A-level Further Mathematics₇₃₆₇

Specification7367

7367

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A-level Further Mathematics Specification for first teaching in 2017

20 Oct 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).

	Knowledge/skill
B8	Understand de Moivre’s theorem and use it to find multiple angle formulae and sums of series.

	Content
B9	Know and use the definition $e^{i θ} = \cos θ + i \sin θ$ and the form $z = {r e}^{i θ}$

	Content
B10	Find the $n$ distinct $n^{}$ th roots of ${r e}^{i θ}$ for $r \neq 0$ and know that they form the vertices of a regular $n$ -gon in the Argand diagram.

	Content
B11	Use complex roots of unity to solve geometric problems.

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 and 3 $\times$ 3 matrices and interpret as scale factors, including the effect on orientation.

	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 and 3 $\times$ 3 matrices.

	Content
C7	Solve three linear simultaneous equations in three variables by use of the inverse matrix.

	Content
C8	Interpret geometrically the solution and failure of solution of three simultaneous linear equations.

	Content
C9	Factorisation of determinants using row and column operations.

	Content
C10	Find eigenvalues and eigenvectors of 2 $\times$ 2 and 3 $\times$ 3 matrices. Find and use the characteristic equation. Understand the geometrical significance of eigenvalues and eigenvectors.

	Content
C11	Diagonalisation of matrices; M = UDU ^-1 ; M $^{n}$ = UD $^{n}$ U ^-1 ; when eigenvalues are real.

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 including use of partial fractions.

	Content
D5	Find the Maclaurin series of a function including the general term.

	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
D7	Evaluation of limits using Maclaurin series or l'Hôpital's rule.

	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
D10	Modulus of functions and associated inequalities.

	Content
D11	Graphs of $y = \|f (x)\|$ , $y = \frac{1}{f (x)}$ for given $y = f (x)$

	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; oblique asymptotes.

	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 equations $y^{2} = 4 a x$ , $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ , $x y = c^{2}$ 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 $y = \pm x$ . Extend to composite transformations including rotations and enlargements.

3.2.5 E: Further calculus

	Content
E1	Evaluate improper integrals where either the integrand is undefined at a value in the range of integration or the range of integration extends to infinity.

	Content
E2	Derive formulae for and calculate volumes of revolution.

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

	Content
E4	Integrate using partial fractions (extend to quadratic factors $a x^{2} + c$ in the denominator).

	Content
E5	Differentiate inverse trigonometric functions.

	Content
E6	Integrate functions of the form ${(a^{2} - x^{2})}^{- \frac{1}{2}}$ and ${(a^{2} + x^{2})}^{- 1}$ and be able to choose trigonometric substitutions to integrate associated functions.

	Content
E7	Arc length and area of surface of revolution for curves expressed in Cartesian or parametric coordinates.

	Content
E8	Derivation and use of reduction formulae for integration.

	Content
E9	The limits $\lim_{x \to \infty} (x^{k} e^{- x})$ and $\lim_{x \to 0} (x^{k} \ln x)$ where $k > 0$ , applied to improper integrals

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
F2	Understand and use the vector and Cartesian forms of the equation of a plane.

	Content
F3	Calculate the scalar product and use it to calculate the angle between two lines, to express the equation of a plane, and to calculate the angle between two planes and the angle between a line and a plane.

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

	Content
F5	Calculate and understand the properties of the vector product. Understand and use the equation of a straight line in the form ( r – a ) × b = 0. Use vector products to find the area of a triangle.

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

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.

	Content
G3	Find the area enclosed by a polar curve.

3.2.8 H: Hyperbolic functions

	Content
H1	Understand the definitions of hyperbolic functions sinh $x$ , cosh $x$ and tanh $x$ , including their domains and ranges, and be able to sketch their graphs. Understand the definitions of hyperbolic functions sech $x$ , cosech $x$ and coth $x$ , including their domains and ranges.

	Content
H2	Differentiate and integrate hyperbolic functions.

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

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

	Content
H5	Integrate functions of the form ${(x^{2} + a^{2})}^{- \frac{1}{2}}$ and ${(x^{2} - a^{2})}^{- \frac{1}{2}}$ and be able to choose substitutions to integrate associated functions.

	Content
H6	Understand and use $\tanh x \equiv \frac{\sinh x}{\cosh x}$ Understand and use $\cosh^{2} x - \sinh^{2} x \equiv 1$ ; ${sech}^{2} x \equiv 1 - \tanh^{2} x$ and ${c o s e c h}^{2} x \equiv \coth^{2} x - 1$ , $\cosh 2 x \equiv \cosh^{2} x + \sinh^{2} x$ , $\sinh 2 x \equiv 2 \sinh x \cosh x$

	Content
H7	Construct proofs involving hyperbolic functions and identities.

3.2.9 I: Differential equations

	Content
I1	Find and use an integrating factor to solve differential equations of the form $\frac{d y}{d x} + P (x) y = Q (x)$ and recognise when it is appropriate to do so.

	Content
I2	Find both general and particular solutions of differential equations.

	Content
I3	Use differential equations in modelling in kinematics and in other contexts.

	Content
I4	Solve differential equations of the form $y " + a y' + b y = 0$ where $a$ and $b$ are constants, by using the auxiliary equation.

	Content
I5	Solve differential equations of the form $y " + a y' + b y = f (x)$ where $a$ and $b$ are constants by solving the homogeneous case and adding a particular integral to the complementary function (in cases where $f (x)$ is a polynomial, exponential or trigonometric function).

	Content
I6	Understand and use the relationship between the cases when the discriminant of the auxiliary equation is positive, zero and negative and the form of solution of the differential equation.

	Content
I7	Solve the equation for simple harmonic motion $\ddot{x} = {- ω}^{2} x$ and relate the solution to the motion.

	Content
I8	Model damped oscillations using 2nd order differential equations and interpret their solutions. Understand light, critical and heavy damping and be able to determine when each will occur.

	Content
I9	Analyse and interpret models of situations with one independent variable and two dependent variables as a pair of coupled 1st order simultaneous equations and be able to solve them, for example predator-prey models.

	Content
I10	Use of Hooke’s Law with $T = k x$ to formulate a differential equation for simple harmonic motion, where $k$ is a constant.

	Content
I11	Use models for damped motion where the damping force is proportional to the velocity.

3.2.10 J: Numerical methods

	Content
J1	Mid-ordinate rule and Simpson’s rule for integration.

	Content
J2	Euler’s step by step method for solving first order differential equations.

	Content
J3	Improved Euler method for solving first order differential equations. $y_{r + 1} = y_{r - 1} + 2 h f (x_{r} {, y}_{r}), x_{r + 1} = x_{r} + h$

3.1 Overarching themes

3.3 Optional application 1 – mechanics