TMUA Algebra and Functions — Practice Questions by Topic

These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.

MM1Algebra and functions
182 questions129 on Paper 1 · 53 on Paper 2
9 free solutionsThe rest show the correct letter only

Covers: quadratics and the discriminant, linear and quadratic inequalities, polynomials and the factor and remainder theorems, composite and inverse functions, modulus equations and inequalities, binomial expansions, surds and indices, simultaneous equations.

Paper 1 (129)Paper 2 (53)Practise this topic

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Question 1Algebra and functions

The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.

What this topic tests

Algebra and functions, as set out for this test, covers quadratics and the discriminant, inequalities, the factor and remainder theorems, composite and inverse functions, the modulus, binomial expansions, surds and indices, and simultaneous equations.

How it is assessed

This is Section 1, so both papers can test it. This set has questions on Paper 1 and on Paper 2. Each question has five options. A paper is 20 questions in 75 minutes, with no calculator.

This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.

Key methods

The discriminant

For ax² + bx + c = 0, the discriminant b² − 4ac tells you how many real roots there are. An inequality in a parameter often becomes a statement about that discriminant.

Factor and remainder

The remainder when a polynomial is divided by x − a is the value of the polynomial at a. If that value is 0, then x − a is a factor.

A modulus is two cases

An equation with a modulus splits where the inside changes sign. Dropping the modulus and solving one linear equation keeps only one of the cases.

Common mistakes

Dividing away a root

Dividing an equation by x, or by x − a, is only valid when that factor is not zero. The lost root is often one of the options.

The wrong binomial coefficient

The term in x^r inside (a + b)^n uses nCr, not n × r. The powers of a and b must still add to n.

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.

A. −81
B. −27
C. −9
D. −3
E. 9

Answer: C. −9

Worked solution. Expand (2x + c)³ = 8x³ + 12c x² + 6c² x + c³. The x² coefficient gives 12c = −36, so c = −3. Then 6c² = 6 × 9 = 54, which matches the given x coefficient. The constant term is c³ = −27, and this equals 3k, so k = −9.

Why the other options look right. A sets k = 3c³ = −81. B reports the constant term c³ = −27 and does not divide by 3. D reports c = −3. E takes c = 3 from the unsigned x² coefficient, then computes c³/3 = 9.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

Q1. The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.
Free · worked solution included
Answer and worked solution
Answer: C. −9
Worked solution. Expand (2x + c)³ = 8x³ + 12c x² + 6c² x + c³. The x² coefficient gives 12c = −36, so c = −3. Then 6c² = 6 × 9 = 54, which matches the given x coefficient. The constant term is c³ = −27, and this equals 3k, so k = −9.
Why the other options look right. A sets k = 3c³ = −81. B reports the constant term c³ = −27 and does not divide by 3. D reports c = −3. E takes c = 3 from the unsigned x² coefficient, then computes c³/3 = 9.
Q2. The cubic x³ − 3x² − 6x + c, where c is a constant, has (x − 1) as a factor. Which product is a complete factorisation of this cubic?
Free · worked solution included
Answer and worked solution
Answer: A. (x − 1)(x − 4)(x + 2)
Worked solution. The factor theorem gives 1 − 3 − 6 + c = 0, so c = 8. Dividing x³ − 3x² − 6x + 8 by (x − 1) leaves x² − 2x − 8. The pair of numbers multiplying to −8 and adding to −2 is −4 and 2, so x² − 2x − 8 = (x − 4)(x + 2). The complete factorisation is (x − 1)(x − 4)(x + 2).
Why the other options look right. B uses the pair −8 and 1, which add to −7. C factors x² − 2x − 8 with the pair 4 and −2, which add to +2. D uses the pair 8 and −1, which add to 7, and so repeats (x − 1). E uses the pair −4 and −2, which multiply to +8 rather than −8.
Q3. The roots of x³ − 5x² + px + q = 0 have product −5, and the sum of the products of the roots taken two at a time is −1. Find p + q.
Free · worked solution included
Answer and worked solution
Answer: C. 4
Worked solution. For x³ − (sum)x² + (pairwise sum)x − (product) = 0, the pairwise sum is p and the product is −q. Thus p = −1 and −q = −5, so q = 5. Therefore p + q = −1 + 5 = 4. (The cubic is x³ − 5x² − x + 5 = (x + 1)(x − 1)(x − 5); its roots −1, 1 and 5 have sum 5, product −5 and pairwise sum −1.)
Why the other options look right. A takes the product to be q rather than −q, so q = −5 and p + q = −1 + (−5) = −6. B gives p = −1 only. D gives q = 5 only. E takes p to be the sum of the roots, 5, and computes 5 + 5 = 10.
Q4. The equation x² − kx + (2k − 4) = 0 has one root equal to twice the other. Find the sum of the possible values of k.
Free · worked solution included
Answer and worked solution
Answer: D. 9
Worked solution. Let the roots be t and 2t. Their sum is 3t = k and their product is 2t² = 2k − 4. Substituting k = 3t gives 2t² = 6t − 4, so t² − 3t + 2 = 0, hence (t − 1)(t − 2) = 0. The solutions are t = 1, giving k = 3 (roots 1 and 2), and t = 2, giving k = 6 (roots 2 and 4). Both give real roots of the stated form. The sum of the possible values is 3 + 6 = 9.
Why the other options look right. A reports only k = 3, from the roots 1 and 2. B reads the condition as the two roots being equal and sets the discriminant k² − 4(2k − 4) = (k − 4)² to zero, giving k = 4. C reports only k = 6, from the roots 2 and 4. E multiplies the two possible values instead of adding them, 3 × 6 = 18.
Q5. The numbers α and β are the roots of x² − 4x − 1 = 0. Find α³ + β³.
Free · worked solution included
Answer and worked solution
Answer: E. 76
Worked solution. From the quadratic, α + β = 4 and αβ = −1. Then α³ + β³ = (α + β)((α + β)² − 3αβ) = 4(16 − 3(−1)) = 4(16 + 3) = 4 × 19 = 76.
Why the other options look right. A uses the product of the roots as +1, giving 4(16 − 3) = 52. B assumes α³ + β³ = (α + β)³, dropping the cross term −3αβ(α + β), and gets 4³ = 64. C replaces 3αβ by αβ, giving 4(16 − (−1)) = 68. D computes (α + β)(α² + β²) = 4(16 + 2) = 72.
Q6. Let f(x) = 3x − 1 and g(x) = x² + 2x. Find g(f(x)).
Free · worked solution included
Answer and worked solution
Answer: E. 9x² − 1
Worked solution. Substitute f(x) into g: g(3x − 1) = (3x − 1)² + 2(3x − 1) = 9x² − 6x + 1 + 6x − 2. The linear terms cancel, leaving 9x² − 1.
Why the other options look right. A is (3x − 1)², with the term 2f(x) omitted. B expands (3x − 1)² as 9x² + 6x + 1 and then adds 2(3x − 1), getting 9x² + 12x − 1. C subtracts 2f(x) instead of adding it, getting 9x² − 12x + 3. D is f(g(x)) = 3(x² + 2x) − 1.
Q7. Find every real x for which (x² − x − 6)/(x − 3) is at least 2.
Free · worked solution included
Answer and worked solution
Answer: D. x ≥ 0 and x ≠ 3
Worked solution. For x ≠ 3 the numerator factors as (x − 3)(x + 2), so the quotient equals x + 2. The inequality becomes x + 2 ≥ 2, that is x ≥ 0, and x = 3 is excluded because the original denominator is zero. The solution is x ≥ 0 and x ≠ 3.
Why the other options look right. A cancels the factor x − 3 and then leaves x = 3 in the solution. B keeps only the part of x ≥ 0 on which the denominator is positive, namely x > 3. C multiplies both sides by x − 3 with no sign check, obtaining x(x − 3) ≥ 0, and then excludes x = 3, which gives x ≤ 0 or x > 3. E reverses the simplified inequality to x + 2 ≤ 2, which gives x ≤ 0.
Q8. The polynomial p(x) = x³ − 9x² + ax + b has roots r, s and t. Given that r + s = 5 and rs + rt + st = 26, find a and b.
Free · worked solution included
Answer and worked solution
Answer: D. a = 26, b = −24
Worked solution. For x³ − 9x² + ax + b, Vieta's formulas give r + s + t = 9 and rs + rt + st = a. Thus a = 26 and t = 9 − 5 = 4. Then rs + t(r + s) = 26 becomes rs + 20 = 26, so rs = 6. The product of the roots is 6 × 4 = 24, and b = −(rst) = −24. So a = 26, b = −24.
Why the other options look right. A uses r + s = 5 as the coefficient a. B uses the sum of all three roots, 9, as the coefficient a. C takes b to be the product rst rather than −rst. E sets b = −rs and forgets to multiply by t.
Q9. What is the largest real root of x³ − 5x² − 8x + 12 = 0?
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Answer
Answer: D. 6
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Q10. For a real number of hours x, the cost in pounds of running a kiln is C(x) = 2x² − 12x + 25. Find the minimum cost.
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Answer
Answer: D. 7
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Q11. Solve the inequality |x² − 4x + 1| < 2.
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Answer
Answer: E. 2 − √5 < x < 1 or 3 < x < 2 + √5
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Q12. Find the number of real solutions of the equation |x + 3| = 7 − x − 2x².
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Answer
Answer: D. 2
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Q13. The graph of y = 2x² + x − 6 lies on or below the x-axis for which values of x?
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Answer
Answer: D. −2 ≤ x ≤ 3/2
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Q14. The roots of x³ − 5x² + 2x + 8 = 0 are all nonzero. Find the sum of the reciprocals of the roots.
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Answer
Answer: C. −1/4
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Q15. Which list gives every real solution of |3x − 4| = |x + 6|?
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Answer
Answer: B. −1/2 and 5
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Q16. The number z is nonzero and z + 1/z = 3. What is z³ + 1/z³?
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Answer
Answer: B. 18
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Q17. The graph of y = p(x − 2)² + q has its vertex at (2, −3) and passes through (4, 5). What is p + q?
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Answer
Answer: B. −1
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Q18. Find the complete set of real x for which (x² − 4)/(x − 3) < 0.
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Answer
Answer: A. x < −2 or 2 < x < 3
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Q19. The polynomial f(x) = x³ − 4x² + x + k passes through the point (3, 4). What is k?
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Answer
Answer: E. 10
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Q20. Write floor(t) for the greatest integer less than or equal to t. What is the sum of all real solutions of x² = floor(x) + 2?
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Answer
Answer: A. 1 + √3
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Q21. The expansion of (ax + b)³ is 8x³ − q x² + 24x − 8, where a, b and q are real and a > 0. Find q.
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Answer
Answer: D. 24
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Q22. The number a is chosen so that x + 3 is a factor of 2x³ + 3x² − 11x + a. Which product is a complete factorisation of this cubic?
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Answer
Answer: A. (x + 3)(x − 2)(2x + 1)
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Q23. The coefficient of x³ in the expansion of (1 + 2x + 3x²)⁵ is twice the coefficient of x⁴ in the expansion of (1 − a x²)⁵. Find every possible value of the constant a.
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Answer
Answer: C. ±√10
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Q24. S is the set of real x that satisfy both x² − 7x + 10 < 0 and 3x − 1 > 8. Which single inequality describes S?
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Answer
Answer: C. x² − 8x + 15 < 0
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Q25. The solution set of x² + bx + c < 0 is p < x < q, where b, c, p and q are real constants with c < 0 and p < q. What is the solution set of cx² + bx + 1 < 0?
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Answer
Answer: C. x < 1/p or x > 1/q
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Q26. The curves y = x² − 4x + a and y = 2x − x² meet at two distinct points, and both points have positive x-coordinates. Here a is a real constant. What is the complete set of possible values of a?
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Answer
Answer: E. 0 < a < 9/2
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Q27. The polynomial f(x) = x³ + ax² + bx + c uses each of 2, 3 and 4 once as a, b and c. Let R be the remainder on division by x + 1, and let S be the remainder on division by x + 2. What is the largest possible value of R − S?
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Answer
Answer: C. 5
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Q28. The real numbers x and y satisfy |x − 1| ≤ 3 and |y + 1| ≤ 5. What is the greatest possible value of |xy|?
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Answer
Answer: D. 24
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Q29. The quadratic f has a turning point at (2, −3) and the graph of y = f(x) passes through (5, 15). Which expression is f(x)?
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Answer
Answer: B. 2x² − 8x + 5
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Q30. Find the complete set of real k for which x² + kx + (k + 3) is positive for every real x.
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Answer
Answer: C. −2 < k < 6
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Q31. (2x − 1) and (x + 3) are factors of 2x³ + px² + q, which has no x term. What is the value of 3p + q?
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Answer
Answer: E. 84/5
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Q32. Find the complete set of values of x for which (x + 6)(x + 1)(3 − x) > 0 and (x + 4)(x − 2) < 0.
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Answer
Answer: A. −1 < x < 2
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Q33. The curve y = px² + 8x − q, where p and q are constants, has a line of symmetry x = −1/3 and touches the x-axis at exactly one point. What is the value of p + 3q?
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Answer
Answer: D. 8
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Q34. The function f is defined for all real x by f(x) = (p − x)(x + 3), where p is a real constant. Find the complete set of values of p for which the maximum value of f(x) is less than p + 6.
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Answer
Answer: A. −5 < p < 3
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Q35. The quadratic x² − 8x + 4 factorises as (x − α)(x − β), where α and β are positive real numbers. Which quadratic factorises as (x − √α)(x − √β)?
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Answer
Answer: C. x² − √12 x + 2
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Q36. What is the coefficient of x⁵ in the expansion of (1 + x)⁹ (1 − x)⁸?
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Answer
Answer: A. 28
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Q37. What is the lowest positive integer n for which n² − 30n − 30 is positive?
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Answer
Answer: C. 31
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Q38. For how many real values of a does the equation (x − a)(x² − 4x + a) = 0 have exactly two distinct real solutions?
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Answer
Answer: D. 3
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Q39. The function f satisfies f(mn) = f(m)f(n) whenever mn is a multiple of 5, and f(mn) = mn otherwise, for all positive integers m and n. Given that f(25) + f(4) − f(20) = 0, what is f(5)?
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Answer
Answer: D. 2
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Q40. In the expansion of (a + bx)ⁿ, where a and b are positive and n is a positive integer, the third term in ascending powers of x is 90x², the fourth term in ascending powers of x is 240x³, and the fourth term in descending powers of x is also 240x³. What is (a/b)²?
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Answer
Answer: E. 1/4
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Q41. Let a > 0 and suppose that a + 4/a = 5. What is the value of a² + 16/a²?
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Answer
Answer: B. 17
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Q42. The real numbers a and b are nonzero and satisfy (a³ + 4/b³)(4/a³ − b³) = 6. What is the least possible value of ab?
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Answer
Answer: B. −2
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Q43. The line y = x − 2 meets the curve x² + xy − y² = 7 at two points. What is the sum of the x-coordinates of those points?
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Answer
Answer: C. −2
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Q44. What is the complete set of real x for which (x² − 4)(x − 1) < 0?
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Answer
Answer: E. x < −2 or 1 < x < 2
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Q45. The linear factor x − 1 divides x³ + m x² − x(m − 2)² + 4. What is the sum of the possible values of m?
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Answer
Answer: D. 5
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Q46. The roots of 3x² − 10x + k = 0 differ by 1. What is k?
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Answer
Answer: C. 91/12
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Q47. How many real roots does (x² + 3x)² = 6 have?
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Answer
Answer: C. 2
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Q48. For which non-zero real numbers k does the quadratic equation k x² + 4x + k − 3 = 0 have two distinct real roots?
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Answer
Answer: B. −1 < k < 4 and k ≠ 0
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Q49. S is the set of real x that satisfy both x² − 6x + 5 < 0 and x² − 2x − 3 < 0. Which single inequality describes S?
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Answer
Answer: C. x² − 4x + 3 < 0
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Q50. For integers a and b, which condition is enough to guarantee that 2^a × 3^b / (6^{a − b} × 9^b) is an integer?
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Answer
Answer: C. a ≤ 0 and b ≥ 0
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Q51. The polynomial (2x² − x + 3)(ax + 1) is divided by x − 1 and the remainder is 24. Find a.
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Answer
Answer: B. 5
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Q52. The inequality |x − 3| + |x + 1| ≤ 8 holds. What is the greatest possible value of |x|?
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Answer
Answer: C. 5
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Q53. Solve 5x − 7 = 2x + 8.
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Answer and worked solution
Answer: C. 5
Worked solution. Subtract 2x from both sides: 3x − 7 = 8. Add 7: 3x = 15. Divide by 3: x = 5. Check: 5(5) − 7 = 18 and 2(5) + 8 = 18.
Why the other options look right. A moves −7 to the right without changing its sign, giving 3x = 8 − 7 = 1 and hence x = 1/3. B collects the x terms as 5x − 2x = 3x and then reports the coefficient 3 as the answer, without solving 3x = 15. D is 15, the value of 3x before dividing by 3. E collects the x terms as 2x − 5x = −3x and still adds 8 and 7, so −3x = 15 and x = −5.
Q54. A function f is defined for every real x except x = 3, and it satisfies 3f(x) − 4f((3x + 4)/(x − 3)) = (52 − 7x)/4. Find f(4).
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Answer
Answer: E. 6
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Q55. a and b are non-zero integers. Dividing x² − ax − a² by x − b leaves remainder −5. Also, 2x − a is a factor of 2bx² − 3x − 9. Find a + b.
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Answer
Answer: C. −2
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Q56. Let n be an odd positive integer. What is the remainder when (x + 1)ⁿ + (x − 1)ⁿ is divided by x² − 1?
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Answer
Answer: C. 2ⁿ x
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Q57. The graph of a quadratic f crosses the x-axis at x = −1 and x = 5 and passes through (1, 16). Which expression equals f(x)?
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Answer
Answer: D. −2x² + 8x + 10
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Q58. Which set is exactly the set of real x that satisfy both x² − 2x − 24 < 0 and x² − 9 > 0?
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Answer
Answer: A. −4 < x < −3 or 3 < x < 6
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Q59. The cubic x³ − 2x² + 5x + k has remainder 4 on division by x − 1. What is the remainder on division by x − 2?
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Answer
Answer: B. 10
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Q60. Solve the inequality |4x + 2| ≤ 10.
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Answer
Answer: A. −3 ≤ x ≤ 2
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Q61. The real numbers x and y satisfy x² + y² = 34 and x + y = 8, with x > y. Find x − y.
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Answer
Answer: B. 2
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Q62. The function f is defined by f(x) = (4x − 1)/(x + 2) for x ≠ −2. Find f⁻¹(3).
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Answer
Answer: C. 7
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Q63. How many distinct real roots does x⁴ − 2x² − 8 = 0 have?
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Answer
Answer: C. 2
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Q64. Evaluate 8 to the power 2/3, plus 25 to the power −1/2.
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Answer
Answer: A. 21/5
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Q65. Which inequality describes every real x that satisfies both (x − 1)(x − 10) < 0 and (x − 4)(x − 13) < 0?
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Answer
Answer: B. 4 < x < 10
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Q66. The polynomial x³ − 3x + 4 leaves remainder 2 when it is divided by x − 1. What is the remainder when it is divided by x + 1?
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Answer
Answer: E. 6
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Q67. Which inequality describes the solution of |2x + 3| ≤ 11?
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Answer
Answer: C. −7 ≤ x ≤ 4
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Q68. The real numbers x and y satisfy x² + y² = 65 and x + y = 11, with x > y. Find x − y.
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Answer
Answer: B. 3
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Q69. The function f is defined by f(x) = (4x − 6)/(x − 2) for x not equal to 2. Find the value of the inverse of f at 3.
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Answer: D. 0
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Q70. When 1/(√10 − 2) is written with a rational denominator, which expression is obtained?
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Answer: E. (√10 + 2)/6
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Q71. How many distinct real roots does the equation x⁴ − 7x² + 12 = 0 have?
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Answer: A. 4
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Q72. Which set is exactly the set of real x that satisfy (x − 1)/(x − 4) ≥ 2?
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Answer: E. 4 < x ≤ 7
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Q73. Dividing x³ − 3x² + 4x + 1 by x − 1 leaves remainder 3. Find the remainder on division by x + 1.
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Answer
Answer: B. −7
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Q74. Which inequality is equivalent to |2x − 8| ≤ 6?
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Answer: B. 1 ≤ x ≤ 7
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Q75. Suppose x and y are real numbers with x² + y² = 52, x + y = 10 and x > y. Find x − y.
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Answer: E. 2
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Q76. The function f is given by f(x) = (4x + 1)/(x − 2), where x is not 2. Find the value of the inverse of f at 5.
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Answer: A. 11
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Q77. Write 1/(√13 − 3) with a rational denominator.
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Answer: D. (√13 + 3)/4
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Q78. How many distinct real solutions does x⁴ − 11x² + 18 = 0 have?
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Answer: E. 4
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Q79. Evaluate 32 to the power 3/5, minus 27 to the power −2/3.
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Answer: D. 71/9
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Q80. Which statement describes every real x that satisfies both (x + 4)(x − 5) < 0 and (x + 1)(x − 3) > 0?
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Answer: E. −4 < x < −1 or 3 < x < 5
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Q81. Find the remainder when 2x³ − 5x² + 3x + 7 is divided by x + 2.
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Answer: B. −35
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Q82. Which inequality is equivalent to |3x + 6| < 15?
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Answer: A. −7 < x < 3
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Q83. The real numbers x and y satisfy x + y = 9 and x² + y² = 53, with x > y. Find x − y.
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Answer: D. 5
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Q84. Which of these numbers is greatest?
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Answer: C. √77
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Q85. The function f is given by f(x) = (2x + 5)/(x − 3), where x is not 3. Find the value of the inverse of f at 4.
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Answer: A. 17/2
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Q86. The number (√5 + √2)/(√5 − √2) can be written as p + q√10, where p and q are rational. What is p + q?
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Answer: E. 3
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Q87. How many distinct real solutions does x⁴ − 8x² − 9 = 0 have?
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Answer: B. 2
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Q88. Both (x − 1)(x − 9) < 0 and (x − 2)(x − 5) < 0. Which of these describes every such real x?
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Answer: C. 2 < x < 5
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Q89. Dividing x³ + 3x² − 2x + 4 by x − 1 leaves remainder 6. What remainder is left when the same cubic is divided by x + 2?
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Answer: E. 12
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Q90. Solve the inequality |x − 2| > |2x + 1|.
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Answer: D. −3 < x < 1/3
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Q91. Suppose x and y are real, with x² + y² = 85, x + y = 11 and x > y. Find x − y.
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Answer: C. 7
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Q92. Let f(x) = (3x − 2)/(x + 4), where x ≠ −4. Find f⁻¹(5).
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Answer: A. −11
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Q93. How many distinct real solutions does x⁴ − 29x² + 100 = 0 have?
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Answer: C. 4
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Q94. Both (x − 1)(x − 8) < 0 and (x − 4)(x − 10) < 0 hold. Which inequality describes exactly those real x?
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Answer: B. 4 < x < 8
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Q95. Dividing x³ − x² + 3x − 5 by x − 1 leaves remainder −2. Find the remainder on division by x + 1.
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Answer
Answer: E. −10
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Q96. Solve the inequality |x² − 4| < 3x.
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Answer: A. 1 < x < 4
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Q97. The function f is defined by f(x) = (kx + 5)/(2x − 3) for x ≠ 3/2, where k is a constant. For which k does f(f(x)) = x hold for every x in the domain of f with f(x) ≠ 3/2?
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Answer: A. k = 3
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Q98. How many distinct real solutions does x⁶ − 7x³ − 8 = 0 have?
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Answer: D. 2
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Q99. Which statement describes every real x that satisfies x² − 5|x| + 4 < 0?
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Answer
Answer: E. −4 < x < −1 or 1 < x < 4
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Q100. What remainder does the polynomial 2x³ − x² + 5x − 3 leave when it is divided by x − 2?
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Answer: B. 19
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Q101. Find the set of real x satisfying |3x − 4| ≤ 8.
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Answer: E. −4/3 ≤ x ≤ 4
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Q102. The real numbers x and y satisfy x² + y² = 58 and x + y = 10, with x > y. What is x − y?
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Answer: B. 4
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Q103. The function f is defined by f(x) = (3x + 2)/(x − 4) for x ≠ 4. What is the value of the inverse of f at 2?
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Answer
Answer: E. −10
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Q104. How many distinct real roots does x⁴ − 26x² + 25 = 0 have?
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Answer
Answer: B. 4
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Q105. What is the maximum value of −2x² + 12x − y² + 8y − 30?
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Answer: C. 4
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Q106. Find the sum of the real solutions of the equation x² + 2√(x² + 8x) = 15 − 8x.
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Answer
Answer: A. −8
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Q107. The expansion of (√2 − x)⁶ is 8 − b x + 60 x² − d x³ + 30 x⁴ − f x⁵ + x⁶, where b, d and f are positive. What is b + d + f?
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Answer
Answer: D. 70√2
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Q108. The function f is defined for every real x except x = 2, and 2 f(x) − 3 f((2x + 6)/(x − 2)) = 5x + 1. What is f(5)?
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Answer
Answer: B. −27
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Q109. The coefficient of x⁴ in the expansion of (3 + x²)⁵ is equal to 5 times the coefficient of x² in the expansion of (1 + ax)⁴. What are the possible values of a?
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Answer
Answer: B. ±3
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Q110. The coefficient of x³ in the expansion of (3 + bx)⁵ is 6 times the coefficient of x² in the expansion of (1 + bx)⁴. Given that b is not 0, what is b?
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Answer
Answer: C. 2/5
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Q111. The numbers r and s are non-zero integers. Which condition guarantees that (24^r × 3^s) / 36^r is an integer?
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Answer: C. 0 < r ≤ s
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Q112. Suppose u = 5⁶ and v = 6⁵. Which expression is equal to 30³⁰?
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Answer
Answer: E. u⁵ v⁶
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Q113. Find the set of values of x that satisfy both (5x + 2)/(x − 2) < 4 and (x + 3)(x − 5) > 0.
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Answer
Answer: E. −10 < x < −3
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Q114. Find the set of values of x that satisfy 4/(x + 2) > (x − 3)/x.
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Answer
Answer: E. −2 < x < −1 or 0 < x < 6
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Q115. The number p is a positive constant. Find the set of values of x that satisfy (x + p)/(x + 9p) < p/x.
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Answer
Answer: D. −9p < x < −3p or 0 < x < 3p
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Q116. The curve y = x³ + kx² − 32 has exactly two distinct real x-intercepts for one value of the constant k. Find that value of k.
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Answer
Answer: C. 6
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Q117. The equation 3x² + 8x − k = 0, where k is a constant, has two distinct real roots, and one root is 3 more than the other. Find k.
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Answer
Answer: D. 17/12
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Q118. Find the minimum value of 2(3^{sin x}) − 9^{sin x} + 16/3.
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Answer
Answer: A. 7/3
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Q119. When (x² − 3x + 1) is multiplied by (px + 1), and the product is divided by (x − 2), the remainder is 10. Find p.
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Answer
Answer: C. −11/2
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Q120. The equations x² − xy = 4 and y − 2x = p, where p is a real constant, have two distinct real solutions. Which statement describes every value p can take?
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Answer
Answer: C. p < −4 or p > 4
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Q121. What is the sum of the real solutions of |x| − 6 = |2x + 18|?
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Answer
Answer: D. −20
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Q122. How many distinct real solutions does the equation |x² − 1| = 2|x| have?
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Answer: E. 4
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Q123. The non-zero real numbers a and b satisfy (a³ + 9/b³)(b³ − 9/a³) = 24. What is the least possible value of ab?
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Answer
Answer: C. −3^1/3
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Q124. The function f satisfies 3f(x) + f(−x) = 4x + 12 for every real x. Find f(1).
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Answer
Answer: B. 5
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Q125. The function f satisfies 2f(x) − f((2x + 5)/(x − 2)) = 3x − 3 for every real x except x = 2. What is f(3)?
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Answer
Answer: B. 14
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Q126. The function f is given by f(x) = (x − k)/(x² − 6x − k), for the real x at which it is defined, where k is a constant. The range of f is all of the real numbers. What are the possible values of k?
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Answer
Answer: D. 0 < k < 7
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Q127. The curve y = (x − 1)(x − 2)(x − 4)(x + 2)(x + 5)(x − 6) and the line y = 5x − 8 meet at exactly six points. What is the sum of the x-coordinates of these six points?
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Answer
Answer: C. 6
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Q128. A cubic polynomial f satisfies f(1) = 1, f(4) = 16 and f(−2) = 4, and the coefficient of x³ is 3. What is f(3)?
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Answer
Answer: E. −21
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Q129. The polynomial f has degree 1. The composition f(x² − 6x + 13) has exactly two roots, at x = 1 and x = 5, and its minimum value is −12. Which statement about f(x² + 4x + 8) is true?
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Answer
Answer: B. The roots are x = −4 and x = 0, and the minimum value is −12.
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Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

Q1. The quadratic y = f(x) has a turning point at (−1, 5) and passes through (2, 14). What is f(x)?
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Answer
Answer: C. x² + 2x + 6
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Q2. The function f is odd and defined for all real x, and f(x) = x² − 4x for every x > 0. How many real solutions does the equation f(x) = 3 have?
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Answer: C. 3
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Q3. The real numbers x and y satisfy 0 < x < y. Which one of the following must be true?
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Answer: E. x/(x + 1) < y/(y + 1)
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Q4. Let f(x) = x² − 2ax + 3, where a is a real constant. For which values of a is f(x) > 0 for every x with 0 ≤ x ≤ 2?
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Answer
Answer: C. a < √3
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Q5. Find the constant term in the expansion of (x⁵ − x⁻³)⁸.
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Answer
Answer: B. −56
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Q6. The equation √(x + p) − √x = p has at least one real solution for x, where p is a real constant. What is the complete set of possible values of p?
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Answer
Answer: B. −1 ≤ p ≤ 1
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Q7. Find the coefficient of x³ in the expansion of x(3x + 1/x)⁴.
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Answer: D. 108
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Q8. (x − 1) and (2x + 3) are factors of 2x³ + p x² + q. There is no x term. What is p + q?
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Answer
Answer: A. −2
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Q9. a, b and c are real numbers with a < b < c < 0. Which of the following must be true? I. a/c > 1 II. a + b < c III. ac < bc
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Answer: C. I and II only
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Q10. The quadratic y = p x² + q x + q, with p > 0, meets the x-axis at two distinct points. Which condition describes the complete set of possible pairs (p, q)?
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Answer: E. q < 0 or q > 4p
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Q11. Consider a|x| + 1 ≤ |x − 4|, where a is a real constant. Which description gives the complete set of a such that the inequality holds for every real x?
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Answer
Answer: C. a ≤ −1/4
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Q12. Find the value of √(4 − 2√3) + √(7 − 4√3).
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Answer: A. 1
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Q13. Find the complete set of values of k for which the line y = x − 1 crosses or touches the curve y = x² + kx + 5.
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Answer
Answer: D. k ≤ 1 − 2√6 or k ≥ 1 + 2√6
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Q14. The real numbers a, b and c satisfy both 0 < a + 2b < c and 0 < a + 2c < b. Which of the following must be true? I: a < 0. II: b < c. III: a + b + c > 0.
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Answer: D. I and III only
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Q15. Let f(x) = ax³ + bx² + cx + d and g(x) = px³ + qx² + rx + s. If f(x) ≥ g(x) for every x ≥ 0, which of the following must be true? I: a ≥ p. II: d ≥ s. III: c ≥ r.
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Answer: C. I and II only
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Q16. Which of the following statements about a polynomial f with real coefficients are sufficient for f(x) = 0 to have a real solution? I: f(0) < 0 and the leading coefficient of f is positive. II: f(x) = x⁴ + ax + b for some real numbers a and b. III: f'(x) > 0 for every real x.
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Answer: B. I and III only
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Q17. A region R consists of all points (x, y) with x² ≤ y ≤ x + 2. Which of the following is true for every point of R? I: 0 ≤ y ≤ 4. II: xy ≥ −1. III: y ≥ x.
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Answer: D. I and II only
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Q18. For how many real x does there exist a real y such that (x² − 1)y + 1 is negative?
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Answer: D. all real x except exactly two
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Q19. Which describes the set of real x for which both |x + 2| < |x + 8| and |x + 8| < |x − 4| are true?
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Answer: C. an interval of length 3
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Q20. Given that 1/(√x − 4) − 1/(√x + 4) = 2/9, and √x > 4, what is the value of x?
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Answer
Answer: E. 52
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Q21. For which values of m does x⁴ − (m + 1)x² + m = 0 have four distinct real roots?
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Answer: A. m > 0 and m ≠ 1
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Q22. A polynomial p(x) satisfies p(1) = 4 and p(−2) = −5. Which of the following must be true?
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Answer: A. p(x) = (x − 1)(x + 2) q(x) + 3x + 1 for some polynomial q(x).
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Q23. The real numbers p and q satisfy p ≥ q. Consider these statements: 1. −q ≥ −p 2. pq ≥ q² 3. p + q ≥ 2q Which of them must be true?
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Answer: D. 1 and 3 only
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Q24. The positive real numbers a, b and c are such that x³ − a x² + b x − c = 0 has three real roots, all positive. Which statement describes the real roots of x³ + a x² + b x + c = 0?
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Answer
Answer: B. It has three real roots, all negative.
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Q25. For which real values of k does the equation |x − 1| = kx + 2 have exactly two distinct real solutions?
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Answer: C. −1 < k < 1
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Q26. Let p(x) = x⁴ + ax³ + bx² + cx + 3, where a, b and c are nonzero integers, a is positive, and |b| > |c| > |a|. Dividing p(x) by x − 1 leaves remainder R, and dividing by x + 1 leaves remainder S. Given that S = 2R and S − R = 6, what is c?
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Answer
Answer: E. −4
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Q27. Find the value of √(31 + 12√3) − √(19 − 8√3).
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Answer
Answer: A. 4√3 − 2
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Q28. In the expansion of (2x³ + x⁻¹)⁶, which term has the greatest coefficient?
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Answer
Answer: B. 240x¹⁰
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Q29. Let n be a positive integer. For which n is x² − 1 a factor of x³ⁿ − 2xⁿ + 1?
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Answer
Answer: C. Every even positive integer n
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Q30. Dividing x⁴ + ax³ + bx² + cx + 2 by x − 1 leaves remainder R, and dividing it by x + 1 leaves remainder S. If S − R = 10 and S = 3R, find b.
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Answer
Answer: A. 7
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Q31. Exactly one of the following claims has a counterexample among ordered pairs of real numbers (x, y). Which claim is it?
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Answer: E. If x > y, then x² > y²
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Q32. The functions f and g are defined by f(x) = x² − 4 and g(x) = x². How many distinct real solutions does g(f(x)) = 16 have?
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Answer
Answer: D. 3
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Q33. The polynomial p(x) = x⁴ + ax³ + bx² + cx + 5 leaves remainder R when divided by x − 1 and remainder S when divided by x + 1. If S − R = 12 and S = 4R, find b.
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Answer: E. 4
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Q34. A rectangle has perimeter 40, and one of its sides has length at least 14. What is the greatest possible area of the rectangle?
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Answer: B. 84
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Q35. The functions f and g are defined by f(x) = x² − 9 and g(x) = |x| − 2. For which values of the constant c does the equation f(g(x)) = c have exactly four real solutions?
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Answer
Answer: C. −9 < c < −5
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Q36. Dividing x⁴ + ax³ + bx² + cx + 2 by x − 1 leaves remainder R, and dividing it by x + 1 leaves remainder S. If S = 3R and S exceeds R by 8, what is b?
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Answer: D. 5
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Q37. Let f(x) = x² and let g(x) = x² − 3. How many real numbers x satisfy f(g(x)) = 1?
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Answer: D. 4
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Q38. Suppose r and s are positive integers satisfying 2r + s = 20. What is the greatest possible value of r² × s?
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Answer: A. 294
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Q39. The polynomial f(x) = x⁴ + px³ + qx² + rx + 4 leaves remainder R on division by x − 1 and remainder S on division by x + 1. If S = 3R and S − R = 16, what is q?
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Answer: A. 11
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Q40. The functions f and g are defined for all real x by f(x) = 2x + 3 and g(x) = x² + c, where c is a constant. For which value of c does the equation f(g(x)) = g(f(x)) have exactly one real solution?
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Answer
Answer: B. −12
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Q41. The polynomial p(x) = x³ + ax² + bx − 6 has x − 2 as a factor and leaves remainder −12 when divided by x + 1. Find a.
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Answer
Answer: B. −2
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Q42. The functions f and g are given by f(x) = x³ − x and g(x) = x + 1. How many real solutions does f(g(x)) = 0 have?
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Answer
Answer: D. 3
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Q43. Dividing p(x) = x⁴ + ax³ + bx² + cx + 5 by x − 1 leaves remainder R, and dividing it by x + 1 leaves remainder S. Given that S − R = 4 and S = 3R, find b.
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Answer
Answer: E. −2
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Q44. Let f(x) = x² − 2x. How many distinct real numbers x satisfy f(f(x)) = 3?
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Answer: B. 3
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Q45. The polynomial p(x) = x⁴ + ax³ + bx² + cx + 3 leaves remainder R on division by x − 1 and remainder S on division by x + 1. Given that S − R = 6 and S = 3R, what is b?
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Answer
Answer: D. 2
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Q46. The functions f and g are defined by f(x) = x² and g(x) = x² − 5. How many real solutions does f(g(x)) = 36 have?
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Answer: D. 2
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Q47. For how many positive integers n is n² < 9n + 52?
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Answer: E. 12
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Q48. The polynomial x³ + a x² + b x − 6 has (x − 1) and (x + 2) as factors, where a and b are constants. What is the value of a?
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Answer
Answer: D. 4
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Q49. For how many integers k does the equation x² − kx + 2k = 0 have real solutions that are all integers?
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Answer: A. 4
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Q50. For which real numbers x is x/(x² + 1) ≤ 1/4?
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Answer: D. x ≤ 2 − √3 or x ≥ 2 + √3
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Q51. For which real values of a does the equation ax² − 2x + (a − 1) = 0 have at least one real solution? When a = 0, read the equation as the remaining linear equation.
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Answer
Answer: B. (1 − √5)/2 ≤ a ≤ (1 + √5)/2
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Q52. Which term has the greatest coefficient in the expansion of (3x² + 2/x)⁶?
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Answer: C. 4860x⁶
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Q53. The curve y = √x and the line y = (x + 3)/4 intersect at two points. What is the sum of the x-coordinates of those points?
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Answer: C. 10
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The expansion of (2x + c)3 is 8x3 − 36x2 + 54x + 3k, where c and k are real constants. Find k.

What this topic tests

How it is assessed

Key methods

The discriminant

Factor and remainder

A modulus is two cases

Common mistakes

Dividing away a root

The wrong binomial coefficient

Worked example

The expansion of (2x + c)3 is 8x3 − 36x2 + 54x + 3k, where c and k are real constants. Find k.

Paper 1 questions

Q1. The expansion of (2x + c)3 is 8x3 − 36x2 + 54x + 3k, where c and k are real constants. Find k.

Q2. The cubic x3 − 3x2 − 6x + c, where c is a constant, has (x − 1) as a factor. Which product is a complete factorisation of this cubic?

Q3. The roots of x3 − 5x2 + px + q = 0 have product −5, and the sum of the products of the roots taken two at a time is −1. Find p + q.

Q4. The equation x2 − kx + (2k − 4) = 0 has one root equal to twice the other. Find the sum of the possible values of k.

Q5. The numbers α and β are the roots of x2 − 4x − 1 = 0. Find α3 + β3.

Q6. Let f(x) = 3x − 1 and g(x) = x2 + 2x. Find g(f(x)).

Q7. Find every real x for which (x2 − x − 6)/(x − 3) is at least 2.

Q8. The polynomial p(x) = x3 − 9x2 + ax + b has roots r, s and t. Given that r + s = 5 and rs + rt + st = 26, find a and b.

Q9. What is the largest real root of x3 − 5x2 − 8x + 12 = 0?

Q10. For a real number of hours x, the cost in pounds of running a kiln is C(x) = 2x2 − 12x + 25. Find the minimum cost.

Q11. Solve the inequality |x2 − 4x + 1| < 2.

Q12. Find the number of real solutions of the equation |x + 3| = 7 − x − 2x2.

Q13. The graph of y = 2x2 + x − 6 lies on or below the x-axis for which values of x?

Q14. The roots of x3 − 5x2 + 2x + 8 = 0 are all nonzero. Find the sum of the reciprocals of the roots.

Q15. Which list gives every real solution of |3x − 4| = |x + 6|?

Q16. The number z is nonzero and z + 1/z = 3. What is z3 + 1/z3?

Q17. The graph of y = p(x − 2)2 + q has its vertex at (2, −3) and passes through (4, 5). What is p + q?

Q18. Find the complete set of real x for which (x2 − 4)/(x − 3) < 0.

Q19. The polynomial f(x) = x3 − 4x2 + x + k passes through the point (3, 4). What is k?

Q20. Write floor(t) for the greatest integer less than or equal to t. What is the sum of all real solutions of x2 = floor(x) + 2?

Q21. The expansion of (ax + b)3 is 8x3 − q x2 + 24x − 8, where a, b and q are real and a > 0. Find q.

Q22. The number a is chosen so that x + 3 is a factor of 2x3 + 3x2 − 11x + a. Which product is a complete factorisation of this cubic?

Q23. The coefficient of x3 in the expansion of (1 + 2x + 3x2)5 is twice the coefficient of x4 in the expansion of (1 − a x2)5. Find every possible value of the constant a.

Q24. S is the set of real x that satisfy both x2 − 7x + 10 < 0 and 3x − 1 > 8. Which single inequality describes S?

Q25. The solution set of x² + bx + c < 0 is p < x < q, where b, c, p and q are real constants with c < 0 and p < q. What is the solution set of cx² + bx + 1 < 0?

Q26. The curves y = x² − 4x + a and y = 2x − x² meet at two distinct points, and both points have positive x-coordinates. Here a is a real constant. What is the complete set of possible values of a?

Q27. The polynomial f(x) = x3 + ax2 + bx + c uses each of 2, 3 and 4 once as a, b and c. Let R be the remainder on division by x + 1, and let S be the remainder on division by x + 2. What is the largest possible value of R − S?

Q28. The real numbers x and y satisfy |x − 1| ≤ 3 and |y + 1| ≤ 5. What is the greatest possible value of |xy|?

Q29. The quadratic f has a turning point at (2, −3) and the graph of y = f(x) passes through (5, 15). Which expression is f(x)?

Q30. Find the complete set of real k for which x² + kx + (k + 3) is positive for every real x.

Q31. (2x − 1) and (x + 3) are factors of 2x3 + px2 + q, which has no x term. What is the value of 3p + q?

Q32. Find the complete set of values of x for which (x + 6)(x + 1)(3 − x) > 0 and (x + 4)(x − 2) < 0.

Q33. The curve y = px2 + 8x − q, where p and q are constants, has a line of symmetry x = −1/3 and touches the x-axis at exactly one point. What is the value of p + 3q?

Q34. The function f is defined for all real x by f(x) = (p − x)(x + 3), where p is a real constant. Find the complete set of values of p for which the maximum value of f(x) is less than p + 6.

Q35. The quadratic x2 − 8x + 4 factorises as (x − α)(x − β), where α and β are positive real numbers. Which quadratic factorises as (x − √α)(x − √β)?

Q36. What is the coefficient of x5 in the expansion of (1 + x)9 (1 − x)8?

Q37. What is the lowest positive integer n for which n2 − 30n − 30 is positive?

Q38. For how many real values of a does the equation (x − a)(x2 − 4x + a) = 0 have exactly two distinct real solutions?

Q39. The function f satisfies f(mn) = f(m)f(n) whenever mn is a multiple of 5, and f(mn) = mn otherwise, for all positive integers m and n. Given that f(25) + f(4) − f(20) = 0, what is f(5)?

Q40. In the expansion of (a + bx)n, where a and b are positive and n is a positive integer, the third term in ascending powers of x is 90x2, the fourth term in ascending powers of x is 240x3, and the fourth term in descending powers of x is also 240x3. What is (a/b)2?

Q41. Let a > 0 and suppose that a + 4/a = 5. What is the value of a² + 16/a²?

Q42. The real numbers a and b are nonzero and satisfy (a3 + 4/b3)(4/a3 − b3) = 6. What is the least possible value of ab?

Q43. The line y = x − 2 meets the curve x2 + xy − y2 = 7 at two points. What is the sum of the x-coordinates of those points?

Q44. What is the complete set of real x for which (x2 − 4)(x − 1) < 0?

Q45. The linear factor x − 1 divides x3 + m x2 − x(m − 2)2 + 4. What is the sum of the possible values of m?

Q46. The roots of 3x2 − 10x + k = 0 differ by 1. What is k?

Q47. How many real roots does (x2 + 3x)2 = 6 have?

Q48. For which non-zero real numbers k does the quadratic equation k x2 + 4x + k − 3 = 0 have two distinct real roots?

Q49. S is the set of real x that satisfy both x² − 6x + 5 < 0 and x² − 2x − 3 < 0. Which single inequality describes S?

Q50. For integers a and b, which condition is enough to guarantee that 2a × 3b / (6a − b × 9b) is an integer?

Q51. The polynomial (2x² − x + 3)(ax + 1) is divided by x − 1 and the remainder is 24. Find a.

Q52. The inequality |x − 3| + |x + 1| ≤ 8 holds. What is the greatest possible value of |x|?

Q53. Solve 5x − 7 = 2x + 8.

Q54. A function f is defined for every real x except x = 3, and it satisfies 3f(x) − 4f((3x + 4)/(x − 3)) = (52 − 7x)/4. Find f(4).

Q55. a and b are non-zero integers. Dividing x² − ax − a² by x − b leaves remainder −5. Also, 2x − a is a factor of 2bx² − 3x − 9. Find a + b.

Q56. Let n be an odd positive integer. What is the remainder when (x + 1)n + (x − 1)n is divided by x2 − 1?

Q57. The graph of a quadratic f crosses the x-axis at x = −1 and x = 5 and passes through (1, 16). Which expression equals f(x)?

Q58. Which set is exactly the set of real x that satisfy both x² − 2x − 24 < 0 and x² − 9 > 0?

Q59. The cubic x³ − 2x² + 5x + k has remainder 4 on division by x − 1. What is the remainder on division by x − 2?

Q60. Solve the inequality |4x + 2| ≤ 10.

Q61. The real numbers x and y satisfy x² + y² = 34 and x + y = 8, with x > y. Find x − y.

Q62. The function f is defined by f(x) = (4x − 1)/(x + 2) for x ≠ −2. Find f⁻¹(3).

Q63. How many distinct real roots does x⁴ − 2x² − 8 = 0 have?

Q64. Evaluate 8 to the power 2/3, plus 25 to the power −1/2.

Q65. Which inequality describes every real x that satisfies both (x − 1)(x − 10) < 0 and (x − 4)(x − 13) < 0?

Q66. The polynomial x³ − 3x + 4 leaves remainder 2 when it is divided by x − 1. What is the remainder when it is divided by x + 1?

Q67. Which inequality describes the solution of |2x + 3| ≤ 11?

The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.

The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.

Q1. The expansion of (2x + c)³ is 8x³ − 36x² + 54x + 3k, where c and k are real constants. Find k.

Q2. The cubic x³ − 3x² − 6x + c, where c is a constant, has (x − 1) as a factor. Which product is a complete factorisation of this cubic?

Q3. The roots of x³ − 5x² + px + q = 0 have product −5, and the sum of the products of the roots taken two at a time is −1. Find p + q.

Q4. The equation x² − kx + (2k − 4) = 0 has one root equal to twice the other. Find the sum of the possible values of k.

Q5. The numbers α and β are the roots of x² − 4x − 1 = 0. Find α³ + β³.

Q6. Let f(x) = 3x − 1 and g(x) = x² + 2x. Find g(f(x)).

Q7. Find every real x for which (x² − x − 6)/(x − 3) is at least 2.

Q8. The polynomial p(x) = x³ − 9x² + ax + b has roots r, s and t. Given that r + s = 5 and rs + rt + st = 26, find a and b.

Q9. What is the largest real root of x³ − 5x² − 8x + 12 = 0?

Q10. For a real number of hours x, the cost in pounds of running a kiln is C(x) = 2x² − 12x + 25. Find the minimum cost.

Q11. Solve the inequality |x² − 4x + 1| < 2.

Q12. Find the number of real solutions of the equation |x + 3| = 7 − x − 2x².

Q13. The graph of y = 2x² + x − 6 lies on or below the x-axis for which values of x?

Q14. The roots of x³ − 5x² + 2x + 8 = 0 are all nonzero. Find the sum of the reciprocals of the roots.

Q16. The number z is nonzero and z + 1/z = 3. What is z³ + 1/z³?

Q17. The graph of y = p(x − 2)² + q has its vertex at (2, −3) and passes through (4, 5). What is p + q?

Q18. Find the complete set of real x for which (x² − 4)/(x − 3) < 0.

Q19. The polynomial f(x) = x³ − 4x² + x + k passes through the point (3, 4). What is k?

Q20. Write floor(t) for the greatest integer less than or equal to t. What is the sum of all real solutions of x² = floor(x) + 2?

Q21. The expansion of (ax + b)³ is 8x³ − q x² + 24x − 8, where a, b and q are real and a > 0. Find q.

Q22. The number a is chosen so that x + 3 is a factor of 2x³ + 3x² − 11x + a. Which product is a complete factorisation of this cubic?

Q23. The coefficient of x³ in the expansion of (1 + 2x + 3x²)⁵ is twice the coefficient of x⁴ in the expansion of (1 − a x²)⁵. Find every possible value of the constant a.

Q24. S is the set of real x that satisfy both x² − 7x + 10 < 0 and 3x − 1 > 8. Which single inequality describes S?

Q27. The polynomial f(x) = x³ + ax² + bx + c uses each of 2, 3 and 4 once as a, b and c. Let R be the remainder on division by x + 1, and let S be the remainder on division by x + 2. What is the largest possible value of R − S?

Q31. (2x − 1) and (x + 3) are factors of 2x³ + px² + q, which has no x term. What is the value of 3p + q?

Q33. The curve y = px² + 8x − q, where p and q are constants, has a line of symmetry x = −1/3 and touches the x-axis at exactly one point. What is the value of p + 3q?

Q35. The quadratic x² − 8x + 4 factorises as (x − α)(x − β), where α and β are positive real numbers. Which quadratic factorises as (x − √α)(x − √β)?

Q36. What is the coefficient of x⁵ in the expansion of (1 + x)⁹ (1 − x)⁸?

Q37. What is the lowest positive integer n for which n² − 30n − 30 is positive?

Q38. For how many real values of a does the equation (x − a)(x² − 4x + a) = 0 have exactly two distinct real solutions?

Q40. In the expansion of (a + bx)ⁿ, where a and b are positive and n is a positive integer, the third term in ascending powers of x is 90x², the fourth term in ascending powers of x is 240x³, and the fourth term in descending powers of x is also 240x³. What is (a/b)²?

Q42. The real numbers a and b are nonzero and satisfy (a³ + 4/b³)(4/a³ − b³) = 6. What is the least possible value of ab?

Q43. The line y = x − 2 meets the curve x² + xy − y² = 7 at two points. What is the sum of the x-coordinates of those points?

Q44. What is the complete set of real x for which (x² − 4)(x − 1) < 0?

Q45. The linear factor x − 1 divides x³ + m x² − x(m − 2)² + 4. What is the sum of the possible values of m?

Q46. The roots of 3x² − 10x + k = 0 differ by 1. What is k?

Q47. How many real roots does (x² + 3x)² = 6 have?

Q48. For which non-zero real numbers k does the quadratic equation k x² + 4x + k − 3 = 0 have two distinct real roots?

Q50. For integers a and b, which condition is enough to guarantee that 2^a × 3^b / (6^{a − b} × 9^b) is an integer?

Q56. Let n be an odd positive integer. What is the remainder when (x + 1)ⁿ + (x − 1)ⁿ is divided by x² − 1?

Q95. Dividing x³ − x² + 3x − 5 by x − 1 leaves remainder −2. Find the remainder on division by x + 1.

Q98. How many distinct real solutions does x⁶ − 7x³ − 8 = 0 have?

Q105. What is the maximum value of −2x² + 12x − y² + 8y − 30?

Q106. Find the sum of the real solutions of the equation x² + 2√(x² + 8x) = 15 − 8x.

Q107. The expansion of (√2 − x)⁶ is 8 − b x + 60 x² − d x³ + 30 x⁴ − f x⁵ + x⁶, where b, d and f are positive. What is b + d + f?

Q109. The coefficient of x⁴ in the expansion of (3 + x²)⁵ is equal to 5 times the coefficient of x² in the expansion of (1 + ax)⁴. What are the possible values of a?

Q110. The coefficient of x³ in the expansion of (3 + bx)⁵ is 6 times the coefficient of x² in the expansion of (1 + bx)⁴. Given that b is not 0, what is b?

Q111. The numbers r and s are non-zero integers. Which condition guarantees that (24^r × 3^s) / 36^r is an integer?

Q112. Suppose u = 5⁶ and v = 6⁵. Which expression is equal to 30³⁰?

Q116. The curve y = x³ + kx² − 32 has exactly two distinct real x-intercepts for one value of the constant k. Find that value of k.

Q117. The equation 3x² + 8x − k = 0, where k is a constant, has two distinct real roots, and one root is 3 more than the other. Find k.

Q118. Find the minimum value of 2(3^{sin x}) − 9^{sin x} + 16/3.

Q119. When (x² − 3x + 1) is multiplied by (px + 1), and the product is divided by (x − 2), the remainder is 10. Find p.

Q120. The equations x² − xy = 4 and y − 2x = p, where p is a real constant, have two distinct real solutions. Which statement describes every value p can take?

Q122. How many distinct real solutions does the equation |x² − 1| = 2|x| have?

Q123. The non-zero real numbers a and b satisfy (a³ + 9/b³)(b³ − 9/a³) = 24. What is the least possible value of ab?

Q126. The function f is given by f(x) = (x − k)/(x² − 6x − k), for the real x at which it is defined, where k is a constant. The range of f is all of the real numbers. What are the possible values of k?

Q2. The function f is odd and defined for all real x, and f(x) = x² − 4x for every x > 0. How many real solutions does the equation f(x) = 3 have?

Q5. Find the constant term in the expansion of (x⁵ − x⁻³)⁸.