IB DP Mathematical Studies : Past Years Question Bank with Solution Paper – 2

Paper 2

Topic 1 – Number and algebra

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.

Leave a Reply