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Admissions tests / TMUA / Paper 2

20 questions in 75 minutes. Multiple choice. No calculator.

TMUA Paper 2: Mathematical Reasoning

Paper 2 is the paper that decides most TMUA scores, because it is the one nothing at A level prepares you for. It assumes all of the Paper 1 content and adds Section 2: the logic of arguments, mathematical proof, and identifying errors in purported proofs. You are asked to reason about arguments rather than to execute a method.

168 original practice questions across 14 sets, grouped by the strand of the published specification each one covers. Every question carries a full worked solution.

  • Section 2 content is assessed on Paper 2 only, but Paper 1 content is assessed on both papers.
  • Candidates are not expected to recognise or use symbolic logic notation. The reasoning is tested in words.

The logic of arguments

True and false, and, inclusive or, not, conditional statements in all their English forms, converse and contrapositive, necessary and sufficient, quantifiers, and negation.

What the specification says (12 points)
  • Arg1 (Paper 2 only) Understand and be able to use mathematical logic in simple situations: the terms true and false.
  • Arg1 (Paper 2 only) The terms and, or (meaning inclusive or), not.
  • Arg1 (Paper 2 only) Statements of the form 'if A then B'.
  • Arg1 (Paper 2 only) Statements of the form 'A if B'.
  • Arg1 (Paper 2 only) Statements of the form 'A only if B'.
  • Arg1 (Paper 2 only) Statements of the form 'A if and only if B'.
  • Arg1 (Paper 2 only) The converse of a statement.
  • Arg1 (Paper 2 only) The contrapositive of a statement.
  • Arg1 (Paper 2 only) The relationship between the truth of a statement and its converse and its contrapositive. NOTE: candidates will NOT be expected to recognise or use symbolic notation for any of these terms, nor will they be expected to complete formal truth tables.
  • Arg2 (Paper 2 only) Understand and use the terms necessary and sufficient.
  • Arg3 (Paper 2 only) Understand and use the terms for all, for some (meaning for at least one), and there exists.
  • Arg4 (Paper 2 only) Be able to negate statements that use any of the above terms.

Mathematical proof

Direct deductive proof, proof by cases, proof by contradiction, disproof by counterexample, deducing implications, conjecturing from small cases, and ordering a scrambled proof.

What the specification says (8 points)
  • Prf1 (Paper 2 only) Follow, and in simple cases construct, a direct deductive proof ('Since A, therefore B, therefore C, ..., therefore Z, which is what we wanted to prove.').
  • Prf1 (Paper 2 only) Follow, and in simple cases construct, a proof by cases (for example, by considering even and odd cases separately).
  • Prf1 (Paper 2 only) Follow, and in simple cases construct, a proof by contradiction.
  • Prf1 (Paper 2 only) Follow, and in simple cases construct, a disproof by counterexample.
  • Prf2 (Paper 2 only) Deduce implications from given statements.
  • Prf3 (Paper 2 only) Make conjectures based on small cases, and then justify these conjectures.
  • Prf4 (Paper 2 only) Rearrange a sequence of statements into the correct order to give a proof for a statement.
  • Prf5 (Paper 2 only) Problems requiring a sophisticated chain of reasoning to solve.

Identifying errors in proofs

Finding the exact step at which a purported proof fails, and recognising the standard fallacies: dividing by a quantity that may be zero, squaring or taking roots without justification, and assuming the converse.

What the specification says (2 points)
  • Err1 (Paper 2 only) Identifying errors in purported proofs.
  • Err2 (Paper 2 only) Be aware of common mathematical errors in purported proofs; for example, claiming 'if ab = ac, then b = c' or assuming 'if sin A = sin B, then A = B', neither of which are valid deductions.

Reasoning with algebra and number

Section 2 reasoning applied to the Section 1 content: statements about divisibility, parity, inequalities, sequences and functions, where the work is deciding what follows rather than computing.

What the specification says (46 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.
  • MM1.6a Algebraic manipulation of polynomials: expanding brackets and collecting like terms.
  • MM1.6b Algebraic manipulation of polynomials: factorisation and simple algebraic division (by a linear polynomial, including those of the form ax + b, and by quadratics, including those of the form ax^2 + bx + c).
  • MM1.6c Algebraic manipulation of polynomials: 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.
  • M2.1 Order positive and negative integers, decimals and fractions. Understand and use the symbols =, !=, <, >, <=, >=.
  • M2.2 Apply the four operations (addition, subtraction, multiplication and division) to integers, decimals, simple fractions (proper and improper) and mixed numbers - any of which could be positive and negative. Understand and use place value.
  • M2.3 Use the concepts and vocabulary of prime numbers, factors (divisors), multiples, common factors, common multiples, highest common factor, lowest common multiple, and prime factorisation (including use of product notation and the unique factorisation theorem).
  • M2.4 Recognise and use relationships between operations, including inverse operations. Use cancellation to simplify calculations and expressions. Understand and use the convention for priority of operations, including brackets, powers, roots and reciprocals.
  • M2.5 Apply systematic listing strategies (for instance, if there are m ways of doing one task and for each of these there are n ways of doing another task, the total number of ways the two tasks can be done in order is m x n).
  • M2.6 Use and understand the terms square, positive and negative square root, cube and cube root.
  • M2.7 Use index laws to simplify numerical expressions, and for multiplication and division of integer, fractional and negative powers.
  • M2.8 Interpret, order and calculate with numbers written in standard index form (standard form); numbers are written in standard form as a x 10^n, where 1 <= a < 10 and n is an integer.
  • M2.9 Convert between terminating decimals, percentages and fractions. Convert between recurring decimals and their corresponding fractions.
  • M2.10 Use fractions, decimals and percentages interchangeably in calculations. Understand equivalent fractions.
  • M2.11 Calculate exactly with fractions, surds and multiples of pi. Simplify surd expressions involving squares and rationalise denominators.
  • M2.12 Calculate with upper and lower bounds, and use in contextual problems.
  • M2.13 Round numbers and measures to an appropriate degree of accuracy, e.g. to a specified number of decimal places or significant figures. Use inequality notation to specify simple error intervals due to truncation or rounding.
  • M2.14 Use approximation to produce estimates of calculations, including expressions involving pi or surds.
  • M4.1 Understand, use and interpret algebraic notation; for instance ab in place of a x b, 3y in place of y + y + y, a^2 in place of a x a, a^2 b in place of a x a x b, a/b in place of a divided by b.
  • M4.2 Use index laws in algebra for multiplication and division of integer, fractional, and negative powers.
  • M4.3 Substitute numerical values into formulae and expressions, including scientific formulae. Understand and use the concepts and vocabulary: expressions, equations, formulae, identities, inequalities, terms and factors.
  • M4.4 Collect like terms, multiply a single term over a bracket, take out common factors, and expand products of two or more binomials.
  • M4.5 Factorise quadratic expressions of the form x^2 + bx + c, including the difference of two squares. Factorise quadratic expressions of the form ax^2 + bx + c, including the difference of two squares.
  • M4.6 Simplify expressions involving sums, products and powers, including the laws of indices. Simplify rational expressions by cancelling, or factorising and cancelling. Use the four rules on algebraic rational expressions.
  • M4.7 Rearrange formulae to change the subject.
  • M4.8 Understand the difference between an equation and an identity. Argue mathematically to show that algebraic expressions are equivalent.
  • M4.9 Work with coordinates in all four quadrants.
  • M4.10 Identify and interpret gradients and intercepts of linear functions (y = mx + c) graphically and algebraically. Identify pairs of parallel lines and pairs of perpendicular lines, including the relationships between gradients. Find the equation of the line through two given points, or through one point with a given gradient.
  • M4.11 Identify and interpret roots, intercepts and turning points of quadratic functions graphically. Deduce roots algebraically, and turning points by completing the square.
  • M4.12 Recognise, sketch and interpret graphs of: linear functions; quadratic functions; simple cubic functions; the reciprocal function y = 1/x with x != 0; the exponential function y = k^x for positive values of k; trigonometric functions (with arguments in degrees) y = sin x, y = cos x, y = tan x for angles of any size.
  • M4.13 Interpret graphs (including reciprocal graphs and exponential graphs) and graphs of non-standard functions in real contexts to find approximate solutions to problems, such as simple kinematic problems involving distance, speed and acceleration.
  • M4.14 Calculate or estimate gradients of graphs and areas under graphs (including quadratic and other non-linear graphs), and interpret results in cases such as distance-time graphs, speed-time graphs and graphs in financial contexts.
  • M4.15 Set up and solve, both algebraically and graphically, simple equations including simultaneous equations involving two unknowns; this may include one linear and one quadratic equation. Solve two simultaneous equations in two variables (linear/linear or linear/quadratic) algebraically. Find approximate solutions using a graph. Translate simple situations or procedures into algebraic expressions or formulae.
  • M4.16 Solve quadratic equations (including those that require rearrangement) algebraically by factorising, by completing the square, and by using the quadratic formula. Know the quadratic formula. Find approximate solutions of quadratic equations using a graph.
  • M4.17 Solve linear inequalities in one or two variables. Represent the solution set on a number line, or on a graph, or in words.
  • M4.18 Generate terms of a sequence using term-to-term or position-to-term rules.
  • M4.19 Deduce expressions to calculate the nth term of linear or quadratic sequences.

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