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CAPE Pure Mathematics Unit 1 · May/June 2026 · Paper 2

42 questions and parts from this paper. Open one to see it in full, then practise it on Quelpr and get it marked against the mark scheme.

Practise this paper
  1. 1(a)(i)4 marksComplete the truth table for p, q, r, p ⇒ q, q ⇒ r, (p ⇒ q) ∧ (q ⇒ r), p ∨ q, and (p ∨ q) ⇒ r.
  2. 1(a)(ii)2 marksDetermine whether (p ⇒ q) ∧ (q ⇒ r) is logically equivalent to (p ∨ q) ⇒ r. Justify your response.
  3. 1(b)3 marksRationalize the denominator of (√3 + √7) / (√5 + √2), expressing your answer in surd form.
  4. 1(c)8 marksProve, by mathematical induction, that for all natural numbers n, 1 + 5 + 9 + 13 + ... + (4n - 3) = n(2n - 1).
  5. 1(d)(i)2 marksGiven that x + 3 is a factor of x³ - 7x + q, determine the value of q.
  6. 1(d)(ii)4 marksHence, factorize the expression x³ - 7x + q.
  7. 1(e)2 marksDetermine the remainder when x⁴ - 2x² + x + 1 is divided by 2x - 1.
  8. 2(a)(i)2 marksComplete the table of values for h(x) = 5 - 2^(x+1) for x = 0, 1, 2, 3, 4.
  9. 2(a)(ii)2 marksOn the grid provided on page 9, plot the graph of h(x) = 5 - 2^(x+1).
  10. 2(b)(i)2 marksOn the grid provided on page 10, sketch the graph of y = |f(x)|.
  11. 2(b)(ii)2 marksOn the grid provided on page 10, sketch the graph of y = |g(x)|.
  12. 2(c)(i)4 marksDetermine whether f is injective.
  13. 2(c)(ii)5 marksAssuming that f is a bijective function, determine an expression for f⁻¹(x).
  14. 2(d)(i)3 marksSolve for x: 3^(4x + 1) = 177 147.
  15. 2(d)(ii)5 marksSolve for x: log₂(x + 6) + log₂(x + 2) = 5.
  16. 3(a)3 marksShow that 1 + tan² θ = sec² θ.
  17. 3(b)(i)3 marksDerive the identity for cos 2θ in terms of cos θ only.
  18. 3(b)(ii)6 marksHence, solve cos 2θ - 3 cos θ = 1 for 0 ≤ θ ≤ 2π.
  19. 3(c)5 marksDetermine the equation of the circle that has (1, 0) and (3, 2) as endpoints of a diameter.
  20. 3(d)8 marksDetermine the points of intersection of the curves 2x² - y - 11 = 0 and x² - 4x - y + 10 = 0.
  21. 4(a)8 marksDetermine the equation of the normal to the curve x = t², y = t + 1/t at the point on the curve where t = 2.
  22. 4(b)(i)2 marksCalculate a · b.
  23. 4(b)(ii)4 marksHence, calculate the angle between the vectors a and b.
  24. 4(c)(i)2 marksDetermine the displacement vector AB in terms of i, j and k.
  25. 4(c)(ii)2 marksDetermine a unit vector parallel to AB in terms of i, j and k.
  26. 4(c)(iii)3 marksDetermine the Cartesian equation of the line passing through A and B.
  27. 4(d)4 marksDetermine the vector equation of the plane that passes through the point (1, -1, 2) and that is perpendicular to the vector 2i + 3j - k.
  28. 5(a)(i)a)1 markDetermine the value of lim_{x → -5⁺} f(x).
  29. 5(a)(i)b)1 markDetermine the value of lim_{x → 5} f(x).
  30. 5(a)(ii)1 markState the value of k such that lim_{x → k} f(x) = -2.
  31. 5(b)(i)a)1 markDetermine g(1).
  32. 5(b)(i)b)2 marksDetermine lim_{x → 1} g(x).
  33. 5(b)(ii)2 marksHence, state whether g is continuous at x = 1. Give a reason for your answer.
  34. 5(c)5 marksGiven that y = (2x³ + 4)⁷ and x = 1 - 2t, determine dy/dt.
  35. 5(d)6 marksDetermine f'(x), using first principles, given that f(x) = sin x.
  36. 5(e)6 marksDetermine ∫₂⁴ x(x - 4)⁵ dx using the substitution u = x - 4.
  37. 6(a)(i)4 marksDetermine the coordinates of the points where the curve y = f(x) intersects the x-axes and y-axes.
  38. 6(a)(ii)a)4 marksDetermine the coordinates of the stationary points on y = f(x).
  39. 6(a)(ii)b)3 marksClassify EACH of the stationary points on y = f(x) as a maximum point, minimum point or point of inflection.
  40. 6(a)(iii)3 marksHence, sketch a carefully labelled graph of y = f(x).
  41. 6(b)6 marksCalculate the area of the shaded region.
  42. 6(c)5 marksThe gradient at any point (x, y) on a curve is equal to (5x + 1)². Given that the curve passes through the point (1, 0.2), determine the equation of the curve.

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