Quelpr

CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2

35 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 marksFind the exact values of x such that e^x + 7e^{-x} = 8.
  2. 1(b)7 marksGiven that u = e^{2x} + e^{-2x} and v = e^{2x} - e^{-2x}, show that…
  3. 1(c)(i)4 marksDifferentiate (x \ln x) \sin^{-1} 2x with respect to x.
  4. 1(c)(ii)a)3 marksShow that \frac{\mathrm{d}y}{\mathrm{d}x} = t + \frac{1}{t}.
  5. 1(c)(ii)b)6 marksShow that C has points of inflexion at (8, 8) and (8, -8).
  6. 2(a)(i)3 marksFind \int \frac{1}{x} \ln x \, \mathrm{d}x.
  7. 2(a)(ii)5 marksSolve the differential equation x^2 \frac{\mathrm{d}y}{\mathrm{d}x} + xy = \ln x.
  8. 2(b)(i)5 marksFind the values of the constants m and n, given that y = m \cos x + n \sin x satisfies the differential equation \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 4\frac{\mathrm{d}y}{\mathrm{d}x} + 3y = 10 \sin x.
  9. 2(b)(ii)3 marksHence, find the general solution of the differential equation.
  10. 2(c)(i)6 marksExpress \frac{2 + 3x - x^2}{(x - 1)(x^2 + 1)} in partial fractions.
  11. 2(c)(ii)3 marksHence, find \int \frac{2 + 3x - x^2}{(x - 1)(x^2 + 1)} \, \mathrm{d}x.
  12. 3(a)(i)a)2 marksShow that T = S.
  13. 3(a)(i)b)3 marksDeduce that S = \frac{1}{2} n(n + 1).
  14. 3(a)(ii)7 marksUse the principle of mathematical induction to prove that \sum_{r=1}^n r^2 = \frac{1}{6} n(n + 1)(2n + 1).
  15. 3(a)(iii)4 marksHence, prove that \sum_{r=1}^n 2r(3r + 1) = 2n(n + 1)^2.
  16. 3(b)(i)2 marksShow that the equation x^3 + 3x^2 + 6x - 3 = 0 has a root \alpha between 0 and 1.
  17. 3(b)(ii)3 marksProve that \alpha is the only real root.
  18. 3(b)(iii)4 marksUsing TWO iterations of the Newton-Raphson method, find \alpha correct to 2 decimal places.
  19. 4(a)(i)2 marksFind a_2 and a_3.
  20. 4(a)(ii)2 marksExpress a_{n+1} - 2 in terms of a_n.
  21. 4(a)(iii)a)3 marksShow that a_{n+1} < 2.
  22. 4(a)(iii)b)6 marksShow that a_n < a_{n+1}.
  23. 4(b)6 marksFind the term independent of x in the binomial expansion of (x^2 - \frac{6}{x^3})^{15}. [You may leave your answer in the form of factorials and powers, for example, \frac{15!}{2!} \times 8^5.]
  24. 4(c)6 marksUse the binomial theorem to find the difference between 2^{10} and (2.002)^{10} correct to 5 decimal places.
  25. 5(a)(i)a)2 marksHow many 4-digit numbers can be formed if the digits 1, 2, 3, 4, 7, 9 can all be repeated?
  26. 5(a)(i)b)2 marksHow many 4-digit numbers can be formed if none of the digits 1, 2, 3, 4, 7, 9 can be repeated?
  27. 5(a)(ii)3 marksCalculate the probability that a 4-digit number formed without repetition is even.
  28. 5(b)6 marksA father and son practise shooting at basketball, and score when the ball hits the basket. The son scores 75\% of the time and the father scores 4 out of 7 tries. If EACH takes one shot at the basket, calculate…
  29. 5(c)(i)6 marksFind the values of h, k \in \mathbb{R} such that 3 + 4\mathrm{i} is a root of the quadratic equation z^2 + hz + k = 0.
  30. 5(c)(ii)6 marksUse De Moivre's theorem for (\cos \theta + \mathrm{i} \sin \theta)^3 to show that \cos 3\theta = 4\cos^3 \theta - 3\cos \theta.
  31. 6(a)12 marksSolve for x the equation \begin{vmatrix} 1 & 1 & 1 \\ x & 2 & 1 \\ x^3 & 8 & 1 \end{vmatrix} = 0.
  32. 6(b)(i)3 marksExpress the information above as a matrix equation AX = Y, where A is a 3 \times 3 matrix, and X and Y are 3 \times 1 matrices with X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.
  33. 6(b)(ii)a)3 marksCalculate AB.
  34. 6(b)(ii)b)3 marksDeduce the inverse A^{-1} of A.
  35. 6(b)(iii)4 marksHence, or otherwise, determine the number of cars and buses used in the 34\text{ km} tours.

More CAPE Pure Mathematics Unit 2 papers