10 free TMUA-style practice questions, with worked solutions
Five questions in the style of Paper 1 and five in the style of Paper 2. Allow 35–40 minutes and no calculator, which is about the real pace. Click an answer to see whether you're right and how to solve it.
These are original questions written for Admit Maths, matched to the official TMUA content specification. They are not past papers.
Question 1 · Paper 1 styleBinomial expansion
What is the coefficient of x3 in the expansion of (2 − x)5?
The term in x3 is 5C3 · 22 · (−x)3 = 10 · 4 · (−1)x3 = −40x3. Trap: forgetting the sign of (−x)3 gives 40.
Question 2 · Paper 1 styleExponential equations
What is the sum of all real solutions of 9x − 4 · 3x + 3 = 0?
Let y = 3x, so 9x = y2. Then y2 − 4y + 3 = 0, so y = 1 or 3, giving x = 0 or 1. The sum is 1. Trap: adding the values of y gives 4.
Question 3 · Paper 1 styleIntegration and area
Let A be the total area enclosed between the curve y = x2 − 4x + 3, the x-axis and the lines x = 0 and x = 3. What is A?
The curve crosses the axis at x = 1 and x = 3. With F(x) = x³3 − 2x2 + 3x: from 0 to 1 the integral is 43 (above the axis); from 1 to 3 it is 0 − 43 = −43 (below the axis). Area = 43 + 43 = 83. Trap: integrating straight from 0 to 3 gives 0, because the two parts cancel.
Question 4 · Paper 1 styleCircles
The circle C has equation x2 + y2 − 6x + 4y − 12 = 0. What is the shortest distance from the origin to a point on C?
Completing the square: (x − 3)2 + (y + 2)2 = 25, so the centre is (3, −2) and the radius is 5. The origin is √13 ≈ 3.6 from the centre, which is less than 5, so it lies inside the circle. The nearest point of the circle is 5 − √13 away. Trap: √13 − 5 is negative, so it can't be a distance.
Question 5 · Paper 1 styleTrigonometric equations
How many solutions does 2 sin2x = sin x have in the interval 0 ≤ x < 2π?
Rearrange to sin x (2 sin x − 1) = 0. Either sin x = 0, giving x = 0 or π, or sin x = ½, giving x = π/6 or 5π/6. That is 4 solutions. Trap: dividing both sides by sin x loses the two solutions where sin x = 0.
Question 6 · Paper 2 styleContrapositive
Which statement is the contrapositive of: “If n2 is even, then n is even”?
The contrapositive of “if A then B” is “if not B then not A”. Here A is “n2 is even” and B is “n is even”, so the contrapositive is “if n is odd, then n2 is odd”. A statement and its contrapositive are always both true or both false. Option A is the converse; option B is the inverse.
Question 7 · Paper 2 styleNecessary and sufficient
For a real number x, the condition “x > 2” is ______ for “x2 > 4”.
If x > 2 then certainly x2 > 4, so it is sufficient. But x = −3 has x2 = 9 > 4 without x > 2, so it is not necessary.
Question 8 · Paper 2 styleNegating quantifiers
Which is the negation of: “For every positive integer n, there is a prime p with n < p < 2n”?
To negate, swap “for every” and “there exists”, then negate the condition at the end. “For every n, there exists p with P” becomes “there exists n such that for every p, not P”. And not(n < p < 2n) is “p ≤ n or p ≥ 2n”. (The original is Bertrand's postulate, which is true for n ≥ 2 but fails at n = 1, so the negation happens to be true!)
Question 9 · Paper 2 styleCounterexamples
Which pair (a, b) is a counterexample to the claim: “If a and b are both irrational, then ab is irrational”?
A counterexample needs both numbers irrational but the product rational. √3 and √12 = 2√3 are both irrational, and √3 × √12 = √36 = 6, which is rational. The others give √6, √2 + 2 and √10, all irrational.
Question 10 · Paper 2 styleErrors in proofs
Here is a “proof” that 1 = 2. (I) Let a = b. (II) Then a2 = ab, so a2 − b2 = ab − b2. (III) Factorising, (a + b)(a − b) = b(a − b). (IV) So a + b = b. (V) Since a = b, 2b = b, so 2 = 1. Which step contains the first error?
Step (IV) divides both sides by a − b, which is 0 because a = b. This is the error the specification names: from xy = xz you cannot conclude y = z unless x ≠ 0. (Step (V) also divides by b, which could be 0, but (IV) comes first.)
Score: 0 / 0
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