# IB DP Mathematical Studies : Past Years Question Bank with Solution Paper – 1

### Paper 1

Topic 1 – Number and algebra

Topic 2 – Descriptive statistics

Topic 3 – Logic, sets and probability

Topic 4 – Statistical applications

Topic 5 – Geometry and trigonometry

• Topic 5.0
• Basic geometric concepts: point, line, plane, angle
• Simple two-dimensional 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
• Topic 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, $${m_1} \times {m_2} = – 1$$ .
• Topic 5.2
• Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
• Angles of elevation and depression.
• Topic 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.
• Topic 5.4
• Geometry of three-dimensional 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.
• Topic 5.5
• Volume and surface areas of the three-dimensional solids defined in 5.4.

Topic 6 – Mathematical models

• Topic 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.
• Topic 6.2
• Linear models.
• Linear functions and their graphs, $$f\left( x \right) = mx + c$$
• Topic 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, $$x = \ – \frac{b}{{2a}}$$.
• Topic 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.
• Topic 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.
• Topic 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.
• Topic 6.7
• Use of a GDC to solve equations involving combinations of the functions above.

Topic 7 – Introduction to differential calculus

• Topic 7.1
• Concept of the derivative as a rate of change.
• Tangent to a curve.
• Topic 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.
• Topic 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).
• Topic 7.4
• Increasing and decreasing functions.
• Graphical interpretation of $$f’\left( x \right) > 0$$, $$f’\left( x \right) = 0$$ and $$f’\left( x \right) < 0$$.
• Topic 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.
• Topic 7.6
• Optimization problems.