Cambridge International AS & A Level Mathematics (9709) – Normal Distribution, Related Topics and Full Syllabus Overview
1. Syllabus Mapping – Where Each Topic Belongs
| Paper |
Unit (Syllabus Code) |
Key Sub‑topics (Core) |
| Paper 1 |
Pure Mathematics 1 (P1) |
- Quadratics – completing the square, factor theorem, roots
- Functions – domain, range, composition, inverse
- Coordinate geometry – straight line, circle, parabola
- Circular measure – radians, arc length, sector area
- Trigonometry – identities, solving equations, sine/cosine rule
- Series – arithmetic & geometric progressions
- Differentiation – basic rules, tangent, normal
- Integration – elementary antiderivatives, area under a curve
|
| Paper 2 |
Pure Mathematics 2 (P2) |
- Algebra – simultaneous equations, factorisation, inequalities
- Logarithms & exponentials – change of base, solving equations
- Advanced trigonometry – multiple‑angle, sum‑to‑product, R‑formula
- Differentiation – product, quotient, chain rule, optimisation
- Integration – substitution, integration by parts (restricted), definite integrals
- Numerical methods – Newton‑Raphson, iteration, trapezium rule
|
| Paper 3 |
Pure Mathematics 3 (P3) |
- Vectors – notation, scalar product, vector product (2‑D), applications
- Complex numbers – Cartesian & polar forms, De Moivre’s theorem
- First‑order differential equations – separable, linear, modelling
- Partial fractions – simple & repeated linear factors
- Integration techniques – completing the square, trigonometric substitution (restricted)
- Series – binomial expansion, Maclaurin series (up to x³ term)
|
| Paper 4 |
Mechanics (M) |
- Forces & equilibrium – free‑body diagrams, resultant, moments
- Kinematics – constant acceleration, projectile motion, relative velocity
- Momentum – impulse, conservation, collisions (elastic & inelastic)
- Newton’s laws – applications, friction, tension
- Work, energy & power – kinetic & potential energy, work‑energy theorem
|
| Paper 5 |
Probability & Statistics 1 (PS1) |
- Data representation – stem‑and‑leaf, histogram, box‑plot
- Measures of central tendency & dispersion – mean, median, mode, range, IQR, standard deviation
- Basic probability – sample spaces, complementary, addition & multiplication rules
- Counting techniques – permutations, combinations, binomial coefficients
- Discrete random variables – binomial distribution (definition, mean, variance)
- Continuous random variables – normal distribution (definition, properties, standardisation, Z‑tables, Empirical rule, percentiles, normal approximation to the binomial with continuity correction)
|
| Paper 6 |
Probability & Statistics 2 (PS2) |
- Poisson distribution – definition, mean = variance, applications
- Linear combinations of independent normal variables
- Sampling distribution of the mean – Central Limit Theorem (CLT)
- Point & interval estimation – confidence intervals for a mean (known & unknown σ)
- Hypothesis testing – one‑sample z‑test for a mean (two‑tailed and one‑tailed)
|
2. Pure Mathematics 1 (Paper 1)
2.1 Quadratics
- Standard form: \(ax^{2}+bx+c=0\)
- Completing the square: \(a\bigl(x+\tfrac{b}{2a}\bigr)^{2}= \tfrac{b^{2}-4ac}{4a}\)
- Roots: \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)
- Factor theorem: if \(f(r)=0\) then \((x-r)\) is a factor.
2.2 Functions & Graphs
- Domain / range, composition \((f\circ g)(x)=f(g(x))\), inverse \(f^{-1}(x)\).
- Key shapes: linear, quadratic, cubic, reciprocal, absolute‑value.
2.3 Coordinate Geometry
- Line: \(y=mx+c\) or \(ax+by+d=0\); slope \(m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
- Circle: \((x-h)^{2}+(y-k)^{2}=r^{2}\).
- Parabola (standard): \(y=ax^{2}+bx+c\) or \((x-h)^{2}=4p(y-k)\).
2.4 Trigonometry
- Fundamental identities: \(\sin^{2}\theta+\cos^{2}\theta=1\), \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
- Solving equations: e.g. \(\sin\theta=0.6\) ⇒ \(\theta=36.87^{\circ}\) or \(180^{\circ}-36.87^{\circ}\).
- R‑formula: \(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha)\) where \(R=\sqrt{a^{2}+b^{2}}\).
2.5 Differentiation & Integration (Core)
Typical exam question: find the gradient of \(y=3x^{2}-5x+2\) at \(x=1\).
- Derivative: \(\dfrac{dy}{dx}=6x-5\); at \(x=1\), gradient \(=1\).
- Integral: \(\displaystyle\int (3x^{2}-5x+2)\,dx = x^{3}-\tfrac{5}{2}x^{2}+2x+C\).
3. Pure Mathematics 2 (Paper 2)
3.1 Algebraic Techniques
- Simultaneous linear equations – substitution or elimination.
- Inequalities – solving quadratic inequalities by sign chart.
- Logarithms: \(\log_{a}b=c\iff a^{c}=b\); change of base \(\log_{a}b=\dfrac{\log b}{\log a}\).
3.2 Advanced Trigonometry
- Multiple‑angle: \(\sin2\theta=2\sin\theta\cos\theta\), \(\cos3\theta=4\cos^{3}\theta-3\cos\theta\).
- Sum‑to‑product: \(\sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\).
3.3 Differentiation – Chain, Product & Quotient Rules
\[
\frac{d}{dx}[u v]=u'v+uv',\qquad
\frac{d}{dx}\!\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}},\qquad
\frac{d}{dx}[f(g(x))]=f'(g(x))\,g'(x)
\]
3.4 Integration – Substitution & Definite Integrals
Example: \(\displaystyle\int 2x\sqrt{x^{2}+1}\,dx\).
- Let \(u=x^{2}+1\), \(du=2x\,dx\).
- Integral becomes \(\int \sqrt{u}\,du = \frac{2}{3}u^{3/2}+C = \frac{2}{3}(x^{2}+1)^{3/2}+C\).
3.5 Numerical Methods
- Newton‑Raphson: \(x_{n+1}=x_{n}-\dfrac{f(x_{n})}{f'(x_{n})}\).
- Trapezium rule for \(\displaystyle\int_{a}^{b}f(x)dx\): \(\approx\frac{b-a}{2}[f(a)+f(b)]\) (or with more sub‑intervals).
4. Pure Mathematics 3 (Paper 3)
4.1 Vectors (2‑D)
- Position vector \(\mathbf{r}=x\mathbf{i}+y\mathbf{j}\).
- Scalar product: \(\mathbf{a}\cdot\mathbf{b}=|a||b|\cos\theta = ax+by\).
- Applications: finding angle between two lines, projection of a vector.
4.2 Complex Numbers
- Cartesian form: \(z=a+bi\).
- Polar form: \(z=r(\cos\theta+i\sin\theta)=re^{i\theta}\) where \(r=\sqrt{a^{2}+b^{2}}\), \(\theta=\tan^{-1}\frac{b}{a}\).
- De Moivre: \((\cos\theta+i\sin\theta)^{n}=\cos n\theta+i\sin n\theta\).
4.3 First‑Order Differential Equations (Separable)
General form \(\dfrac{dy}{dx}=g(x)h(y)\).
- Separate: \(\dfrac{1}{h(y)}dy=g(x)dx\).
- Integrate both sides and apply the constant of integration.
4.4 Partial Fractions (Restricted)
Example: \(\displaystyle\frac{3x+5}{(x-1)(x+2)} = \frac{A}{x-1}+\frac{B}{x+2}\).
- Solve for \(A,B\): \(3x+5=A(x+2)+B(x-1)\) ⇒ \(A=2,\;B=1\).
4.5 Series Expansions (Binomial, up to \(x^{3}\))
\[
(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+O(x^{4})
\]
5. Mechanics (Paper 4)
5.1 Forces & Equilibrium
- Resultant of concurrent forces: vector sum \(\mathbf{R}=\sum\mathbf{F}_{i}\).
- Equilibrium conditions: \(\sum\mathbf{F}=0\) and \(\sum\text{moments}=0\).
5.2 Kinematics (Constant Acceleration)
\[
v = u+at,\qquad s = ut+\tfrac12 at^{2},\qquad v^{2}=u^{2}+2as
\]
5.3 Momentum & Collisions
- Momentum: \(\mathbf{p}=m\mathbf{v}\).
- Impulse: \(\mathbf{J}=\Delta\mathbf{p}= \int \mathbf{F}\,dt\).
- Conservation in isolated system: \(\sum\mathbf{p}_{\text{initial}}=\sum\mathbf{p}_{\text{final}}\).
5.4 Work, Energy & Power
- Work done by a constant force: \(W=F\,s\cos\theta\).
- Kinetic energy: \(KE=\frac12 mv^{2}\).
- Work‑energy theorem: \(W_{\text{net}}=\Delta KE\).
- Power: \(P=\frac{W}{t}=Fv\cos\theta\).
5.5 Example – Projectile Motion
Find the maximum height of a projectile launched with speed \(20\text{ m s}^{-1}\) at \(30^{\circ}\) to the horizontal.
- Vertical component \(u_{y}=20\sin30^{\circ}=10\text{ m s}^{-1}\).
- Maximum height \(h=\frac{u_{y}^{2}}{2g}= \frac{10^{2}}{2\times9.8}\approx5.10\text{ m}\).
6. Probability & Statistics 1 (Paper 5) – Core Concepts & Normal Distribution
6.1 Visualising Data
- Stem‑and‑leaf plot – retains raw values, shows shape.
- Histogram – class intervals, frequency density; symmetry hints at normality.
- Box‑plot – median, quartiles, possible outliers.
6.2 Measures of Central Tendency & Dispersion
\[
\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i},\qquad
s=\sqrt{\frac{\sum (x_{i}-\bar{x})^{2}}{n-1}}
\]
- Median – middle value (or average of two middle values).
- Mode – most frequent value.
- Range = max – min; IQR = Q3 – Q1.
6.3 Basic Probability Rules
- Complementary: \(P(A^{c})=1-P(A)\).
- Addition (mutually exclusive): \(P(A\cup B)=P(A)+P(B)\).
- Multiplication (independent): \(P(A\cap B)=P(A)P(B)\).
6.4 Counting Techniques
- Permutations: \({}^{n}P_{r}= \dfrac{n!}{(n-r)!}\).
- Combinations: \(\displaystyle{n\choose r}= \dfrac{n!}{r!(n-r)!}\).
6.5 Binomial Distribution (Discrete Random Variable)
For \(n\) independent trials with success probability \(p\):
\[
P(B=k)=\binom{n}{k}p^{k}(1-p)^{\,n-k},\qquad k=0,1,\dots ,n
\]
Mean \(\mu=np\), variance \(\sigma^{2}=np(1-p)\).
6.6 The Normal Distribution (Continuous Random Variable)
6.6.1 Definition & Key Properties
\[
f(x)=\frac{1}{\sigma\sqrt{2\pi}}\;
\exp\!\left[-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right],\qquad -\infty
\(\mu\) = mean = median = mode.
\(\sigma>0\) = standard deviation.
Symmetric about \(x=\mu\); total area = 1.
Empirical (68‑95‑99.7) rule – see §6.6.3.
6.6.2 Standard Normal Distribution
When \(\mu=0,\;\sigma=1\) we write \(Z\sim N(0,1)\) with density
\[
f_{Z}(z)=\frac{1}{\sqrt{2\pi}}e^{-z^{2}/2}.
\]
Standardisation:
\[
Z=\frac{X-\mu}{\sigma}\qquad\text{so that }X=\mu+\sigma Z.
\]
6.6.3 Using a Z‑Table (Cambridge format)
| z | \(P(Z\le z)\) |
|---|
| 0.00 | 0.5000 |
| 0.50 | 0.6915 |
| 1.00 | 0.8413 |
| 1.28 | 0.8997 |
| 1.64 | 0.9495 |
| 1.96 | 0.9750 |
| 2.33 | 0.9901 |
| 2.58 | 0.9950 |
- Upper tail: \(P(Z>z)=1-P(Z\le z)\).
- Negative \(z\): \(P(Z\le -z)=1-P(Z\le z)\) (symmetry).
- Two‑tailed: \(P(|Z|>z)=2[1-P(Z\le z)]\).
6.6.4 Empirical (68‑95‑99.7) Rule
- \(P(\mu-\sigma
- \(P(\mu-2\sigma
- \(P(\mu-3\sigma
6.6.5 Example – Finding a Probability
Let \(X\sim N(120,10^{2})\). Find \(P(115
- Standardise: \(z_{1}=\frac{115-120}{10}=-0.5,\;z_{2}=\frac{130-120}{10}=1.0.\)
- From the table: \(P(Z\le0.5)=0.6915,\;P(Z\le1.0)=0.8413.\)
- Use symmetry: \(P(Z\le-0.5)=1-0.6915=0.3085.\)
- Probability required: \(0.8413-0.3085=0.5328.\)
6.6.6 Finding Percentiles (Inverse Normal)
- Locate the required cumulative probability \(p\) in the Z‑table to obtain \(z_{p}\).
- Transform back: \(x=\mu+\sigma z_{p}\).
Example: 90th percentile of \(N(50,4^{2})\).
\(z_{0.90}=1.28\) ⇒ \(x=50+4(1.28)=55.12.\)
6.6.7 Normal Approximation to the Binomial (with Continuity Correction)
- Check validity: \(np\ge5\) and \(n(1-p)\ge5\).
- Compute \(\mu=np,\;\sigma=\sqrt{np(1-p)}\).
- Replace the discrete bound by a continuous one (±0.5). Example: \(P(B\ge k)=P(B\ge k-0.5).\)
- Standardise and use the Z‑table.
Worked Example: \(B\sim\text{Bin}(40,0.30)\), find \(P(B\ge12)\).
- Validity: \(np=12,\;n(1-p)=28\) – both ≥ 5.
- \(\mu=12,\;\sigma=\sqrt{40\cdot0.3\cdot0.7}=2.90.\)
- Continuity correction: \(P(B\ge12)=P(B\ge11.5).\)
- Standardise: \(z=\dfrac{11.5-12}{2.90}=-0.17.\)
- From table \(P(Z\le0.17)=0.5675\); because the z‑value is negative, \(P(B\ge12)=P(Z\ge-0.17)=0.5675.\)
- Result ≈ 0.57 (to 2 d.p.).
7. Probability & Statistics 2 (Paper 6) – Advanced Topics
7.1 Poisson Distribution
\[
P(P=k)=\frac{e^{-\lambda}\lambda^{k}}{k!},\qquad k=0,1,2,\dots
\]
- Mean = Variance = \(\lambda\).
- Good approximation to \(\text{Bin}(n,p)\) when \(n\) large, \(p\) small, \(np=\lambda\le5\).
Example: Average of 3 calls per minute; probability of exactly 5 calls in a minute.
\[
P(P=5)=\frac{e^{-3}3^{5}}{5!}\approx0.1008.
\]
7.2 Linear Combinations of Independent Normal Variables
If \(X\sim N(\mu_{X},\sigma_{X}^{2})\) and \(Y\sim N(\mu_{Y},\sigma_{Y}^{2})\) are independent, then for constants \(a,b\):
\[
Z=aX+bY\;\sim\;N\bigl(a\mu_{X}+b\mu_{Y},\;a^{2}\sigma_{X}^{2}+b^{2}\sigma_{Y}^{2}\bigr).
\]
Useful for questions involving sums or differences of measurements (e.g. total weight of two independent items).
7.3 Sampling Distribution of the Mean – Central Limit Theorem (CLT)
- If a population has mean \(\mu\) and standard deviation \(\sigma\), the sample mean \(\bar{X}\) from a random sample of size \(n\) satisfies
\[
\bar{X}\;\approx\;N\!\left(\mu,\;\frac{\sigma^{2}}{n}\right)\qquad\text{(for sufficiently large }n\text{).}
\]
7.4 Point & Interval Estimation
- Point estimate of a population mean: \(\hat{\mu}=\bar{x}\).
- 95 % confidence interval (known \(\sigma\)):
\[
\bar{x}\;\pm\;z_{0.975}\frac{\sigma}{\sqrt{n}},\qquad z_{0.975}=1.96.
\]
- If \(\sigma\) is unknown, replace \(z\) by the appropriate \(t\)-value (not required for the core 9709 syllabus but appears in extensions).
Example: \(n=36,\;\bar{x}=78,\;\sigma=12\).
Margin of error \(=1.96\frac{12}{\sqrt{36}}=3.92\).
95 % CI: \((74.08,\;81.92)\).
7.5 Hypothesis Testing – One‑Sample z‑Test for a Mean
- State hypotheses (e.g. \(H_{0}:\mu=50\), \(H_{A}:\mueq50\)).
- Compute test statistic:
\[
z=\frac{\bar{x}-\mu_{0}}{\sigma/\sqrt{n}}.
\]
- Determine critical region from the Z‑table (e.g. two‑tailed 5 % ⇒ reject if \(|z|>1.96\)).
- Conclusion – “reject \(H_{0}\)” or “fail to reject \(H_{0}\)”.
Worked Example: Sample of 64 observations, \(\bar{x}=102\), known \(\sigma=15\). Test \(H_{0}:\mu=100\) at 5 % two‑tailed.
- \(z=\dfrac{102-100}{15/\sqrt{64}}=\dfrac{2}{1.875}=1.07.\)
- \(|z|=1.07<1.96\) ⇒ fail to reject \(H_{0}\).
8. Summary Checklist for Exam Questions
- Read the question carefully – identify the paper and the required variable type (discrete vs continuous).
- Check which syllabus block the question belongs to (Pure 1‑3, Mechanics, PS 1, PS 2).
- For normal‑distribution problems:
- Write down \(\mu\) and \(\sigma\); standardise to a Z‑score.
- Apply continuity correction when approximating a binomial.
- Use the Empirical rule for quick estimates, otherwise read the exact value from the Z‑table.
- For binomial questions:
- Check the \(np\) and \(n(1-p)\) criteria.
- Compute \(\mu\) and \(\sigma\), apply continuity correction, then use the Z‑table.
- For Poisson questions:
- Identify the mean rate \(\lambda\) and use the Poisson formula.
- For confidence‑interval or hypothesis‑testing items:
- Confirm whether \(\sigma\) is known.
- Choose the correct critical value (z or t) and compute the statistic.
- Mechanics:
- Draw a clear free‑body diagram.
- Write down equilibrium or motion equations before substituting numbers.
- Pure‑math algebra/trigonometry:
- State the relevant identity or theorem before manipulation.
- Check units and significant figures in the final answer.
Keep this checklist handy during the exam – it helps ensure that no step is missed and that the answer is presented in the format expected by Cambridge assessors.