Further Pure 2 — A-Level Further Mathematics Study Guide

For: A-Level Further Mathematics candidates sitting Further Mathematics.

Covers: All core Further Pure 2 topics: hyperbolic functions and their inverses, differentiation and integration of inverse trigonometric and hyperbolic functions, first and second-order differential equations, Maclaurin and Taylor series, and De Moivre's theorem with nth roots of unity.

You should already know: Strong A-Level Mathematics Pure Mathematics 1, 2, and 3 foundation.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the A-Level Further Mathematics style for educational use. They are not reproductions of past Cambridge International examination papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official Cambridge mark schemes for grading conventions.

1. What Is Further Pure 2?

Further Pure 2 (FP2) is the second advanced pure mathematics unit for A-Level Further Mathematics, extending core pure concepts from the A-Level Mathematics syllabus to solve complex mathematical problems applicable in engineering, physics, and quantitative finance. It builds on algebraic manipulation, calculus, and complex number fundamentals, with questions typically worth 10 to 15 marks each on the FP2 exam paper, accounting for 25% of your total Further Mathematics grade. Examiners frequently combine multiple subtopics in a single question, so you will need to draw connections across the content covered in this guide to score full marks.

2. Hyperbolic functions and inverses

Hyperbolic functions are exponential analogs of standard trigonometric functions, defined using the unit hyperbola rather than the unit circle. They satisfy identities similar to trigonometric identities, with only sign differences in most cases.

Core Definitions

$$\sinh x=\frac{e^x-e^{-x}}{2}\cosh x=\frac{e^x+e^{-x}}{2}\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}$$

Inverse Hyperbolic Functions

Each inverse function has strict domain and range rules you must memorize for exams:

Worked Example: Find the exact value of $\text{arcosh }3$.

1. Substitute $x=3$ into the explicit form for $\text{arcosh }x$: $$\text{arcosh }3=\ln \left(3+\sqrt{3^2-1}\right)=\ln \left(3+\sqrt{8}\right)=\ln \left(3+2\sqrt{2}\right)$$ Examiners will penalize decimal approximations here, so always leave answers in exact logarithmic or radical form unless stated otherwise.

3. Differentiation and integration of inverse trig and hyperbolic functions

The derivatives of inverse trigonometric and hyperbolic functions follow directly from implicit differentiation of their original function definitions, and their integral forms are used to solve a wide range of calculus problems on the FP2 exam.

Core Derivatives

$$\frac{d}{dx}\arcsin x=\frac{1}{\sqrt{1-x^2}}\frac{d}{dx}\arctan x=\frac{1}{1+x^2}$$ $$\frac{d}{dx}\text{arsinh }x=\frac{1}{\sqrt{x^2+1}}\frac{d}{dx}\text{arcosh }x=\frac{1}{\sqrt{x^2-1}}(x>1)$$

Standard Integral Forms

Reverse the derivatives above to get these high-yield integral identities, which you can apply directly without rederiving in exams: $$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \left(\frac{x}{a}\right)+C\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \left(\frac{x}{a}\right)+C$$ $$\int \frac{1}{\sqrt{x^2+a^2}}dx=\text{arsinh}\left(\frac{x}{a}\right)+C\int \frac{1}{\sqrt{x^2-a^2}}dx=\text{arcosh}\left(\frac{x}{a}\right)+C(x>a)$$

Worked Example: Evaluate ${\int }_0^3\frac{1}{x^2+9}dx$.

1. Match to the arctangent integral form with $a=3$: $${\int }_0^3\frac{1}{x^2+3^2}dx=\frac{1}{3}{\left[\arctan \left(\frac{x}{3}\right)\right]}_0^3$$
2. Substitute bounds: $$\frac{1}{3}\left(\arctan (1)-\arctan (0)\right)=\frac{1}{3}\left(\frac{\pi }{4}-0\right)=\frac{\pi }{12}$$

4. First and second-order differential equations

Differential equations (DEs) are one of the highest-weighted topics on FP2, with questions often requiring you to combine calculus and algebraic manipulation to find general or particular solutions.

First Order Linear DEs

These take the form $\frac{dy}{dx}+P(x)y=Q(x)$. To solve:

1. Calculate the integrating factor (IF): $IF=e^{\int P(x)dx}$ (you can ignore the constant of integration for the IF)
2. Multiply both sides of the DE by the IF: the left-hand side simplifies to $\frac{d}{dx}\left(IF\times y\right)$
3. Integrate both sides and rearrange to solve for $y$

Second Order Linear DEs (Constant Coefficients)

Homogeneous form: $a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=0$

1. Write the auxiliary equation: $a{m}^2+bm+c=0$
2. Find the general solution (complementary function, CF) based on roots of the auxiliary equation:

• Real distinct roots $m_1,m_2$: $y=A{e}^{m_{1}x}+B{e}^{m_{2}x}$
• Repeated real root $m$: $y=(Ax+B)e^{mx}$
• Complex conjugate roots $\alpha \pm i\beta$: $y=e^{\alpha x}(A\cos \beta x+B\sin \beta x)$ For non-homogeneous DEs $a\frac{d^{2}y}{d{x}^2}+b\frac{dy}{dx}+cy=f(x)$, add a particular integral (PI, a trial function matching the form of $f(x)$) to the CF to get the full general solution.

Worked Example: Find the general solution of $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=2e^x$

1. Auxiliary equation: $m^2-2m+1=0$, repeated root $m=1$, so CF is $y=(Ax+B)e^x$
2. Standard trial PI for $2e^x$ is $C{e}^x$, but this matches the CF, so multiply by $x^2$: $PI=C{x}^2{e}^x$
3. Substitute PI into the DE to find $C=1$, so full general solution: $y=(Ax+B)e^x+x^2{e}^x$

5. Maclaurin / Taylor series

Maclaurin and Taylor series are expansions of non-polynomial functions as infinite power series, used to approximate function values, find limits, and solve non-analytic differential equations.

• Maclaurin series: Taylor series centered at $x=0$, general form: $$f(x)=f(0)+f^{\prime }(0)x+\frac{f^{\prime \prime }(0)}{2!}{x}^2+\frac{f^{\prime \prime \prime }(0)}{3!}{x}^3+...+\frac{f^{(n)}(0)}{n!}{x}^n+...$$
• Taylor series: centered at $x=a$, general form: $$f(x)=f(a)+f^{\prime }(a)(x-a)+\frac{f^{\prime \prime }(a)}{2!}(x-a{)}^2+...+\frac{f^{(n)}(a)}{n!}(x-a{)}^n+...$$

Worked Example: Find the first 3 non-zero terms of the Maclaurin series for $f(x)={\cos }^{2}x$

1. Rewrite using double angle identity: ${\cos }^{2}x=\frac{1+\cos 2x}{2}$
2. Use the standard Maclaurin series for $\cos t=1-\frac{t^2}{2!}+\frac{t^4}{4!}-...$, substitute $t=2x$: $$\cos 2x=1-\frac{(2x{)}^2}{2}+\frac{(2x{)}^4}{24}-...=1-2x^2+\frac{2x^4}{3}-...$$
3. Substitute back: $${\cos }^{2}x=\frac{1}{2}\left(1+1-2x^2+\frac{2x^4}{3}\right)=1-x^2+\frac{x^4}{3}$$

6. Complex numbers — De Moivre, $n$-th roots of unity

De Moivre's theorem connects complex number powers to trigonometric identities, and is used to find powers and roots of complex numbers efficiently.

De Moivre's Theorem

For any integer $n$: $$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$$ In exponential form ($z=r{e}^{i\theta }$) this simplifies to $(r{e}^{i\theta }{)}^n=r^n{e}^{in\theta }$.

$n$-th Roots of Unity

The solutions to $z^n=1$ are given by: $$z=e^{\frac{2\pi ki}{n}}=\cos \left(\frac{2\pi k}{n}\right)+i\sin \left(\frac{2\pi k}{n}\right)\text{for }k=0,1,2,...,n-1$$ These form a regular $n$-gon on the Argand diagram centered at the origin with radius 1, and their sum is always 0.

Worked Example: Find all 3rd roots of unity.

1. $n=3$, so $k=0,1,2$:

• $k=0$: $\cos 0+i\sin 0=1$
• $k=1$: $\cos \left(\frac{2\pi }{3}\right)+i\sin \left(\frac{2\pi }{3}\right)=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$
• $k=2$: $\cos \left(\frac{4\pi }{3}\right)+i\sin \left(\frac{4\pi }{3}\right)=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$

7. Common Pitfalls (and how to avoid them)

• Wrong move: Forgetting domain restrictions for $\text{arcosh }x$ and $\text{artanh }x$, leading to undefined results. Why students do it: They mix up inverse hyperbolic domain rules with trigonometric inverses. Correct move: Always check $x\ge 1$ for $\text{arcosh }x$ and $|x|<1$ for $\text{artanh }x$ before applying formulas, and state the domain explicitly if asked.
• Wrong move: Using a trial particular integral that matches a term in the complementary function for second order DEs. Why students do it: They memorize trial PI forms without checking for overlap. Correct move: If your trial PI is a multiple of a CF term, multiply it by $x$ (or $x^2$ for repeated roots) before substituting into the DE.
• Wrong move: Writing Taylor series with $(x+a)$ instead of $(x-a)$ when centered at $x=a$. Why students do it: They mix up the sign of the center. Correct move: Remember the Taylor series uses $(x-a)$, so for a center at $x=2$ you use $(x-2)$ not $(x+2)$.
• Wrong move: Starting $k$ at 1 instead of 0 when calculating $n$-th roots of unity, missing the real positive root. Why students do it: They assume roots start at the first non-real value. Correct move: $k$ always runs from 0 to $n-1$ to get exactly $n$ distinct roots.
• Wrong move: Omitting the constant of integration when solving differential equations for a general solution. Why students do it: They focus on the PI and CF and forget the arbitrary constant. Correct move: Always include constants $A$ and $B$ for second order DEs, and a single constant for first order DEs unless you are given boundary conditions to find their values.

8. Practice Questions (A-Level Further Mathematics Style)

Question 1

(a) Find the exact value of $\text{artanh}\left(\frac{1}{2}\right)$. [2 marks] (b) Evaluate $\int \frac{1}{\sqrt{4x^2+25}}dx$. [3 marks]

Solution

(a) Use the explicit form for $\text{artanh }x$: $$\text{artanh}\left(\frac{1}{2}\right)=\frac{1}{2}\ln \left(\frac{1+1/2}{1-1/2}\right)=\frac{1}{2}\ln 3$$ (b) Rewrite the integrand to match the $\text{arsinh}$ integral form with $a=5$, $u=2x$, $du=2dx$: $$\int \frac{1}{\sqrt{(2x{)}^2+5^2}}dx=\frac{1}{2}\text{arsinh}\left(\frac{2x}{5}\right)+C$$

Question 2

Solve the differential equation $\frac{dy}{dx}+\frac{2}{x}y=x^3$ given that $y(1)=\frac{1}{6}$. [6 marks]

Solution

1. This is a first order linear DE with $P(x)=\frac{2}{x}$, $Q(x)=x^3$
2. Calculate integrating factor: $IF=e^{\int \frac{2}{x}dx}=e^{2\ln x}=x^2$
3. Multiply both sides by IF: $\frac{d}{dx}(x^{2}y)=x^5$
4. Integrate both sides: $x^{2}y=\frac{x^6}{6}+C$
5. Apply boundary condition $x=1,y=1/6$: $\frac{1}{6}=\frac{1}{6}+C⟹C=0$
6. Final solution: $y=\frac{x^4}{6}$

Question 3

Use De Moivre's theorem to show that $\sin 3\theta =3\sin \theta -4{\sin }^3\theta$. [5 marks]

Solution

1. By De Moivre's theorem: $(\cos \theta +i\sin \theta {)}^3=\cos 3\theta +i\sin 3\theta$
2. Expand left side with binomial theorem: $${\cos }^3\theta +3i{\cos }^2\theta \sin \theta +3i^2\cos \theta {\sin }^2\theta +i^3{\sin }^3\theta$$
3. Take the imaginary parts (equal to $\sin 3\theta$): $$3{\cos }^2\theta \sin \theta -{\sin }^3\theta$$
4. Substitute ${\cos }^2\theta =1-{\sin }^2\theta$: $$3(1-{\sin }^2\theta )\sin \theta -{\sin }^3\theta =3\sin \theta -3{\sin }^3\theta -{\sin }^3\theta =3\sin \theta -4{\sin }^3\theta$$ as required.

9. Quick Reference Cheatsheet

10. What's Next

The concepts covered in this FP2 guide are foundational for the rest of the A-Level Further Mathematics syllabus. Hyperbolic functions and advanced calculus are used extensively in Further Mechanics to model motion in resistive media and calculate work done against variable forces, while differential equations are core to both Further Mechanics and Further Statistics for modeling dynamic systems and probability distributions. Complex number theorems are extended in Further Pure 3 to cover complex loci and contour integration, and series expansions are used to approximate solutions to non-analytic differential equations in advanced applied units.

If you are stuck on any of the concepts, practice questions, or exam techniques covered in this guide, you can ask Ollie, our AI tutor, for personalized explanations, extra practice problems, or step-by-step walkthroughs tailored to your learning pace at any time by visiting the homepage. Make sure to also practice with official A-Level Further Mathematics past papers to get familiar with the exam structure, mark scheme conventions, and common question patterns ahead of your assessment.

Aligned with the Cambridge International AS & A Level Further Mathematics 9231 syllabus. OwlsAi is not affiliated with Cambridge Assessment International Education.