IB Math AA HL — Master Practice Quiz

T1 Sequences & Series — Arithmetic ▾

Arithmetic sequence: \(u_n = u_1 + (n-1)d\)

\(S_n = \dfrac{n}{2}(2u_1 + (n-1)d) = \dfrac{n}{2}(u_1 + u_n)\)

📌 Memorise: Sum formula uses first + last or 2×first + (n−1)d

Example

Find \(S_{10}\) of the AP: \(3, 7, 11, \ldots\)

\(u_1=3,\; d=4,\; S_{10}=\frac{10}{2}(6+36)=5\times42=210\)

Answer: 210

T2 Binomial Theorem ▾

\((a+b)^n = \displaystyle\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r\)

General term: \(T_{r+1} = \binom{n}{r}a^{n-r}b^r\)

📌 To find a specific term, set the power of \(b\) equal to \(r\) and solve.

Example

Coefficient of \(x^3\) in \((2+x)^5\)

\(T_4 = \binom{5}{3}(2)^2(x)^3 = 10\cdot 4\cdot x^3 = 40x^3\)

Coefficient: 40

T3 Functions — Inverses & Transformations ▾

Inverse: Swap \(x\) and \(y\), then solve for \(y\).

Domain of \(f^{-1}\) = Range of \(f\); Range of \(f^{-1}\) = Domain of \(f\).

\(f(x)=\dfrac{ax+b}{cx+d} \implies f^{-1}(x)=\dfrac{dx-b}{-cx+a}\)

Example

Find \(f^{-1}\) where \(f(x)=\dfrac{2x+1}{x-3}\)

Swap: \(x=\dfrac{2y+1}{y-3}\Rightarrow y=\dfrac{3x+1}{x-2}\)

\(f^{-1}(x) = \dfrac{3x+1}{x-2}\)

T4 Trigonometry — Identities & Equations ▾

\(\sin 2x = 2\sin x\cos x\quad\cos 2x = \cos^2\!x - \sin^2\!x\)

Pythagorean: \(\sin^2\theta+\cos^2\theta=1\)

📌 For equations, factorise — don't divide both sides by a trig function (you lose solutions).

Example

Solve \(\sin 2x = \sin x\) for \(x \in [0, 2\pi)\)

\(2\sin x\cos x - \sin x = 0 \Rightarrow \sin x(2\cos x - 1)=0\)

\(x = 0,\; \tfrac{\pi}{3},\; \pi,\; \tfrac{5\pi}{3}\)

T5 Complex Numbers & De Moivre's Theorem ▾

\(z = r e^{i\theta} = r(\cos\theta + i\sin\theta)\)

De Moivre: \((r e^{i\theta})^n = r^n e^{in\theta}\)

Modulus-argument: \(|z|=r,\; \arg(z)=\theta\)

Example

Compute \((1+i)^4\)

\(|1+i|=\sqrt{2},\; \theta=\tfrac{\pi}{4}\Rightarrow(\sqrt{2})^4 e^{i\pi}=4\cdot(-1)=-4\)

\((1+i)^4 = -4\)

T6 Differentiation — Product, Chain, Integration by Parts ▾

\(\dfrac{d}{dx}[uv]=u'v+uv'\qquad \int u\,dv = uv - \int v\,du\)

Chain rule: \(\dfrac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)

Example

\(\dfrac{d}{dx}[x^2 \ln x] = 2x\ln x + x^2\cdot\tfrac{1}{x} = 2x\ln x + x\)

\(2x\ln x + x\)

T7 Statistics — Normal Distribution & Probability ▾

\(Z = \dfrac{X - \mu}{\sigma}\qquad X\sim N(\mu,\sigma^2)\)

Bayes' Theorem: \(P(A|B)=\dfrac{P(B|A)\,P(A)}{P(B)}\)

📌 \(P(Z>2)\approx 0.0228\); \(P(Z>1)\approx 0.1587\); \(P(Z>1.645)\approx 0.05\)

Example

\(X\sim N(50,16)\). Find \(P(X>58)\).

\(Z=\tfrac{58-50}{4}=2 \Rightarrow P(Z>2)=0.0228\)

0.0228

T8 Differential Equations & Maclaurin Series ▾

\(e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\)

Separable DE: \(\dfrac{dy}{dx}=f(x)g(y)\Rightarrow \displaystyle\int\dfrac{dy}{g(y)}=\int f(x)\,dx\)

Example

Solve \(\dfrac{dy}{dx}=xy\) with \(y(0)=C\)

\(\displaystyle\int\dfrac{dy}{y}=\int x\,dx\Rightarrow \ln|y|=\dfrac{x^2}{2}+C_1\Rightarrow y=Ce^{x^2/2}\)

\(y = Ce^{x^2/2}\)

Analysis & Approaches HL