About AS Level
Paper ClipSAT Chapter Key Topics P1 Pure 1 Algebra — Indices, Surds & Quadratics Laws of indices, surds, completing the square, discriminant Simultaneous Equations & Inequalities Linear/nonlinear systems, modulus inequalities Graphs & Transformations y=f(x) transformations: translations, stretches, reflections Coordinate Geometry Line segments, circles, intersection Circular Measure Radians, arc length, sector area, segment Trigonometry Exact values, identities, solving equations in given range Series & Binomial Theorem AP/GP, sum to infinity, \((a+b)^n\) expansion Differentiation & Integration Gradient, tangent/normal, stationary points, definite integral, area P1 Functions: Composite & Inverse Domain, range, \(fg\), \(f^{-1}\) M1 Mechanics Kinematics SUVAT, velocity-time graphs, calculus kinematics Forces & Equilibrium / Newton's Laws Resultant, components, \(F=ma\), connected particles Energy, Work & Power KE, PE, work-energy principle, power S1 Statistics Representation of Data Histograms, cumulative freq., mean, variance, SD, box plots Permutations & Combinations \(n!\), \(^nP_r\), \(^nC_r\), arrangement with restrictions Discrete Random Variables Probability distribution, \(E(X)\), \(\text{Var}(X)\) Binomial Distribution \(X\sim B(n,p)\), cumulative probabilities, mean & variance Normal Distribution \(X\sim N(\mu,\sigma^2)\), standardisation, inverse normal
AS Level Mathematics is the first year of the A Level, offered by boards such as Cambridge (9709) and Edexcel. It blends Pure Mathematics with an introduction to Statistics and Mechanics.
The Pure core runs through indices and surds, quadratics, coordinate geometry, graphs and transformations, trigonometry, and the first ideas of differentiation and integration. Applied units add probability, data handling and kinematics.
Each chapter pairs a reference panel with worked examples; the explorers make slope, transformations and the gradient idea visual. Finish with the 50-problem set, then export it as a worksheet.
Indices, Surds & Quadratics
📌 Algebra — Indices, Surds & Quadratics P1 · Laws of indices · Surds · Completing the square · Discriminant & nature of roots
The laws of indices, simplifying and rationalising surds, and the full toolkit for quadratics — factorising, completing the square and the formula.
\( a^{m}a^{n}=a^{m+n} \), \( a^{-n}=\tfrac{1}{a^{n}} \), \( a^{1/n}=\sqrt[n]{a} \). Quadratic formula \( x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} \); the discriminant \( b^{2}-4ac \) fixes the number of real roots.
Simplify \( \sqrt{12} \).
Complete the square for \( x^{2}+4x \).
Simultaneous Equations & Inequalities
📌 Simultaneous Equations & Inequalities P1 · Linear & non-linear systems · Quadratic inequalities · Modulus function
Solving linear and linear-with-quadratic systems, and linear and quadratic inequalities.
Solve a linear system by substitution or elimination; for a linear–quadratic pair, substitute the linear equation into the quadratic. A quadratic inequality is solved by finding the roots and testing the sign of each interval.
Solve \( x+y=5 \), \( x-y=1 \). Add: \( 2x=6 \):
Solve \( x^{2}-4<0 \). Roots \( \pm2 \), curve below axis between them:
Graphs & Transformations
📌 Graphs & Transformations P1 · Sketching curves · \(f(x+a)\), \(f(ax)\), \(-f(x)\), \(f(-x)\) · Identifying transformations
Sketching standard curves and applying translations, reflections and stretches.
\( f(x)+a \) shifts up by \( a \); \( f(x-a) \) shifts right by \( a \); \( -f(x) \) reflects in the \( x \)-axis; \( af(x) \) stretches vertically by factor \( a \).
The grey curve is \( y=\sin x \); the indigo curve is \( y=\sin(x-h)+k \). Slide \( h \) and \( k \) to translate it.
| x | sin(x) | translated |
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Inside the function, \( x-h \) moves right; outside, \( +k \) moves up.
Describe \( y=(x-2)^{2} \) relative to \( y=x^{2} \). It is a translation \( 2 \) units to the right.
Describe \( y=-x^{2} \) relative to \( y=x^{2} \). A reflection in the \( x \)-axis.
Coordinate Geometry
📌 Coordinate Geometry P1 · Line equations · Circle equation & properties · Tangent to a circle
Straight lines — gradient, midpoint, distance and equations — and the equation of a circle.
Gradient \( m=\dfrac{y_2-y_1}{x_2-x_1} \); line \( y-y_1=m(x-x_1) \). A circle with centre \( (a,b) \) and radius \( r \) is \( (x-a)^{2}+(y-b)^{2}=r^{2} \).
Adjust gradient \( m \) and intercept \( b \) of \( y=mx+b \); the amber dots are the intercepts.
| x | y |
|---|
Perpendicular gradients satisfy \( m_1m_2=-1 \).
Find the gradient through \( (1,2) \) and \( (4,11) \).
State the centre of \( (x-2)^{2}+(y+3)^{2}=25 \).
Trigonometry
📌 Trigonometry P1 · Exact trig values · \(\sin^2\theta+\cos^2\theta=1\) · Solving trig equations in a range
Exact values, the identity linking sine and cosine, and solving simple trigonometric equations.
\( \sin^{2}\theta+\cos^{2}\theta=1 \) and \( \tan\theta=\dfrac{\sin\theta}{\cos\theta} \). Exact values: \( \sin30^{\circ}=\tfrac12 \), \( \cos30^{\circ}=\tfrac{\sqrt3}{2} \), \( \sin60^{\circ}=\tfrac{\sqrt3}{2} \). The sine and cosine curves have period \( 360^{\circ} \).
Solve \( \sin\theta=\tfrac12 \) for \( 0^{\circ}\le\theta\le180^{\circ} \).
State the exact value of \( \sin60^{\circ} \).
Differentiation & Integration
📌 Differentiation & Integration P1 · Gradient & stationary points · Tangent & normal · Definite integral & area under curve
The gradient of a curve from first principles, the power rule for differentiation, and reversing it to integrate.
\( \dfrac{d}{dx}x^{n}=nx^{n-1} \); the gradient at a point is the value of \( \tfrac{dy}{dx} \) there. Integration reverses this: \( \displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C \) for \( n\neq-1 \).
The amber line is a secant from \( x=1 \) to \( x=1+h \) on \( y=x^{2} \). Shrink \( h \) and watch its slope approach the tangent gradient, \( 2 \).
| x | f(x) |
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The secant gradient is \( 2+h \); as \( h\to0 \) it tends to the tangent gradient \( 2 \).
Differentiate \( y=x^{2}+3x \).
Evaluate \( \displaystyle\int_{0}^{1} 2x\,dx \).
Statistics & Probability
📌 Statistics & Probability S1 · Basic probability · Conditional probability · Tree diagrams · Venn diagrams
Summarising data with averages and spread, and the probability of simple events.
Mean is total over count; the median is the middle of the ordered data; range is largest minus smallest. Probability \( =\dfrac{\text{favourable}}{\text{total}} \).
Find the mean of \( 2,\,4,\,6,\,8 \).
Find the median of \( 3,\,7,\,9,\,12,\,20 \).
Functions: Composite & Inverse
📌 Functions: Composite & Inverse P1 · Domain & range · Composite \(fg(x)\) · Inverse \(f^{-1}(x)\) · Modulus function
Beyond graphs, AS treats functions formally — domain, range, composition and inverses.
\( fg(x)=f(g(x)) \). The inverse \( f^{-1} \) reverses \( f \) and reflects its graph in the line \( y=x \); the range of \( f \) is the domain of \( f^{-1} \).
If \( f(x)=3x-2 \), find \( f^{-1}(x) \).
Given \(f(x)=\sqrt{x+1}\) (\(x\geq-1\)) and \(g(x)=2x-3\), find (a) \(fg(x)\), (b) the range of \(fg\). (a) \(fg(x)=f(2x-3)=\sqrt{2x-3+1}=\sqrt{2x-2}\), domain \(x\geq1\). (b) As \(x\geq1\), \(2x-2\geq0\), so range of \(fg\) is \(fg(x)\geq0\).
Circular Measure
📌 Circular Measure P1 · Radians ↔ degrees · Arc length \(s=r\theta\) · Sector area \(A=\tfrac{1}{2}r^2\theta\) · Segment area
Angles measured in radians give clean formulas for arc length and sector area.
\( 180^{\circ}=\pi \) radians. Arc length \( s=r\theta \); sector area \( A=\tfrac12 r^{2}\theta \), with \( \theta \) in radians.
Find the arc length when \( r=5 \) and \( \theta=\tfrac{\pi}{3} \).
A sector OAB has radius 8 cm and angle \(\angle AOB=1.2\) rad. Find the area of the minor segment cut off by chord AB.
Series & the Binomial Theorem
📌 Series & Binomial Theorem P1 · AP & GP formulas · Sum to infinity · \((a+b)^n\) expansion · General term
Arithmetic and geometric progressions, and the binomial expansion of \( (a+b)^{n} \).
AP sum \( S_n=\tfrac{n}{2}\big(2a+(n-1)d\big) \); GP sum \( S_n=\tfrac{a(1-r^{n})}{1-r} \); binomial \( (1+x)^{n}=1+nx+\tfrac{n(n-1)}{2}x^{2}+\cdots \).
Find the sum of the first \( 10 \) terms of \( 2,5,8,\dots \)
A geometric series has first term 12 and common ratio \(\tfrac{2}{3}\). Find (a) the 5th term, (b) \(S_\infty\). (a) \(u_5=12\cdot(\tfrac{2}{3})^4=12\cdot\tfrac{16}{81}=\tfrac{64}{27}\). (b) \(S_\infty=\dfrac{12}{1-\tfrac{2}{3}}=\dfrac{12}{\tfrac{1}{3}}=36\).
Kinematics
📌 Kinematics M1 · SUVAT equations · Velocity–time graphs · Calculus kinematics
Mechanics begins with motion in a straight line under constant acceleration.
For constant acceleration: \( v=u+at \), \( s=ut+\tfrac12 at^{2} \), \( v^{2}=u^{2}+2as \). Velocity is the derivative of displacement; acceleration the derivative of velocity.
With \( u=0 \), \( a=2 \), \( t=3 \), find \( v \).
A particle moves so that its velocity at time \(t\) s is \(v=3t^2-12t+9\) m/s. Find (a) the acceleration at \(t=2\), (b) when the particle is momentarily at rest. (a) \(a=\dfrac{dv}{dt}=6t-12\). At \(t=2\): \(a=0\) m/s². (b) \(v=0\Rightarrow3(t-1)(t-3)=0\Rightarrow t=1\) s or \(t=3\) s.
Forces & Equilibrium
📌 Forces & Equilibrium M1 · Resultant force · Resolving into components · Lami's theorem · Friction \(F=\mu R\)
Types of Forces
Weight \(W=mg\), Normal reaction \(N\), Tension \(T\), Friction \(F\leq\mu N\). Forces are vectors — resolve horizontally and vertically.
Equilibrium
A particle is in equilibrium when the resultant force is zero: \(\sum F_x=0\) and \(\sum F_y=0\). Use triangle/polygon of forces.
Resolving Forces
Force \(F\) at angle \(\theta\): horizontal component \(F\cos\theta\), vertical component \(F\sin\theta\).
Friction
Limiting friction: \(F=\mu N\). When moving: \(F=\mu N\). Static: \(F\leq\mu N\). \(\mu\) is the coefficient of friction.
Lami's Theorem
For three concurrent forces in equilibrium: \(\dfrac{F_1}{\sin\alpha}=\dfrac{F_2}{\sin\beta}=\dfrac{F_3}{\sin\gamma}\)
Inclined Planes
On slope at angle \(\alpha\): component along slope \(=mg\sin\alpha\), perpendicular \(=mg\cos\alpha\).
A force is a vector (magnitude + direction). The resultant of forces is found by vector addition; for equilibrium the resultant is zero. Resolve forces into horizontal and vertical components: \(F_x=F\cos\theta\), \(F_y=F\sin\theta\). Friction force satisfies \(F\leq\mu R\); limiting equilibrium: \(F=\mu R\).
A particle is in equilibrium under three forces: 10 N acting due East, 8 N acting due North, and a third force \(F\). Find the magnitude and direction of \(F\). For equilibrium: \(F_x=-10\) N, \(F_y=-8\) N.
A block of mass 5 kg lies on a rough plane inclined at 30° to the horizontal (\(\mu=0.4\), \(g=10\) m/s²). Find the friction force acting when the block is on the point of sliding down. Normal reaction: \(R=5g\cos30°=50\times\tfrac{\sqrt3}{2}\approx43.3\) N. Friction (limiting, up the slope): \(F=0.4\times43.3\approx17.3\) N. Check: Gravity component down slope \(=50\sin30°=25\) N \(>F\), so block would slide — friction acts up slope.
Newton's Laws of Motion
📌 Newton's Laws of Motion M1 · \(F=ma\) · Connected particles & pulleys · Tension & reaction forces
Newton's First Law
A body remains at rest or in uniform motion unless acted upon by a resultant force. Inertia.
Newton's Second Law
\(F=ma\) where \(F\) is resultant force (N), \(m\) is mass (kg), \(a\) is acceleration (m/s²).
Newton's Third Law
For every action there is an equal and opposite reaction. Forces occur in equal pairs on different bodies.
Connected Particles
Particles connected by strings over pulleys: apply \(F=ma\) to each particle separately, then solve simultaneously.
Acceleration on Slopes
Net force along slope \(=mg\sin\alpha - F_{friction}\). Apply \(F=ma\) to find acceleration.
Momentum
Momentum \(p=mv\). Newton's 2nd law: \(F=\dfrac{d(mv)}{dt}\). Conservation: \(m_1u_1+m_2u_2=m_1v_1+m_2v_2\).
N1: A body remains at rest or in uniform motion unless acted on by a net force. N2: \(F=ma\) (net force = mass × acceleration). N3: For every action there is an equal and opposite reaction. For connected particles (e.g. Atwood's machine), consider each particle separately; the tension \(T\) is the same throughout an inextensible string over a smooth pulley.
A force of 30 N acts on a 6 kg box on a rough horizontal surface (\(\mu=0.3\), \(g=10\) m/s²). Find the acceleration.
Two particles of mass 4 kg and 6 kg are connected by a light inextensible string over a smooth pulley. Find the acceleration and the tension in the string (\(g=10\) m/s²). Taking down as positive for the 6 kg particle:
Energy, Work & Power
📌 Energy, Work & Power M1 · Kinetic energy · Gravitational PE · Work-energy principle · Power \(P=Fv\)
Work Done
\(W=Fd\cos\theta\) where \(F\) is force, \(d\) distance, \(\theta\) angle between force and displacement. Units: Joules (J).
Kinetic Energy
\(KE=\tfrac{1}{2}mv^2\). Work-energy theorem: \(W_{net}=\Delta KE=\tfrac{1}{2}mv^2-\tfrac{1}{2}mu^2\).
Potential Energy
Gravitational PE: \(GPE=mgh\). Change in GPE: \(\Delta GPE=mg\Delta h\).
Conservation of Energy
\(KE+PE=\text{constant}\) (no friction). With friction: \(\Delta KE+\Delta PE+W_{friction}=0\).
Power
\(P=\dfrac{W}{t}=Fv\). Units: Watts (W). Maximum speed when driving force = resistance: \(P=F_{resistance}\cdot v_{max}\).
Efficiency
\(\text{efficiency}=\dfrac{\text{useful power output}}{\text{power input}}\times100\%\)
Kinetic energy: \(\text{KE}=\tfrac{1}{2}mv^2\). Gravitational PE: \(\text{GPE}=mgh\). Work-energy principle: net work done = change in KE. Work done by a force: \(W=Fd\cos\theta\). Power: \(P=\dfrac{W}{t}=Fv\). Conservation of energy: \(\text{KE}_1+\text{GPE}_1=\text{KE}_2+\text{GPE}_2\) (no friction).
A 2 kg ball rolls down a smooth slope of height 5 m from rest. Find its speed at the bottom (\(g=10\) m/s²).
A car of mass 900 kg moves at constant speed 30 m/s against a resistance of 500 N. Find the power output of the engine. At constant speed, driving force = resistance = 500 N.
Representation of Data
📌 Representation of Data S1 · Histograms & frequency density · Cumulative frequency · Mean, variance & SD · Box-and-whisker plots
Stem-and-Leaf Diagrams
Display raw data in order. Back-to-back stem plots compare two datasets. Median and quartiles easy to read.
Box-and-Whisker Plots
Shows minimum, Q1, median, Q3, maximum. IQR = Q3 − Q1. Outliers: values outside \(Q_1-1.5\times IQR\) or \(Q_3+1.5\times IQR\).
Histograms
Frequency density = frequency ÷ class width. Area of bar represents frequency. Used for continuous grouped data.
Cumulative Frequency
Ogive (cumulative frequency curve). Read median at \(n/2\), Q1 at \(n/4\), Q3 at \(3n/4\).
Measures of Location
Mean \(\bar{x}=\dfrac{\sum fx}{\sum f}\), Median (middle value), Mode (most frequent). For grouped data use midpoint.
Measures of Spread
Range, IQR, Variance \(\sigma^2=\dfrac{\sum fx^2}{n}-\bar{x}^2\), Standard deviation \(\sigma\).
For grouped data with \(n\) values: mean \(\bar{x}=\dfrac{\sum fx}{\sum f}\); variance \(\sigma^2=\dfrac{\sum fx^2}{\sum f}-\bar{x}^2\); standard deviation \(\sigma=\sqrt{\text{Var}}\). In a histogram (unequal class widths), plot frequency density \(=\text{freq}/\text{class width}\). In a cumulative frequency graph, the median is the \(\tfrac{n}{2}\)th value, \(Q_1\) is the \(\tfrac{n}{4}\)th, \(Q_3\) is the \(\tfrac{3n}{4}\)th.
Scores: 1 (freq 2), 2 (freq 5), 3 (freq 8), 4 (freq 3), 5 (freq 2). Find the mean and standard deviation. \(\sum f=20\), \(\sum fx=2+10+24+12+10=58\). Mean \(=58/20=2.9\). \(\sum fx^2=2+20+72+48+50=192\). Var \(=192/20-2.9^2=9.6-8.41=1.19\). SD \(\approx1.09\).
Data range 0–50. Cumulative frequencies at 10,20,30,40,50 are 5,18,34,44,50. Estimate the median and IQR. Median at \(n/2=25\)th value: read from cf graph at cf=25 → \(\approx27\). \(Q_1\) at cf=12.5 → \(\approx18\). \(Q_3\) at cf=37.5 → \(\approx33\). IQR\(\approx15\).
Permutations & Combinations
📌 Permutations & Combinations S1 · Factorial · \(^nP_r\) · \(^nC_r\) · Arrangements with identical items or restrictions
Factorial
\(n! = n\times(n-1)\times\cdots\times2\times1\). Convention: \(0!=1\).
Permutations
Ordered arrangements: \(^nP_r=\dfrac{n!}{(n-r)!}\). Order matters. \(n\) objects in a line: \(n!\) ways.
Combinations
Unordered selections: \(^nC_r=\dbinom{n}{r}=\dfrac{n!}{r!(n-r)!}\). Order does not matter.
Restrictions
Always together: treat as one unit, then \((n-1)!\times r!\). Never together: total − (always together).
Circular Arrangements
\((n-1)!\) ways to arrange \(n\) objects in a circle (one position is fixed).
Identical Objects
Arrangements of \(n\) objects with repeats: \(\dfrac{n!}{a!\,b!\,c!\cdots}\)
Permutation (order matters): \(^nP_r=\dfrac{n!}{(n-r)!}\). Combination (order does not matter): \(^nC_r=\dbinom{n}{r}=\dfrac{n!}{r!\,(n-r)!}\). Arrangements with identical items: divide by the factorial of each repeated item. With restrictions: fix the restricted items first, then arrange the rest.
A committee of 4 is chosen from 5 men and 6 women. Find the number of ways if the committee must contain at least 2 women. Exactly 2W: \(\binom{6}{2}\binom{5}{2}=15\times10=150\). Exactly 3W: \(\binom{6}{3}\binom{5}{1}=20\times5=100\). Exactly 4W: \(\binom{6}{4}\binom{5}{0}=15\). Total: \(265\).
How many different arrangements are there of the letters of the word BANANA? 6 letters; N appears 2 times, A appears 3 times: \(\dfrac{6!}{3!\,2!}=\dfrac{720}{12}=60\).
Discrete Random Variables
📌 Discrete Random Variables S1 · Probability distribution table · \(E(X)\) · \(\text{Var}(X)\) · Linear transforms \(E(aX+b)\)
Probability Distribution
Table of \(x\) values and corresponding \(P(X=x)\). Must satisfy: \(\sum P(X=x)=1\) and \(P(X=x)\geq0\).
Expectation
\(E(X)=\sum x\cdot P(X=x)=\mu\). Also \(E(aX+b)=aE(X)+b\).
Variance
\(\text{Var}(X)=E(X^2)-[E(X)]^2\) where \(E(X^2)=\sum x^2P(X=x)\). Also \(\text{Var}(aX+b)=a^2\text{Var}(X)\).
Standard Deviation
\(\sigma=\sqrt{\text{Var}(X)}\). Measures spread around the mean.
Uniform Distribution
Each value equally likely. If \(X\) takes values \(1,2,\ldots,n\): \(E(X)=\dfrac{n+1}{2}\), \(\text{Var}(X)=\dfrac{n^2-1}{12}\).
Combined Variables
\(E(X+Y)=E(X)+E(Y)\). If independent: \(\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)\).
A discrete random variable \(X\) has a probability distribution with \(\sum P(X=x)=1\). Expectation: \(E(X)=\sum x\,P(X=x)\). Variance: \(\text{Var}(X)=E(X^2)-[E(X)]^2\) where \(E(X^2)=\sum x^2 P(X=x)\). For linear transforms: \(E(aX+b)=aE(X)+b\); \(\text{Var}(aX+b)=a^2\text{Var}(X)\).
A biased die shows: \(P(1)=0.1,P(2)=0.2,P(3)=0.3,P(4)=0.3,P(5)=0.1\). Find \(E(X)\) and \(\text{Var}(X)\).
\(X\) takes values 0, 1, 2 with \(P(X=0)=k\), \(P(X=1)=2k\), \(P(X=2)=3k\). Find \(k\) and \(E(X)\).
Binomial Distribution
📌 Binomial Distribution S1 · \(X\sim B(n,p)\) · \(P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}\) · Mean \(np\) · Variance \(np(1-p)\)
Conditions
Fixed number of trials \(n\); each trial has two outcomes (success/failure); constant probability \(p\); trials independent. Then \(X\sim B(n,p)\).
Probability Formula
\(P(X=r)=\dbinom{n}{r}p^r(1-p)^{n-r}\) for \(r=0,1,2,\ldots,n\).
Mean and Variance
\(E(X)=np\), \(\text{Var}(X)=np(1-p)\), \(\sigma=\sqrt{np(1-p)}\).
Cumulative Probabilities
Use tables or calculator: \(P(X\leq r)\). Note: \(P(X\geq r)=1-P(X\leq r-1)\).
Normal Approximation
If \(n\) large and \(p\) close to 0.5: \(X\approx N(np,\,np(1-p))\) with continuity correction.
Finding Unknown n or p
Set up equations using given probabilities and solve. May require trial and error or logarithms for \(n\).
If \(X\sim B(n,p)\): \(n\) independent trials, each with probability \(p\) of success. \(P(X=r)=\dbinom{n}{r}p^r(1-p)^{n-r}\). Mean: \(E(X)=np\). Variance: \(\text{Var}(X)=np(1-p)\). Use the binomial distribution when: fixed \(n\), trials independent, constant \(p\), only 2 outcomes (success/failure).
\(X\sim B(8,0.3)\). Find (a) \(P(X=3)\), (b) \(P(X\leq1)\). (a) \(P(X=3)=\binom{8}{3}(0.3)^3(0.7)^5=56\times0.027\times0.16807\approx0.2541\). (b) \(P(X=0)=(0.7)^8\approx0.0576\); \(P(X=1)=8(0.3)(0.7)^7\approx0.1977\). \(P(X\leq1)\approx0.2553\).
The mean of \(X\sim B(n,0.4)\) is 6. Find \(n\) and the standard deviation.
Normal Distribution
📌 Normal Distribution S1 · \(X\sim N(\mu,\sigma^2)\) · Standardising \(Z=\tfrac{X-\mu}{\sigma}\) · Tables & symmetry · Finding \(\mu\) or \(\sigma\)
The Normal Curve
Bell-shaped, symmetric about mean \(\mu\). \(X\sim N(\mu,\sigma^2)\). Approximately 68%/95%/99.7% within 1/2/3 standard deviations.
Standardising
\(Z=\dfrac{X-\mu}{\sigma}\sim N(0,1)\). Use \(Z\)-tables (standard normal) to find probabilities.
Finding Probabilities
\(P(X<a)=\Phi\!\left(\dfrac{a-\mu}{\sigma}\right)\). Use symmetry: \(P(Z<-z)=1-P(Z<z)\).
Inverse Normal
Given probability, find the value. \(P(X<a)=p\Rightarrow a=\mu+z\sigma\) where \(z=\Phi^{-1}(p)\).
Finding μ or σ
Set up two equations from given probabilities, solve simultaneously for unknown parameters.
Normal Approximation
To Binomial \(B(n,p)\): if \(n\) large, \(np>5\), \(n(1-p)>5\): use \(N(np,np(1-p))\) with continuity correction \(\pm0.5\).
If \(X\sim N(\mu,\sigma^2)\): standardise with \(Z=\dfrac{X-\mu}{\sigma}\) where \(Z\sim N(0,1)\). Use tables (or calculator) for \(\Phi(z)=P(Z\leq z)\). Key symmetry: \(P(Z\leq -z)=1-P(Z\leq z)\). For inverse normal: given a probability, find the corresponding \(x\) value using \(x=\mu+z\sigma\).
\(X\sim N(50,16)\) so \(\sigma=4\). Find \(P(44\leq X\leq58)\).
\(X\sim N(\mu,25)\). Given \(P(X>43)=0.2119\), find \(\mu\). \(P(Z>\tfrac{43-\mu}{5})=0.2119\Rightarrow\tfrac{43-\mu}{5}=0.8\) (from tables). \(\mu=43-4=39\).
Practice set
Fifty AS-style problems across the Pure core plus statistics, tagged by difficulty. Click any problem for a full worked solution.
Simplify \( x^{3}\cdot x^{4} \).
BasicEvaluate \( 2^{-3} \).
IntermediateSimplify \( \sqrt{12} \).
IntermediateRationalise \( \dfrac{1}{\sqrt{2}} \).
AdvancedMultiply by \( \tfrac{\sqrt2}{\sqrt2} \):
\[ \tfrac{\sqrt{2}}{2}. \]Simplify \( (x^{2})^{5} \).
IntermediateEvaluate \( 27^{1/3} \).
AdvancedSolve \( x^{2}=16 \).
BasicFactor \( x^{2}+5x+6 \).
IntermediateSolve \( x^{2}-6x+8=0 \).
Intermediate\( (x-2)(x-4)=0 \):
\[ x=2,\ 4. \]Complete the square for \( x^{2}+4x \).
IntermediateHow many real roots does \( x^{2}+x+1=0 \) have?
AdvancedDiscriminant \( =1-4=-3<0 \):
\[ \text{none.} \]Find the sum of the roots of \( x^{2}-5x+6=0 \).
IntermediateSolve \( 2x^{2}-5x-3=0 \).
Advanced\( (2x+1)(x-3)=0 \):
\[ x=3,\ -\tfrac12. \]Find the vertex of \( y=x^{2}-4x+3 \).
Intermediate\( (x-2)^{2}-1 \):
\[ (2,\,-1). \]Solve \( x+y=5 \), \( x-y=1 \).
IntermediateAdd: \( 2x=6 \):
\[ x=3,\ y=2. \]Solve \( 2x+y=8 \), \( y=x-1 \).
Intermediate\( 2x+x-1=8 \Rightarrow x=3 \):
\[ x=3,\ y=2. \]Solve \( y=x^{2} \), \( y=x+2 \).
Advanced\( x^{2}-x-2=0\Rightarrow(x-2)(x+1)=0 \):
\[ (2,4),\ (-1,1). \]Solve \( 2x-3>5 \).
Intermediate\( 2x>8 \):
\[ x>4. \]Solve \( 3x+1\le10 \).
Intermediate\( 3x\le9 \):
\[ x\le3. \]Solve \( x^{2}-4<0 \).
AdvancedFind the gradient through \( (1,2) \) and \( (4,11) \).
IntermediateFind the midpoint of \( (2,3) \) and \( (8,7) \).
IntermediateFind the equation of the line with gradient \( 2 \) through \( (1,3) \).
Advanced\( y-3=2(x-1) \):
\[ y=2x+1. \]Find the distance between \( (1,1) \) and \( (4,5) \).
IntermediateState the centre of \( (x-2)^{2}+(y+3)^{2}=25 \).
AdvancedState the radius of \( x^{2}+y^{2}=49 \).
IntermediateDescribe \( y=x^{2}+3 \) relative to \( y=x^{2} \).
IntermediateA translation \( 3 \) units up.
Describe \( y=(x-2)^{2} \) relative to \( y=x^{2} \).
IntermediateA translation \( 2 \) units to the right.
Describe \( y=-x^{2} \) relative to \( y=x^{2} \).
AdvancedA reflection in the \( x \)-axis.
How does \( y=f(x)+k \) transform the graph of \( y=f(x) \)?
IntermediateA vertical translation by \( k \).
Evaluate \( \sin30^{\circ} \).
IntermediateEvaluate \( \cos0^{\circ} \).
IntermediateEvaluate \( \tan45^{\circ} \).
IntermediateSolve \( \sin\theta=\tfrac12 \) for \( 0^{\circ}\le\theta\le180^{\circ} \).
AdvancedSimplify \( \sin^{2}\theta+\cos^{2}\theta \).
IntermediateState the exact value of \( \sin60^{\circ} \).
AdvancedEvaluate \( \cos90^{\circ} \).
IntermediateState the period of \( y=\sin x \).
AdvancedDifferentiate \( x^{2} \).
IntermediateDifferentiate \( x^{3} \).
IntermediateDifferentiate \( 5x \).
IntermediateFind \( \dfrac{dy}{dx} \) for \( y=x^{2}+3x \).
AdvancedFind \( \displaystyle\int 2x\,dx \).
IntermediateFind \( \displaystyle\int x^{2}\,dx \).
AdvancedFind the gradient of \( y=x^{2} \) at \( x=3 \).
Intermediate\( \tfrac{dy}{dx}=2x=6 \):
\[ 6. \]Evaluate \( \displaystyle\int_{0}^{1} 2x\,dx \).
AdvancedFind the mean of \( 2,\,4,\,6,\,8 \).
IntermediateFind the median of \( 3,\,7,\,9,\,12,\,20 \).
IntermediateA fair die is rolled. Find \( P(\text{even}) \).
AdvancedFind the range of \( 5,\,9,\,2,\,11 \).
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Based on Cambridge International AS Level Mathematics 9709 Paper 1: Pure Mathematics 1. Structured questions covering algebra, coordinate geometry, trigonometry, calculus, and vectors.
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