Further Pure Mathematics 2

Board Coverage

Board	Paper	Notes
AQA	Paper 2	Groups, further complex, Maclaurin series, further DEs
Edexcel	FP2	Series, further complex, 1st/2nd order DEs, conics
OCR (A)	Pure Core 2	Groups, Maclaurin, polar area, vectors, conics
CIE (9709)	Paper 3	Complex, vectors, further calculus, conics

:::info This is advanced content in most cases appearing in the second year of Further Mathematics. Ensure full mastery of Further Pure 1 before proceeding.

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1. Further Complex Numbers

1.1 Euler’s Relation

Euler’s relation connects the exponential and trigonometric functions:

$e^{i\theta} = \cos\theta + i\sin\theta$

From which:

$\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \qquad \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$

Euler’s identity: $e^{i\pi} + 1 = 0$ (connects $e$, $i$, $\pi$, $1$, and $0$).

1.2 Exponential Form

Any non-zero complex number can be written as:

$z = re^{i\theta}$

where $r = |z|$ and $\theta = \arg(z)$.

This form is particularly convenient for:

• Raising to powers: $z^n = r^n e^{in\theta}$
• Taking roots: $z^{1/n} = r^{1/n} e^{i(\theta + 2k\pi)/n}$
• Multiplication and division (rules are the same as in modulus-argument form)

1.3 Solving Equations with Complex Numbers

Linear equations: straightforward algebraic manipulation.

Quadratic with complex coefficients: the quadratic formula still applies.

$az^2 + bz + c = 0 \implies z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Even when $a, b, c$ are complex, the formula gives both roots.

Example. Solve $z^2 = i$. In exponential form: $z = e^{i(\pi/2 + 2k\pi)/2} = e^{i\pi/4}$ or $e^{i5\pi/4}$, so $z = \frac{1}{\sqrt{2}}(1 + i)$ or $\frac{1}{\sqrt{2}}(-1 - i)$.

1.4 Geometry in the Complex Plane

Rotation about a point $z_0$ by angle $\alpha$: multiply by $e^{i\alpha}$ and add $z_0$.

$w - z_0 = e^{i\alpha}(z - z_0)$

Reflection in a line: express the line as $\arg(z - a) = \theta$ and use the formula:

$w = a + e^{2i\theta}(\bar{z} - \bar{a})$

Distance between two points: $|z_1 - z_2|$.

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2. Groups

2.1 Axioms

A group $(G, *)$ is a set $G$ with a binary operation $*$ satisfying:

1. Closure: for all $a, b \in G$, $a * b \in G$
2. Associativity: for all $a, b, c \in G$, $(a * b) * c = a * (b * c)$
3. Identity: there exists $e \in G$ such that $a * e = e * a = a$ for all $a$
4. Inverse: for each $a \in G$, there exists $a^{-1} \in G$ such that $a * a^{-1} = a^{-1} * a = e$

An abelian group also satisfies commutativity: $a * b = b * a$ for all $a, b$.

2.2 Subgroups

$(H, *)$ is a subgroup of $(G, *)$ if $H \subseteq G$ and $(H, *)$ is itself a group.

Subgroup test: $H$ is a subgroup of $G$ if and only if for all $a, b \in H$: $a * b^{-1} \in H$.

Lagrange’s theorem: For a finite group, the order of any subgroup divides the order of the group. $|H|$ divides $|G|$.

2.3 Cyclic Groups

A group is cyclic if every element is a power of a single element (the generator).

If $G = \langle g \rangle$, then $G = \{e, g, g^2, g^3, \ldots\}$.

Example. $(\mathbb{Z}_6, +)$ is cyclic: $\langle 1 \rangle = \{0, 1, 2, 3, 4, 5\}$.

Subgroups of a cyclic group: If $G = \langle g \rangle$ with $|G| = n$, then for each divisor $d$ of $n$, there is exactly one subgroup of order $d$, generated by $g^{n/d}$.

2.4 Isomorphism

An isomorphism $\phi: G \to H$ is a bijection that preserves the group operation:

$\phi(a * b) = \phi(a) \circ \phi(b)$

If an isomorphism exists between $G$ and $H$, they are isomorphic ($G \cong H$) and have the same group structure (same Cayley table up to relabelling).

Isomorphic groups share: order, number of elements of each order, abelian or non-abelian, subgroup structure, number of subgroups of each order.

2.5 Group Homomorphisms

A homomorphism $\phi: G \to H$ is a function preserving the operation (but not necessarily bijective):

$\phi(a * b) = \phi(a) \circ \phi(b)$

Kernel: $\ker(\phi) = \{g \in G : \phi(g) = e_H\}$. The kernel is always a subgroup of $G$.

Image: $\text{im}(\phi) = \{\phi(g) : g \in G\}$. The image is always a subgroup of $H$.

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3. Further Calculus

3.1 Maclaurin Series

The Maclaurin series of $f(x)$ is its Taylor expansion about $x = 0$:

$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \cdots = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

Standard Maclaurin series:

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad \text{for } |x| < 1$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots \quad \text{(general binomial)}$

3.2 Taylor Series

Expanding about a general point $x = a$:

$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$

Useful technique: let $u = x - a$, then expand in powers of $u$.

3.3 Differential Equations: Series Solutions

For equations that cannot be solved by standard methods, assume a power series solution:

$y = \sum_{n=0}^{\infty} a_n x^n$

Substitute into the differential equation, equate coefficients of each power of $x$, and solve the resulting recurrence relation for $a_n$.

Example. For $y' = y$ with $y(0) = 1$: substituting $y = a_0 + a_1 x + a_2 x^2 + \cdots$ gives $a_1 = a_0$, $2a_2 = a_1$, $3a_3 = a_2$, and so $a_n = a_0/n!$, yielding $y = \sum x^n/n! = e^x$.

3.4 Reduction Formulae

A reduction formula expresses an integral $I_n$ (depending on $n$) in terms of $I_{n-1}$ or $I_{n-2}$.

Method: Use integration by parts, choosing $u$ and $dv$ so that the resulting integral involves $n - 1$ or $n - 2$.

Example. For $I_n = \int_0^{\pi/2} \sin^n x\,dx$:

$I_n = \frac{n-1}{n}I_{n-2}$

with $I_0 = \pi/2$ and $I_1 = 1$.

Another common example: $I_n = \int x^n e^{ax}\,dx$:

$I_n = \frac{1}{a}x^n e^{ax} - \frac{n}{a}I_{n-1}$

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4. Vectors and 3D Geometry

4.1 Scalar Product

The scalar (dot) product of vectors $\mathbf{a}$ and $\mathbf{b}$:

$\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\cos\theta = a_1 b_1 + a_2 b_2 + a_3 b_3$

where $\theta$ is the angle between them.

Perpendicular vectors: $\mathbf{a} \cdot \mathbf{b} = 0$.

Angle between vectors: $\cos\theta = \dfrac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}$.

4.2 Vector Product

The vector (cross) product in 3D:

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

$|\mathbf{a} \times \mathbf{b}| = |\mathbf{a}|\,|\mathbf{b}|\sin\theta$

Properties:

• $\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$
• $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$ (anti-commutative)
• $\mathbf{a} \times \mathbf{a} = \mathbf{0}$
• $\mathbf{a} \times \mathbf{b}$ gives the area of the parallelogram formed by $\mathbf{a}$ and $\mathbf{b}$
• $|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|$ = volume of the parallelepiped

4.3 Lines in 3D

Parametric form through point $\mathbf{a}$ with direction $\mathbf{d}$:

$\mathbf{r} = \mathbf{a} + t\mathbf{d} \qquad (t \in \mathbb{R})$

Cartesian (symmetric) form:

$\frac{x - a_1}{d_1} = \frac{y - a_2}{d_2} = \frac{z - a_3}{d_3} = t$

Two lines are parallel if their direction vectors are proportional.

Two lines intersect if there exists a parameter value $t_1 = t_2$ satisfying all three component equations simultaneously.

Angle between two lines: $\cos\theta = \dfrac{|\mathbf{d}_1 \cdot \mathbf{d}_2|}{|\mathbf{d}_1|\,|\mathbf{d}_2|}$.

4.4 Planes

Equation of a plane: $\mathbf{r} \cdot \mathbf{n} = \mathbf{a} \cdot \mathbf{n}$, or in Cartesian form:

$n_1 x + n_2 y + n_3 z = d$

where $\mathbf{n} = (n_1, n_2, n_3)$ is the normal vector and $d = \mathbf{a} \cdot \mathbf{n}$ for a known point $\mathbf{a}$ on the plane.

Plane through three points: Find two direction vectors from the points, take their cross product for the normal, then substitute.

Angle between two planes: angle between their normal vectors.

Line of intersection: find two points satisfying both plane equations, then write the parametric line through them.

4.5 Shortest Distance

Point to line: $d = \dfrac{|(\mathbf{a} - \mathbf{p}) \times \mathbf{d}|}{|\mathbf{d}|}$

where $\mathbf{p}$ is a point on the line with direction $\mathbf{d}$, and $\mathbf{a}$ is the external point.

Point to plane: $d = \dfrac{|\mathbf{n} \cdot \mathbf{a} - d|}{|\mathbf{n}|}$

where $\mathbf{a}$ is the point and the plane is $\mathbf{r} \cdot \mathbf{n} = d$.

Skew lines (shortest distance between):

$d = \frac{|(\mathbf{a}_2 - \mathbf{a}_1) \cdot (\mathbf{d}_1 \times \mathbf{d}_2)|}{|\mathbf{d}_1 \times \mathbf{d}_2|}$

where $\mathbf{a}_1, \mathbf{a}_2$ are points on the respective lines and $\mathbf{d}_1, \mathbf{d}_2$ are direction vectors.

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5. Conics

5.1 Focus-Directrix Properties

A conic section is the locus of points $P$ such that:

$\frac{\text{distance from } P \text{ to focus } F}{\text{distance from } P \text{ to directrix}} = e \quad (\text{eccentricity})$

Conic	Eccentricity
Circle	$e = 0$
Ellipse	$0 < e < 1$
Parabola	$e = 1$
Hyperbola	$e > 1$

5.2 Parametric Equations

Ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b$):

$x = a\cos\theta, \quad y = b\sin\theta$

Focus: $(\pm ae, 0)$ where $e = \sqrt{1 - b^2/a^2}$.

Hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$:

$x = a\sec\theta, \quad y = b\tan\theta$

Focus: $(\pm ae, 0)$ where $e = \sqrt{1 + b^2/a^2}$. Asymptotes: $y = \pm \dfrac{b}{a}x$.

Parabola $y^2 = 4ax$:

$x = at^2, \quad y = 2at$

Focus: $(a, 0)$. Directrix: $x = -a$.

5.3 Cartesian Equations

Rectangular hyperbola: $xy = c^2$. Parametric form: $x = ct$, $y = c/t$. Asymptotes are the coordinate axes. Eccentricity $e = \sqrt{2}$.

Tangent to a parabola $y^2 = 4ax$ at $(at^2, 2at)$:

$ty = x + at^2$

Normal to a parabola $y^2 = 4ax$ at $(at^2, 2at)$:

$y = -tx + 2at + at^3$

Tangent to ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ at $(a\cos\theta, b\sin\theta)$:

$\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1$

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6. Common Mistakes

1. Incorrect branch of $\cosh^{-1}$. $\cosh^{-1} x$ is defined only for $x \geq 1$ and returns non-negative values. Do not forget the domain restriction.
2. Wrong sign in the hyperbolic identity. It is $\cosh^2 x - \sinh^2 x = 1$ (minus, not plus). A sign error here propagates through to derivatives and integrals.
3. Forgetting closure or identity when checking group axioms. All four axioms (closure, associativity, identity, inverses) must be verified; omission of any one is a common error.
4. Confusing scalar and vector products. $\mathbf{a} \cdot \mathbf{b}$ is a scalar; $\mathbf{a} \times \mathbf{b}$ is a vector. Know which to use: scalar for angles and projections, vector for areas and perpendicular directions.
5. Incorrect shortest distance formula. The formula for point-to-line distance uses the cross product (3D); the formula for point-to-plane distance uses the dot product with the normal. Do not mix them up.
6. Wrong parametric form for hyperbola. For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, use $x = a\sec\theta$, $y = b\tan\theta$. Do not use $\cosh$ and $\sinh$ unless the question specifies the rectangular hyperbola $xy = c^2$.
7. Series solution recurrence with wrong initial terms. When equating coefficients in series solutions, carefully determine $a_0$ and $a_1$ from initial conditions before deriving the recurrence for $a_n$.

Worked Examples

Example 1: Finding a Reduction Formula

Problem: Find a reduction formula for $I_n = \int_0^{\pi/2} \cos^n x\,dx$ where $n \geq 0$. Solution: Use integration by parts with $u = \cos^{n-1}x$ and $dv = \cos x\,dx$: $u = \cos^{n-1}x \implies du = -(n-1)\cos^{n-2}x\sin x\,dx, \quad v = \sin x$ $I_n = \left[\cos^{n-1}x\sin x\right]_0^{\pi/2} + (n-1)\int_0^{\pi/2}\cos^{n-2}x\sin^2 x\,dx$ The boundary term vanishes. Replace $\sin^2 x = 1 - \cos^2 x$: $I_n = (n-1)\int_0^{\pi/2}\cos^{n-2}x\,dx - (n-1)\int_0^{\pi/2}\cos^n x\,dx = (n-1)I_{n-2} - (n-1)I_n$ $nI_n = (n-1)I_{n-2} \implies I_n = \frac{n-1}{n}I_{n-2}$ With $I_0 = \pi/2$ and $I_1 = 1$.

Example 2: Proving a Group Isomorphism

Problem: Show that the group $(\{1, -1, i, -i\}, \times)$ is isomorphic to the cyclic group $(\mathbb{Z}_4, +)$. Solution: Define $\phi: \mathbb{Z}_4 \to \{1, -1, i, -i\}$ by $\phi(k) = i^k$. Check: $\phi$ is bijective and $\phi(a + b) = i^{a+b} = i^a \cdot i^b = \phi(a)\phi(b)$. The multiplication table of $\{1, -1, i, -i\}$ matches the addition table of $\mathbb{Z}_4$ under this mapping, confirming the isomorphism.

Common Pitfalls

• Omitting axioms when checking groups: All four axioms (closure, associativity, identity, inverses) must be verified. Associativity is often assumed but should be checked or stated.
• Wrong reduction formula boundary terms: Always evaluate $[uv]$ at the limits. If the boundary term is non-zero, it must be included in the formula.
• Confusing $\cosh^{-1}$ domain: $\cosh^{-1} x$ is defined only for $x \geq 1$. Applying it to values outside this domain produces an error.

Summary

Further Pure 2 extends FP1 material through further complex numbers (Euler’s relation, geometry in the complex plane), group theory (axioms, subgroups, cyclic groups, isomorphism, homomorphisms), further calculus (Maclaurin and Taylor series, series solutions, reduction formulae), vectors in 3D (scalar and vector products, lines, planes, shortest distances), and conics (focus-directrix properties, parametric and Cartesian equations, tangents and normals).

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