OCR A Level Mathematics: Pure Mathematics — Revision Notes

Condensed for the final weeks. For the full explanation, use the Pure Mathematics study guide.

Proof

Method	Approach
Deduction	Argue directly from known results
Exhaustion	Check every case
Counter-example	One case disproves a general statement
Contradiction	Assume the negation, derive an impossibility

Contradiction is the one examined most. The classic results are the irrationality of √2 and the infinitude of primes. The structure must be explicit: assume the opposite, derive a contradiction, conclude the original statement.

A single counter-example disproves a statement, but no number of examples proves one. Stating that explicitly is often worth a mark.

Functions

• Domain — the set of permitted inputs. Range — the set of outputs.
• fg(x) means “do g first”. The order is the most common error.
• Inverse f⁻¹ exists only if f is one-to-one; the graph of f⁻¹ is the reflection of f in y = x, and the domain and range swap.
• Modulus: |f(x)| reflects negative parts above the axis; f(|x|) reflects the right-hand side into the left.

Transformations:

	Effect
f(x) + a	Up by a
f(x + a)	Left by a
af(x)	Vertical stretch, factor a
f(ax)	Horizontal stretch, factor 1/a

Changes inside the bracket affect x and behave opposite to expectation. That is the trap.

Sequences and series

arithmetic:  u_n = a + (n-1)d       S_n = (n/2)[2a + (n-1)d]
geometric:   u_n = ar^(n-1)         S_n = a(1-r^n)/(1-r)
sum to infinity:  S = a/(1-r)       ONLY if |r| < 1

The convergence condition is a mark in itself. A sum to infinity exists only when |r| < 1; stating the answer without the condition is incomplete.

Binomial expansion:

(1 + x)^n = 1 + nx + n(n-1)x^2/2! + ...

For negative or fractional n the expansion is infinite and valid only for |x| < 1. Stating the validity range is required.

Trigonometry

sin^2 + cos^2 = 1      1 + tan^2 = sec^2      1 + cot^2 = cosec^2
sin2A = 2 sinA cosA    cos2A = cos^2A - sin^2A = 2cos^2A - 1 = 1 - 2sin^2A
R form:  a sinx + b cosx = R sin(x + alpha)

Radians: π rad = 180°. Arc length s = rθ; sector area = ½r²θ. These formulae only work in radians — using degrees is the standard error.

Small angle approximations (θ in radians): sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2.

Differentiation

chain:     dy/dx = dy/du x du/dx
product:   (uv)' = u'v + uv'
quotient:  (u/v)' = (u'v - uv')/v^2

d/dx(e^x) = e^x        d/dx(ln x) = 1/x
d/dx(sin x) = cos x    d/dx(cos x) = -sin x

Trigonometric derivatives require radians.

Implicit differentiation: differentiate both sides with respect to x, applying the chain rule to every y term, so d/dx(y²) = 2y·dy/dx.

Integration

integral of 1/x dx = ln|x| + c
integral of e^x = e^x + c
by parts:  integral of u dv = uv - integral of v du

Choosing u for parts: pick the term that simplifies on differentiation — logs first, then polynomials. Choosing the wrong way round makes the integral harder rather than easier.

Substitution: change the variable and the limits, or convert back before evaluating. Forgetting to change the limits is a routine loss.

Numerical methods

Locating a root: if f(x) changes sign over an interval (one value positive, one negative) and f is continuous, a root lies between them.

Iteration: rearrange f(x) = 0 into x = g(x) and iterate x_(n+1) = g(x_n) from a starting value. A staircase diagram shows convergence to the root; a cobweb diagram shows convergence that spirals in from both sides. Some rearrangements diverge — this must be checked, not assumed.

Newton-Raphson:

x_(n+1) = x_n - f(x_n)/f'(x_n)

Fails or converges to the wrong root if the starting value is chosen near a stationary point, where f'(x_n) is close to zero.

Trapezium rule approximates the area under a curve using n trapezia of equal width h = (b-a)/n:

integral ≈ (h/2)[y_0 + y_n + 2(y_1 + y_2 + ... + y_(n-1))]

For a convex (concave up) curve the rule over-estimates the true area; for a concave (concave down) curve it under-estimates. Stating which way the error runs, not just quoting the formula, is what the “comment on accuracy” mark requires.

Vectors

magnitude  |a| = sqrt(x^2 + y^2 + z^2)
unit vector = a / |a|

Vectors are parallel if one is a scalar multiple of the other. Position vectors describe points; direction vectors describe lines.

Exam traps

• Doing f before g in fg(x).
• Getting f(x + a) the wrong way round.
• Omitting the |r| < 1 condition for a sum to infinity.
• Omitting the validity range for a binomial expansion with fractional n.
• Using degrees in arc length, sector area, or calculus.
• Forgetting to change limits after a substitution.
• Omitting + c.

Self-test

1. Give the structure of a proof by contradiction.
2. In fg(x), which function is applied first?
3. When does a geometric series have a sum to infinity?
4. Which term should you choose as u in integration by parts, and why?
5. Why must arc length and sector area formulae use radians?

Answers: 1. Assume the opposite of the statement, derive a logical contradiction, and conclude that the original statement must be true. 2. g. 3. Only when |r| < 1. 4. The one that simplifies when differentiated — logarithms first, then polynomials — because the aim is to make the remaining integral easier. 5. The formulae s = rθ and ½r²θ are derived from the definition of the radian as the angle subtending an arc equal in length to the radius, so they are only valid with θ in radians.