Examius
🇬🇧 Engineering and Science Admissions Test (ESAT) · subject
Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) Syllabus
The Mathematics 2 (Advanced Mathematics Module) syllabus for Engineering and Science Admissions Test (ESAT) in full: 5 chapters, 17 topics and 39 sub-topics, every one listed below, plus 51 Mathematics 2 (Advanced Mathematics Module) flashcards with answers.
Mathematics 2 (Advanced Mathematics Module) syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
-
Algebra, Functions and Proof
4 topics- Advanced Algebraic Techniques
- Partial fractions with repeated and quadratic factors
- Algebraic and modulus inequalities
- The Binomial Expansion
- Expansion for positive integer powers
- Expansion for fractional and negative indices; range of validity
- Mathematical Proof
- Proof by deduction and exhaustion
- Disproof by counterexample
- Proof by contradiction
- Functions and Transformations
- Composite and inverse functions
- Modulus function equations and graphs
- Advanced Algebraic Techniques
-
Differentiation
3 topics- Differentiation from First Principles and Standard Results
- Derivatives of powers, exponentials, logarithms and trig functions
- Rules of Differentiation
- Product and quotient rules
- Chain rule and related rates
- Implicit and parametric differentiation
- Applications of Differentiation
- Tangents, normals and stationary points
- Maxima, minima and points of inflection
- Optimisation problems
- Differentiation from First Principles and Standard Results
-
Integration
4 topics- Indefinite and Definite Integration
- Standard integrals of powers, exponentials and trig functions
- Definite integrals and area under a curve
- Techniques of Integration
- Integration by substitution
- Integration by parts
- Integration using partial fractions and trig identities
- Applications of Integration
- Areas between curves
- Volumes of revolution
- Differential Equations
- First-order separable differential equations
- Modelling with differential equations
- Indefinite and Definite Integration
-
Trigonometry and Further Functions
3 topics- Trigonometric Identities and Formulae
- Compound and double-angle formulae
- The R cos(x+a) / R sin(x+a) form
- Reciprocal and inverse trigonometric functions
- Parametric Equations
- Converting between parametric and Cartesian forms
- Sequences, Series and Iteration
- Recurrence relations
- Sigma notation
- Numerical solution of equations by iteration
- Trigonometric Identities and Formulae
-
Vectors, Graphs and Numerical Methods
3 topics- Vectors in Two and Three Dimensions
- Vector arithmetic and magnitude
- Position vectors and geometric applications
- Unit vectors and direction
- Graphs of Functions
- Curve sketching using calculus
- Asymptotes and behaviour at infinity
- Numerical Methods
- Locating roots by sign change
- Trapezium rule for numerical integration
- Vectors in Two and Three Dimensions
Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) flashcards — 51 questions with answers
State the binomial expansion of $(1+x)^{n}$ for any real $n$, valid for $|x|<1$.
$$(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+\cdots$$ valid for $|x|<1$ when $n$ is not a non-negative integer.
What is the general term in the expansion of $(a+b)^{n}$ for positive integer $n$?
The term is $\binom{n}{r}a^{n-r}b^{r}$, where $\binom{n}{r}=\frac{n!}{r!(n-r)!}$, for $r=0,1,\dots,n$.
How do you split $\frac{4x+1}{(x-1)(x+2)}$ into partial fractions (state the form)?
Write it as $\frac{A}{x-1}+\frac{B}{x+2}$, then solve for $A$ and $B$ by equating numerators. Here $A=\frac{5}{3}$, $B=\frac{7}{3}$.
What partial-fraction form is used for a repeated linear factor, e.g. $\frac{f(x)}{(x-2)^{2}}$?
$$\frac{A}{x-2}+\frac{B}{(x-2)^{2}}$$ A separate term is needed for each power up to the multiplicity.
In a proof by contradiction, what is the logical structure?
Assume the negation of the statement is true, derive a logical contradiction (an impossibility), and conclude the original statement must be true.
What does proof by exhaustion involve?
Breaking the statement into a finite number of cases and verifying the result holds in every case.
What single counterexample is needed to disprove the statement 'for all integers $n$, $n^{2}+n+41$ is prime'?
Take $n=41$: then $41^{2}+41+41=41\times43$, which is composite. A single counterexample disproves a universal claim.
Differentiate $f(x)=x^{2}$ from first principles.
$$f'(x)=\lim_{h\to 0}\frac{(x+h)^{2}-x^{2}}{h}=\lim_{h\to 0}(2x+h)=2x$$
State the formal (first principles) definition of the derivative of $f(x)$.
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
State the standard derivatives of $\sin x$, $\cos x$ and $e^{x}$.
$\frac{d}{dx}\sin x=\cos x$, $\quad\frac{d}{dx}\cos x=-\sin x$, $\quad\frac{d}{dx}e^{x}=e^{x}$.
What is the derivative of $\ln x$ and of $\tan x$?
$\frac{d}{dx}\ln x=\frac{1}{x}$ and $\frac{d}{dx}\tan x=\sec^{2}x$.
State the product rule for differentiation.
If $y=uv$ then $$\frac{dy}{dx}=u\frac{dv}{dx}+v\frac{du}{dx}$$
State the quotient rule for differentiation.
If $y=\dfrac{u}{v}$ then $$\frac{dy}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^{2}}$$
State the chain rule for $y=f(g(x))$.
$$\frac{dy}{dx}=f'(g(x))\,g'(x),\quad\text{or}\quad\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$$
How do you find the derivative $\frac{dy}{dx}$ when $x$ and $y$ are given parametrically as functions of $t$?
$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{\dot{y}}{\dot{x}}\quad(\dot{x}\neq 0)$$
How are stationary points classified using the second derivative?
At a stationary point ($f'(x)=0$): if $f''(x)>0$ it is a local minimum; if $f''(x)<0$ a local maximum; if $f''(x)=0$ the test is inconclusive (check the sign of $f'$).
What condition identifies a point of inflection?
$f''(x)=0$ and $f''(x)$ changes sign through that point (the concavity changes).
What is the equation of the tangent to $y=f(x)$ at the point $(a,f(a))$?
$$y-f(a)=f'(a)(x-a)$$
How are the gradients of perpendicular lines (e.g. a tangent and its normal) related?
Their product is $-1$; the normal gradient is $-\frac{1}{f'(a)}$ where $f'(a)$ is the tangent gradient.
State the standard result $\int x^{n}\,dx$ and note the exception.
$$\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C\quad(n\neq -1)$$ When $n=-1$, $\int x^{-1}\,dx=\ln|x|+C$.
Show the other 31 Mathematics 2 (Advanced Mathematics Module) flashcards
State $\int e^{x}\,dx$, $\int \sin x\,dx$ and $\int \cos x\,dx$.
$\int e^{x}\,dx=e^{x}+C$, $\quad\int\sin x\,dx=-\cos x+C$, $\quad\int\cos x\,dx=\sin x+C$.
State the Fundamental Theorem of Calculus linking definite integration and antiderivatives.
If $F'(x)=f(x)$ then $$\int_{a}^{b}f(x)\,dx=F(b)-F(a)$$
State the formula for integration by parts.
$$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx$$
For the substitution $u=g(x)$, how does $\int f(g(x))g'(x)\,dx$ transform?
It becomes $\int f(u)\,du$, since $du=g'(x)\,dx$. For definite integrals, also change the limits to $u$-values.
What is the integral $\int \frac{f'(x)}{f(x)}\,dx$?
$$\int\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C$$
What integral gives the area between the curve $y=f(x)$, the $x$-axis, and the lines $x=a$, $x=b$?
$$A=\int_{a}^{b}f(x)\,dx$$ (taking the modulus of any region where $f(x)<0$).
State the formula for the volume of revolution about the $x$-axis.
$$V=\pi\int_{a}^{b}y^{2}\,dx$$
How do you solve a separable differential equation $\frac{dy}{dx}=f(x)g(y)$?
Separate variables and integrate both sides: $$\int\frac{1}{g(y)}\,dy=\int f(x)\,dx$$ then include a constant and apply any initial condition.
What is the general solution of $\frac{dy}{dx}=ky$, and what does it model?
$y=Ae^{kx}$. It models exponential growth ($k>0$) or decay ($k<0$).
What does a 'general solution' versus a 'particular solution' of a differential equation mean?
The general solution contains arbitrary constant(s); a particular solution fixes those constants using given boundary or initial conditions.
State the Pythagorean trigonometric identities.
$\sin^{2}\theta+\cos^{2}\theta=1$, $\quad 1+\tan^{2}\theta=\sec^{2}\theta$, $\quad 1+\cot^{2}\theta=\csc^{2}\theta$.
State the compound angle formulae for $\sin(A\pm B)$ and $\cos(A\pm B)$.
$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$; $\quad\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$.
State the double angle formulae for $\sin 2\theta$ and the three forms of $\cos 2\theta$.
$\sin 2\theta=2\sin\theta\cos\theta$; $\quad\cos 2\theta=\cos^{2}\theta-\sin^{2}\theta=2\cos^{2}\theta-1=1-2\sin^{2}\theta$.
How is $a\sin\theta+b\cos\theta$ written in the form $R\sin(\theta+\alpha)$?
$R=\sqrt{a^{2}+b^{2}}$ and $\tan\alpha=\frac{b}{a}$ (with $R>0$, $\alpha$ acute).
How do you convert parametric equations $x=f(t)$, $y=g(t)$ to Cartesian form?
Eliminate the parameter $t$ — solve one equation for $t$ (or use an identity) and substitute into the other to get a relation between $x$ and $y$.
Give the parametric equations of a circle of radius $r$ centred at the origin.
$x=r\cos t$, $\quad y=r\sin t$, for $0\leq t<2\pi$ (since $x^{2}+y^{2}=r^{2}$).
State the formula for the $n$th term of an arithmetic sequence and the sum of the first $n$ terms.
$u_{n}=a+(n-1)d$; $\quad S_{n}=\frac{n}{2}\big(2a+(n-1)d\big)=\frac{n}{2}(a+l)$.
State the $n$th term and sum to $n$ terms of a geometric sequence.
$u_{n}=ar^{n-1}$; $\quad S_{n}=\frac{a(1-r^{n})}{1-r}$ for $r\neq 1$.
When does an infinite geometric series converge, and to what sum?
It converges when $|r|<1$, to $$S_{\infty}=\frac{a}{1-r}$$
What does an iterative formula $x_{n+1}=g(x_{n})$ converge to, if it converges?
A root of $x=g(x)$ (a fixed point), which is a solution of the equation rearranged into the form $x=g(x)$.
How do you find the magnitude of the vector $\vec{a}=\begin{pmatrix}x\\y\\z\end{pmatrix}$?
$$|\vec{a}|=\sqrt{x^{2}+y^{2}+z^{2}}$$
State the scalar (dot) product formula and the condition for perpendicular vectors.
$\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}$. Vectors are perpendicular iff $\vec{a}\cdot\vec{b}=0$.
How is the angle between two vectors found from their dot product?
$$\cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}$$
State the vector equation of a straight line through point $\vec{a}$ with direction $\vec{d}$.
$$\vec{r}=\vec{a}+\lambda\vec{d},\quad \lambda\in\mathbb{R}$$
How does the graph of $y=f(x-a)+b$ relate to $y=f(x)$?
It is translated by the vector $\begin{pmatrix}a\\b\end{pmatrix}$: $a$ units in the positive $x$-direction and $b$ units in the positive $y$-direction.
Describe the transformations $y=f(ax)$ and $y=af(x)$.
$y=f(ax)$ is a horizontal stretch with scale factor $\frac{1}{a}$; $y=af(x)$ is a vertical stretch with scale factor $a$.
What transformation maps $y=f(x)$ to $y=|f(x)|$ versus $y=f(|x|)$?
$y=|f(x)|$ reflects any part below the $x$-axis up above it. $y=f(|x|)$ keeps $x\geq 0$ and reflects it in the $y$-axis.
What is the condition for a function to have an inverse, and how do their graphs relate?
It must be one-to-one (injective) on its domain. The graph of $y=f^{-1}(x)$ is the reflection of $y=f(x)$ in the line $y=x$.
State the Newton-Raphson iterative formula for solving $f(x)=0$.
$$x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}$$
How does the change-of-sign method locate a root of $f(x)=0$?
If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, then there is at least one root in the interval $(a,b)$.
State the trapezium rule for estimating $\int_{a}^{b}y\,dx$ with $n$ strips of width $h$.
$$\int_{a}^{b}y\,dx\approx\frac{h}{2}\Big(y_{0}+y_{n}+2(y_{1}+y_{2}+\cdots+y_{n-1})\Big),\quad h=\frac{b-a}{n}$$
Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) FAQ
What is in the Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) syllabus?
Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module) is split into 5 chapters: Algebra, Functions and Proof, Differentiation, Integration, Trigonometry and Further Functions and Vectors, Graphs and Numerical Methods. Together they contain 17 topics and 39 sub-topics.
How many chapters are in Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module)?
5 chapters, containing 17 topics and 39 sub-topics. The largest chapter is Algebra, Functions and Proof, with 4 topics.
Are there flashcards for Engineering and Science Admissions Test (ESAT) Mathematics 2 (Advanced Mathematics Module)?
Yes — a 51-card Mathematics 2 (Advanced Mathematics Module) deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.