A-level Further Mathematics₇₃₆₇

Know and use the definition ${\mathrm{e}}^{\mathrm{i}\mathit{\theta }}\mathsf{=}\mathsf{cos}\mathit{\theta }\mathsf{+}\mathrm{i}\mathsf{sin}\mathit{\theta }$ and the form $\mathit{z}\mathsf{=}{\mathit{r}\mathrm{e}}^{\mathrm{i}\mathit{\theta }}$

Find the $\mathit{n}$ distinct ${\mathit{n}}^{}$ th roots of ${\mathit{r}\mathrm{e}}^{\mathrm{i}\mathit{\theta }}$ for $\mathit{r}\mathsf{\ne }\mathsf{0}$ and know that they form the vertices of a regular $\mathit{n}$ -gon in the Argand diagram.

Understand and use singular and non-singular matrices; properties of inverse matrices.

Calculate and use the inverse of non-singular 2 $\mathsf{\times }$ 2 matrices and 3 $\mathsf{\times }$ 3 matrices.

Find eigenvalues and eigenvectors of 2 $\mathsf{\times }$ 2 and 3 $\mathsf{\times }$ 3 matrices.

Find and use the characteristic equation.

Understand the geometrical significance of eigenvalues and eigenvectors.

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

Graphs of $\mathit{y}\mathsf{=}\left|\mathrm{f}\left(\mathit{x}\right)\right|$ , $\mathit{y}\mathsf{=}\frac{\mathsf{1}}{\mathrm{f}\left(\mathit{x}\right)}$ for given $\mathit{y}\mathsf{=}\mathrm{f}\left(\mathit{x}\right)$

Single transformations of curves involving translations, stretches parallel to coordinate axes and reflections in the coordinate axes and the lines $\mathit{y}\mathsf{=}\mathsf{\pm }\mathit{x}$ . Extend to composite transformations including rotations and enlargements.

The limits $\underset{\mathit{x}\mathsf{\to }\mathit{\infty }}{\mathsf{lim}}\mathsf{}\mathsf{\left(}{\mathit{x}}^{\mathit{k}}{\mathrm{e}}^{\mathsf{-}\mathit{x}}\mathsf{\right)}$ and $\underset{\mathit{x}\mathsf{\to }\mathsf{0}}{\mathsf{lim}}\mathsf{}\mathsf{\left(}{\mathit{x}}^{\mathit{k}}\mathsf{ln}\mathsf{}\mathit{x}\mathsf{\right)}$ where $\mathit{k}\mathsf{>}\mathsf{0}$ , applied to improper integrals

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.

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.

Understand the definitions of hyperbolic functions sinh $\mathit{x}$ , cosh $\mathit{x}$ and tanh $\mathit{x}$ , including their domains and ranges, and be able to sketch their graphs.

Understand the definitions of hyperbolic functions sech $\mathit{x}$ , cosech $\mathit{x}$ and coth $\mathit{x}$ , including their domains and ranges.

Understand and use $\mathsf{tanh}\mathsf{}\mathit{x}\mathsf{\equiv }\frac{\mathsf{sinh}\mathsf{}\mathit{x}}{\mathsf{cosh}\mathsf{}\mathit{x}}$

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

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

Solve differential equations of the form $\mathit{y}\mathsf{"}\mathsf{+}\mathit{a}\mathit{y}\mathsf{\text{'}}\mathsf{+}\mathit{b}\mathit{y}\mathsf{=}\mathrm{f}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ where $\mathit{a}$ and $\mathit{b}$ are constants by solving the homogeneous case and adding a particular integral to the complementary function (in cases where $\mathrm{f}\mathsf{\left(}\mathit{x}\mathsf{\right)}$ is a polynomial, exponential or trigonometric function).

Solve the equation for simple harmonic motion $\ddot{\mathit{x}}\mathsf{=}{\mathsf{-}\mathit{\omega }}^{\mathsf{2}}\mathit{x}$ and relate the solution to the motion.

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.

Use of Hooke’s Law with $\mathit{T}\mathsf{=}\mathit{k}\mathit{x}$ to formulate a differential equation for simple harmonic motion, where $\mathit{k}$ is a constant.