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DP Mathematical Studies Questionbank
Syllabus
Section name
Topic 1  Number and algebra
1.0
Basic use of the four operations of arithmetic, using integers, decimals and fractions, including order of operations
Prime numbers, factors and multiples.
Simple applications of ratio, percentage and proportion.
Basic manipulation of simple algebraic expressions, including factorization and expansion
Rearranging formulae
Evaluating expressions by substitution.
Solving linear equations in one variable.
Solving systems of linear equations in two variables.
Evaluating exponential expressions with integer values
Use of inequalities \( < \), \( \leqslant \), \( > \), \( \geqslant \). Intervals on the real number line
Intervals on the real number line
Solving linear inequalities.
Familiarity with commonly accepted world currencies
1.1
Natural numbers, \(\mathbb{N}\) ; integers, \(\mathbb{Z}\) ; rational numbers, \(\mathbb{Q}\) ; and real numbers, \(\mathbb{R}\) .
1.2
Approximation: decimal places, significant figures.
Percentage errors.
Estimation.
1.3
Expressing numbers in the form \(a \times {10^k}\) , where \(1 \le a < 10\) and \(k\) is an integer.
Operations with numbers in this form.
1.4
SI (Système International) and other basic units of measurement: for example, kilogram (\({\text{kg}}\)), metre (\({\text{m}}\)), second (\({\text{s}}\)), litre (\({\text{l}}\)), metre per second (\({\text{m}}{{\text{s}}^{  1}}\)), Celsius scale.
1.5
Currency conversions.
1.6
Use of a GDC to solve pairs of linear equations in two variables.
Use of a GDC to solve quadratic equations.
1.7
Arithmetic sequences and series, and their applications.
Use of the formulae for the \(n\)th term and the sum of the first \(n\) terms of the sequence.
1.8
Geometric sequences and series.
Use of the formulae for the \(n\)th term and the sum of the first \(n\) terms of the sequence.
1.9
Financial applications of geometric sequences and series: compound interest.
Financial applications of geometric sequences and series: annual depreciation.
Topic 2  Descriptive statistics
2.0
The collection of data and its representation in bar charts, pie charts and pictograms.
2.1
Classification of data as discrete or continuous.
2.2
Simple discrete data: frequency tables.
2.3
Grouped discrete or continuous data: frequency tables; midinterval values; upper and lower boundaries.
Frequency histograms.
2.4
Cumulative frequency tables for grouped discrete data and for grouped continuous data; cumulative frequency curves, median and quartiles
Boxandwhisker diagram.
2.5
Measures of central tendency.
For simple discrete data: mean; median; mode.
For grouped discrete and continuous data: estimate of a mean; modal class.
2.6
Measures of dispersion: range, interquartile range, standard deviation.
Topic 3  Logic, sets and probability
3.1
Basic concepts of symbolic logic: definition of a proposition; symbolic notation of propositions.
3.2
Compound statements: implication, \( \Rightarrow \) ; equivalence, \( \Leftrightarrow \) ; negation, \(\neg \) ; conjunction, \( \wedge \) ; disjunction, \( \vee \) ; exclusive disjunction, \(\underline \vee \) .
Translation between verbal statements and symbolic form.
3.3
Truth tables: concepts of logical contradiction and tautology.
3.4
Converse, inverse, contrapositive.
Logical equivalence.
Testing the validity of simple arguments through the use of truth tables.
3.5
Basic concepts of set theory: elements \(x \in A\), subsets \(A \subset B\); intersection \(A\mathop \cap \nolimits B\); union \(A\mathop \cup \nolimits B\); complement \({A'}\) .
Venn diagrams and simple applications.
3.6
Sample space; event \(A\); complementary event, \({A'}\) .
Probability of an event.
Probability of a complementary event.
Expected value.
3.7
Probability of combined events, mutually exclusive events, independent events.
Use of tree diagrams, Venn diagrams, sample space diagrams and tables of outcomes.
Probability using “with replacement” and “without replacement”.
Conditional probability.
Topic 4  Statistical applications
4.1
The normal distribution.
The concept of a random variable; of the parameters \(\mu \) and \(\sigma \) ; of the bell shape; the symmetry about \(x = \mu \) .
Diagrammatic representation.
Normal probability calculations.
Expected value.
Inverse normal calculations.
4.2
Bivariate data: the concept of correlation.
Scatter diagrams; line of best fit, by eye, passing through the mean point.
Pearson’s product–moment correlation coefficient, \(r \).
Interpretation of positive, zero and negative, strong or weak correlations.
4.3
The regression line for \(y \) on \(x \).
Use of the regression line for prediction purposes.
4.4
The \({\chi ^2}\) test for independence: formulation of null and alternative hypotheses; significance levels; contingency tables; expected frequencies; degrees of freedom; \(p\)values.
Topic 5  Geometry and trigonometry
5.0
Basic geometric concepts: point, line, plane, angle
Simple twodimensional shapes and their properties, including perimeters and areas of circles, triangles, quadrilaterals and compound shapes.
SI units for length and area.
Pythagoras’ theorem
Coordinates in two dimensions.
Midpoints, distance between points
5.1
Equation of a line in two dimensions: the forms \(y = mx + c\) and \(ax + by + d = 0\) .
Gradient; intercepts.
Points of intersection of lines.
Lines with gradients, \({m_1}\) and \({m_2}\) .
Parallel lines \({m_1} = {m_2}\).
Perpendicular lines, \({m_1} \times {m_2} =  1\) .
5.2
Use of sine, cosine and tangent ratios to find the sides and angles of rightangled triangles.
Angles of elevation and depression.
5.3
Use of the sine rule: \(\frac{a}{{\sin A}} = \frac{b}{{\sin B}} = \frac{c}{{\sin C}}\).
Use of the cosine rule: \({a^2} = {b^2} + {c^2}  2bc\cos A\) ; \(\cos A = \frac{{{b^2} + {c^2}  {a^2}}}{{2bc}}\).
Use of the area of a triangle \( = \frac{1}{2}ab\sin C\).
Construction of labelled diagrams from verbal statements.
5.4
Geometry of threedimensional solids: cuboid; right prism; right pyramid; right cone; cylinder; sphere; hemisphere; and combinations of these solids.
The distance between two points; eg between two vertices or vertices with midpoints or midpoints with midpoints.
The size of an angle between two lines or between a line and a plane.
5.5
Volume and surface areas of the threedimensional solids defined in 5.4.
Topic 6  Mathematical models
6.1
Concept of a function, domain, range and graph.
Function notation, eg \(f\left( x \right)\), \(v\left( t \right)\), \(C\left( n \right)\) .
Concept of a function as a mathematical model.
6.2
Linear models.
Linear functions and their graphs, \(f\left( x \right) = mx + c\) .
6.3
Quadratic models.
Quadratic functions and their graphs (parabolas): \(f\left( x \right) = a{x^2} + bx + c\) ; \(a \ne 0\)
Properties of a parabola: symmetry; vertex; intercepts on the \(x\)axis and \(y\)axis.
Equation of the axis of symmetry, \(x = \  \frac{b}{{2a}}\).
6.4
Exponential models.
Exponential functions and their graphs: \(f\left( x \right) = k{a^x} + c\); \(a \in {\mathbb{Q}^ + }\), \(a \ne 1\), \(k \ne 0\) .
Exponential functions and their graphs: \(f\left( x \right) = k{a^{  x}} + c\); \(a \in {\mathbb{Q}^ + }\), \(a \ne 1\), \(k \ne 0\) .
Concept and equation of a horizontal asymptote.
6.5
Models using functions of the form \(f\left( x \right) = a{x^m} + b{x^n} + \ldots \); \(m,n \in \mathbb{Z}\) .
Functions of this type and their graphs.
The \(y\)axis as a vertical asymptote.
6.6
Drawing accurate graphs.
Creating a sketch from information given.
Transferring a graph from GDC to paper.
Reading, interpreting and making predictions using graphs.
Included all the functions above and additions and subtractions.
6.7
Use of a GDC to solve equations involving combinations of the functions above.
Topic 7  Introduction to differential calculus
7.1
Concept of the derivative as a rate of change.
Tangent to a curve.
7.2
The principle that \(f\left( x \right) = a{x^n} \Rightarrow f'\left( x \right) = an{x^{n  1}}\) .
The derivative of functions of the form \(f\left( x \right) = a{x^n} + b{x^{n  1}} + \ldots \), where all exponents are integers.
7.3
Gradients of curves for given values of \(x\).
Values of \(x\) where \(f'\left( x \right)\) is given.
Equation of the tangent at a given point.
Equation of the line perpendicular to the tangent at a given point (normal).
7.4
Increasing and decreasing functions.
Graphical interpretation of \(f'\left( x \right) > 0\), \(f'\left( x \right) = 0\) and \(f'\left( x \right) < 0\).
7.5
Values of x where the gradient of a curve is zero.
Solution of \(f'\left( x \right) = 0\).
Stationary points.
Local maximum and minimum points.
7.6
Optimization problems.