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CAPE Pure Mathematics Unit 1 · May/June 2023 · Paper 2

29 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)3 marksComplete the truth table for p \to q, q \to p, and (p \to q) \wedge (q \to p).
  2. 1(a)(ii)2 marksHence, state whether the statements q \to p and (p \to q) \wedge (q \to p) are logically equivalent. Justify your response.
  3. 1(b)3 marksLet x and y be negative real numbers and let z be any real number. Use a counter example to show that the statement "if x > y then xz > yz" is false.
  4. 1(c)9 marksThe expression f(x) = 6x^3 + px^2 + qx + 2 is divisible by 2x - 1 and has a remainder of 2 when divided by x - 1. Calculate the values of p and q.
  5. 1(d)(i)4 marksSolve the logarithmic equation \log_3(x^2 - 9) - \log_3(x + 3) = 3.
  6. 1(d)(ii)4 marksShow that \sqrt{320x^3} + \sqrt{125x^3} simplifies to 13x\sqrt{5x}.
  7. 2(a)5 marksLet f(x) = 7x + 2. Prove that f is bijective.
  8. 2(b)8 marksThe roots of the cubic equation 3x^3 - x^2 - 2x + 1 = 0 are \alpha, \beta and \gamma. Determine the equation whose roots are 1/\alpha, 1/\beta and 1/\gamma.
  9. 2(c)(i)3 marksOn the axes provided, sketch and label the graph of g(x) = |x^2 + 6x + 8|.
  10. 2(c)(ii)5 marksOn the same axes, sketch and label the inverse of f for x \ge -3.
  11. 2(d)4 marksGiven that g(x) = (2x + 3)/(x + 3), prove that g^{-1}(2) does not exist.
  12. 3(a)9 marksProve that (1 - \sin\theta)/(1 + \sin\theta) = (\sec\theta - \tan\theta)^2.
  13. 3(b)9 marksSolve the equation 2\cos^2 x - 3\sin x = 3 for 0 \le x \le 2\pi.
  14. 3(c)4 marksShow that \cos(\pi/2 + x) = -\sin x.
  15. 3(d)3 marksA and B are acute angles such that \sin A = 3/5 and \cos B = 5/13. Calculate, without using tables or calculators, the EXACT value of \cos(A - B).
  16. 4(a)5 marksObtain the Cartesian equation of the curve given in parametric form x = 3\cos t and y = 4\sin t.
  17. 4(b)6 marksThe equation of a line is x = 2 + t, y = 1 - 3t and z = 4 + t, and the equation of a plane is x + 2y + z = 12. Determine the point of intersection of the line and the plane.
  18. 4(c)(i)4 marksDetermine the vector equation of the plane which passes through (1, 5, -1) and which is perpendicular to the vector \begin{pmatrix}2\\4\\3\end{pmatrix}.
  19. 4(c)(ii)3 marksHence, determine the coordinates of the point in the plane where y = 3 and z = 1.
  20. 4(d)7 marksGiven that a line is parallel to the vector u = \begin{pmatrix}1\\-3\\1\end{pmatrix} and that the vector v = \begin{pmatrix}1\\2\\1\end{pmatrix} is normal to the plane, calculate the angle between the line and the plane.
  21. 5(a)5 marksDetermine \lim_{x \to \infty} \frac{2x^3 - 4x + 1}{3x^4 + x^2 - 2}.
  22. 5(b)8 marksHow fast is the water in the cup rising when the height is 4 cm?
  23. 5(c)(i)2 marksShow that the volume of the container is 4x^3 - 58x^2 + 208x.
  24. 5(c)(ii)9 marksUsing the method of the second derivative, determine the height, x, that will maximize the volume of the container.
  25. 5(c)(iii)1 markDetermine the maximum volume of the container.
  26. 6(a)7 marksCalculate the volume of the solid generated by revolving the region bounded by the line y = 3x - 6 and the parabola y = x^2 + 3x - 2, on the interval [0, 1] about the x-axis.
  27. 6(b)7 marksThe diagram below shows the curves y = \cos x and y = \sin x. Determine the area bounded by the curves between x = \pi/4 and x = 3\pi/2.
  28. 6(c)7 marksDetermine the velocity and height of the rocket 5 seconds after launch.
  29. 6(d)4 marksDetermine \int \cos^3 2x \sin 2x \, dx.

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