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9231 Syllabus Content 2016 | Matrix (Mathematics) | Eigenvalues And Eigenvectors

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Further Mathematics syllabus for Cambridge A Levels (2016)
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  Syllabus content 10Cambridge International A Level Further Mathematics 9231. Syllabus for examination in 2016. 5. Syllabus content 5.1 Paper 1 Knowledge of the syllabus for Pure Mathematics (units P1 and P3) in Mathematics 9709 is assumed, and candidates may need to apply such knowledge in answering questions. Theme or topicCurriculum objectives1. Polynomials and rational functions Candidates should be able to:  ã recall and use the relations between the roots and coefficients of polynomial equations, for equations of degree 2, 3, 4 only;ã use a given simple substitution to obtain an equation whose roots are related in a simple way to those of the srcinal equation;ã sketch graphs of simple rational functions, including the determination of oblique asymptotes, in cases where the degree of the numerator and the denominator are at most 2 (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as turning points, asymptotes and intersections with the axes). 2. Polar coordinates ã understand the relations between cartesian and polar coordinates (using the convention r   0), and convert equations of curves from cartesian to polar form and vice versa ;ã sketch simple polar curves, for 0 θ    <  2 π    or − π    <   θ    π    or a subset of either of these intervals (detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as symmetry, the form of the curve at the pole and least/greatest values of r );ã recall the formula 21    β α  ∫  r    2   d θ    for the area of a sector, and use this formula in simple cases.  Syllabus content 11Cambridge International A Level Further Mathematics 9231. Syllabus for examination in 2016. 3. Summation of series ã use the standard results for ∑ r  , ∑ r    2 ,   ∑ r    3   to find related sums;ã use the method of differences to obtain the sum of a finite series, e.g. by expressing the general term in partial fractions;ã recognise, by direct consideration of a sum to n  terms, when a series is convergent, and find the sum to infinity in such cases. 4. Mathematical induction ã use the method of mathematical induction to establish a given result (questions set may involve divisibility tests and inequalities, for example);ã recognise situations where conjecture based on a limited trial followed by inductive proof is a useful strategy, and carry this out in simple cases, e.g. find the n th derivative of x  e x  . 5. Differentiation and integration ã obtain an expression for d 2 y  d x    2  in cases where the relation between y   and x   is defined implicitly or parametrically;ã derive and use reduction formulae for the evaluation of definite integrals in simple cases;ã use integration to find:  ○ mean values and centroids of two- and three-dimensional figures (where equations are expressed in cartesian coordinates, including the use of a parameter), using strips, discs or shells as appropriate,  ○ arc lengths (for curves with equations in cartesian coordinates, including the use of a parameter, or in polar coordinates),  ○ surface areas of revolution about one of the axes (for curves with equations in cartesian coordinates, including the use of a parameter, but not for curves with equations in polar coordinates).  Syllabus content 12Cambridge International A Level Further Mathematics 9231. Syllabus for examination in 2016. 6. Differential equations ã recall the meaning of the terms ‘complementary function’ and ‘particular integral’ in the context of linear differential equations, and recall that the general solution is the sum of the complementary function and a particular integral;ã find the complementary function for a second order linear differential equation with constant coefficients;ã recall the form of, and find, a particular integral for a second order linear differential equation in the cases where a polynomial or e bx   or a  cos px    +  b   sin px   is a suitable form, and in other simple cases find the appropriate coefficient(s) given a suitable form of particular integral;ã use a substitution to reduce a given differential equation to a second order linear equation with constant coefficients;ã use initial conditions to find a particular solution to a differential equation, and interpret a solution in terms of a problem modelled by a differential equation. 7. Complex numbers ã understand de Moivre’s theorem, for a positive integral exponent, in terms of the geometrical effect of multiplication of complex numbers;ã prove de Moivre’s theorem for a positive integral exponent;ã use de Moivre’s theorem for positive integral exponent to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle;ã use de Moivre’s theorem, for a positive or negative rational exponent:  ○ in expressing powers of sin θ    and cos θ    in terms of multiple angles,  ○ in the summation of series,  ○ in finding and using the n th roots of unity.  Syllabus content 13Cambridge International A Level Further Mathematics 9231. Syllabus for examination in 2016. 8. Vectors ã use the equation of a plane in any of the forms ax    +  by +   cz =   d   or r.n   =   p   or  r  =  a   +    λ b   +   µ  c , and convert equations of planes from one form to another as necessary in solving problems;ã recall that the vector product a   ×   b  of two vectors can be expressed either as |a| |b|  sin θ    n ˆ , where n ˆ is a unit vector, or in component form as ( a 2 b  3  – a 3 b  2 ) i   +  ( a 3 b  1  – a 1 b  3 )  j   +  ( a 1 b  2  – a 2 b  1 ) k ;ã use equations of lines and planes, together with scalar and vector products where appropriate, to solve problems concerning distances, angles and intersections, including:  ○ determining whether a line lies in a plane, is parallel to a plane or intersects a plane, and finding the point of intersection of a line and a plane when it exists,  ○ finding the perpendicular distance from a point to a plane,  ○ finding the angle between a line and a plane, and the angle between two planes,  ○ finding an equation for the line of intersection of two planes,  ○ calculating the shortest distance between two skew lines,  ○ finding an equation for the common perpendicular to two skew lines. 9. Matrices and linear spaces ã recall and use the axioms of a linear (vector) space (restricted to spaces of finite dimension over the field of real numbers only);ã understand the idea of linear independence, and determine whether a given set of vectors is dependent or independent;ã understand the idea of the subspace spanned by a given set of vectors;ã recall that a basis for a space is a linearly independent set of vectors that spans the space, and determine a basis in simple cases;ã recall that the dimension of a space is the number of vectors in a basis;ã understand the use of matrices to represent linear transformations from o n   →   o m .
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