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Admissions tests / ESAT / Maths 2

27 questions in 40 minutes. Multiple choice. No calculator.

ESAT Mathematics 2

Mathematics 2 is AS-level pure mathematics: algebra and functions, sequences and series, coordinate geometry, trigonometry, exponentials and logarithms, differentiation, integration and graphs. It assumes all of the Mathematics 1 content, and it shares its specification with the TMUA's Section 1 Part 1, so preparation for one is genuine preparation for the other.

240 original practice questions across 16 sets, grouped by the strand of the published specification each one covers. Every question carries a full worked solution.

  • Each module is separately timed. Time not used on one module is not carried into the next.
  • No calculator. An erasable booklet is provided for working.

Algebra, functions and sequences

Laws of indices, surds, quadratics and the discriminant, completing the square, simultaneous equations, inequalities, polynomials, the factor theorem, sequences, series and the binomial expansion.

What the specification says (11 points)
  • MM1.1 Laws of indices for all rational exponents.
  • MM1.2 Use and manipulation of surds; simplifying expressions that contain surds, including rationalising the denominator.
  • MM1.3 Quadratic functions and their graphs; the discriminant of a quadratic function; completing the square; solution of quadratic equations.
  • MM1.4 Simultaneous equations: analytical solution by substitution, e.g. of one linear and one quadratic equation.
  • MM1.5 Solution of linear and quadratic inequalities.
  • MM1.6 Algebraic manipulation of polynomials, including expanding brackets and collecting like terms; factorisation and simple algebraic division (by a linear polynomial of the form ax + b, and by quadratics of the form ax^2 + bx + c); use of the Factor Theorem and the Remainder Theorem.
  • MM1.7 Qualitative understanding that a function is a many-to-one (or sometimes just a one-to-one) mapping. Familiarity with the properties of common functions, including f(x) = sqrt(x) (which always means the 'positive square root') and f(x) = |x|.
  • MM2.1 Sequences, including those given by a formula for the nth term and those generated by a simple recurrence relation of the form x_(n+1) = f(x_n).
  • MM2.2 Arithmetic series, including the formula for the sum of the first n natural numbers.
  • MM2.3 The sum of a finite geometric series. The sum to infinity of a convergent geometric series, including the use of |r| < 1.
  • MM2.4 Binomial expansion of (1 + x)^n for positive integer n, and for expressions of the form (a + f(x))^n for positive integer n and simple f(x). The notations n! and nCr.

Coordinate geometry and trigonometry

Straight lines, circles, tangents and normals, intersections, the sine and cosine rules, exact values, trigonometric graphs, identities and equations.

What the specification says (9 points)
  • MM3.1 Equation of a straight line, including y - y1 = m(x - x1) and ax + by + c = 0. Conditions for two straight lines to be parallel or perpendicular to each other. Finding equations of straight lines given information in various forms.
  • MM3.2 Coordinate geometry of the circle, using the equation of a circle in the forms (x - a)^2 + (y - b)^2 = r^2 and x^2 + y^2 + cx + dy + e = 0.
  • MM3.3 Use of circle properties: the perpendicular from the centre to a chord bisects the chord; the tangent at any point on a circle is perpendicular to the radius at that point; the angle subtended by an arc at the centre is twice the angle subtended at the circumference; the angle in a semicircle is a right angle; angles in the same segment are equal; opposite angles in a cyclic quadrilateral add to 180 degrees; the angle between the tangent and chord at the point of contact equals the angle in the alternate segment.
  • MM4.1 The sine and cosine rules, and the area of a triangle in the form (1/2)ab sin C. The sine rule includes an understanding of the 'ambiguous' case (angle-side-side). Problems might be set in 2 or 3 dimensions.
  • MM4.2 Radian measure, including use for arc length and area of sector and segment.
  • MM4.3 The values of sine, cosine and tangent for the angles 0, 30, 45, 60 and 90 degrees.
  • MM4.4 The sine, cosine and tangent functions; their graphs, symmetries, and periodicity.
  • MM4.5 Knowledge and use of the equations tan(theta) = sin(theta)/cos(theta) and sin^2(theta) + cos^2(theta) = 1.
  • MM4.6 Solution of simple trigonometric equations in a given interval (this may involve the use of the identities in MM4.5).

Exponentials and logarithms

Exponential graphs, the laws of logarithms, solving exponential equations, and using logarithms to linearise a relationship.

What the specification says (3 points)
  • MM5.1 y = a^x and its graph, for simple positive values of a.
  • MM5.2 Laws of logarithms, including the special cases log_a(1/x) = -log_a(x) and log_a(a) = 1. Questions requiring knowledge of the change of base formula will NOT be set.
  • MM5.3 The solution of equations of the form a^x = b, and equations which can be reduced to this form, including those that need prior algebraic manipulation.

Differentiation and integration

The derivative as a gradient, differentiating powers of x, tangents, normals, stationary points, increasing and decreasing functions, indefinite and definite integration, and areas.

What the specification says (9 points)
  • MM6.1 The derivative of f(x) as the gradient of the tangent to the graph y = f(x) at a point; interpretation of a derivative as a rate of change; second-order derivatives; knowledge of the notation dy/dx, d2y/dx2, f'(x) and f''(x). Differentiation from first principles is EXCLUDED.
  • MM6.2 Differentiation of x^n for rational n, and related sums and differences. This might require some simplification before differentiating.
  • MM6.3 Applications of differentiation to gradients, tangents, normals, stationary points (maxima and minima only), strictly increasing functions and strictly decreasing functions. Points of inflexion will NOT be examined, although a qualitative understanding of points of inflexion in the curves of simple polynomial functions is expected.
  • MM7.1 Definite integration as related to the 'area between a curve and an axis'. The difference between finding a definite integral and finding the area between a curve and an axis is expected to be understood.
  • MM7.2 Finding definite and indefinite integrals of x^n for n rational, n != -1, and related sums and differences, including expressions which require simplification prior to integrating.
  • MM7.3 An understanding of the Fundamental Theorem of Calculus and its significance to integration.
  • MM7.4 Combining integrals with either equal or contiguous ranges.
  • MM7.5 Approximation of the area under a curve using the trapezium rule; determination of whether this constitutes an overestimate or an underestimate.
  • MM7.6 Solving differential equations of the form dy/dx = f(x).

Graphs of functions

Sketching the common function families, transformations, asymptotes, and reading the number and location of solutions off a sketch.

What the specification says (7 points)
  • MM8.1 Recognise and be able to sketch the graphs of common functions that appear in this specification: lines, quadratics, cubics, trigonometric functions, logarithmic functions, exponential functions, square roots, and the modulus function.
  • MM8.2 Knowledge of the effect of simple transformations on the graph of y = f(x): y = a f(x), y = f(x) + a, y = f(x + a) and y = f(ax). Compositions of these transformations. Knowledge and use of the notation f(g(x)).
  • MM8.3 Understand how altering the values of m and c affects the graph of y = mx + c.
  • MM8.4 Understand how altering the values of a, b and c in y = a(x + b)^2 + c affects the corresponding graph.
  • MM8.5 Use differentiation to help determine the shape of the graph of a given function, including finding stationary points (excluding inflexions) and when the graph is increasing or decreasing.
  • MM8.6 Use algebraic techniques to determine where the graph of a function intersects the coordinate axes; appreciate the possible numbers of real roots that a general polynomial can possess.
  • MM8.7 Geometric interpretation of algebraic solutions of equations; relationship between the intersections of two graphs and the solutions of the corresponding simultaneous equations.

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