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CAPE Pure Mathematics Unit 1 · May/June 2024 · Paper 2

27 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)5 marksDetermine the domain of the composite function f(g(x)).
  2. 1(b)4 marksSolve the inequality -3|2x - 5| + 2 >= -4.
  3. 1(c)8 marksSolve the logarithmic equation log_5(x) - 4 log_x(5) = -3.
  4. 1(d)8 marksProve that the function f(x) = 3x - 2 is bijective.
  5. 2(a)8 marksDetermine the cubic equation with roots (alpha - 1), (beta - 1) and (gamma - 1), given that (alpha - 1)(beta - 1) + (alpha - 1)(gamma - 1) + (beta - 1)(gamma - 1) = 12.
  6. 2(b)9 marksCalculate the values of p and q.
  7. 2(c)8 marksProvide a proof by contradiction for this proposition, by assuming n is odd and showing the assumption is incorrect.
  8. 3(a)(i)6 marksExpress 5 sin theta + 12 cos theta in the form r sin(theta + alpha), where r > 0 and 0 < alpha < 2*pi.
  9. 3(a)(ii)3 marksHence, or otherwise, show that 5 sin theta + 12 cos theta + 7 <= 20.
  10. 3(b)(i)5 marksShow that 2 sin x + 3 cos x = 3 may be written as 13 cos^2 x - 18 cos x + 5 = 0.
  11. 3(b)(ii)5 marksHence or otherwise, solve the equation 2 sin x + 3 cos x = 3 for -pi <= x <= pi.
  12. 3(c)6 marksShow that (2 cos^2 x - 1)^2 / (cos^4 x - sin^4 x) == 1 - 2 sin^2 x.
  13. 4(a)(i)4 marksShow that the equation of the plane is 3x - 4y + 2z - 5 = 0.
  14. 4(a)(ii)6 marksCalculate the angle, in radians, between P and the plane with equation 7x - 4y + 3z = 5.
  15. 4(b)(i)5 marksDetermine the centre and radius of the circle.
  16. 4(b)(ii)5 marksDetermine the equation of the tangent to the circle at the point (11, 4).
  17. 4(c)5 marksA point, Q, moves in the x-y plane such that it is equidistant from A(2, -5) and B(-2, 3). Determine the locus of Q.
  18. 5(a)(i)1 markDetermine f(-2).
  19. 5(a)(ii)5 marksDetermine lim_{x -> -2} f(x).
  20. 5(a)(iii)1 markHence, or otherwise, determine whether f is continuous at x = -2.
  21. 5(b)4 marksEvaluate lim_{theta -> 0} (sin theta / sin 4 theta).
  22. 5(c)(i)9 marksDetermine the stationary points of g and the nature of the stationary points.
  23. 5(c)(ii)5 marksHence, or otherwise, sketch the graph of g showing its intercepts, stationary points and any other important features on the provided grid.
  24. 6(a)7 marksCalculate the volume of the solid generated by revolving the region bounded by the graphs of x = sqrt(y) and y = 2x about the y-axis.
  25. 6(b)7 marksCalculate the area between the line and the curve.
  26. 6(c)5 marksGiven that f is an even function, show that integral from -a to a of f(x) dx = 2 * integral from 0 to a of f(x) dx.
  27. 6(d)6 marksSolve the differential equation x^3 y' = 2 - x^4, given that at x = 1, y = 2.

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