ESAT Mathematics 2: Revision Notes

MM1.1

Laws of indices provide a concise notation and a set of rules for manipulating expressions involving powers, roots, and reciprocals. For the ESAT, they are fundamental for simplifying complex algebraic terms efficiently without a calculator.

MM1.2

Surds are expressions involving irrational roots, like sqrt(3), used to maintain exact mathematical values. For the ESAT, you must be proficient in simplifying expressions with surds and, most importantly, in rewriting fractions to remove surds from the denominator.

MM1.3

This topic covers quadratic functions, expressed as y = ax^2 + bx + c. Mastering the links between their algebraic forms and graphical properties (parabolas) is essential for solving equations and inequalities efficiently without a calculator.

MM1.4

This topic covers finding the common solutions to a pair of equations, specifically a linear and a quadratic one. This is equivalent to finding the coordinate(s) where the graphs of a straight line and a parabola intersect.

MM1.5

This topic covers the methods for solving inequalities involving linear and quadratic expressions. Unlike equations, inequalities define a range of possible values, and the rules for manipulating them, especially with multiplication and division, are stricter.

MM1.6

This topic covers the core skills for manipulating polynomials, which are expressions like 3x³ - x + 4. Mastering these techniques—expanding, factorising, and dividing—is essential for solving higher-order equations and analysing function behaviour in the ESAT.

MM1.7

This topic covers the fundamental definition of a function as a mapping from inputs to outputs, where each input has only one output. It focuses on distinguishing between one-to-one and many-to-one functions and understanding the properties of the square root and modulus functions.

MM2.1

This topic covers sequences, which are ordered lists of numbers. ESAT questions will test your ability to generate terms from either an explicit formula for the nth term or from a recurrence relation, and then spot patterns to find a specific term or a sum.

MM2.2

This topic covers arithmetic sequences, where numbers increase or decrease by a constant amount. For ESAT, you must be able to find any term in a sequence and calculate the sum of a series efficiently without a calculator.

MM2.3

This topic covers the summation of geometric series, which are sequences where each term is a constant multiple of the previous one. Mastery is crucial for ESAT as it frequently tests algebraic manipulation and logical application of conditions, particularly for infinite sums.

MM2.4

The Binomial Theorem provides a powerful and quick method for expanding brackets raised to a positive integer power, of the form (a + b)^n. It allows you to find any specific term or coefficient without the lengthy process of manual multiplication.

MM3.1

This topic covers the fundamental algebraic representation of straight lines. Mastering how to find and manipulate their equations based on points, gradients, and parallel/perpendicular relationships is essential for solving a wide range of geometry problems.

MM3.2

This topic covers the algebraic equations of circles on the (x, y)-plane. Mastering the two standard forms allows you to quickly determine a circle's centre and radius, or to find its equation from these properties.

MM3.3

Circle theorems are fundamental geometric rules that describe relationships between angles, chords, tangents, and arcs in a circle. They are essential for solving complex geometry problems by breaking them down into simpler steps, a key skill for the non-calculator ESAT.

MM4.1

This topic covers essential rules for solving any triangle, not just right-angled ones. These formulae allow you to find unknown sides and angles when given sufficient information and are applicable in both 2D and 3D problems.

MM4.2

This topic introduces radians as an alternative to degrees for measuring angles, which is the standard in higher-level mathematics. For the ESAT, you must be able to use radians to calculate the arc length, sector area, and segment area of a circle without a calculator.

MM4.3

This topic covers the exact trigonometric values for key angles (0°, 30°, 45°, 60°, 90°), which you must know or be able to derive instantly. These values are fundamental building blocks for solving non-calculator problems in geometry, mechanics, and other scientific contexts.

MM4.4

This topic covers the fundamental properties of the sine, cosine, and tangent functions, focusing on their graphical shapes, repeating patterns (periodicity), and symmetries. Mastering these is crucial for solving trigonometric equations and analysing wave-like phenomena without a calculator.

MM4.5

This topic covers two fundamental trigonometric identities that link sine, cosine, and tangent. They are essential tools for simplifying trigonometric expressions and solving equations without a calculator.

MM4.6

This topic covers finding all angle solutions to trigonometric equations within a specific range. It's a test of algebraic manipulation, knowledge of trigonometric identities, and careful, systematic work to ensure no solutions are missed.

MM5.1

This topic covers the fundamental shape and properties of exponential functions, y = a^x. Understanding these graphs is crucial for modelling rapid growth or decay and forms the visual basis for logarithms.

MM5.2

Logarithms are the inverse of exponential functions, used to find the power to which a base must be raised to produce a given number. Mastery of the logarithm laws is crucial for solving equations involving unknown powers and for simplifying expressions without a calculator.

MM5.3

This topic covers solving equations where the unknown variable is in the exponent, such as 3^x = 81. The key technique is using logarithms to manipulate the equation and isolate the variable.

MM6.1

Differentiation is the process of finding the instantaneous rate of change of a function. This is geometrically interpreted as finding the gradient of the tangent to the function's graph at any given point.

MM6.2

This topic covers finding the derivative (gradient function) of expressions with powers of x. For the ESAT, the main challenge is not the differentiation rule itself, but algebraically simplifying complex expressions into a simple sum of powers before you can apply it.

MM6.3

This topic covers using differentiation to analyse the properties of a function's graph, such as its slope, turning points, and intervals where it rises or falls. These are fundamental calculus tools for understanding and sketching curves.

MM7.1

This topic covers the use of definite integration to calculate the area between a function's curve and the x-axis. It is crucial to distinguish between a definite integral, which can be negative and represents 'net area', and the total geometric area, which must always be positive.

MM7.2

This topic covers reversing the process of differentiation for expressions involving powers of x. The key skill for ESAT is algebraically simplifying complex expressions, like expanded brackets or fractions, into a sum of simple `x^n` terms before applying the standard integration rule.

MM7.3

The Fundamental Theorem of Calculus (FTC) establishes the critical link between differentiation and integration, showing they are inverse processes. It provides the standard method for evaluating definite integrals, which is essential for calculating quantities like the area under a curve.

MM7.4

This topic covers the algebraic rules for manipulating definite integrals. These rules allow you to combine or split up integrals, which often simplifies complex expressions into something much easier to evaluate without a calculator.

MM7.5

The trapezium rule is a method to estimate the area under a curve by dividing the region into vertical strips and summing the areas of the trapezia formed. This is useful when direct integration is too complex or when you only have discrete data points.

MM7.6

This topic covers how to reverse the process of differentiation. Given a function for the gradient of a curve (dy/dx), you will use integration to find the equation of the original curve (y).

MM8.1

This topic covers the fundamental skill of sketching and recognising common mathematical functions without a calculator. A strong visual understanding of these graphs is essential for solving a wide range of problems, including finding roots and solving inequalities.

MM8.2

This topic covers how the graph of a function y = f(x) is altered by simple, predictable changes to its equation. Understanding these rules allows for rapid sketching and analysis of related functions without needing to plot individual points, a key skill for non-calculator exams.

MM8.3

The equation y = mx + c describes any straight line. Understanding the roles of the gradient 'm' and the y-intercept 'c' allows you to instantly interpret, sketch, and transform linear graphs.

MM8.4

This topic explores the 'vertex form' of a quadratic equation, y = a(x + b)² + c, which directly reveals the graph's key features. Understanding how the parameters a, b, and c transform the basic y = x² parabola is crucial for quickly sketching graphs and identifying their turning points without a calculator.

MM8.5

This topic uses differentiation as a tool to analyse function behaviour without plotting every point. By finding the derivative (gradient), you can pinpoint exactly where a graph has peaks and troughs (stationary points) and identify the intervals where it is rising or falling.

MM8.6

This topic covers finding where a function's graph crosses the coordinate axes, a key step in sketching. It also involves understanding the maximum and minimum possible number of x-intercepts (real roots) a polynomial can have based on its degree.

MM8.7

This topic connects algebra and geometry by showing that the solutions to simultaneous equations are the coordinates of the points where their graphs intersect. Understanding this visual link allows you to determine the number of solutions to an equation by simply sketching the corresponding graphs.