Pure Mathematics Flashcards

A-Level Pure Mathematics — Flashcard Deck

15 flashcards with spaced repetition. Press Space to flip, then rate your recall (1—4).

What These Flashcards Cover

These flashcards test your knowledge of the core pure mathematics topics that underpin the A-Level specification. You should be confident with algebraic manipulation, trigonometric identities, sequences and series, and vector operations.

Key areas:

• Algebra and Proof: Factorising quadratics and cubics. The discriminant $b^{2} - 4ac$ determines the nature of roots. Proof by contradiction (e.g. proving $\sqrt{2}$ is irrational) and proof by induction (used for summing series and divisibility results).
• Trigonometry: The three trigonometric ratios and their reciprocals. Key identities: $\sin^{2}\theta + \cos^{2}\theta = 1$, $\tan\theta = \frac{\sin\theta}{\cos\theta}$. Addition formulae: $\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$. Double angle formulae. Solving trigonometric equations in given intervals.
• Sequences and Series: Arithmetic sequences: $u_{n} = a + (n-1)d$. Sum: $S_{n} = \frac{n}{2}(2a + (n-1)d)$. Geometric sequences: $u_{n} = ar^{n-1}$. Sum: $S_{n} = \frac{a(1-r^{n})}{1-r}$. Convergent geometric series: $S_{\infty} = \frac{a}{1-r}$ for $|r| < 1$.
• Logarithms and Exponentials: $y = \ln x \iff x = e^{y}$. Laws: $\ln(ab) = \ln a + \ln b$, $\ln\left(\frac{a}{b}\right) = \ln a - \ln b$, $\ln(a^{n}) = n\ln a$. Solving equations like $2^{x} = 5$ by taking logarithms of both sides.
• Differentiation and Integration (Pure): Differentiate $\sin x$, $\cos x$, $e^{x}$, $\ln x$, $\tan x$. Integrate $\cos x$, $\sin x$, $e^{x}$, $\sec^{2}x$, $\tan x$. Use substitution and integration by inspection.
• Vectors: 2D and 3D vectors. Magnitude $|\mathbf{a}| = \sqrt{x^{2}+y^{2}+z^{2}}$. Dot product $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta$. Vector equations of lines: $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$. Intersection of lines.

Intuition

Pure mathematics is about structure and pattern. Algebra gives you the tools to manipulate abstract expressions. Trigonometry connects geometry (triangles, circles) to algebra (equations, identities). Sequences and series describe patterns that repeat or accumulate.

Think of proof as the difference between “it looks true” and “it must be true.” Empirical evidence (checking a few cases) is not the same as a logical proof that covers all cases.

Common Pitfalls

1. Losing solutions when squaring both sides. Squaring can introduce extraneous solutions. Always check your answers in the original equation.
2. Forgetting that $\ln x$ is undefined for $x \leq 0$. When solving logarithmic equations, any solution that gives a negative or zero argument must be rejected.
3. Confusing arithmetic and geometric series. Arithmetic series have a common difference ($d$); geometric series have a common ratio ($r$). Using the wrong formula gives incorrect sums.
4. Misapplying vector direction. The vector from $A$ to $B$ is $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, not $\mathbf{a} - \mathbf{b}$. Getting the direction wrong flips the sign of your answer.

Cross-References

• Calculus: Calculus is a core pure maths topic
• Mechanics: Pure maths supports mechanics
• Statistics: Pure maths underpins statistics

Advanced Content

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Derivations and Proofs

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Extended Examples

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

Research Connections

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Prerequisites

Ensure you have mastered the prerequisite material before attempting this advanced content.