IB Math AA SL — Core Concept Practice

T1–2 Sequences & Series

Arithmetic: \(a_n = a_1 + (n-1)d\) \(S_n = \tfrac{n}{2}(2a_1+(n-1)d)\)

Geometric: \(a_n = a_1 r^{n-1}\) \(S_n = \tfrac{a_1(r^n-1)}{r-1}\) \(S_\infty = \tfrac{a_1}{1-r},\;|r|<1\)

Sum of GP: \(S_6 = \frac{a_1(r^6-1)}{r-1}\)

EXAMPLE Q

Find \(a_8\) if \(a_1=5,\,d=3\)

→ \(5+7(3)=26\)

EXAMPLE Q

GP: \(a_1=1,r=2\). Find \(S_5\)

→ \(\frac{1(2^5-1)}{1}=31\)

T2 Algebra & Functions

Quadratic discriminant: \(\Delta = b^2-4ac\) Two real roots if \(\Delta > 0\)

Laws of exponents: \(a^m\cdot a^n=a^{m+n}\) \((a^m)^n=a^{mn}\) \(\frac{a^m}{a^n}=a^{m-n}\)

Logarithms: \(\log_a(xy)=\log_a x+\log_a y\) \(\log_a(x^n)=n\log_a x\)

EXAMPLE Q

Solve \(\log_2 x + \log_2 4 = 5\)

→ \(\log_2 4x=5\Rightarrow x=8\)

EXAMPLE Q

Simplify \(\dfrac{3^5}{3^2}\)

→ \(3^3=27\)

T3 Trigonometry

Key values: \(\sin\frac{\pi}{6}=\frac{1}{2}\), \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), \(\cos\frac{\pi}{3}=\frac{1}{2}\)

CAST rule: S (Q2), A (Q1), T (Q3), C (Q4) — tells you where each ratio is positive

For \(\sin\theta=k\) in \([0,2\pi]\): solutions are \(\theta=\arcsin k\) and \(\pi-\arcsin k\)

EXAMPLE Q

Solve \(\sin\theta=\frac{1}{2},\;0\le\theta\le2\pi\)

→ \(\theta=\frac{\pi}{6},\frac{5\pi}{6}\)

EXAMPLE Q

Exact value of \(\cos\frac{2\pi}{3}\)

→ \(-\frac{1}{2}\)

T4–5 Calculus (Differentiation & Integration)

Power rule: \(\frac{d}{dx}(x^n)=nx^{n-1}\) Integral: \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)

Definite integral: \(\int_a^b f(x)\,dx = [F(x)]_a^b = F(b)-F(a)\)

Optimisation: At max/min: \(f'(x)=0\) · Check \(f''(x)<0\) (max) or \(f''(x)>0\) (min)

EXAMPLE Q

Differentiate \(5x^3-2x\)

→ \(15x^2-2\)

EXAMPLE Q

\(\int_0^1 3x^2\,dx\)

→ \([x^3]_0^1=1\)

T6–7 Probability & Statistics · Vectors · Binomial

Probability: \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)

Normal distribution: \(Z=\frac{X-\mu}{\sigma}\) — standardise, then use z-table

Binomial: \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\) where \(X\sim B(n,p)\)

Vector magnitude: \(|\mathbf{v}|=\sqrt{v_1^2+v_2^2}\) Dot product: \(\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2\)

Binomial theorem: Coefficient of \(x^r\) in \((x+a)^n\) is \(\binom{n}{r}a^{n-r}\)

EXAMPLE Q

\(X\sim B(5,0.4)\), find \(P(X=2)\)

→ \(\binom{5}{2}(0.4)^2(0.6)^3\approx0.346\)

EXAMPLE Q

\(X\sim N(20,9)\), find \(P(X<23)\)

→ \(z=1\Rightarrow P\approx0.841\)