CAPE Pure Mathematics Unit 2 · May/June 2010 · Paper 2

39 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.

1. 1(a)(i)2 marksDetermine the initial temperature of the water in the tank.
2. 1(a)(ii)2 marksDetermine the temperature at which the water in the tank will eventually stabilize.
3. 1(a)(iii)4 marksDetermine the time when the temperature of the water in the tank is 70^\circ\text{ C}.
4. 1(b)(i)4 marksGiven that y = e^{\tan^{-1}(2x)}, where -\frac{1}{2}\pi < \tan^{-1}(2x) < \frac{1}{2}\pi, show that (1 + 4x^2)\frac{\mathrm{d}y}{\mathrm{d}x} = 2y.
5. 1(b)(ii)4 marksHence, show that (1 + 4x^2)^2\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 4y(1 - 4x).
6. 1(c)(i)6 marksDetermine \int \frac{4}{e^x + 1}\,\mathrm{d}x by using the substitution u = e^x.
7. 1(c)(ii)3 marksDetermine \int \frac{4}{e^x + 1}\,\mathrm{d}x by first multiplying both the numerator and denominator of the integrand by e^{-x} before integrating.
8. 2(a)(i)4 marksGiven that n is a positive integer, find \frac{\mathrm{d}}{\mathrm{d}x}[x(\ln x)^n].
9. 2(a)(ii)4 marksHence, or otherwise, derive the reduction formula I_n = x(\ln x)^n - n I_{n-1}, where I_n = \int (\ln x)^n\,\mathrm{d}x.
10. 2(a)(iii)6 marksUse the reduction formula in (a)(ii) to determine \int (\ln x)^3\,\mathrm{d}x.
11. 2(b)(i)7 marksUsing a suitable integrating factor, show that the general solution of this differential equation is y = t + 10 + \frac{c}{(t+10)^2}, where c is an arbitrary constant.
12. 2(b)(ii)4 marksGiven that the tank initially contains 5\text{ kg} of salt in the liquid, calculate the amount of salt that dissolves in the tank of water at t = 15.
13. 3(a)(i)2 marksExpress, in terms of r, the r^{\text{th}} term of the sequence.
14. 3(a)(ii)5 marksIf S_n denotes the series formed by summing the first n terms of the sequence, find S_n in terms of n.
15. 3(b)6 marksThe 9^{\text{th}} term of an A.P. is three times the 3^{\text{rd}} term and the sum of the first 10 terms is 110. Find the first term a and the common difference d.
16. 3(c)(i)5 marksUse the binomial theorem to expand (1 + 2x)^{\frac{1}{2}} as far as the term in x^3, stating the values of x for which the expansion is valid.
17. 3(c)(ii)4 marksProve that \frac{x}{1 + x + \sqrt{1 + 2x}} = \frac{1}{x}(1 + x - \sqrt{1 + 2x}) for x > 0.
18. 3(c)(iii)3 marksHence, or otherwise, show that, if x is small so that the term in x^3 and higher powers of x can be neglected, the expansion in (c)(ii) above is approximately equal to \frac{1}{2}x(1 - x).
19. 4(a)(i)6 marksBy expressing ^n\mathrm{C}_r and ^n\mathrm{C}_{r-1} in terms of factorials, prove that ^n\mathrm{C}_r + {}^n\mathrm{C}_{r-1} = {}^{n+1}\mathrm{C}_r.
20. 4(a)(ii)a)3 marksGiven that r is a positive integer and f(r) = \frac{1}{r!}, show that f(r) - f(r + 1) = \frac{r}{(r + 1)!}.
21. 4(a)(ii)b)5 marksHence, or otherwise, find the sum S_n = \sum_{r=1}^{n} \frac{r}{(r + 1)!}.
22. 4(a)(ii)c)2 marksDeduce the sum to infinity of S_n in (ii) b) above.
23. 4(b)(i)5 marksShow that the function f(x) = x^3 - 6x + 4 has a root x in the closed interval [0, 1].
24. 4(b)(ii)4 marksBy taking 0.6 as a first approximation of x_1 in the interval [0, 1], use the Newton-Raphson method to obtain a second approximation x_2 in the interval [0, 1].
25. 5(a)(i)3 marksCalculate the number of different permutations of the 8 letters of the word SYLLABUS.
26. 5(a)(ii)5 marksCalculate the number of different selections of 5 letters which can be made from the letters of the word SYLLABUS.
27. 5(b)(i)3 marksFind P(A \cap B).
28. 5(b)(ii)a)3 marksStating a reason, determine whether or not the events A and B are mutually exclusive.
29. 5(b)(ii)b)3 marksStating a reason, determine whether or not the events A and B are independent.
30. 5(c)(i)4 marksExpress the complex number (2 + 3i) + \frac{i - 1}{i + 1} in the form a + ib, where a and b are both real numbers.
31. 5(c)(ii)4 marksGiven that 1 - i is the root of the equation z^3 + z^2 - 4z + 6 = 0, find the remaining roots.
32. 6(a)(i)2 marksWrite the augmented matrix of the system.
33. 6(a)(ii)3 marksReduce the augmented matrix to echelon form.
34. 6(a)(iii)2 marksDeduce the value of k for which the system is consistent.
35. 6(a)(iv)4 marksFind ALL solutions corresponding to the value of k obtained in (iii) above.
36. 6(b)(i)a)4 marksFind A^2.
37. 6(b)(i)b)4 marksFind B = 3I + A - A^2.
38. 6(b)(ii)4 marksCalculate AB.
39. 6(b)(iii)2 marksDeduce the inverse, A^{-1}, of the matrix A.