ$, no amplitude, asymptotes at $x = \pm 90^\circ, 270^\circ, \dots$

5.7 Trigonometric Identities

1. $\tan(x) \equiv \frac{\sin(x)}{\cos(x)}$

2. $\sin^2(x) + \cos^2(x) \equiv 1$

Chapter 6: Series (Pages 155–185)

6.1 Binomial Expansion

For positive integer $n$: $(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}b^n$ Where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$.

6.3 Arithmetic Progressions (AP)

• nth term: $u_n = a + (n - 1)d$

• Sum of n terms: $S_n = \frac{n}{2}(a + l) = \frac{n}{2}[2a + (n - 1)d]$.

6.4 Geometric Progressions (GP)

• nth term: $u_n = ar^{n-1}$

• Sum of n terms: $S_n = \frac{a(1 - r^n)}{1 - r}$.

• Sum to infinity: $S_\infty = \frac{a}{1 - r}$, convergent only if -1 < r < 1.

Chapter 7 & 8: Differentiation (Pages 190–237)

7.1 Derivatives

 Differentiation finds the exact gradient of a curve.

• Power Rule: If $y = x^n$, then $\frac{dy}{dx} = nx^{n-1}$.

• Chain Rule: $\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}$.

7.3 Tangents and Normals

• Gradient of Tangent ($m_T$) is $\frac{dy}{dx}$ at point $(x_1, y_1)$.

• Gradient of Normal ($m_N$) is $-\frac{1}{m_T}$.

8.1 Increasing and Decreasing Functions

• Increasing: \frac{dy}{dx} > 0

• Decreasing: \frac{dy}{dx} < 0

8.2 Stationary Points

Stationary points occur when $\frac{dy}{dx} = 0$.

• Minimum point: \frac{d^2y}{dx^2} > 0

• Maximum point: \frac{d^2y}{dx^2} < 0

• Point of inflexion: $\frac{d^2y}{dx^2} = 0$ (requires first derivative test check for sign change).

Chapter 9: Integration (Pages 238–283)

9.1 Rules of Integration

Integration is the reverse of differentiation.

• Power Rule: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + c$ (for $n \neq -1$).

• Definite Integral: $\int_a^b f(x)\,dx = [F(x)]_a^b = F(b) - F(a)$.

9.6 Area Under a Curve

• Area between curve and x-axis: $A = \int_a^b y\,dx$.

• Area between curve and y-axis: $A = \int_a^b x\,dy$.

• Area between two curves: $A = \int_a^b [f(x) - g(x)]\,dx$.

9.9 Volumes of Revolution

Region rotated $360^\circ$ about an axis:

• Rotation about x-axis: $V = \int_a^b \pi y^2\,dx$.

• Rotation about y-axis: $V = \int_a^b \pi x^2\,dy$.