About A2 Level
Paper ClipSAT Chapter Key Topics P3 Pure 3 Functions: Modulus & Inverse Modulus function, inverse, domain restriction Exponentials & Logarithms \(e^x\), \(\ln x\), solving exponential equations, log graphs Advanced Trigonometry Compound angle, double angle, R formula \(a\sin\theta+b\cos\theta\) Differentiation Implicit, parametric, related rates, product/quotient/chain Integration By parts, substitution, partial fractions, volumes of revolution Sequences & Series Maclaurin series, binomial expansion for fractional/negative powers Vectors 3D vectors, dot product, angle between lines, equation of a plane Algebra: Partial Fractions & Division Partial fractions (all cases), polynomial long division Complex Numbers Cartesian form, Argand diagram, modulus-argument, De Moivre Numerical Methods Change of sign, iterative sequences, Newton-Raphson Differential Equations Separable DEs, integrating factor, forming DEs S2 Statistics 2 Poisson Distribution \(X\sim\text{Po}(\lambda)\), mean = variance = \(\lambda\), approximation to binomial Continuous Random Variables pdf, cdf, \(E(X)\), \(\text{Var}(X)\), median, quartiles Sampling & Estimation Unbiased estimators, CLT, confidence intervals for \(\mu\) Hypothesis Testing One/two-tailed tests for \(\mu\), Poisson mean, Binomial \(p\)
A2 is the second year of the A Level, completing courses such as Cambridge 9709 and Edexcel. It deepens Pure Mathematics and extends the applied content.
Pure 2 & 3 brings modulus, composite and inverse functions, exponentials and logarithms, advanced trigonometry, the full rules of differentiation and integration, sequences and series (including the binomial expansion), and vectors.
Reference panels and worked examples sit beside three explorers — exponential behaviour, the tangent gradient and vector addition. Finish with the 50-problem set, then export it as a worksheet.
Functions: Modulus & Inverse
📌 Functions: Modulus & Inverse P3 · Modulus graph · Inverse (domain restriction) · Composite functions · Graph of \(|f(x)|\)
The modulus function, composite functions, inverses, and domain and range.
\( |x|=c \) gives \( x=\pm c \). A composite \( fg(x)=f(g(x)) \) is applied inside-out. The inverse \( f^{-1} \) reverses \( f \); swap \( x \) and \( y \) and solve. The range of \( f \) is the domain of \( f^{-1} \).
Find the inverse of \( f(x)=2x+1 \). Let \( y=2x+1 \), swap and solve:
Solve \( |x-1|=3 \).
Exponentials & Logarithms
📌 Exponentials & Logarithms P3 · \(e^x\) and \(\ln x\) · Log laws · Solving \(e^{ax+b}=k\) · Log-linear graph to find unknowns
The exponential function \( e^{x} \), logarithms and their laws, and solving exponential equations.
\( \log_a(xy)=\log_a x+\log_a y \); \( \log_a(x^{n})=n\log_a x \). The natural log inverts the exponential: \( \ln(e^{x})=x \). To solve \( a^{x}=b \), take logs of both sides.
The curve is \( y=e^{kx} \): a positive \( k \) grows, a negative \( k \) decays. Slide \( k \) through zero to see it flip.
| x | y |
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All curves pass through \( (0,1) \); the sign of \( k \) sets growth versus decay.
Evaluate \( \log_2 8 \).
Solve \( 3^{x}=81 \).
Advanced Trigonometry
📌 Advanced Trigonometry P3 · Compound angle formulae · Double angle · \(R\sin(\theta+\alpha)\) form · Solving equations in a range
Reciprocal ratios, the double-angle formulae, and solving trigonometric equations.
\( \sec\theta=\dfrac{1}{\cos\theta} \). Double angle: \( \sin2\theta=2\sin\theta\cos\theta \) and \( \cos2\theta=2\cos^{2}\theta-1 \). The tangent curve has period \( 180^{\circ} \).
Write \( \cos2\theta \) in terms of \( \cos\theta \).
Solve \( \cos\theta=0 \) for \( 0^{\circ}\le\theta\le180^{\circ} \).
Differentiation
📌 Differentiation P3 · Implicit differentiation · Parametric differentiation · Product/quotient rule · Related rates
Differentiating standard functions and applying the chain, product and quotient rules.
\( \tfrac{d}{dx}e^{x}=e^{x} \), \( \tfrac{d}{dx}\ln x=\tfrac1x \), \( \tfrac{d}{dx}\sin x=\cos x \). Chain: \( \tfrac{dy}{dx}=\tfrac{dy}{du}\tfrac{du}{dx} \). Product: \( (uv)'=u'v+uv' \). Quotient: \( \left(\tfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^{2}} \).
Slide the point along \( y=x^{3}-3x \). The amber line is the tangent; its gradient is \( 3x^{2}-3 \).
| x | f(x) | tangent y |
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The gradient is zero at the turning points, where \( 3x^{2}-3=0 \), i.e. \( x=\pm1 \).
Differentiate \( y=(2x+1)^{3} \). Chain rule:
Differentiate \( y=\dfrac{x}{x+1} \).
Integration
📌 Integration P3 · Integration by parts · Substitution · Partial fractions · Volumes of revolution
Integrating standard functions, definite integrals, and the area under a curve.
\( \displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C \) (\( n\neq-1 \)); \( \displaystyle\int e^{x}\,dx=e^{x}+C \); \( \displaystyle\int\tfrac1x\,dx=\ln|x|+C \). A definite integral gives the signed area: \( F(b)-F(a) \).
Find \( \displaystyle\int 6x^{2}\,dx \).
Find the area under \( y=x^{2} \) from \( 0 \) to \( 3 \).
Sequences & Series
📌 Sequences & Series P3 · Maclaurin series · Binomial expansion (fractional & negative) · Convergence condition \(|x|<1\)
Arithmetic and geometric progressions, sums to infinity, and the binomial expansion.
AP term \( a+(n-1)d \); GP term \( ar^{\,n-1} \). A convergent GP \( (|r|<1) \) sums to \( \dfrac{a}{1-r} \). The sum of the first \( n \) integers is \( \dfrac{n(n+1)}{2} \).
Find the 10th term of \( 2,5,8,\dots \) \( a=2,\ d=3 \):
Find the sum to infinity of \( 1+\tfrac12+\tfrac14+\dots \)
Vectors
📌 Vectors P3 · 3D column vectors · Dot product · Angle between lines · Vector equation of a plane
Vector components, addition and scalar multiples, and the magnitude of a vector.
Add componentwise: \( (a_1,a_2)+(b_1,b_2)=(a_1+b_1,\,a_2+b_2) \). A scalar multiple stretches: \( k(a_1,a_2)=(ka_1,ka_2) \). Magnitude \( |\mathbf{v}|=\sqrt{a_1^{2}+a_2^{2}} \).
Vector \( \mathbf{a}=(3,1) \) is fixed (indigo). Slide the components of \( \mathbf{b} \) (amber); the dashed line is the resultant \( \mathbf{a}+\mathbf{b} \).
| vector | (x, y) | magnitude |
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The resultant is the diagonal of the parallelogram formed by \( \mathbf{a} \) and \( \mathbf{b} \).
Find \( (1,2)+(3,1) \).
Find the magnitude of \( (6,8) \).
Algebra: Partial Fractions & Division
📌 Algebra: Partial Fractions & Division P3 · Partial fractions (proper, improper, repeated, quadratic denominator) · Polynomial long division
Polynomial division and partial fractions break complicated algebraic fractions into simpler pieces.
A proper fraction \( \tfrac{px+q}{(x+a)(x+b)} \) splits as \( \tfrac{A}{x+a}+\tfrac{B}{x+b} \). If the fraction is improper, divide first.
Express \( \dfrac{1}{x(x+1)} \) in partial fractions.
Express \(\dfrac{x^3+2x}{x^2-1}\) in partial fractions (after division). Degree of numerator ≥ denominator, so divide first: \(x^3+2x=(x^2-1)\cdot x+3x\). So: Cover-up: \(A=\tfrac{3}{2}\), \(B=\tfrac{3}{2}\). Result: \(x+\dfrac{3/2}{x-1}+\dfrac{3/2}{x+1}\).
Complex Numbers
📌 Complex Numbers P3 · Cartesian \(a+bi\) · Argand diagram · Modulus-argument form · Roots of polynomials
The imaginary unit \( i \) extends the reals; complex numbers add, multiply, and have a modulus and conjugate.
\( i^{2}=-1 \); \( z=a+bi \) has modulus \( |z|=\sqrt{a^{2}+b^{2}} \) and conjugate \( \bar{z}=a-bi \), with \( z\bar{z}=|z|^{2} \).
Find \( |3+4i| \).
Write \(z=-1+\sqrt{3}\,i\) in modulus-argument form and find \(z^4\) using De Moivre's theorem. \(|z|=\sqrt{1+3}=2\). \(\arg z=\pi-\arctan\sqrt{3}=\pi-\tfrac{\pi}{3}=\tfrac{2\pi}{3}\). \(z=2\bigl(\cos\tfrac{2\pi}{3}+i\sin\tfrac{2\pi}{3}\bigr)\). By De Moivre: \(z^4=2^4\bigl(\cos\tfrac{8\pi}{3}+i\sin\tfrac{8\pi}{3}\bigr)=16\bigl(\cos\tfrac{2\pi}{3}+i\sin\tfrac{2\pi}{3}\bigr)=16(-\tfrac{1}{2}+\tfrac{\sqrt3}{2}i)=-8+8\sqrt3\,i\).
Numerical Methods
📌 Numerical Methods P3 · Interval bisection / change of sign · Iterative formula \(x_{n+1}=g(x_n)\) · Newton-Raphson
When an equation cannot be solved exactly, sign changes locate roots and iteration refines them.
A sign change of \( f \) across \( [a,b] \) traps a root. Iteration \( x_{n+1}=g(x_n) \) converges when it settles; the Newton–Raphson step is \( x_{n+1}=x_n-\tfrac{f(x_n)}{f'(x_n)} \).
For \( f(x)=x^{2}-2 \), between which integers does a root lie?
Use Newton-Raphson to find the root of \(f(x)=x^3-5\) near \(x_0=1.7\). Do one iteration. \(f(x)=x^3-5\), \(f'(x)=3x^2\).
Differential Equations
📌 Differential Equations P3 · Separable variables · Integrating factor · Forming a DE from context
A differential equation relates a function to its rate of change; separable equations integrate directly.
Write \( \tfrac{dy}{dx}=f(x)g(y) \) as \( \tfrac{1}{g(y)}\,dy=f(x)\,dx \) and integrate both sides; an initial condition fixes the constant.
Solve \( \dfrac{dy}{dx}=y \).
Solve \(\dfrac{dy}{dx}+2y=e^x\). Integrating factor: \(\mu=e^{\int2\,dx}=e^{2x}\). Multiply through: \(\dfrac{d}{dx}(ye^{2x})=e^{3x}\).
Poisson Distribution
📌 Poisson Distribution S2 · \(X\sim\text{Po}(\lambda)\) · \(P(X=r)=e^{-\lambda}\lambda^r/r!\) · Mean = Var = \(\lambda\) · Poisson approx. to Binomial
Conditions
Events occur independently, at a constant average rate \(\lambda\), and singly in time/space. \(X\sim\text{Po}(\lambda)\).
Probability Formula
\(P(X=r)=\dfrac{e^{-\lambda}\lambda^r}{r!}\) for \(r=0,1,2,\ldots\)
Mean and Variance
\(E(X)=\text{Var}(X)=\lambda\). The mean equals the variance — a key identifying feature.
Cumulative Tables
Use Poisson tables for \(P(X\leq r)\). Note \(P(X\geq r)=1-P(X\leq r-1)\). Additive: \(X+Y\sim\text{Po}(\lambda_1+\lambda_2)\).
Changing the Interval
If \(X\sim\text{Po}(\lambda)\) per hour, then in \(t\) hours: \(X\sim\text{Po}(\lambda t)\).
Normal Approximation
If \(\lambda>15\): \(\text{Po}(\lambda)\approx N(\lambda,\lambda)\) with continuity correction.
If events occur at a constant average rate \(\lambda\) independently, \(X\sim\text{Po}(\lambda)\) with: \(P(X=r)=\dfrac{e^{-\lambda}\lambda^r}{r!}\) for \(r=0,1,2,\ldots\) · \(E(X)=\lambda\), \(\text{Var}(X)=\lambda\). Poisson approximation to Binomial: if \(n\) is large, \(p\) is small, \(np=\lambda\) is moderate (\(n>50, p<0.1\)), then \(B(n,p)\approx\text{Po}(\lambda)\).
Faults in a wire occur at a rate of 2 per metre. Find the probability of (a) exactly 3 faults in 1 m, (b) fewer than 2 faults in 0.5 m. (a) \(\lambda=2\): \(P(X=3)=\dfrac{e^{-2}\cdot8}{6}\approx0.1804\). (b) \(\lambda=1\): \(P(X<2)=P(0)+P(1)=e^{-1}+e^{-1}=2e^{-1}\approx0.7358\).
\(X\sim B(200,0.02)\). Use a Poisson approximation to find \(P(X=5)\). \(\lambda=200\times0.02=4\). \(P(X=5)=\dfrac{e^{-4}\cdot4^5}{120}=\dfrac{1024e^{-4}}{120}\approx0.1563\).
Continuous Random Variables
📌 Continuous Random Variables S2 · Probability density function · CDF \(F(x)\) · \(E(X)\) & \(\text{Var}(X)\) · Median & quartiles
Probability Density Function
\(f(x)\geq0\) for all \(x\) and \(\displaystyle\int_{-\infty}^{\infty}f(x)\,dx=1\). \(P(a\leq X\leq b)=\displaystyle\int_a^b f(x)\,dx\).
Cumulative Distribution Function
\(F(x)=P(X\leq x)=\displaystyle\int_{-\infty}^{x}f(t)\,dt\). Differentiating: \(f(x)=F'(x)\).
Mean (Expectation)
\(E(X)=\displaystyle\int_{-\infty}^{\infty}xf(x)\,dx\). \(E(g(X))=\displaystyle\int g(x)f(x)\,dx\).
Variance
\(\text{Var}(X)=E(X^2)-[E(X)]^2\) where \(E(X^2)=\displaystyle\int x^2f(x)\,dx\).
Median and Percentiles
Median \(m\): \(F(m)=0.5\). \(p\)-th percentile \(x_p\): \(F(x_p)=p/100\).
Uniform Distribution
\(X\sim U(a,b)\): \(f(x)=\dfrac{1}{b-a}\), \(E(X)=\dfrac{a+b}{2}\), \(\text{Var}(X)=\dfrac{(b-a)^2}{12}\).
A CRV \(X\) has a probability density function \(f(x)\geq0\) with \(\int_{-\infty}^{\infty}f(x)\,dx=1\). CDF: \(F(x)=P(X\leq x)=\int_{-\infty}^{x}f(t)\,dt\). Mean: \(E(X)=\int x\,f(x)\,dx\). Variance: \(\text{Var}(X)=\int x^2f(x)\,dx-[E(X)]^2\). Median \(m\): solve \(F(m)=0.5\). Mode: maximum of \(f(x)\).
\(f(x)=kx(2-x)\) for \(0\leq x\leq2\), zero otherwise. Find \(k\), then \(E(X)\).
With the same \(f(x)=\tfrac{3}{4}x(2-x)\) on \([0,2]\), find the CDF \(F(x)\) and the median. Median: \(F(m)=\tfrac{1}{2}\Rightarrow\tfrac{3m^2}{4}-\tfrac{m^3}{4}=\tfrac{1}{2}\Rightarrow m^3-3m^2+2=0\). Try \(m=1\): \(1-3+2=0\) ✓. So median \(=1\).
Sampling & Estimation
📌 Sampling & Estimation S2 · Random sampling · Unbiased estimators \(\bar{x}\) and \(s^2\) · Central Limit Theorem · Confidence interval for \(\mu\)
Populations and Samples
Population: entire group of interest. Sample: subset used to make inferences. Random sampling ensures unbiased estimates.
Sample Mean Distribution
If \(X\sim N(\mu,\sigma^2)\) then \(\bar{X}\sim N\!\left(\mu,\dfrac{\sigma^2}{n}\right)\). Standard error \(=\dfrac{\sigma}{\sqrt{n}}\).
Unbiased Estimators
\(\bar{X}\) is unbiased for \(\mu\). Unbiased variance: \(s^2=\dfrac{1}{n-1}\sum(x_i-\bar{x})^2\).
Confidence Intervals
95% CI for \(\mu\): \(\bar{x}\pm z_{0.025}\cdot\dfrac{\sigma}{\sqrt{n}}\) where \(z_{0.025}=1.96\).
Interpretation
A 95% CI means: 95% of all such intervals constructed would contain the true \(\mu\). Not a probability statement about \(\mu\).
Sample Size
Width of CI \(\propto 1/\sqrt{n}\). To halve width, multiply \(n\) by 4. Required \(n\geq\left(\dfrac{z\sigma}{E}\right)^2\).
An unbiased estimator of \(\mu\) is \(\bar{X}\). An unbiased estimator of \(\sigma^2\) is \(s^2=\dfrac{\sum(x_i-\bar{x})^2}{n-1}=\dfrac{\sum x_i^2-n\bar{x}^2}{n-1}\). CLT: For large \(n\), \(\bar{X}\sim N\!\left(\mu,\dfrac{\sigma^2}{n}\right)\) approximately. A 95% confidence interval for \(\mu\) (known \(\sigma\)): \(\bar{x}\pm1.96\dfrac{\sigma}{\sqrt{n}}\). Use \(z=2.576\) for 99% CI.
A sample of 36 light bulbs has mean lifetime \(\bar{x}=1200\) hours. The population SD is known to be \(\sigma=60\) hours. Find a 95% CI for the population mean. 95% CI: \((1180.4,\;1219.6)\) hours.
A sample of 5 values: 12, 15, 11, 14, 13. Find unbiased estimates of \(\mu\) and \(\sigma^2\). \(\bar{x}=65/5=13\). \(\sum x^2=144+225+121+196+169=855\). \(s^2=\dfrac{855-5(169)}{4}=\dfrac{855-845}{4}=\dfrac{10}{4}=2.5\).
Hypothesis Testing
📌 Hypothesis Testing S2 · Null & alternative hypothesis · Critical region · One- & two-tailed tests · Type I & Type II errors
Hypotheses
\(H_0\) (null hypothesis): assumed true. \(H_1\) (alternative): what we test for. One-tailed or two-tailed test.
Test Statistic
\(Z=\dfrac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}\sim N(0,1)\) under \(H_0\). Compare with critical value \(z_\alpha\).
Critical Region
Range of test statistic that leads to rejection of \(H_0\). For two-tailed 5%: \(|Z|>1.96\). One-tailed 5%: \(Z>1.645\) or \(Z<-1.645\).
p-value
Probability of observing a result as extreme as the sample, assuming \(H_0\). Reject \(H_0\) if \(p\text{-value}<\alpha\).
Type I and II Errors
Type I: reject \(H_0\) when true (probability = significance level \(\alpha\)). Type II: fail to reject \(H_0\) when false (probability \(\beta\)).
Testing Poisson/Binomial
Find \(P(X\geq x_0)\) (or \(P(X\leq x_0)\)) under \(H_0\). If \(<\alpha\), reject \(H_0\). State conclusion in context.
Steps: (1) State \(H_0\) and \(H_1\). (2) Specify significance level \(\alpha\). (3) Identify test statistic and its distribution under \(H_0\). (4) Find the critical region or \(p\)-value. (5) Compare and conclude in context.
One-tailed: \(H_1:\mu>\mu_0\) or \(\mu<\mu_0\). Two-tailed: \(H_1:\mu\neq\mu_0\) (split \(\alpha\) equally).
Type I error: reject \(H_0\) when it is true (probability = \(\alpha\)). Type II error: fail to reject \(H_0\) when it is false.
Bags of flour are labelled 500 g. A sample of 25 bags gives \(\bar{x}=497\) g. Population SD is 8 g. Test at 5% whether there is evidence that bags are underweight. \(H_0:\mu=500\), \(H_1:\mu<500\) (one-tailed). \(Z=\dfrac{497-500}{8/5}=\dfrac{-3}{1.6}=-1.875\). Critical value: \(-1.645\). Since \(-1.875<-1.645\), reject \(H_0\). There is significant evidence at 5% that bags are underweight.
Road accidents used to occur at a rate of 4 per week. After safety measures, only 1 accident is observed in one week. Test at 5% whether the rate has decreased. \(H_0:\lambda=4\), \(H_1:\lambda<4\). \(P(X\leq1)=e^{-4}(1+4)=5e^{-4}\approx0.0916\). Since \(0.0916>0.05\), do not reject \(H_0\). Insufficient evidence that the rate has decreased.
Practice set
Fifty A2-style problems across functions, logarithms, trigonometry, calculus, series and vectors, tagged by difficulty. Click any problem for a full worked solution.
Find the inverse of \( f(x)=2x+1 \).
BasicIf \( f(x)=x^{2} \) and \( g(x)=x+1 \), find \( fg(2) \).
Intermediate\( g(2)=3 \), then \( f(3)=9 \):
\[ 9. \]Solve \( |x|=4 \).
IntermediateSolve \( |x-1|=3 \).
Advanced\( x-1=\pm3 \):
\[ x=4,\ -2. \]If \( f(x)=x^{2} \) and \( g(x)=x+1 \), find \( gf(x) \).
Intermediate\( g(f(x))=f(x)+1 \):
\[ x^{2}+1. \]State the range of \( f(x)=x^{2}+1 \).
AdvancedSolve \( 2^{x}=8 \).
BasicEvaluate \( \log_2 8 \).
IntermediateEvaluate \( \log_{10}1000 \).
IntermediateSolve \( e^{x}=1 \).
AdvancedEvaluate \( \ln e \).
IntermediateSolve \( \log_{10}x=2 \).
AdvancedWrite \( \log a+\log b \) as a single logarithm.
IntermediateSolve \( 3^{x}=81 \).
AdvancedState the formula for \( \sin2\theta \).
IntermediateEvaluate \( \cos0^{\circ} \).
IntermediateSolve \( \cos\theta=0 \) for \( 0^{\circ}\le\theta\le180^{\circ} \).
AdvancedEvaluate \( \tan45^{\circ} \).
IntermediateState the exact value of \( \sin45^{\circ} \).
AdvancedExpress \( \sec\theta \) in terms of \( \cos\theta \).
IntermediateWrite \( \cos2\theta \) in terms of \( \cos\theta \).
AdvancedState the period of \( y=\tan x \).
IntermediateDifferentiate \( x^{4} \).
IntermediateDifferentiate \( \sin x \).
IntermediateDifferentiate \( e^{x} \).
IntermediateDifferentiate \( \ln x \).
IntermediateDifferentiate \( x^{2}e^{x} \).
AdvancedProduct rule:
\[ 2xe^{x}+x^{2}e^{x}=(x^{2}+2x)e^{x}. \]Differentiate \( (2x+1)^{3} \).
IntermediateChain rule:
\[ 6(2x+1)^{2}. \]Differentiate \( \cos x \).
AdvancedFind the gradient of \( y=x^{3} \) at \( x=2 \).
Intermediate\( \tfrac{dy}{dx}=3x^{2}=12 \):
\[ 12. \]Differentiate \( \dfrac{x}{x+1} \).
AdvancedQuotient rule:
\[ \frac{1}{(x+1)^{2}}. \]Find the second derivative of \( x^{3} \).
Intermediate\( \tfrac{dy}{dx}=3x^{2} \), again:
\[ 6x. \]Find \( \displaystyle\int x^{3}\,dx \).
IntermediateFind \( \displaystyle\int e^{x}\,dx \).
IntermediateFind \( \displaystyle\int \cos x\,dx \).
IntermediateFind \( \displaystyle\int \tfrac{1}{x}\,dx \).
AdvancedEvaluate \( \displaystyle\int_{0}^{1} x\,dx \).
IntermediateEvaluate \( \displaystyle\int_{0}^{\pi} \sin x\,dx \).
AdvancedFind \( \displaystyle\int 6x^{2}\,dx \).
IntermediateFind the area under \( y=x^{2} \) from \( 0 \) to \( 3 \).
AdvancedState the sum of the first \( n \) integers.
IntermediateFind the 10th term of the AP \( 2,5,8,\dots \)
Intermediate\( a=2,\ d=3 \):
\[ 2+9(3)=29. \]Find the 5th term of the GP \( 3,6,12,\dots \)
Advanced\( a=3,\ r=2 \):
\[ 3\cdot2^{4}=48. \]Expand \( (1+x)^{2} \).
IntermediateFind the sum \( 1+2+\dots+100 \).
AdvancedFind the sum to infinity of a GP with \( a=1 \), \( r=\tfrac12 \).
IntermediateFind the magnitude of \( (3,4) \).
IntermediateFind \( (1,2)+(3,1) \).
IntermediateFind \( 2(1,3) \).
AdvancedFind the magnitude of \( (6,8) \).
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Based on Cambridge International A Level Mathematics 9709 Paper 3: Pure Mathematics 3. Advanced topics: complex numbers, series, differential equations, vectors, numerical methods.
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