📐 Sequences & Series
- AP: \(u_n = u_1 + (n-1)d\)
- AP sum: \(S_n = \tfrac{n}{2}(2u_1 + (n-1)d)\)
- GP: \(u_n = u_1 \cdot r^{n-1}\)
- GP sum: \(S_n = \frac{u_1(r^n-1)}{r-1}\)
- Infinite GP \((|r|<1)\): \(S_\infty = \frac{u_1}{1-r}\)
Test: AP → constant difference; GP → constant ratio
📊 Algebra & Functions
- Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
- Discriminant \(\Delta = b^2 - 4ac\)
- Domain of \(\sqrt{f(x)}\): set \(f(x) \ge 0\)
- Inverse: swap \(x\) & \(y\), solve for \(y\)
- Composite: \((f \circ g)(x) = f(g(x))\)
Logs: \(\log_a b = \frac{\ln b}{\ln a}\)
📐 Binomial Theorem
- \((a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)
- \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)
- Term \((r+1)\): \(\binom{n}{r}a^{n-r}b^r\)
Pascal's triangle or nCr calculator
📏 Trigonometry
- \(\sin^2\theta + \cos^2\theta = 1\)
- \(\cos 2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1\)
- \(\sin 150° = \frac{1}{2}\), \(\cos 60° = \frac{1}{2}\)
- Sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B}\)
- Cosine rule: \(a^2 = b^2 + c^2 - 2bc\cos A\)
CAST diagram for quadrant signs
∫ Calculus
- Power rule: \(\frac{d}{dx}x^n = nx^{n-1}\)
- Integration: \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\)
- Definite integral = signed area
- Stationary: \(f'(x)=0\); max if \(f''(x)<0\)
\(\int_a^b f(x)\,dx = [F(x)]_a^b = F(b)-F(a)\)
🎲 Probability & Statistics
- \(P(A \cup B) = P(A)+P(B)-P(A \cap B)\)
- Normal: standardise \(z = \frac{x - \mu}{\sigma}\)
- Binomial: \(P(X=r) = \binom{n}{r}p^r(1-p)^{n-r}\)
Independent: \(P(A \cap B) = P(A) \cdot P(B)\)
↗ Vectors
- Magnitude: \(|\mathbf{v}| = \sqrt{v_1^2 + v_2^2}\)
- Dot product: \(\mathbf{u}\cdot\mathbf{v} = u_1v_1 + u_2v_2\)
- Angle: \(\cos\theta = \frac{\mathbf{u}\cdot\mathbf{v}}{|\mathbf{u}||\mathbf{v}|}\)
Perpendicular ⟺ dot product = 0
📉 Exponential & Logs
- \(e^{a+b} = e^a \cdot e^b\)
- \(\ln(e^x) = x\) and \(e^{\ln x} = x\)
- Growth: \(N(t) = N_0 e^{kt}\)
- \(\log_2 8 = 3\) (since \(2^3=8\))
Change of base: \(\log_a b = \frac{\log b}{\log a}\)
Worked Example — Calculus
Find the local maximum of \(f(x) = x^3 - 3x\).
Step 1 Differentiate: \(f'(x) = 3x^2 - 3 = 3(x-1)(x+1)\)
Step 2 Set \(f'(x)=0\): \(x = \pm 1\)
Step 3 Second derivative test: \(f''(x)=6x\)
• \(f''(-1) = -6 < 0\) → local maximum at \(x=-1\)
• \(f(-1) = -1+3 = 2\) → max point is \((-1,\,2)\)