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Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) Syllabus
The Mathematics 1 (Core, Compulsory Module) syllabus for Engineering and Science Admissions Test (ESAT) in full: 5 chapters, 20 topics and 50 sub-topics, every one listed below, plus 50 Mathematics 1 (Core, Compulsory Module) flashcards with answers.
Mathematics 1 (Core, Compulsory Module) syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
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Number, Ratio and Numerical Reasoning
4 topics- Integers, Fractions, Decimals and Surds
- Operations with fractions and recurring decimals
- Simplifying and rationalising surds
- Estimation, rounding and significant figures
- Ratio, Proportion and Percentages
- Dividing quantities in a given ratio
- Direct and inverse proportion
- Percentage change, reverse percentages and compound interest
- Indices and Standard Form
- Laws of indices including fractional and negative powers
- Calculations in standard form
- Factors, Multiples and Primes
- HCF, LCM and prime factorisation
- Integers, Fractions, Decimals and Surds
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Algebra and Manipulation
5 topics- Algebraic Expressions and Identities
- Expanding brackets and factorising
- Algebraic fractions and partial fractions
- Completing the square
- Linear and Quadratic Equations
- Solving linear equations and rearranging formulae
- Quadratic formula and discriminant
- Roots: sum and product relationships
- Simultaneous Equations and Inequalities
- Linear and one-linear-one-quadratic systems
- Linear and quadratic inequalities
- Representing inequalities on number lines and regions
- Polynomials
- Factor and remainder theorems
- Polynomial division
- Sequences and Series
- Arithmetic sequences and series
- Geometric sequences, series and sum to infinity
- Algebraic Expressions and Identities
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Functions, Graphs and Coordinate Geometry
4 topics- Functions and Graph Sketching
- Domain, range and composite/inverse functions
- Sketching polynomial, reciprocal and modulus graphs
- Graph transformations: translations, stretches, reflections
- Straight Line Geometry
- Gradient, midpoint and distance
- Equations of lines; parallel and perpendicular conditions
- Circles
- Equation of a circle and completing the square form
- Tangents, chords and intersections
- Exponentials and Logarithms
- Laws of logarithms and change of base
- Solving exponential and logarithmic equations
- Exponential growth and decay models
- Functions and Graph Sketching
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Geometry, Mensuration and Trigonometry
4 topics- Plane Geometry and Angles
- Angle rules, polygons and parallel lines
- Circle theorems
- Mensuration
- Area and perimeter of plane shapes
- Surface area and volume of solids
- Arc length and sector area
- Trigonometry
- Right-angled trigonometry and Pythagoras
- Sine and cosine rules; area of a triangle
- Trigonometric graphs and exact values
- Identities and solving trigonometric equations
- Radians
- Conversion between degrees and radians
- Plane Geometry and Angles
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Statistics and Probability
3 topics- Data Handling and Summary Statistics
- Mean, median, mode and range
- Quartiles, interquartile range and outliers
- Frequency tables and grouped data
- Statistical Diagrams
- Histograms, box plots and cumulative frequency
- Scatter graphs and correlation
- Probability
- Sample spaces and the addition/multiplication rules
- Tree diagrams and Venn diagrams
- Conditional probability and independence
- Data Handling and Summary Statistics
Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) flashcards — 50 questions with answers
What is a surd, and what makes an expression a surd?
A surd is an irrational root that cannot be simplified to remove the root, e.g. $\sqrt{2}$ or $\sqrt[3]{5}$. Its value is an irrational number (a non-terminating, non-repeating decimal).
How do you rationalise the denominator of $\dfrac{1}{\sqrt{a}}$ and of $\dfrac{1}{b+\sqrt{a}}$?
Multiply by $\dfrac{\sqrt{a}}{\sqrt{a}}$ to get $\dfrac{\sqrt{a}}{a}$. For $\dfrac{1}{b+\sqrt{a}}$, multiply by the conjugate $\dfrac{b-\sqrt{a}}{b-\sqrt{a}}$ to get $\dfrac{b-\sqrt{a}}{b^{2}-a}$.
State the surd laws for multiplication and division of square roots.
$\sqrt{a}\,\sqrt{b}=\sqrt{ab}$ and $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$ (for $a,b\geq 0$, $b\neq 0$). Note $\sqrt{a+b}\neq\sqrt{a}+\sqrt{b}$ in general.
How do you convert a recurring decimal such as $0.\overline{27}$ to a fraction?
Let $x=0.\overline{27}$. Multiply by $100$: $100x=27.\overline{27}$. Subtract: $99x=27$, so $x=\dfrac{27}{99}=\dfrac{3}{11}$. Multiply by $10^{n}$ where $n$ is the length of the repeating block.
What is the difference between direct and inverse proportion, and their equations?
Direct: $y\propto x$, so $y=kx$ (as $x$ increases, $y$ increases proportionally). Inverse: $y\propto\dfrac{1}{x}$, so $y=\dfrac{k}{x}$ (as $x$ increases, $y$ decreases). $k$ is the constant of proportionality.
How do you increase or decrease a quantity by a percentage using a multiplier?
Increase by $p\%$: multiply by $\left(1+\dfrac{p}{100}\right)$. Decrease by $p\%$: multiply by $\left(1-\dfrac{p}{100}\right)$. For repeated changes over $n$ steps, raise the multiplier to the power $n$.
What is the formula for percentage change, and how do you reverse a percentage increase?
$\text{percentage change}=\dfrac{\text{new}-\text{old}}{\text{old}}\times 100\%$. To reverse an increase of $p\%$, divide the new amount by $\left(1+\dfrac{p}{100}\right)$ to recover the original.
State the laws of indices for $a^{m}\times a^{n}$, $\dfrac{a^{m}}{a^{n}}$, $(a^{m})^{n}$, $a^{0}$, $a^{-n}$ and $a^{m/n}$.
$a^{m}a^{n}=a^{m+n}$, $\dfrac{a^{m}}{a^{n}}=a^{m-n}$, $(a^{m})^{n}=a^{mn}$, $a^{0}=1$, $a^{-n}=\dfrac{1}{a^{n}}$, and $a^{m/n}=\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}$.
What is standard form (scientific notation) and its required form?
A number written as $a\times 10^{n}$ where $1\leq |a|<10$ and $n$ is an integer. Large numbers have positive $n$; small numbers (less than 1) have negative $n$. Example: $0.0042=4.2\times 10^{-3}$.
How do you find the HCF and LCM of two numbers from their prime factorisations?
HCF: take each shared prime to its lowest power appearing in both. LCM: take each prime to its highest power appearing in either. Also $\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$.
Define a prime number and the Fundamental Theorem of Arithmetic.
A prime is a natural number greater than $1$ with exactly two factors: $1$ and itself ($1$ is not prime). The Fundamental Theorem of Arithmetic states every integer greater than $1$ has a unique prime factorisation (up to order).
Expand and state the three key algebraic identities: $(a+b)^{2}$, $(a-b)^{2}$, and $(a+b)(a-b)$.
$(a+b)^{2}=a^{2}+2ab+b^{2}$; $(a-b)^{2}=a^{2}-2ab+b^{2}$; $(a+b)(a-b)=a^{2}-b^{2}$ (difference of two squares).
How do you complete the square for $ax^{2}+bx+c$ (with $a=1$)?
Write $x^{2}+bx+c=\left(x+\dfrac{b}{2}\right)^{2}-\left(\dfrac{b}{2}\right)^{2}+c$. This reveals the vertex form, giving the turning point at $x=-\dfrac{b}{2}$.
State the quadratic formula and the discriminant, and what the discriminant tells you.
$x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. The discriminant is $\Delta=b^{2}-4ac$: $\Delta>0$ gives two distinct real roots, $\Delta=0$ a repeated root, $\Delta<0$ no real roots.
For $ax^{2}+bx+c=0$ with roots $\alpha,\beta$, what are the sum and product of the roots?
$\alpha+\beta=-\dfrac{b}{a}$ and $\alpha\beta=\dfrac{c}{a}$. A quadratic with these roots can be written $x^{2}-(\alpha+\beta)x+\alpha\beta=0$.
What happens to an inequality when you multiply or divide both sides by a negative number?
The inequality sign reverses. E.g. from $-2x<6$ dividing by $-2$ gives $x>-3$. Multiplying/dividing by a positive number keeps the sign unchanged.
Describe the elimination method for solving simultaneous linear equations.
Scale one or both equations so a variable has matching coefficients, then add or subtract the equations to eliminate that variable. Solve for the remaining variable, then back-substitute to find the other.
How do you solve a linear–quadratic pair of simultaneous equations such as a line and a curve?
Rearrange the linear equation for one variable, substitute into the quadratic, solve the resulting quadratic, then back-substitute. The number of real solutions indicates how many intersection points (two, one tangent, or none).
State the Remainder Theorem and the Factor Theorem for a polynomial $p(x)$.
Remainder Theorem: the remainder when $p(x)$ is divided by $(x-a)$ is $p(a)$. Factor Theorem: $(x-a)$ is a factor of $p(x)$ if and only if $p(a)=0$.
What is the general shape and end behaviour of a cubic $y=ax^{3}+\dots$?
For $a>0$ the curve falls from bottom-left to top-right (as $x\to-\infty$, $y\to-\infty$; as $x\to+\infty$, $y\to+\infty$). For $a<0$ it is reversed. A cubic has up to two turning points and up to three real roots.
Show the other 30 Mathematics 1 (Core, Compulsory Module) flashcards
Give the formulas for the $n$th term and sum of an arithmetic series with first term $a$ and common difference $d$.
$n$th term: $u_{n}=a+(n-1)d$. Sum of first $n$ terms: $S_{n}=\dfrac{n}{2}\left[2a+(n-1)d\right]=\dfrac{n}{2}(a+l)$, where $l$ is the last term.
Give the formulas for the $n$th term and sum of a geometric series with first term $a$ and ratio $r$.
$n$th term: $u_{n}=ar^{n-1}$. Sum: $S_{n}=\dfrac{a(1-r^{n})}{1-r}$ for $r\neq 1$. Sum to infinity (if $|r|<1$): $S_{\infty}=\dfrac{a}{1-r}$.
What conditions make a relation a function, and what are domain and range?
A function maps each input to exactly one output (passes the vertical line test). The domain is the set of allowed inputs; the range is the set of resulting outputs.
Describe the graph transformations of $y=f(x)$ given by $f(x)+a$, $f(x+a)$, $af(x)$ and $f(ax)$.
$f(x)+a$: shift up by $a$. $f(x+a)$: shift left by $a$. $af(x)$: vertical stretch by factor $a$. $f(ax)$: horizontal stretch by factor $\dfrac{1}{a}$.
How do you find an inverse function $f^{-1}(x)$, and what is the graphical relationship?
Set $y=f(x)$, swap $x$ and $y$, then solve for $y$. The graph of $f^{-1}$ is the reflection of $f$ in the line $y=x$. The inverse exists only if $f$ is one-to-one.
State the gradient and midpoint formulas for points $(x_1,y_1)$ and $(x_2,y_2)$.
Gradient $m=\dfrac{y_2-y_1}{x_2-x_1}$. Midpoint $=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$. Distance $=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$.
What is the gradient relationship for parallel and perpendicular lines?
Parallel lines have equal gradients ($m_1=m_2$). Perpendicular lines have gradients whose product is $-1$, i.e. $m_1 m_2=-1$, so $m_2=-\dfrac{1}{m_1}$.
Give the equation of a straight line in gradient–point form and in general form.
Gradient–point: $y-y_1=m(x-x_1)$. Slope–intercept: $y=mx+c$. General form: $ax+by+c=0$. The $y$-intercept is $c$ in $y=mx+c$.
State the equation of a circle with centre $(a,b)$ and radius $r$.
$(x-a)^{2}+(y-b)^{2}=r^{2}$. Expanded general form is $x^{2}+y^{2}+2gx+2fy+c=0$ with centre $(-g,-f)$ and radius $\sqrt{g^{2}+f^{2}-c}$.
State three key circle theorems relating angles.
The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is $90^{\circ}$. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral sum to $180^{\circ}$.
What is the relationship between a tangent to a circle and the radius at the point of contact?
The tangent is perpendicular to the radius at the point of contact (angle $=90^{\circ}$). Also, two tangents drawn from an external point are equal in length.
State the laws of logarithms for $\log_a(xy)$, $\log_a\!\left(\dfrac{x}{y}\right)$ and $\log_a(x^{k})$.
$\log_a(xy)=\log_a x+\log_a y$; $\log_a\!\left(\dfrac{x}{y}\right)=\log_a x-\log_a y$; $\log_a(x^{k})=k\log_a x$. Also $\log_a a=1$ and $\log_a 1=0$.
What is the relationship between exponentials and logarithms, and the change of base formula?
$y=a^{x}\iff x=\log_a y$ (logarithm is the inverse of exponentiation). Change of base: $\log_a x=\dfrac{\log_b x}{\log_b a}$.
What are the key features of the exponential graph $y=a^{x}$ for $a>1$?
It passes through $(0,1)$, is always positive, increasing, with the $x$-axis ($y=0$) as a horizontal asymptote. For $0<a<1$ the curve is decreasing instead.
State the angle facts: angles on a straight line, around a point, and vertically opposite.
Angles on a straight line sum to $180^{\circ}$. Angles around a point sum to $360^{\circ}$. Vertically opposite angles are equal. The angles in a triangle sum to $180^{\circ}$.
What is the sum of interior angles of a polygon with $n$ sides, and the sum of exterior angles?
Sum of interior angles $=(n-2)\times 180^{\circ}$. Sum of exterior angles $=360^{\circ}$ for any convex polygon. Each interior angle of a regular polygon $=\dfrac{(n-2)\times 180^{\circ}}{n}$.
Give the area and circumference formulas for a circle of radius $r$.
Area $=\pi r^{2}$. Circumference $=2\pi r=\pi d$. Arc length (angle $\theta$ in radians) $=r\theta$; sector area $=\dfrac{1}{2}r^{2}\theta$.
State the volume and surface area formulas for a sphere, cylinder and cone.
Sphere: $V=\dfrac{4}{3}\pi r^{3}$, $A=4\pi r^{2}$. Cylinder: $V=\pi r^{2}h$, curved area $=2\pi r h$. Cone: $V=\dfrac{1}{3}\pi r^{2}h$, curved area $=\pi r l$ ($l$ slant height).
State Pythagoras' theorem and the three basic trigonometric ratios.
Pythagoras: $a^{2}+b^{2}=c^{2}$ (hypotenuse $c$). In a right triangle: $\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$.
State the sine rule and the cosine rule for a general triangle.
Sine rule: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$. Cosine rule: $a^{2}=b^{2}+c^{2}-2bc\cos A$. Triangle area $=\dfrac{1}{2}ab\sin C$.
State the fundamental trig identities $\sin^{2}\theta+\cos^{2}\theta$ and $\tan\theta$ in terms of sine and cosine.
$\sin^{2}\theta+\cos^{2}\theta=1$ and $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$. Dividing the first identity by $\cos^{2}\theta$ gives $1+\tan^{2}\theta=\sec^{2}\theta$.
Give the exact trig values for $30^{\circ}$, $45^{\circ}$ and $60^{\circ}$ (sine and cosine).
$\sin 30^{\circ}=\dfrac{1}{2}$, $\cos 30^{\circ}=\dfrac{\sqrt{3}}{2}$; $\sin 45^{\circ}=\cos 45^{\circ}=\dfrac{\sqrt{2}}{2}$; $\sin 60^{\circ}=\dfrac{\sqrt{3}}{2}$, $\cos 60^{\circ}=\dfrac{1}{2}$.
How do you convert between degrees and radians?
$\pi$ radians $=180^{\circ}$. To convert degrees to radians multiply by $\dfrac{\pi}{180}$; to convert radians to degrees multiply by $\dfrac{180}{\pi}$. For example $90^{\circ}=\dfrac{\pi}{2}$ rad.
What is the period and range of $\sin x$, $\cos x$ and $\tan x$?
$\sin x$ and $\cos x$ have period $2\pi$ and range $[-1,1]$. $\tan x$ has period $\pi$, range all real numbers, with asymptotes at $x=\dfrac{\pi}{2}+n\pi$.
How do you calculate the mean, median and mode of a data set?
Mean $=\dfrac{\sum x}{n}$ (sum divided by count). Median is the middle value when ordered (average of the two middle values if $n$ is even). Mode is the most frequently occurring value.
Define range, interquartile range (IQR), and how to find the quartiles.
Range $=\text{max}-\text{min}$. IQR $=Q_3-Q_1$ (upper quartile minus lower quartile), measuring the spread of the middle $50\%$ and resistant to outliers. Quartiles split the ordered data into four equal parts.
Give the formula for variance and standard deviation of a data set.
Variance $\sigma^{2}=\dfrac{\sum (x-\bar{x})^{2}}{n}=\dfrac{\sum x^{2}}{n}-\bar{x}^{2}$. Standard deviation $\sigma=\sqrt{\sigma^{2}}$, measuring average spread about the mean.
What does a box plot display, and how is an outlier commonly identified?
A box plot shows the minimum, $Q_1$, median, $Q_3$ and maximum (five-number summary). An outlier is typically a value below $Q_1-1.5\times\text{IQR}$ or above $Q_3+1.5\times\text{IQR}$.
State the probability rules for the complement, mutually exclusive events, and independent events.
Complement: $P(\text{not }A)=1-P(A)$. Mutually exclusive: $P(A\cup B)=P(A)+P(B)$. Independent: $P(A\cap B)=P(A)\times P(B)$. General addition: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
State the conditional probability formula and what tree diagrams represent.
$P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$. A tree diagram shows successive outcomes with probabilities on each branch; multiply along branches for combined outcomes and add across separate branches for an 'or' result. Branches from a node sum to $1$.
Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) FAQ
What is in the Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) syllabus?
Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) is split into 5 chapters: Number, Ratio and Numerical Reasoning, Algebra and Manipulation, Functions, Graphs and Coordinate Geometry, Geometry, Mensuration and Trigonometry and Statistics and Probability. Together they contain 20 topics and 50 sub-topics.
How many chapters are in Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module)?
5 chapters, containing 20 topics and 50 sub-topics. The largest chapter is Algebra and Manipulation, with 5 topics.
Are there flashcards for Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module)?
Yes — a 50-card Mathematics 1 (Core, Compulsory 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.