Pearson Edexcel International A Level Pure Mathematics 4 Study Notes

CHAPTER 1: PROOF

1.1 Proof by Contradiction

• Definition of Contradiction: A disagreement between two statements, meaning both cannot be true simultaneously.
• Methodology:
  1. Negation: Start by assuming the statement you wish to prove is not true.
  2. Logical Deduction: Use logical steps to show that this assumption leads to something impossible.
  3. Contradiction: This result will either contradict the initial assumption or a known mathematical fact.
  4. Conclusion: Conclude that the assumption was incorrect, therefore the original statement must be true.
• Notation: The negation of a statement is the assertion of its falsehood.
• Rational and Irrational Numbers:
  • Rational Numbers ($\mathbb{Q}$): Can be written in the form $\frac{a}{b}$ where $a$ and $b$ are integers.
  • Irrational Numbers: Cannot be expressed in the form $\frac{a}{b}$ with integer components.

Examples and Applications

• Greatest Odd Integer: Proved false by assuming a greatest odd integer $n$ exists; however, $n + 2$ is also an integer, greater than $n$, and odd ($\text{odd} + \text{even} = \text{odd}$). This contradicts the assumption.
• Even Squares: If $n^2$ is even, then $n$ must be even. To prove this, assume $n$ is odd ($n = 2k + 1$). Then $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$, which is odd. This contradicts the given fact that $n^2$ is even.
• Irrationality of $\sqrt{2}$: Assume $\sqrt{2} = \frac{a}{b}$ in simplest form. Then $2b^2 = a^2$, implying $a$ is even ($a = 2n$). Substituting gives $2b^2 = 4n^2$, so $b^2 = 2n^2$, implying $b$ is also even. If both are even, they share a factor of 2, contradicting the "simplest form" assumption.
• Infinite Prime Numbers: Assume a finite set of primes $P_1, P_2, …, P_n$. Consider $N = (P_1 \times P_2 \times … \times P_n) + 1$. Dividing $N$ by any listed prime results in a remainder of 1. Thus, $N$ is either prime or has a prime factor not in the list, contradicting the finite assumption.

CHAPTER 2: PARTIAL FRACTIONS

2.1 Linear Factors

• The Concept: Splitting a single fraction with distinct linear factors in the denominator into a sum of two or more separate fractions.
• General Form: $\frac{5}{(x+1)(x-4)} = \frac{A}{x+1} + \frac{B}{x-4}$.
• Methods to Find Constants:
  1. Substitution: Choosing values for $x$ that eliminate one or more terms (usually the roots of the linear factors).
  2. Equating Coefficients: Expanding the numerator and matching the coefficients of $x^n$ on both sides of the identity.

2.2 Repeated Factors

• Requirement: If a linear factor is squared in the denominator, it must be represented twice: once with the linear power and once with the squared power.
• General Form: $\frac{2x+9}{(x-5)(x+3)^2} = \frac{A}{x-5} + \frac{B}{x+3} + \frac{C}{(x+3)^2}$.

2.3 Improper Fractions

• Definition: A fraction where the degree of the numerator is greater than or equal to the degree of the denominator.
• Procedure: Perform algebraic division or use a structural identity to convert the fraction into a polynomial plus a proper fraction before splitting into partial fractions.
• Structural Forms:
  • Degree 2 over Degree 2: $A + \frac{B}{linear} + \frac{C}{linear}$.
  • Degree 3 over Degree 2: $Ax + B + \frac{C}{linear} + \frac{D}{linear}$.

CHAPTER 3: COORDINATE GEOMETRY

3.1 Parametric Equations

• Definitions: Coordinates $x$ and $y$ are expressed as functions of a third variable, the parameter (usually $t$ or $\theta$).
  • $x = p(t)$
  • $y = q(t)$
• Cartesian Conversion: Eliminate the parameter by rearranging one equation for $t$ and substituting into the other.
• Domain and Range:
  • The domain of the Cartesian function $y = f(x)$ is the range of the parametric function for $x$.
  • The range of the Cartesian function $y = f(x)$ is the range of the parametric function for $y$.

3.2 Trigonometric Identities in Parametric Forms

• Parameters involving trig functions often require identities for elimination:
  • Use $\sin^2(t) + \cos^2(t) = 1$ for circles/ellipses.
  • Use $1 + \tan^2(t) = \sec^2(t)$ and $1 + \cot^2(t) = \csc^2(t)$.
  • Double angle identities: $\sin(2t) = 2\sin(t)\cos(t)$.
• Shapes: A specific common form is $x = a\sin(t) + h$ and $y = a\cos(t) + k$, which represents a circle $(x - h)^2 + (y - k)^2 = a^2$.

CHAPTER 4: BINOMIAL EXPANSION

4.1 Expanding $(1 + x)^n$

• The Formula (for rational $n$):
  $(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + …$
• Validity: The expansion is an infinite series and only converges (is valid) if |x| < 1.
• Generalizing for $(1 + bx)^n$: Valid when |bx| < 1, or |x| < \frac{1}{|b|}.

4.2 Expanding $(a + bx)^n$

• Transformation: You must factor out $a^n$ to make the first term inside the bracket 1:
  $(a + bx)^n = a^n \left( 1 + \frac{b}{a}x \right)^n$
• Validity: Valid when |\frac{b}{a}x| < 1, or |x| < |\frac{a}{b}|.

4.3 Using Partial Fractions

• For complex rational expressions, first use partial fractions to decompose the expression into terms of the form $A(1 + kx)^{-n}$, then expand each term individually.

CHAPTER 5: DIFFERENTIATION

5.1 Parametric Differentiation

• Chain Rule: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.
• Used to find the gradient of a curve defined parametrically without needing the Cartesian form.

5.2 Implicit Differentiation

• Differentiating equations where $y$ is not the subject (e.g., $x^2 + y^2 = 25$).
• Key Rules:
  • $\frac{d}{dx}(f(y)) = f'(y) \frac{dy}{dx}$
  • Product Rule: $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$

5.3 Rates of Change and Differential Equations

• Chain Rule for Rates: $\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$.
• Formulating Equations:
  • "Rate of change is proportional to…": $\frac{dx}{dt} = kx$.
  • "Rate of loss/decrease": Includes a negative sign, e.g., $\frac{d\theta}{dt} = -k(\theta - \theta_0)$.

CHAPTER 6: INTEGRATION

6.1 Parametric Area and Volume

• Area: $A = \int_{x_1}^{x_2} y \,dx = \int_{t_1}^{t_2} g(t) f'(t) \,dt$.
• Volume (Rotation about x-axis): $V = \pi \int_{x_1}^{x_2} y^2 \,dx = \pi \int_{t_1}^{t_2} [g(t)]^2 f'(t) \,dt$.

6.2 Advanced Methods

• Substitution: Letting $u = f(x)$ to simplify the integral. Remember to differentiate the substitution to find $dx$ in terms of $du$ and adjust the limits.
• Integration by Parts: $\int u \frac{dv}{dx} \,dx = uv - \int v \frac{du}{dx} \,dx$.
  • Order of priority for $u$: Logarithms, Algebraic, Trigonometric, Exponentials (LATE).
• Partial Fractions: Integrate rational functions by decomposing them into simpler linear denominators.

CHAPTER 7: VECTORS

7.1 Arithmetic and Representation

• Triangle Law: $\mathbf{AB} + \mathbf{BC} = \mathbf{AC}$.
• Magnitudes: In 3D, for $\mathbf{a} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$, the magnitude is $|\mathbf{a}| = \sqrt{x^2 + y^2 + z^2}$.
• Unit Vectors: $\mathbf{\hat{a}} = \frac{\mathbf{a}}{|\mathbf{a}|}$.

7.2 Lines in 3D

• Equation of a Line: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$.
  • $\mathbf{a}$: Position vector of a point on the line.
  • $\mathbf{b}$: Direction vector parallel to the line.
• Intersection: Set equations for two lines equal; solve the resulting simultaneous equations for parameters $\lambda$ and $\mu$. If they satisfy the third coordinate equation, the lines intersect.

7.3 Scalar Product (Dot Product)

• Definition: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta)$.
• Component Form: $\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$.
• Properties:
  • $\mathbf{a} \cdot \mathbf{b} = 0$ if and only if $\mathbf{a}$ and $\mathbf{b}$ are perpendicular.
  • $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$.