Pure Mathematics 4

Worked examples (6)

• 1Proof by deduction — algebraic identity (4 marks, P4)
• 2Proof by deduction — divisibility (5 marks, P4)
• 3Proof by exhaustion (4 marks, P4)
• 4Disproof by counter-example (3 marks, P4)
• 5Proof by contradiction — $\\sqrt{2}$ is irrational (6 marks, P4)
• 6Proof by contradiction — infinitely many primes (Euclid, 6 marks, P4)

Worked examples (5)

• 1Distinct linear factors — cover-up method (4 marks, P4)
• 2Three distinct linear factors (5 marks, P4)
• 3Repeated linear factor (6 marks, P4)
• 4Irreducible quadratic factor (6 marks, P4)
• 5Improper fraction — divide first (6 marks, P4)

Worked examples (5)

• 1$(1 + x)^{1/2}$ to four terms (4 marks, P4)
• 2$(1 - 2x)^{-3}$ to three terms (5 marks, P4)
• 3$(4 + x)^{1/2}$ — factor out the constant (6 marks, P4)
• 4Partial fractions + binomial (8 marks, P4)
• 5Approximation application (5 marks, P4)

Worked examples (6)

• 1Implicit differentiation — circle (4 marks, P4)
• 2Implicit differentiation with a product term (6 marks, P4)
• 3Tangent to an implicit curve (6 marks, P4)
• 4Differentiating $a^{x}$ (4 marks, P4)
• 5Connected rates of change — implicit (6 marks, P4)
• 6Derivative of $\\arctan x$ (4 marks, P4)

Worked examples (7)

• 1Integration via partial fractions (6 marks, P4)
• 2Integration by substitution (6 marks, P4)
• 3Integration by parts (5 marks, P4)
• 4Integration by parts twice (6 marks, P4)
• 5Volume of revolution (6 marks, P4)
• 6Separable differential equation (7 marks, P4)
• 7First-order linear DE — integrating factor (8 marks, P4)

Worked examples (7)

• 1Magnitude and unit vector (4 marks, P4)
• 2Dot product and angle (5 marks, P4)
• 3Perpendicularity (4 marks, P4)
• 4Vector equation of a line through two points (5 marks, P4)
• 5Showing two lines intersect — find the point (7 marks, P4)
• 6Classify two lines as parallel / intersecting / skew (6 marks, P4)
• 7Acute angle between two lines (4 marks, P4)