Cambridge IGCSE Mathematics (0580) Syllabus

What does the discriminant $b^{2}-4ac$ tell you about the roots of a quadratic?

If $b^{2}-4ac>0$ there are two distinct real roots; if $b^{2}-4ac=0$ there is one repeated (equal) real root; if $b^{2}-4ac<0$ there are no real roots.

When solving an inequality, what operation reverses the inequality sign?

Multiplying or dividing both sides by a negative number reverses the inequality sign, e.g. $-2x<6 \Rightarrow x>-3$.

Give the formula for the $n$th term of an arithmetic sequence with first term $a$ and common difference $d$.

$$u_{n}=a+(n-1)d$$

How do you recognise a geometric sequence and what is its $n$th term?

Each term is multiplied by a constant ratio $r$ (consecutive terms have a common ratio). The $n$th term is $u_{n}=a\,r^{\,n-1}$, where $a$ is the first term.

What is the $n$th-term rule for the square numbers, cube numbers and triangular numbers?

Square numbers: $n^{2}$. Cube numbers: $n^{3}$. Triangular numbers: $\frac{n(n+1)}{2}$.

For a function $f$, what do $f(x)$, the inverse $f^{-1}(x)$ and the composite $fg(x)$ mean?

$f(x)$ is the output for input $x$. The inverse $f^{-1}(x)$ reverses $f$ (swap and make $x$ the subject). The composite $fg(x)=f(g(x))$ means apply $g$ first, then apply $f$ to the result.

How do you find the inverse function $f^{-1}(x)$ of $f(x)$?

Write $y=f(x)$, swap $x$ and $y$, then rearrange to make $y$ the subject; that expression is $f^{-1}(x)$. E.g. for $f(x)=3x+2$, $f^{-1}(x)=\frac{x-2}{3}$.

What are the general shapes/features of the graphs $y=mx+c$, $y=ax^{2}$, $y=\frac{a}{x}$ and $y=a^{x}$?

$y=mx+c$ is a straight line (gradient $m$, intercept $c$). $y=ax^{2}$ is a parabola. $y=\frac{a}{x}$ is a reciprocal/hyperbola with asymptotes on the axes. $y=a^{x}$ is an exponential curve through $(0,1)$.

Give the gradient formula and the equation of a straight line.

Gradient $m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$. Equation: $y=mx+c$, or through a point $y-y_{1}=m(x-x_{1})$.

What are the gradient conditions for two lines to be parallel and to be perpendicular?

Parallel lines have equal gradients: $m_{1}=m_{2}$. Perpendicular lines have gradients whose product is $-1$: $m_{1}\times m_{2}=-1$, i.e. $m_{2}=-\frac{1}{m_{1}}$.

How do you estimate the gradient of a curve at a point graphically?

Draw a tangent line touching the curve at that point, then calculate the gradient of the tangent using $\frac{\text{rise}}{\text{run}}=\frac{\Delta y}{\Delta x}$.

State the rule for differentiating $y=ax^{n}$.

$$\frac{dy}{dx}=an\,x^{\,n-1}$$ The derivative of a constant is $0$.

How do you find and classify the stationary points of a curve using differentiation?

Solve $\frac{dy}{dx}=0$ to find the $x$-coordinates. Classify using the second derivative: if $\frac{d^{2}y}{dx^{2}}>0$ it is a minimum; if $\frac{d^{2}y}{dx^{2}}<0$ it is a maximum.

State the angle facts: angles on a straight line, angles around a point, and angles in a triangle.

Angles on a straight line sum to $180^{\circ}$; angles around a point sum to $360^{\circ}$; the interior angles of a triangle sum to $180^{\circ}$.

State the parallel-line angle relationships for corresponding, alternate and co-interior angles.

Corresponding angles are equal (F-shape); alternate angles are equal (Z-shape); co-interior (allied) angles are supplementary, summing to $180^{\circ}$ (C-shape).

Give the formulas for the sum of interior angles and each exterior angle of a regular polygon with $n$ sides.

Sum of interior angles $=(n-2)\times180^{\circ}$. Each exterior angle of a regular polygon $=\frac{360^{\circ}}{n}$, and interior + exterior $=180^{\circ}$.

State the main circle theorems linking angles at the centre and circumference.

The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is $90^{\circ}$. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to $180^{\circ}$.

State the tangent properties of a circle.

A tangent is perpendicular to the radius at the point of contact. Two tangents drawn from an external point are equal in length. The alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

What are the conditions for two triangles to be congruent?

SSS (three sides equal), SAS (two sides and the included angle), ASA (two angles and a corresponding side), and RHS (right angle, hypotenuse and one other side equal).

For similar shapes with length scale factor $k$, what are the area and volume scale factors?

Area scale factor $=k^{2}$ and volume scale factor $=k^{3}$, where $k$ is the ratio of corresponding lengths.

Give the circumference and area formulas for a circle, and the arc length and sector area for angle $\theta$.

$$C=2\pi r,\quad A=\pi r^{2}$$ $$\text{arc}=\frac{\theta}{360}\times2\pi r,\quad \text{sector area}=\frac{\theta}{360}\times\pi r^{2}$$

State the volume formulas for a prism, cylinder, cone and sphere.

Prism: $V=\text{area of cross-section}\times\text{length}$. Cylinder: $V=\pi r^{2}h$. Cone: $V=\frac{1}{3}\pi r^{2}h$. Sphere: $V=\frac{4}{3}\pi r^{3}$.

State the surface area formulas for a sphere, the curved surface of a cylinder, and the curved surface of a cone.

Sphere: $A=4\pi r^{2}$. Cylinder curved surface: $A=2\pi r h$. Cone curved surface: $A=\pi r l$, where $l$ is the slant height.

Describe the four transformations and the information needed to define each fully.

Translation: a column vector $\begin{pmatrix}x\\y\end{pmatrix}$. Reflection: the mirror line. Rotation: centre, angle and direction. Enlargement: centre and scale factor (a negative factor inverts the image).

How do you find the magnitude (modulus) of a column vector $\begin{pmatrix}x\\y\end{pmatrix}$?

$$|\mathbf{v}|=\sqrt{x^{2}+y^{2}}$$

State Pythagoras' theorem and the three right-angled trig ratios (SOH CAH TOA).

Pythagoras: $a^{2}+b^{2}=c^{2}$ ($c$ = hypotenuse). $$\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad \cos\theta=\frac{\text{adj}}{\text{hyp}},\quad \tan\theta=\frac{\text{opp}}{\text{adj}}$$

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

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

Give the formula for the area of a triangle using two sides and the included angle.

$$\text{Area}=\frac{1}{2}ab\sin C$$ where $a$ and $b$ are two sides and $C$ is the angle between them.

Describe the key features (period and range) of the graphs $y=\sin x$ and $y=\cos x$, and how bearings are measured.

$y=\sin x$ and $y=\cos x$ both have period $360^{\circ}$ and range $-1\leq y\leq1$; $\tan x$ has period $180^{\circ}$. A bearing is measured clockwise from north, written as three digits, e.g. $072^{\circ}$.

What is the difference between discrete and continuous data, and what is a stratified sample?

Discrete data take exact separate values (counted, e.g. number of pets); continuous data can take any value in a range (measured, e.g. height). A stratified sample takes from each group in proportion to the group's size in the population.

How do you estimate the mean from a grouped frequency table?

Use the midpoint $x$ of each class. $$\text{Estimated mean}=\frac{\sum f x}{\sum f}$$ where $f$ is the frequency of each class. It is an estimate because the exact values within classes are unknown.

Define the mean, median, mode and range, and how to read the median and quartiles from a cumulative frequency curve.

Mean $=\frac{\sum x}{n}$; median is the middle value when ordered; mode is the most frequent value; range $=\max-\min$. On a cumulative frequency curve of $n$ values, read the median at $\frac{n}{2}$, lower quartile at $\frac{n}{4}$, upper quartile at $\frac{3n}{4}$; IQR $=Q_{3}-Q_{1}$.

State the basic probability formula and the value range of a probability.

$$P(\text{event})=\frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$$ Probabilities lie between $0$ (impossible) and $1$ (certain), and $P(\text{not }A)=1-P(A)$.

State the AND and OR rules for probability and when each applies.

For independent events (AND, both happen): $P(A\text{ and }B)=P(A)\times P(B)$. For mutually exclusive events (OR, one or the other): $P(A\text{ or }B)=P(A)+P(B)$.

On a tree diagram, how do you find the probability of a sequence of outcomes and of a combined event?

Multiply probabilities along the branches for a particular sequence (AND). Add the results of the different sequences that satisfy the required outcome (OR). The probabilities on each set of branches sum to $1$.