International Baccalaureate Diploma Programme (IB) Group 5: Mathematics (Analysis and Approaches) Syllabus

What does the discriminant $\Delta$ tell you about the roots of a quadratic?

$\Delta > 0$: two distinct real roots; $\Delta = 0$: one repeated real root; $\Delta < 0$: no real roots (two complex roots).

Give the general form and key features of an exponential function $f(x) = a^x$ ($a>0$).

Domain all real $x$, range $f(x) > 0$, horizontal asymptote $y = 0$, passes through $(0,1)$; increasing if $a>1$, decreasing if $0<a<1$.

For the rational function $f(x) = \frac{ax+b}{cx+d}$, give the equations of its vertical and horizontal asymptotes.

Vertical asymptote: $x = -\frac{d}{c}$; horizontal asymptote: $y = \frac{a}{c}$

Describe the effect of the transformation $y = f(x) + k$ and $y = f(x + k)$ on the graph of $y = f(x)$.

$f(x)+k$: vertical translation by $k$ (up if $k>0$); $f(x+k)$: horizontal translation by $-k$ (left if $k>0$).

Describe the transformations $y = p\,f(x)$ and $y = f(qx)$.

$p\,f(x)$: vertical stretch by factor $p$; $f(qx)$: horizontal stretch by factor $\frac{1}{q}$.

What transformations do $y = -f(x)$ and $y = f(-x)$ represent?

$-f(x)$: reflection in the $x$-axis; $f(-x)$: reflection in the $y$-axis.

Define the radian and give the arc length and area of a sector of radius $r$ subtending angle $\theta$ radians.

One radian is the angle subtending an arc equal to the radius. Arc length $l = r\theta$; sector area $A = \frac{1}{2}r^2\theta$.

On the unit circle, what are the coordinates of the point at angle $\theta$, and how do they relate to trig ratios?

The point is $(\cos\theta, \sin\theta)$; the $x$-coordinate is $\cos\theta$ and the $y$-coordinate is $\sin\theta$, with $\tan\theta = \frac{\sin\theta}{\cos\theta}$.

State the exact values of $\sin$, $\cos$ and $\tan$ for $\frac{\pi}{6}$, $\frac{\pi}{4}$ and $\frac{\pi}{3}$.

$\frac{\pi}{6}$: $\frac{1}{2}, \frac{\sqrt{3}}{2}, \frac{1}{\sqrt{3}}$; $\frac{\pi}{4}$: $\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 1$; $\frac{\pi}{3}$: $\frac{\sqrt{3}}{2}, \frac{1}{2}, \sqrt{3}$.

State the Pythagorean identity and its two derived forms.

$\sin^2\theta + \cos^2\theta = 1$; dividing by $\cos^2\theta$ gives $1 + \tan^2\theta = \sec^2\theta$, and by $\sin^2\theta$ gives $1 + \cot^2\theta = \csc^2\theta$.

State the double angle identities for $\sin 2\theta$ and $\cos 2\theta$.

$\sin 2\theta = 2\sin\theta\cos\theta$; $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

State the sine rule and the cosine rule for a triangle.

Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$.

For vectors $\vec{a}$ and $\vec{b}$, give the scalar (dot) product in component and geometric form (HL).

$\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec{a}||\vec{b}|\cos\theta$

How do you test whether two non-zero vectors are perpendicular, and how is the magnitude of $\vec{a}$ found (HL)?

They are perpendicular iff $\vec{a}\cdot\vec{b} = 0$; magnitude $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$.

Give the vector equation of a line through point with position vector $\vec{a}$ in direction $\vec{b}$ (HL).

$\vec{r} = \vec{a} + \lambda\vec{b}$, where $\lambda \in \mathbb{R}$

State the formula for the vector (cross) product magnitude and its geometric meaning (HL).

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, equal to the area of the parallelogram formed by $\vec{a}$ and $\vec{b}$; the result is perpendicular to both.

Distinguish between the mean, median and mode as measures of central tendency.

Mean: arithmetic average of all values. Median: the middle value when data are ordered. Mode: the most frequently occurring value.

Define the interquartile range (IQR) and state a common rule for outliers.

$\text{IQR} = Q_3 - Q_1$. An outlier is any value below $Q_1 - 1.5\,\text{IQR}$ or above $Q_3 + 1.5\,\text{IQR}$.

What does the Pearson product-moment correlation coefficient $r$ measure, and what is its range?

It measures the strength and direction of a linear relationship between two variables; $-1 \leq r \leq 1$, where $\pm 1$ is perfect linear correlation and $0$ is none.

State the addition rule and the definition of mutually exclusive events.

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$. Events are mutually exclusive when $P(A \cap B) = 0$, giving $P(A \cup B) = P(A) + P(B)$.

Define conditional probability and state the condition for two events to be independent.

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$. $A$ and $B$ are independent iff $P(A \cap B) = P(A)P(B)$, equivalently $P(A\mid B) = P(A)$.

For a binomial distribution $X \sim B(n,p)$, give $P(X=r)$, the mean and the variance.

$P(X=r) = \binom{n}{r}p^r(1-p)^{n-r}$; mean $E(X)=np$; variance $\operatorname{Var}(X)=np(1-p)$.

State the key properties of the normal distribution and the empirical 68-95-99.7 rule.

Symmetric, bell-shaped about the mean $\mu$, with mean = median = mode. About $68\%$ of data lie within $1\sigma$, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$ of the mean.

How is a normal variable $X \sim N(\mu, \sigma^2)$ standardised to the $z$-score?

$z = \frac{x - \mu}{\sigma}$, giving the standard normal distribution $Z \sim N(0, 1)$.

State the limit definition of the derivative $f'(x)$.

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

State the product rule, quotient rule and chain rule for differentiation.

Product: $(uv)' = u'v + uv'$. Quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$. Chain: $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$.

Give the derivatives of $\sin x$, $\cos x$, $\mathrm{e}^x$ and $\ln x$.

$\frac{d}{dx}\sin x = \cos x$; $\frac{d}{dx}\cos x = -\sin x$; $\frac{d}{dx}\mathrm{e}^x = \mathrm{e}^x$; $\frac{d}{dx}\ln x = \frac{1}{x}$

How do you classify a stationary point using the second derivative test?

At a stationary point ($f'(x)=0$): if $f''(x) > 0$ it is a local minimum; if $f''(x) < 0$ it is a local maximum; if $f''(x) = 0$ the test is inconclusive (check further).

State the fundamental theorem of calculus linking integration and differentiation.

If $F'(x) = f(x)$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

Give the power rule for integration and the integral of $\frac{1}{x}$.

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$); $\int \frac{1}{x}\,dx = \ln|x| + C$

State the formula for the volume of revolution about the $x$-axis between $x=a$ and $x=b$.

$V = \pi \int_a^b [f(x)]^2\,dx$

What is the method of separation of variables for a differential equation $\frac{dy}{dx} = g(x)h(y)$ (HL)?

Rearrange to $\frac{1}{h(y)}\,dy = g(x)\,dx$, then integrate both sides: $\int \frac{1}{h(y)}\,dy = \int g(x)\,dx$.

State the general Maclaurin series of a function $f(x)$ (HL).

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

Give the Maclaurin series for $\mathrm{e}^x$, $\sin x$ and $\cos x$ (HL).

$\mathrm{e}^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$; $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$; $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

In mathematical modelling, what is the role of validating a model against data?

Validation compares the model's predictions with observed/real data to assess accuracy; large discrepancies indicate the model's assumptions must be refined or the model adjusted.

What is the purpose of the Internal Assessment (IA) mathematical exploration in IB Maths AA?

It is an individual written exploration (about 12-20 pages) of an area of mathematics of the student's choice, assessed on criteria including presentation, mathematical communication, personal engagement, reflection and use of mathematics.