# iGCSE Mathematics (0580) Extended Syllabus: iGCSE Exam Style Practice Questions- Paper 4

### Assessment

All candidates take three papers.
Candidates who have studied the Core subject content, or who are expected to achieve a grade D or below, should be entered for Paper 1, Paper 3 and either Paper 5 or Paper 6. These candidates will be eligible for grades C to G.Candidates who have studied the Extended subject content (Core and Supplement), and who are expected to achieve a grade C or above, should be entered for Paper 2, Paper 4 and either Paper 5 or Paper 6. These candidates will be eligible for grades A* to G.

Extended assessment – Paper 4

• Time: 150 minutes (130 marks)
• Structured questions
• No marks deducted from incorrect answers
• 65% weight
• Externally assessed

### Paper 4: Structured questions (Extended)

E1 Number

• E1.1 Identify and use natural numbers, integers (positive, negative and zero), prime numbers, square and cube numbers, common factors and common multiples, rational and irrational numbers (e.g. $$\pi ,\sqrt{2}$$ ), real numbers, reciprocals.
• E1.2 Use language, notation and Venn diagrams to describe sets and represent relationships between sets.
Definition of sets
e.g. A = {x: x is a natural number}
B = {(x, y): y = mx + c}
C = {x: a ⩽ x ⩽ b}
D = {a, b, c, …}
• E1.3 Calculate with squares, square roots, cubes and cube roots and other powers and roots of numbers.
• E1.4 Use directed numbers in practical situations.
• E1.5 Use the language and notation of simple vulgar and decimal fractions and percentages in appropriate contexts.
Recognize equivalence and convert between these forms.
• E1.6 Order quantities by magnitude and demonstrate familiarity with the symbols
$$=,\neq ,>,<,\geq ,\leqslant$$
• E1.7 Understand the meaning of indices (fractional, negative and zero) and use the rules of indices.
Use the standard form A × 10n where n is a positive or negative integer, and 1 ⩽ A < 10.
• E1.8 Use the four rules for calculations with whole numbers, decimals and fractions (including mixed numbers and improper fractions), including correct ordering of operations and use of brackets.
• E1.9 Make estimates of numbers, quantities and lengths, give approximations to specified numbers of significant figures and decimal places and round off answers to reasonable accuracy in the context of a given problem.
• E1.10 Give appropriate upper and lower bounds for data given to a specified accuracy. Obtain appropriate upper and lower bounds to solutions of simple problems given data to a specified accuracy.
• E1.11 Demonstrate an understanding of ratio and proportion.
Increase and decrease a quantity by a given ratio.
Calculate average speed.
Use common measures of rate.
• E1.12 Calculate a given percentage of a quantity.
Express one quantity as a percentage of another. Calculate percentage increase or decrease.
Carry out calculations involving reverse percentages.
• E1.13 Use a calculator efficiently.
Apply appropriate checks of accuracy.
• E1.14 Calculate times in terms of the 24-hour and 12-hour clock.
• E1.15 Calculate using money and convert from one currency to another
• E1.16 Use given data to solve problems on personal and household finance involving earnings, simple interest and compound interest.
Extract data from tables and charts.
• E1.17 Use exponential growth and decay in relation to population and finance.

E2 Algebra and graphs

• E2.1 Use letters to express generalized numbers and express basic arithmetic processes algebraically.
Substitute numbers for words and letters in complicated formulae.
Construct and rearrange complicated formulae and equations.
• E2.2 Manipulate directed numbers.
Use brackets and extract common factors.
Expand products of algebraic expressions.
Factorise where possible expressions of the form:

$$ax+bx+kay+kby$$
$$a^2x^2-b^2y^2$$
$$a^2+2ab+b^2$$
$$ax^2+bx+c$$

• E2.3 Manipulate algebraic fractions.
Factorise and simplify rational expressions.
• E2.4 Use and interpret positive, negative and zero indices.
Use and interpret fractional indices.
Use the rules of indices.
• E2.5 Derive and solve linear equations in one unknown.
Derive and solve simultaneous linear equations in two unknowns.
Derive and solve simultaneous equations, involving one linear and one quadratic.
Derive and solve quadratic equations by factorization, completing the square and by use of the formula.
Derive and solve linear inequalities.
• E2.6 Represent inequalities graphically and use this representation to solve simple linear programming problems.
• E2.7 Continue a given number sequence.
Recognise patterns in sequences including the term to term rule and relationships between different sequences.
Find and use the nth term of sequences.
• E2.8 Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities.
• E2.9 Use function notation, e.g. f(x) = 3x – 5, f: x ⟼ 3x – 5, to describe simple functions.
Find inverse functions f –1(x).
Form composite functions as defined by gf(x) = g(f(x)).
• E2.10 Interpret and use graphs in practical situations including travel graphs and conversion graphs.
Draw graphs from given data.
Apply the idea of rate of change to simple kinematics involving distance–time and speed–time graphs, acceleration and
deceleration.
Calculate distance travelled as area under a speed–time graph.
• E2.11 Construct tables of values and draw graphs for functions of the form axn (and simple sums of these) and functions of the form abx + c.
Solve associated equations approximately, including finding and interpreting roots by graphical methods.
Draw and interpret graphs representing exponential growth and decay problems.
Recognise, sketch and interpret graphs of functions.
• E2.12 Estimate gradients of curves by drawing tangents.
• E2.13 Understand the idea of a derived function.
Use the derivatives of functions of the form axn, and simple sums of not more than three of these.
Apply differentiation to gradients and turning points (stationary points).
Discriminate between maxima and minima by any method.

E3 Coordinate geometry

• E3.1 Demonstrate familiarity with Cartesian coordinates in two dimensions.3.3 Active transport.
• E3.2 Find the gradient of a straight line.
Calculate the gradient of a straight line from the coordinates of two points on it.
• E3.3 Calculate the length and the coordinates of the midpoint of a straight line from the coordinates of its end points.
• E3.4 Interpret and obtain the equation of a straight line graph.
• E3.5 Determine the equation of a straight line parallel to a given line.
• E3.6 Find the gradient of parallel and perpendicular lines.

E4 Geometry

• E4.1 Use and interpret the geometrical terms: point, line, parallel, bearing, right angle, acute, obtuse and reflex angles, perpendicular, similarity and congruence.
Use and interpret vocabulary of triangles, quadrilaterals, circles, polygons and simple solid figures including nets.
• E4.2 Measure and draw lines and angles.
Construct a triangle given the three sides using a ruler and a pair of compasses only.
• E4.3 Read and make scale drawings.
• E4.4 Calculate lengths of similar figures.
Use the relationships between areas of similar triangles, with corresponding results for similar figures and extension to volumes and surface areas of similar solids.
• E4.5 Use the basic congruence criteria for triangles (SSS, ASA, SAS, RHS).
• E4.6 Recognise rotational and line symmetry (including order of rotational symmetry) in two dimensions.
Recognise symmetry properties of the prism (including cylinder) and the pyramid (including cone).
Use the following symmetry properties of circles:
• equal chords are equidistant from the centre
• the perpendicular bisector of a chord passes through the centre
• tangents from an external point are equal in length.
• E4.7 Calculate unknown angles using the following geometrical properties:
• angles at a point
• angles at a point on a straight line and intersecting straight lines
• angles formed within parallel lines
• angle properties of triangles and quadrilaterals
• angle properties of regular polygons
• angle in a semicircle
• angle between tangent and radius of a circle
• angle properties of irregular polygons
• angle at the centre of a circle is twice the angle at the circumference
• angles in the same segment are equal
• angles in opposite segments are supplementary; cyclic quadrilaterals
• alternate segment theorem.

E5 Mensuration

• E5.1 Use current units of mass, length, area, volume and capacity in practical situations and express quantities in terms of larger or smaller units.
• E5.2 Carry out calculations involving the perimeter and area of a rectangle, triangle, parallelogram and trapezium and compound shapes derived from these.
• E5.3 Carry out calculations involving the circumference and area of a circle.
Solve problems involving the arc length and sector area as fractions of the circumference and area of a circle.
• E5.4 Carry out calculations involving the surface area and volume of a cuboid, prism and cylinder.
Carry out calculations involving the surface area and volume of a sphere, pyramid and cone.
• E5.5 Carry out calculations involving the areas and volumes of compound shapes.

E6 Trigonometry

• E6.1 Interpret and use three-figure bearings.
• E6.2 Apply Pythagoras’ theorem and the sine, cosine and tangent ratios for acute angles to the calculation of a side or of an angle of a right angled triangle.
Solve trigonometric problems in two dimensions involving angles of elevation and depression.
Know that the perpendicular distance from a point to a line is the shortest distance to the line.
• E6.3 Recognise, sketch and interpret graphs of simple trigonometric functions.
Graph and know the properties of trigonometric functions.
Solve simple trigonometric equations for values between 0° and 360°.
• E6.4 Solve problems using the sine and cosine rules
for any triangle and the formula area of triangle $$=\frac{1}{2}absin C$$
• E6.5 Solve simple trigonometrical problems in three dimensions including angle between a line and a plane

E7 Vectors and transformations

• E7.1Describe a translation by using a vector
represented by e.g. $$\binom{x}{y}$$ , $$\vec{AB}$$ or a.
Multiply a vector by a scalar.
• E7.2 Reflect simple plane figures.
Rotate simple plane figures through multiples of 90°.
Construct given translations and enlargements of simple plane figures.
Recognise and describe reflections, rotations, translations and enlargements.
• E7.3 Calculate the magnitude of a vector $$\binom{x}{y}$$ as $$\sqrt{x^2+y^2}$$
Represent vectors by directed line segments.
Use the sum and difference of two vectors to express given vectors in terms of two coplanar vectors.
Use position vectors.

E8 Probability

• E8.1 Calculate the probability of a single event as either a fraction, decimal or percentage.
• E8.2 Understand and use the probability scale from 0 to 1
• E8.3 Understand that the probability of an event occurring = 1 – the probability of the event not occurring.
• E8.4 Understand relative frequency as an estimate of probability.
Expected frequency of occurrences.
• E8.5 Calculate the probability of simple combined events, using possibility diagrams, tree diagrams and Venn diagrams.
• E8.6 Calculate conditional probability using Venn diagrams, tree diagrams and tables.

E9 Statistics

• E9.1 Collect, classify and tabulate statistical data.
• E9.2 Read, interpret and draw inferences from tables and statistical diagrams.
Compare sets of data using tables, graphs and statistical measures.
Appreciate restrictions on drawing conclusions from given data.
• E9.3 Construct and interpret bar charts, pie charts, pictograms, stem-and-leaf diagrams, simple frequency distributions, histograms with equal and unequal intervals and scatter diagrams.
• E9.4 Calculate the mean, median, mode and range for individual and discrete data and distinguish between the purposes for which they are used.
• E9.5 Calculate an estimate of the mean for grouped and continuous data.
Identify the modal class from a grouped frequency distribution.
• E9.6 Construct and use cumulative frequency diagrams.
Estimate and interpret the median, percentiles, quartiles and interquartile range.
Construct and interpret box-and-whisker plots
• E9.7 Understand what is meant by positive, negative and zero correlation with reference to a scatter diagram.
• E9.8 Draw, interpret and use lines of best fit by eye.