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AS Further Mathematics₇₃₆₆

AS Further Mathematics Specification Specification for first teaching in 2017

AS Further Mathematics Specification Specification for first teaching in 2017

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3.2 Compulsory content

3.2.1 A: Proof

	Content
A1	Construct proofs using mathematical induction; contexts include sums of series, divisibility, and powers of matrices.

3.2.2 B: Complex numbers

	Content
B1	Solve any quadratic equation with real coefficients; solve cubic or quartic equations with real coefficients (given sufficient information to deduce at least one root for cubics or at least one complex root or quadratic factor for quartics).

	Content
B2	Add, subtract, multiply and divide complex numbers in the form $x +$ $i y$ with $x$ and $y$ real; understand and use the terms ‘real part’ and ‘imaginary part’.

	Content
B3	Understand and use the complex conjugate; know that non-real roots of polynomial equations with real coefficients occur in conjugate pairs.

	Knowledge/skill
B4	Use and interpret Argand diagrams.

	Content
B5	Convert between the Cartesian form and the modulus-argument form of a complex number (knowledge of radians is assumed).

	Content
B6	Multiply and divide complex numbers in modulus-argument form (knowledge of radians and compound angle formulae is assumed).

	Content
B7	Construct and interpret simple loci in the Argand diagram such as $\|z - a\| > r$ and arg $(z - a) = θ$ (knowledge of radians is assumed).

3.2.3 C: Matrices

	Content
C1	Add, subtract and multiply conformable matrices; multiply a matrix by a scalar.

	Content
C2	Understand and use zero and identity matrices.

	Content
C3	Use matrices to represent linear transformations in 2D; successive transformations; single transformations in 3D (3D transformations confined to reflection in one of $x$ = 0, $y$ = 0, $z$ = 0 or rotation about one of the coordinate axes) (knowledge of 3D vectors is assumed).

	Content
C4	Find invariant points and lines for a linear transformation.

	Content
C5	Calculate determinants of 2 $\times$ 2 matrices.

	Content
C6	Understand and use singular and non-singular matrices; properties of inverse matrices. Calculate and use the inverse of non-singular 2 $\times$ 2 matrices.

3.2.4 D: Further algebra and functions

	Content
D1	Understand and use the relationship between roots and coefficients of polynomial equations up to quartic equations.

	Content
D2	Form a polynomial equation whose roots are a linear transformation of the roots of a given polynomial equation (of at least cubic degree).

	Content
D3	Understand and use formulae for the sums of integers, squares and cubes and use these to sum other series.

	Content
D4	Understand and use the method of differences for summation of series.

	Content
D6	Recognise and use the Maclaurin series for $e^{x}$ , $\ln (1 + x)$ , $\sin x$ , $\cos x$ , and ${(1 + x)}^{n}$ , and be aware of the range of values of $x$ for which they are valid (proof not required).

	Content
D8	Inequalities involving polynomial equations (cubic and quartic).

	Content
D9	Solving inequalities such as $\frac{a x + b}{c x + d} < e x + f$ algebraically.

	Content
D12	Graphs of rational functions of form $\frac{a x + b}{c x + d}$ ; asymptotes, points of intersection with coordinate axes or other straight lines; associated inequalities.

	Content
D13	Graphs of rational functions of form $\frac{a x^{2} + b x + c}{d x^{2} + e x + f}$ , including cases when some of these coefficients are zero; asymptotes parallel to coordinate axes.

	Content
D14	Using quadratic theory (not calculus) to find the possible values of the function and coordinates of the stationary points of the graph for rational functions of form $\frac{a x^{2} + b x + c}{d x^{2} + e x + f}$

	Content
D15	Sketching graphs of curves with equation $y^{2} = 4 a x$ , $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ , $x y = c^{2}$ including intercepts with axes and equations of asymptotes of hyperbolas.

	Content
D16	Single transformations of curves involving translations, stretches parallel to coordinate axes and reflections in the coordinate axes and the lines $y = \pm x$ .

3.2.5 E: Further calculus

	Content
E2	Derive formulae for and calculate volumes of revolution.

	Content
E3	Understand and evaluate the mean value of a function.

3.2.6 F: Further vectors

	Content
F1	Understand and use the vector and Cartesian forms of an equation of a straight line in 3D.

	Content
F3	Calculate the scalar product and use it to calculate the angle between two lines.

	Content
F4	Check whether vectors are perpendicular by using the scalar product.

	Content
F6	Find the intersection of two lines. Calculate the perpendicular distance between two lines and from a point to a line.

3.2.7 G: Polar coordinates

	Content
G1	Understand and use polar coordinates and be able to convert between polar and Cartesian coordinates.

	Content
G2	Sketch curves with $r$ given as a function of $θ$ , including use of trigonometric functions.

3.2.8 H: Hyperbolic functions

	Content
H1	Understand the definitions of hyperbolic functions sinh $x$ , cosh $x$ and tanh $x$ , and be able to sketch their graphs.

	Content
H3	Understand and be able to use the definitions of the inverse hyperbolic functions.

	Content
H4	Derive and use the logarithmic forms of the inverse hyperbolic functions.

	Content
H6	Understand and use $\tanh x \equiv \frac{\sinh x}{\cosh x}$ Understand and use $\cosh^{2} x - \sinh^{2} x \equiv 1$

3.1 Overarching themes

3.3 Optional application 1 – mechanics