Engineering and Science Admissions Test (ESAT) Mathematics 1 (Core, Compulsory Module) Syllabus

Mathematics 1 (Core, Compulsory Module) syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics 1 (Core, Compulsory Module) in Engineering and Science Admissions Test (ESAT), not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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
5. 5

   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

Mathematics 1 (Core, Compulsory Module) flashcards for Engineering and Science Admissions Test (ESAT)

18 of 50 cards from the Mathematics 1 (Core, Compulsory Module) deck — real questions with worked answers.

1. 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).

2. 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 $\dfrac{b-\sqrt{a}}{b^{2}-a}$.

3. 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.

4. 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.

5. 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.

6. 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 $n$.

7. 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.

8. 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 $a^{m/n}=\sqrt[n]{a^{m}}=\left(\sqrt[n]{a}\right)^{m}$.

9. 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: $0.0042=4.2\times 10^{-3}$.

10. 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 $\text{HCF}(a,b)\times\text{LCM}(a,b)=a\times b$.

11. 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).

12. 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).

13. 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 $x=-\dfrac{b}{2}$.

14. 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.

15. 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 $x^{2}-(\alpha+\beta)x+\alpha\beta=0$.

16. 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.

17. 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.

18. 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).

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