Vectors (P4)

What the syllabus asks of you

• Vectors in two and three dimensions.
• Magnitude of a vector.
• Algebraic operations of vector addition and multiplication by scalars, and their geometrical interpretations.
• Position vectors.
• The distance between two points.
• Vector equations of lines.
• The scalar product. Its use for calculating the angle between two lines. Complex numbers; roots of quadratic equations; numerical solution of equations; coordinate systems; matrix algebra; transformations using matrices; series; proof. A knowledge of the specification for P1 and P2, their prerequisites and associated formulae, is assumed and may be tested. It is also necessary for students: • to have a knowledge of location of roots of f(x) = 0 by considering changes of sign of f(x) in an interval in which • f(x) is continuous • to have a knowledge of rotating shapes through any angle about (0, 0) • to be able to divide a cubic polynomial by a quadratic polynomial • to be able to divide a quartic polynomial by a quadratic polynomial. • First assessment: June 2019. • The assessment is 1 hour and 30 minutes. • The assessment is out of 75 marks. • Students must answer all questions. • Calculators may be used in the examination. Please see Appendix 6: Use of calculators. • The booklet Mathematical Formulae and Statistical Tables will be provided for use in the assessments. Students will be expected to understand the symbols outlined in Appendix 7: Notation. Formulae that students are expected to know are given below and will not appear in the booklet Mathematical Formulae and Statistical Tables, which will be provided for use with the paper. Questions will be set in SI units and other units in common usage. This is a list of formulae that students are expected to remember and which will not be included in formulae booklets. Roots of quadratic equations For ax2 + bx + c = 0: α + β = −b Series n =∑ = 1 2 n(n + 1) r r Definition of complex numbers in the form a + ib and r cos θ + i r sin θ. Sum, product and quotient of complex numbers. Geometrical representation of complex numbers in the Argand diagram. Geometrical representation of sums, products and quotients of complex numbers. Complex solutions of quadratic equations with real coefficients. Finding conjugate complex roots and a real root of a cubic equation with integer coefficients. Finding conjugate complex roots and/or real roots of a quartic equation with real coefficients. Sum of roots and product of roots of a quadratic equation. Manipulation of expressions involving the sum of roots and product of roots. Forming quadratic equations with new roots. Equations of the form f(x) = 0 solved numerically by: (i) interval bisection, (ii) linear interpolation, (iii) the Newton-Raphson process. Cartesian equations for the parabola and rectangular hyperbola. Idea of parametric equation for parabola and rectangular hyperbola. The focus-directrix property of the parabola. Tangents and normals to these curves. Addition and subtraction of matrices. Multiplication of a matrix by a scalar. Products of matrices. Evaluation of 2 × 2 determinants. Inverse of 2 × 2 matrices. Linear transformations of column vectors in two dimensions and their matrix representation. Applications of 2 × 2 matrices to represent geometrical transformations. Combinations of transformations. The inverse (when it exists) of a given transformation or combination of transformations.
• Summation of simple finite series.