On-the-Way Revision Reference
[!info] How to Use This Scroll through on the bus. Each subject has a concept map, formula table, and strategy table. ==Highlights== mark exam hot spots. Fold
> [!tip]-blocks for optional deep dives. Last section: the 10 formulas that show up every year.
1. Big Picture — PASUM All 5 Subjects
mindmap
root((PASUM Revision))
FAD1014[Mathematics II]
Integration
Geometry
Binomial Expansion
Power Series
Trig Integration
FAD1018[Chemistry II]
Equilibrium
Organic Chemistry
Thermochemistry
Kinetics
Electrochemistry
FAD1022[Physics II]
Electrostatics
AC Circuits
Magnetism
Semiconductors
Modern Physics
FAC1004[Advanced Math II]
Complex Numbers
Hyperbolic Functions
Inverse Trig
Complex Logarithms
FAD1015[Mathematics III]
Probability
Distributions
Inferential Stats
Matrices
Eigenvalues
2. FAD1014 — Mathematics II
Concept Map
graph TB
subgraph FAD1014["Mathematics II"]
IT[Integration Techniques] --> TS[Trig Substitution]
IT --> IBP[Integration by Parts]
IT --> PF[Partial Fractions]
GEO[Geometry] --> CIR[Circle]
GEO --> PAR[Parabola]
GEO --> ELL[Ellipse]
GEO --> HYP[Hyperbola]
BE[Binomial Expansion] --> GEN[(1+x)^n series]
BE --> VAL[Validity: |x|<1]
PS[Power Series] --> TAY[Taylor Series]
PS --> MAC[Maclaurin Series: a=0]
MAC --> DERIV[Needs derivatives at 0]
TI[Trig Integration] --> PR[Power Reduction]
TI --> WALLIS[Wallis Formula]
TI --> PROD[Product-to-Sum]
end
Quick Formulas
| Topic | Formula | Notes |
|---|---|---|
| Circle | $(x-h)^2+(y-k)^2=r^2$ | Centre $(h,k)$, radius $r$ |
| Parabola | $(x-h)^2=4a(y-k)$ | Vertex $(h,k)$, focal length $a$ |
| Ellipse | $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ | $c^2=a^2-b^2$, foci $(\pm c,0)$ |
| Hyperbola | $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ | Asymptotes $y=\pm\frac{b}{a}x$, $c^2=a^2+b^2$ |
| Binomial | $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\dots$ | Valid for $|x|<1$ |
| Maclaurin | $f(x)=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\dots$ | Taylor centred at 0 |
| $\sin^2x$ | $\frac{1-\cos2x}{2}$ | Power reduction |
| $\cos^2x$ | $\frac{1+\cos2x}{2}$ | Power reduction |
| $\sqrt{a^2-x^2}$ sub | $x=a\sin\theta$ | $dx=a\cos\theta,d\theta$ |
| $\sqrt{a^2+x^2}$ sub | $x=a\tan\theta$ | $dx=a\sec^2\theta,d\theta$ |
| $\sqrt{x^2-a^2}$ sub | $x=a\sec\theta$ | $dx=a\sec\theta\tan\theta,d\theta$ |
| $\int \sin^nx,dx$ | Wallis formula (odd $n$: easy; even $n$: power reduce) | See Integration Techniques |
When to Use What
| Scenario | Technique | Check |
|---|---|---|
| See $\sqrt{a^2-x^2}$ in integral | Trig sub $x=a\sin\theta$ | Domain: $|x|\leq a$ |
| See $\sqrt{a^2+x^2}$ in integral | Trig sub $x=a\tan\theta$ | Remember $\sec^2\theta=1+\tan^2\theta$ |
| See $\sqrt{x^2-a^2}$ in integral | Trig sub $x=a\sec\theta$ | Domain: $|x|\geq a$ |
| Product of different trig functions | Integration by parts or product-to-sum | $\sin A\cos B=\frac{1}{2}[\sin(A+B)+\sin(A-B)]$ |
| Rational function (degree num < denom) | Partial fractions | Factor denominator first |
| $\int \sin^m x \cos^n x$, $m$ or $n$ odd | Split one power, $u$-sub | Let $u=\cos x$ (if $\sin$ odd) or $u=\sin x$ (if $\cos$ odd) |
| $\int \sin^m x \cos^n x$, both even | Power reduction formulas | Reduce to $\cos2x$, $\cos4x$, etc. |
| Given centre and point on circle | Plug into $(x-h)^2+(y-k)^2=r^2$ | Find $r$ from distance |
| Given focus and directrix | Use definition $PF=e\cdot PD$ | $e=1$ for parabola |
| Expanding $(a+bx)^n$, $n$ not integer | Binomial: factor out $a^n$ | Write as $a^n(1+\frac{b}{a}x)^n$ |
| Need $n$th derivative at 0 for Maclaurin | Differentiate repeatedly, evaluate at 0 | Pattern recognition helps |
[!tip] Exam Tip For Maclaurin series of $\sin x$, $\cos x$, $e^x$, $\ln(1+x)$ — ==memorise the standard expansions==. They're almost always asked.
[!warning] Common Mistake Forgetting to check the ==validity interval== for binomial expansion. If $n$ is not a positive integer, the expansion is infinite and only valid for $|x|<1$.
[!tip]- Deep Dive: Wallis Formula $\int_0^{\pi/2} \sin^n x,dx = \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdots\left(\frac{1}{2}\cdot\frac{\pi}{2}\text{ if }n\text{ even, else }1\right)$
3. FAD1018 — Chemistry II
Concept Map
graph TB
subgraph FAD1018["Chemistry II"]
EQ[Chemical Equilibrium] --> LE[Le Chatelier's Principle]
EQ --> KC[Kc and Kp]
EQ --> VANT[van't Hoff Equation]
IE[Ionic Equilibria] --> HH[Henderson-Hasselbalch]
IE --> BUFF[Buffer Solutions]
IE --> KSP[Solubility Product]
PE[Phase Equilibria] --> RAOULT[Raoult's Law]
PE --> COLLIG[Colligative Properties]
STEREO[Stereochemistry] --> CHIRAL[Chirality]
STEREO --> RS[RS Configuration]
STEREO --> ENANT[Enantiomers]
OC[Organic Chemistry] --> ALP[Alcohol & Phenol]
OC --> CARB[Carbonyl Compounds]
OC --> CA[Carboxylic Acids]
OC --> AMINE[Amines & Amino Acids]
OC --> POLY[Polymer Chemistry]
TC[Thermochemistry] --> HESS[Hess's Law]
TC --> BORN[Haber-Born Cycle]
KCHEM[Kinetic Chemistry] --> ARRH[Arrhenius Equation]
KCHEM --> ORDER[Reaction Order]
KCHEM --> MM[Michaelis-Menten]
ELEC[Electrochemistry] --> NER[Nernst Equation]
ELEC --> FAR[Faraday's Law]
ELEC --> CELL[Galvanic Cells]
end
Quick Formulas
| Topic | Formula | Notes |
|---|---|---|
| $K_c$ | $K_c=\frac{[C]^c[D]^d}{[A]^a[B]^b}$ | Products over reactants |
| $K_p$ | $K_p=K_c(RT)^{\Delta n}$ | $\Delta n$ = moles gas products - reactants |
| van't Hoff | $\ln\frac{K_2}{K_1}=-\frac{\Delta H\degree}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$ | Relates $K$ and $T$ |
| Henderson-Hasselbalch | $\text{pH}=\text{p}K_a+\log\frac{[\text{A}^-]}{[\text{HA}]}$ | Buffer pH |
| $K_{sp}$ | $K_{sp}=[\text{M}^{n+}]^m[\text{A}^{m-}]^n$ | Solubility product |
| Arrhenius | $k=Ae^{-E_a/RT}$ | Rate constant vs $T$ |
| Arrhenius (linear) | $\ln k=\ln A-\frac{E_a}{RT}$ | Plot $\ln k$ vs $1/T$, slope $=-E_a/R$ |
| Nernst | $E=E\degree-\frac{RT}{nF}\ln Q$ | At 298K: $E=E\degree-\frac{0.0592}{n}\log Q$ |
| Faraday's Law | $m=\frac{MIt}{nF}$ | Mass deposited, $F=96500$ C/mol |
| Michaelis-Menten | $v=\frac{V_{\max}[S]}{K_m+[S]}$ | Enzyme kinetics |
| Raoult's Law | $P_A=\chi_A P_A\degree$ | Vapour pressure |
| $\Delta T_b$ | $\Delta T_b=iK_b m$ | Boiling point elevation |
| $\Delta T_f$ | $\Delta T_f=iK_f m$ | Freezing point depression |
| $\Pi$ | $\Pi=iMRT$ | Osmotic pressure |
| Degree of dissociation ($\alpha$) | $\alpha=\frac{\Lambda_m}{\Lambda_m\degree}$ | For weak electrolytes |
When to Use What
| Scenario | Technique/Equation | Check |
|---|---|---|
| Need pH of buffer | Henderson-Hasselbalch | Confirm in buffer region ($[\text{HA}]\approx[\text{A}^-]$) |
| Need pH of weak acid | $[\text{H}^+]=\sqrt{K_a C}$ | $C$ is initial concentration |
| Need pH of weak base | $[\text{OH}^-]=\sqrt{K_b C}$, then pOH | Convert to pH: $\text{pH}=14-\text{pOH}$ |
| $K$ at different temperature | van't Hoff equation | $\Delta H\degree$ must be constant |
| Predict equilibrium shift | Le Chatelier's Principle | Check concentration, pressure, temperature |
| Precipitate forms? | Compare $Q_{sp}$ vs $K_{sp}$ | $Q>K$: precipitate; $Q<K$: no precipitate |
| Rate law from data | Method of initial rates | Compare experiments where one $[\ ]$ changes |
| Determine reaction order | Plot $\ln[\ ]$ vs $t$ (1st), $1/[\ ]$ vs $t$ (2nd) | Linear = correct order |
| Cell potential under non-standard conditions | Nernst equation | At equilibrium: $E=0$, $Q=K$ |
| Calculate mass electroplated | Faraday's Law $m=MIt/nF$ | $I$ in A, $t$ in s |
| Assign R/S configuration | CIP priority rules | Rotate so lowest priority is away |
| Identify chiral centre | Look for C with 4 different groups | No plane of symmetry |
| Boiling point of solution | $\Delta T_b=iK_b m$ | $i$ = van't Hoff factor |
| Solubility from $K_{sp}$ | Set up ICE table, solve for $s$ | Watch stoichiometry |
Key Molecular Structures (SMILES)
CC(=O)Oc1ccccc1C(=O)O
O=C(O)c1ccccc1O
NCC(=O)O
C[C@@H](N)C(=O)O
C=Cc1ccccc1
O=CC(O)C(O)C(O)C(O)CO
CC(=O)Nc1ccc(O)cc1
CCO
CC(=O)O
Oc1ccccc1
O=Cc1ccccc1
CC(=O)C
CN
OC(=O)CCCCC(=O)O
OCCO
[!tip] Exam Tip ==The Nernst equation at 298K== ($E=E\degree-\frac{0.0592}{n}\log Q$) is the single most-asked electrochemistry formula. Know when $Q=K$ and $E=0$.
[!warning] Common Mistake Henderson-Hasselbalch only works when ==$\text{pH}\approx\text{p}K_a\pm1$==. Outside that range, use ICE table.
[!info] Key Insight $K_{sp}$ problems: always write the ==dissolution equation first==, then the ICE table. The stoichiometric coefficient becomes the exponent. For $\text{Ag}_2\text{CrO}4$: $K{sp}=(2s)^2(s)=4s^3$.
[!tip]- Deep Dive: Stereochemistry RS Priority CIP rules: (1) Higher atomic number = higher priority. (2) If same atom, go to next. (3) Multiple bonds count as multiple single bonds. Orient lowest-priority group away, then trace 1→2→3. Clockwise = R, anticlockwise = S.
4. FAD1022 — Physics II
Concept Map
graph TB
subgraph FAD1022["Physics II"]
ES[Electrostatics] --> CL[Coulomb's Law]
ES --> EF[Electric Field]
ES --> EP[Electric Potential]
CD[Capacitors & Dielectrics] --> CAP[C = kappa-epsilon0 A/d]
CD --> ENERGY[U = 1/2 CV^2]
CD --> DIEL[Dielectric increases C]
AC[AC Circuits] --> REAC[Xc = 1/omegaC]
AC --> INDREAC[XL = omegaL]
AC --> IMP[Impedance Z]
AC --> RES[Resonance omega0 = 1/sqrt(LC)]
MAG[Magnetism] --> LOR[Lorentz Force F=qvB sin theta]
MAG --> BIOT[Biot-Savart Law]
MAG --> FARADAY[Faraday's Law E = -N dPhi/dt]
IND[Inductance & Transformers] --> SELF[Self-inductance]
IND --> MUT[Mutual inductance]
IND --> XFORM[Vs/Vp = Ns/Np]
SC[Semiconductors & Diodes] --> PN[PN Junction]
SC --> RECT[Rectification]
TR[Transistors & Biasing] --> BJT[BJT: npn/pnp]
TR --> BIAS[Biasing: Q-point]
OP[Operational Amplifiers] --> INV[Inverting Amp: -Rf/Ri]
OP --> NONINV[Non-inverting: 1+Rf/Ri]
ATOM[Atomic Physics] --> BOHR[Bohr Model: En = -13.6/n^2 eV]
ATOM --> SPEC[Hydrogen Spectrum]
NUC[Nuclear Physics] --> DECAY[Radioactive Decay]
NUC --> MCDEF[Mass Defect]
NUC --> EMC[E = mc^2]
MOD[Modern Physics] --> WPD[Wave-Particle Duality]
MOD --> DBB[de Broglie: lambda = h/p]
MOD --> PHOT[Photons: E = hf]
PE[Photoelectric Effect] --> KE[KEmax = hf - phi]
PE --> WF[Work Function]
PE --> THRESH[Threshold Frequency]
end
Quick Formulas
| Topic | Formula | Notes |
|---|---|---|
| Coulomb's Law | $F=\frac{kq_1q_2}{r^2}$ | $k=9\times10^9$ Nm²/C² |
| Electric Field | $E=\frac{F}{q}=\frac{kQ}{r^2}$ | N/C or V/m |
| Electric Potential | $V=\frac{kQ}{r}$ | Scalar (V) |
| Capacitance (parallel plate) | $C=\frac{\kappa\varepsilon_0 A}{d}$ | $\varepsilon_0=8.85\times10^{-12}$ F/m |
| Energy in capacitor | $U=\frac{1}{2}CV^2=\frac{1}{2}QV=\frac{Q^2}{2C}$ | Three forms |
| Capacitors in series | $\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots$ | Reciprocal sum |
| Capacitors in parallel | $C_{eq}=C_1+C_2+\cdots$ | Direct sum |
| Reactance (capacitive) | $X_C=\frac{1}{\omega C}$ | $\omega=2\pi f$ |
| Reactance (inductive) | $X_L=\omega L$ | Increases with $f$ |
| Impedance (RLC series) | $Z=\sqrt{R^2+(X_L-X_C)^2}$ | Phase angle: $\phi=\tan^{-1}\frac{X_L-X_C}{R}$ |
| Resonance | $\omega_0=\frac{1}{\sqrt{LC}}$ | $X_L=X_C$, $Z=R$ (minimum) |
| Faraday's Law | $\mathcal{E}=-N\frac{d\Phi_B}{dt}$ | Induced emf |
| Lorentz Force | $F=qvB\sin\theta$ | $\theta$ = angle between $\vec{v}$ and $\vec{B}$ |
| Force on current-carrying wire | $F=BIL\sin\theta$ | $I$ = current, $L$ = length |
| Transformer | $\frac{V_s}{V_p}=\frac{N_s}{N_p}=\frac{I_p}{I_s}$ | Ideal transformer |
| Bohr energy levels | $E_n=-\frac{13.6}{n^2}$ eV | For hydrogen |
| Photon energy | $E=hf=\frac{hc}{\lambda}$ | $h=6.63\times10^{-34}$ J·s |
| Photoelectric effect | $KE_{\max}=hf-\phi$ | $\phi$ = work function |
| de Broglie wavelength | $\lambda=\frac{h}{p}=\frac{h}{mv}$ | Wave-particle duality |
| Mass-energy | $E=mc^2$ | $c=3\times10^8$ m/s |
| Radioactive decay | $N=N_0 e^{-\lambda t}$ | $t_{1/2}=\frac{\ln 2}{\lambda}$ |
| Op-amp (inverting) | $A_v=-\frac{R_f}{R_i}$ | Negative sign = phase inversion |
| Op-amp (non-inverting) | $A_v=1+\frac{R_f}{R_i}$ | Positive gain |
| Diode current | $I=I_0(e^{eV/\eta kT}-1)$ | Shockley equation |
When to Use What
| Scenario | Technique/Equation | Check |
|---|---|---|
| Two point charges, need force | Coulomb's Law $F=kq_1q_2/r^2$ | Direction: like repel, unlike attract |
| Need electric field from charge distribution | Superposition or Gauss's Law | Symmetry determines method |
| Parallel plate capacitor | $C=\kappa\varepsilon_0 A/d$ | Inserting dielectric multiplies $C$ by $\kappa$ |
| Capacitor network reduction | Series: reciprocal sum; Parallel: direct sum | Redraw after each step |
| RLC circuit — find current | $I=V/Z$, $Z=\sqrt{R^2+(X_L-X_C)^2}$ | Phase: $\phi=\tan^{-1}((X_L-X_C)/R)$ |
| Circuit at resonance | $\omega_0=1/\sqrt{LC}$, $Z=R$ (min), $I=V/R$ (max) | $X_L=X_C$ |
| Need induced emf | Faraday: $\mathcal{E}=-N d\Phi/dt$ | $\Phi=BA\cos\theta$ |
| Charged particle in magnetic field | Lorentz: $F=qvB\sin\theta$ | Direction: right-hand rule |
| Current-carrying wire in B-field | $F=BIL\sin\theta$ | Fleming's left-hand rule |
| Step-up/step-down voltage | Transformer ratio $V_s/V_p=N_s/N_p$ | Power conserved (ideal) |
| Given frequency, need photon energy | $E=hf$ or $E=hc/\lambda$ | Convert between eV and J |
| Electron transition in hydrogen | $\Delta E=E_i-E_f=-13.6(1/n_i^2-1/n_f^2)$ | Emitted photon: $hf=|\Delta E|$ |
| Photoelectric — find $KE_{\max}$ | $KE_{\max}=hf-\phi$ | $KE_{\max}\geq0$, else no emission |
| Radioactive — find remaining nuclei | $N=N_0e^{-\lambda t}$ | $t_{1/2}=\ln2/\lambda$ |
| Mass defect / binding energy | $\Delta m = (Zm_p+Nm_n)-M_{\text{nucleus}}$ | Then $E=\Delta m c^2$ |
| Op-amp gain | Inverting: $-R_f/R_i$; Non-inverting: $1+R_f/R_i$ | Negative feedback required |
| Diode — forward or reverse bias? | Forward: $V_{\text{anode}}>V_{\text{cathode}}$ | Forward bias: current flows |
[!tip] Exam Tip In RLC circuit problems, ==always draw the phasor diagram==. $V_L$ leads $V_R$ by 90°, $V_C$ lags $V_R$ by 90°. This prevents sign errors.
[!warning] Common Mistake For photoelectric effect: $KE_{\max}=hf-\phi$ is the ==maximum== kinetic energy. The stopping potential is $eV_s=KE_{\max}$. Do NOT confuse $\phi$ (work function in eV) with $hf$ (photon energy).
[!info] Key Insight At resonance in a series RLC circuit, the voltage across $L$ or $C$ can be ==much larger== than the source voltage (voltage magnification factor $Q$). $V_L=V_C=QV_{\text{source}}$ where $Q=\omega_0 L/R$.
[!tip]- Deep Dive: Right-Hand Rules
- Lorentz force ($\vec{F}=q\vec{v}\times\vec{B}$): Fingers → $\vec{v}$, curl → $\vec{B}$, thumb → $\vec{F}$ (for +q)
- Straight wire B-field: Thumb → current, curled fingers → B direction
- Solenoid: Curled fingers → current, thumb → N pole (B direction)
5. FAC1004 — Advanced Mathematics II
Concept Map
graph TB
subgraph FAC1004["Advanced Mathematics II"]
CN[Complex Numbers] --> POLAR[Polar Form: r cis theta]
CN --> EULER[Euler: e^i theta = cos theta + i sin theta]
CN --> DEMO[De Moivre's Theorem]
CN --> ROOTS[nth Roots of Unity]
CL[Complex Logarithms] --> LOGDEF[log z = ln|z| + i arg z]
CL --> BRANCH[Principal branch]
LOCI[Complex Loci] --> CIRCLEL[Circle: |z-a|=r]
LOCI --> LINEL[Perp bisector: |z-a|=|z-b|]
LOCI --> RAYL[Ray: arg(z-a)=theta]
ITF[Inverse Trig Functions] --> ARCSIN[arcsin x]
ITF --> ARCCOS[arccos x]
ITF --> ARCTAN[arctan x]
ITF --> DERIVS[Derivatives of inverse trig]
HF[Hyperbolic Functions] --> SINH[sinh x = (e^x - e^-x)/2]
HF --> COSH[cosh x = (e^x + e^-x)/2]
HF --> TANH[tanh x = sinh x/cosh x]
HF --> IDENT[cosh^2 x - sinh^2 x = 1]
IHF[Inverse Hyperbolic] --> ARSINH[arcsinh x = ln(x+sqrt(x^2+1))]
IHF --> ARCOSH[arccosh x = ln(x+sqrt(x^2-1))]
IHF --> ARTANH[artanh x = 1/2 ln((1+x)/(1-x))]
IHINT[Integration of Hyperbolic] --> DIRECT[Direct integration]
IHINT --> SUB[Hyperbolic substitution]
end
Quick Formulas
| Topic | Formula | Notes |
|---|---|---|
| Rectangular → Polar | $z=a+bi$, $r=\sqrt{a^2+b^2}$, $\theta=\arg(z)$ | $\theta=\tan^{-1}(b/a)$ needs quadrant check |
| Euler's formula | $e^{i\theta}=\cos\theta+i\sin\theta$ | $r e^{i\theta}$ is polar exponential form |
| De Moivre | $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$ | Works for integer $n$; extend via roots for rational $n$ |
| $n$th roots | $z_k=r^{1/n}\operatorname{cis}!\left(\frac{\theta+2\pi k}{n}\right)$ | $k=0,1,\dots,n-1$ |
| Complex logarithm | $\log z=\ln|z|+i(\arg z+2\pi k)$ | Principal: $k=0$ |
| $\frac{d}{dx}\arcsin x$ | $\frac{1}{\sqrt{1-x^2}}$ | Domain: $|x|<1$ |
| $\frac{d}{dx}\arccos x$ | $-\frac{1}{\sqrt{1-x^2}}$ | Domain: $|x|<1$ |
| $\frac{d}{dx}\arctan x$ | $\frac{1}{1+x^2}$ | Domain: $\mathbb{R}$ |
| $\cosh^2 x-\sinh^2 x$ | $=1$ | Analogous to $\cos^2+\sin^2=1$ |
| $\frac{d}{dx}\sinh x$ | $\cosh x$ | NO minus sign |
| $\frac{d}{dx}\cosh x$ | $\sinh x$ | NO minus sign |
| $\operatorname{arcsinh} x$ | $\ln(x+\sqrt{x^2+1})$ | Defined for all real $x$ |
| $\operatorname{arccosh} x$ | $\ln(x+\sqrt{x^2-1})$ | $x\geq 1$ |
| $\operatorname{artanh} x$ | $\frac{1}{2}\ln\frac{1+x}{1-x}$ | $|x|<1$ |
| $\int \frac{1}{\sqrt{x^2+a^2}},dx$ | $\operatorname{arcsinh}(x/a)+C$ or $\ln|x+\sqrt{x^2+a^2}|+C$ | Standard integral |
| $\int \frac{1}{\sqrt{x^2-a^2}},dx$ | $\operatorname{arccosh}(x/a)+C$ or $\ln|x+\sqrt{x^2-a^2}|+C$ | $x>a>0$ |
| $\int \frac{1}{a^2-x^2},dx$ | $\frac{1}{a}\operatorname{artanh}(x/a)+C$ | $|x|<a$ |
When to Use What
| Scenario | Technique | Check |
|---|---|---|
| Convert $a+bi$ to polar | $r=\sqrt{a^2+b^2}$, $\theta=\tan^{-1}(b/a)$ | Adjust $\theta$ for quadrant (add $\pi$ for Q2/Q3) |
| Raise complex number to power | De Moivre: convert to polar first | ${r(\cos\theta+i\sin\theta)}^n=r^n(\cos n\theta+i\sin n\theta)$ |
| Find $n$th roots of complex number | $z_k=r^{1/n}\operatorname{cis}((\theta+2\pi k)/n)$ | $n$ distinct roots equally spaced on circle |
| Solve equation with $\cos^n\theta$, $\sin^n\theta$ | Express as cos/sin of multiple angles | Use De Moivre + binomial |
| Evaluate $\log(z)$ for complex $z$ | $\log z=\ln|z|+i\arg z$ | Principal value: $-\pi<\arg z\leq\pi$ |
| Locus $|z-a|=r$ | Circle centre $a$, radius $r$ | Sketch Argand diagram |
| Locus $|z-a|=|z-b|$ | Perpendicular bisector of segment $ab$ | Sketch |
| Locus $\arg(z-a)=\theta$ | Ray from $a$ at angle $\theta$ | Half-line, $a$ excluded |
| Derivative of $\arcsin(x/a)$ | $\frac{1}{\sqrt{a^2-x^2}}$ | Use chain rule |
| Integrate $\frac{1}{\sqrt{a^2-x^2}}$ | $\arcsin(x/a)+C$ | Recognise the pattern |
| Integrate $\frac{1}{a^2+x^2}$ | $\frac{1}{a}\arctan(x/a)+C$ | Complete square if needed |
| Simplify expression with $e^x\pm e^{-x}$ | Recognise as $\sinh x$ or $\cosh x$ | $\sinh x=\frac{e^x-e^{-x}}{2}$, $\cosh x=\frac{e^x+e^{-x}}{2}$ |
| Solve equation with $\sinh$ or $\cosh$ | Convert to exponential form | Or use identities: $\cosh^2x-\sinh^2x=1$ |
| Integral with $\sqrt{x^2+a^2}$ | Use $x=a\sinh t$ (hyperbolic sub) | Alternative to $x=a\tan\theta$ |
| Find $x$ from $\operatorname{arcsinh} x$ | $x=\sinh(\text{value})$ | Or use log form |
| Find $x$ from $\operatorname{arccosh} x$ | $x=\cosh(\text{value})$, $x\geq1$ | Domain restriction |
| Summation using complex numbers | Express $\sum\cos k\theta$ as $\Re\sum e^{ik\theta}$ | Geometric series in complex form |
[!tip] Exam Tip For complex roots, ==draw the roots on the Argand diagram==. They are equally spaced on a circle of radius $r^{1/n}$. This catches arithmetic errors.
[!warning] Common Mistake $\arg(z)$ is not simply $\tan^{-1}(y/x)$ — you must check the ==quadrant==. For $z=-1+i$, $\arg(z)=3\pi/4$ (not $-\pi/4$). Draw the Argand diagram.
[!info] Key Insight Hyperbolic functions behave like trig functions but ==without alternating signs== in derivatives. $\frac{d}{dx}\sinh x=\cosh x$ (no minus), $\frac{d^2}{dx^2}\sinh x=\sinh x$ (no minus). This makes them useful for solving certain ODEs.
[!tip]- Deep Dive: Osborn's Rule To convert a trig identity to a hyperbolic identity: replace $\sin\to i\sinh$, $\cos\to\cosh$, $\tan\to i\tanh$. Then replace any product of two $\sinh$ terms (i.e. $i^2\sinh A\sinh B$) by $-\sinh A\sinh B$. Example: $\cos^2A+\sin^2A=1$ → $\cosh^2A-\sinh^2A=1$.
6. FAD1015 — Mathematics III
Concept Map
graph TB
subgraph FAD1015["Mathematics III"]
CP[Counting & Probability] --> PERM[Permutations nPr]
CP --> COMB[Combinations nCr]
CP --> BAYES[Bayes' Theorem]
DIST[Probability Distributions] --> BIN[Binomial: n, p]
DIST --> POI[Poisson: lambda]
DIST --> NORM[Normal: mu, sigma^2]
DIST --> UNIF[Uniform]
DIST --> EXP[Exponential]
SD[Sampling Distributions] --> CLT[Central Limit Theorem]
SD --> SEXBAR[Distribution of X-bar]
CI[Confidence Intervals] --> CIMEAN[CI for mean: xbar +/- z*sigma/sqrt n]
CI --> CIPROP[CI for proportion]
HT[Hypothesis Testing] --> ZTEST[Z-test: known sigma]
HT --> TTEST[T-test: unknown sigma]
HT --> PVAL[p-value approach]
HT --> ERR[Type I/II Errors]
MAT[Matrices] --> DET[Determinants]
MAT --> INV[Inverse: A^-1]
MAT --> SOLVE[Solving linear systems]
EIG[Eigenvalues] --> CHAREQ[det(A-lambda I)=0]
EIG --> EIGVEC[Eigenvectors]
end
Quick Formulas
| Topic | Formula | Notes |
|---|---|---|
| Permutations | $P(n,r)=\frac{n!}{(n-r)!}$ | Order matters |
| Combinations | $C(n,r)=\binom{n}{r}=\frac{n!}{r!(n-r)!}$ | Order does not matter |
| $P(A\cup B)$ | $P(A)+P(B)-P(A\cap B)$ | For mutually exclusive: $P(A\cap B)=0$ |
| $P(A|B)$ | $\frac{P(A\cap B)}{P(B)}$ | Conditional probability |
| Bayes' Theorem | $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$ | $P(B)=P(B|A)P(A)+P(B|A')P(A')$ |
| Binomial: $P(X=k)$ | $\binom{n}{k}p^k(1-p)^{n-k}$ | $k=0,1,\dots,n$ |
| Binomial: $E(X)$ | $np$ | Variance: $np(1-p)$ |
| Poisson: $P(X=k)$ | $\frac{\lambda^k e^{-\lambda}}{k!}$ | $k=0,1,2,\dots$ |
| Poisson: $E(X)$ | $\lambda$ | Variance: $\lambda$ (mean = variance) |
| Normal: $Z$ | $Z=\frac{X-\mu}{\sigma}$ | Standardize to use $Z$-table |
| Uniform: $f(x)$ | $\frac{1}{b-a}$ for $x\in[a,b]$ | $E(X)=\frac{a+b}{2}$ |
| Exponential: $f(x)$ | $\lambda e^{-\lambda x}$ for $x\geq0$ | $E(X)=1/\lambda$ |
| CI for $\mu$ ($\sigma$ known) | $\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ | 95% CI: $z_{0.025}=1.96$ |
| CI for $\mu$ ($\sigma$ unknown) | $\bar{x}\pm t_{\alpha/2,,n-1}\frac{s}{\sqrt{n}}$ | Use $t$-distribution |
| CI for proportion $p$ | $\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ | Check $n\hat{p}\geq5$, $n(1-\hat{p})\geq5$ |
| $Z$-test statistic | $z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$ | Compare to $z_{\text{crit}}$ |
| $T$-test statistic | $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$ | df $=n-1$ |
| $p$-value | Probability of observing test statistic or more extreme | Reject $H_0$ if $p<\alpha$ |
| Type I Error | Reject $H_0$ when $H_0$ true | $\alpha$ = significance level |
| Type II Error | Fail to reject $H_0$ when $H_0$ false | $\beta$ |
| Power | $1-\beta$ | Probability of correctly rejecting $H_0$ |
| Determinant (2×2) | $\det\begin{pmatrix}a&b\c&d\end{pmatrix}=ad-bc$ | For 3×3: use expansion or Sarrus |
| Inverse (2×2) | $A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$ | Only if $\det(A)\neq0$ |
| Eigenvalues | $\det(A-\lambda I)=0$ | Solve characteristic polynomial |
| Eigenvectors | Solve $(A-\lambda I)\vec{v}=0$ for each $\lambda$ | Non-trivial solutions |
When to Use What
| Scenario | Technique/Formula | Check |
|---|---|---|
| Arrange $r$ items from $n$ (order matters) | $P(n,r)=n!/(n-r)!$ | Permutation |
| Select $r$ items from $n$ (order irrelevant) | $\binom{n}{r}=n!/[r!(n-r)!]$ | Combination |
| "At least one" probability | $1-P(\text{none})$ | Complement rule |
| Conditional probability (reversed) | Bayes' Theorem | Know $P(B|A)$, need $P(A|B)$ |
| Fixed $n$ trials, constant $p$, count successes | Binomial distribution | Trials independent |
| Events occur at constant rate, count occurrences | Poisson distribution | Mean ≈ variance |
| Continuous data, bell-shaped | Normal distribution | Use $Z$-score |
| Equally likely over interval | Uniform distribution | $f(x)=1/(b-a)$ |
| Time until next event | Exponential distribution | Memoryless property |
| Estimate population mean | Confidence interval | $Z$ if $\sigma$ known; $t$ if not |
| Test if sample mean differs from known value | Z-test ($\sigma$ known) or T-test ($\sigma$ unknown) | State $H_0$ and $H_1$ |
| Test for proportion | $z=\frac{\hat{p}-p_0}{\sqrt{p_0(1-p_0)/n}}$ | $np_0\geq5$, $n(1-p_0)\geq5$ |
| Compare two means | Two-sample $t$-test | Check equal/unequal variance |
| Decision: $p$-value vs $\alpha$ | Reject $H_0$ if $p<\alpha$ | Two-tailed: double the tail probability |
| Solve $AX=B$ | $X=A^{-1}B$ if $A$ invertible | Or Gaussian elimination |
| Find eigenvalues | $\det(A-\lambda I)=0$ | Solve characteristic equation |
| Find eigenvectors | $(A-\lambda I)\vec{v}=0$ | Parametric solution, non-zero vector |
| Check if matrix diagonalizable | $n$ linearly independent eigenvectors | Distinct eigenvalues → diagonalizable |
[!tip] Exam Tip For hypothesis testing, ==always write out all 5 steps==: (1) State $H_0$ and $H_1$, (2) Significance level $\alpha$, (3) Test statistic, (4) Critical value / $p$-value, (5) Conclusion in words. Full marks depend on step 5.
[!warning] Common Mistake Poisson approximation to Binomial: use when ==$n$ is large and $p$ is small== ($n>50$, $np<5$). Set $\lambda=np$. Do NOT use Poisson for large $p$.
[!info] Key Insight The Central Limit Theorem: For ==any== population with mean $\mu$ and variance $\sigma^2$, the sampling distribution of $\bar{x}$ is approximately $N(\mu,\sigma^2/n)$ for large $n$ ($n\geq30$). This is why $Z$-tests work even for non-normal populations.
[!tip]- Deep Dive: Bayes' Theorem with Law of Total Probability $P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|A')P(A')}$. Always compute the denominator first — it's just the total probability of $B$. Drawing a tree diagram helps enormously.
7. 5-Minute Pre-Exam Scan
[!info] The Critical 10 — Cover These Before Walking In
| # | Formula/Concept | Subject | Why It's Critical |
|---|---|---|---|
| 1 | Nernst: $E=E\degree-\frac{0.0592}{n}\log Q$ | FAD1018 | Electrochemistry staple — appears every year |
| 2 | Photoelectric: $KE_{\max}=hf-\phi$ | FAD1022 | Direct calculation, easy marks |
| 3 | Trig sub: $\sqrt{a^2-x^2}\to x=a\sin\theta$ | FAD1014 | The most common integration technique |
| 4 | Henderson-Hasselbalch: $\text{pH}=\text{p}K_a+\log\frac{[\text{A}^-]}{[\text{HA}]}$ | FAD1018 | Buffer problems guaranteed |
| 5 | De Moivre: $(r\operatorname{cis}\theta)^n=r^n\operatorname{cis}(n\theta)$ | FAC1004 | Complex number powers and roots |
| 6 | Z-test: $z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$ | FAD1015 | Hypothesis testing backbone |
| 7 | Bohr: $E_n=-13.6/n^2$ eV | FAD1022 | Atomic spectra questions |
| 8 | Bayes: $P(A|B)=\frac{P(B|A)P(A)}{P(B)}$ | FAD1015 | Tree diagram + formula |
| 9 | Binomial: $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\dots$ | FAD1014 | Expansion + approximation |
| 10 | Arrhenius: $\ln k=\ln A-\frac{E_a}{RT}$ | FAD1018 | Kinetics — slope gives $E_a$ |
Speed Checklist (1 min each)
- Integration Techniques — Which trig sub for which radical?
- Chemical Equilibrium — $K_c$ vs $K_p$, Le Chatelier direction
- Capacitors & Dielectrics — Series vs parallel capacitor rules
- Complex Numbers — Convert to polar before raising to powers
- Probability Distributions — Binomial vs Poisson choice criteria
- Stereochemistry — R/S assignment: CIP priority, lowest priority away
- Electrostatics — Coulomb's Law direction: like repel, unlike attract
- Hyperbolic Functions — $\cosh^2x-\sinh^2x=1$, derivative NO minus sign
- Hypothesis Testing — $p<\alpha$ → reject $H_0$
- Phase Equilibria — $\Delta T_b=iK_bm$, $\Pi=iMRT$
[!info] Last Reminder ==Units matter.== Check eV vs J, atm vs Pa, radians vs degrees. Most mark deductions come from unit errors, not conceptual mistakes. Good luck!