CAPE Pure Mathematics Unit 2 · May/June 2011 · Paper 2

46 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)3 marksFind \frac{dy}{dx} if x^2 + y^2 - 2x + 2y - 14 = 0.
2. 1(a)(ii)3 marksFind \frac{dy}{dx} if y = e^{\cos x}.
3. 1(a)(iii)3 marksFind \frac{dy}{dx} if y = \cos^2 6x + \sin^2 8x.
4. 1(b)(i)3 marksShow that x \frac{dy}{dx} = y - \cos \left(\frac{1}{x}\right).
5. 1(b)(ii)3 marksShow that x^4 \frac{d^2y}{dx^2} + y = 0.
6. 1(c)(i)7 marksFind the gradient of the tangent to the curve at the point where t = 4.
7. 1(c)(ii)3 marksFind the equation of the tangent to the curve at the point where t = 4.
8. 2(a)(i)3 marksFind F_0(x) and F_n(0), given that 0! = 1.
9. 2(a)(ii)7 marksShow that F_n(x) = F_{n-1}(x) - \frac{1}{n!} x^n e^{-x}.
10. 2(a)(iii)4 marksHence, show that if M is an integer greater than 1, then e^x F_M(x) = -\left(x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^M}{M!}\right) + (e^x - 1).
11. 2(b)(i)5 marksExpress \frac{2x^2 + 3}{(x^2 + 1)^2} in partial fractions.
12. 2(b)(ii)6 marksHence, find \int \frac{2x^2 + 3}{(x^2 + 1)^2} dx.
13. 3(a)(i)5 marksShow, by mathematical induction, that x_n < \frac{1}{2} for all positive integers n.
14. 3(a)(ii)3 marksBy considering x_{n+1} - x_n, show that x_n < x_{n+1}.
15. 3(b)(i)3 marksFind the constants A and B such that \frac{2 - 3x}{(1 - x)(1 - 2x)} \equiv \frac{A}{1 - x} + \frac{B}{1 - 2x}.
16. 3(b)(ii)4 marksObtain the first FOUR non-zero terms of the expansion of each of (1 - x)^{-1} and (1 - 2x)^{-1} as power series of x in ascending order.
17. 3(b)(iii) a)2 marksFind the range of values of x for which the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)} is valid.
18. 3(b)(iii) b)2 marksFind the coefficient of x^n in the series expansion of \frac{2 - 3x}{(1 - x)(1 - 2x)}.
19. 3(iv)6 marksThe sum, S_n, of the first n terms of a series is given by S_n = n(3n - 4). Show that the series is an Arithmetic Progression (A.P.) with common difference 6.
20. 4(a)(i)4 marksShow that r + 1 + \frac{1}{r} = \frac{13}{3}.
21. 4(a)(ii) a)4 marksHence, find the value of r.
22. 4(a)(ii) b)1 markFind the value of a.
23. 4(a)(ii) c)2 marksFind the sum to infinity of the G.P.
24. 4(b)5 marksExpand \frac{2}{e^x + e^{-x}}, |x| < 1 in ascending powers of x as far as the term in x^4.
25. 4(c)(i)3 marksExpress f(r) - f(r + 1) in terms of r.
26. 4(c)(ii)4 marksHence, or otherwise, find S_n = \sum_{r=1}^n \frac{3}{r(r + 1)(r + 2)}.
27. 4(c)(iii)2 marksDeduce the sum to infinity of the series in (c)(ii).
28. 5(a)(i)2 marksFrom the definition, show that \binom{n}{r} = \binom{n}{n - r}.
29. 5(a)(ii)4 marksFrom the definition, show that \binom{n+1}{r} = \binom{n}{r} + \binom{n}{r - 1}.
30. 5(a)(iii)3 marksHence, prove that \left[ \binom{8}{6} + \binom{8}{5} \right] \times \left[ \binom{8}{3} + \binom{8}{2} \right] is a perfect square.
31. 5(b)(i)3 marksFind the number of 5-digit numbers greater than 30 000 which can be formed with the digits, 1, 3, 5, 6 and 8, if no digit is repeated.
32. 5(b)(ii)5 marksWhat is the probability of one of the numbers chosen in (b)(i) being even?
33. 5(c)(i) a)2 marksShow that (1 - i) is one of the square roots of -2i.
34. 5(c)(i) b)1 markFind the other square root of -2i.
35. 5(c)(ii)5 marksHence, find the roots of the quadratic equation z^2 - (3 + 5i)z + (8i - 4) = 0.
36. 6(a)(i)3 marksShow that |A| = 5.
37. 6(a)(ii) a)2 marksMatrix B is formed by interchanging row 1 and row 2 of matrix A and then interchanging column 1 and column 2 of the resulting matrix. Write down det(B), giving a reason.
38. 6(a)(ii) b)2 marksRow 1 of matrix C is formed by adding row 2 to row 1 of matrix A. The other rows remain unchanged. Write down det(C), giving a reason.
39. 6(a)(ii) c)2 marksMatrix D is formed by multiplying each element of matrix A by 5. Write down det(D), giving a reason.
40. 6(b)(i)3 marksFind AM.
41. 6(b)(ii)2 marksFind the inverse, A^{-1}, of A.
42. 6(c)(i)1 markWrite the system of equations in the form Ax = b.
43. 6(c)(ii)2 marksShow that x = A^{-1}b.
44. 6(c)(iii)2 marksHence, solve the system of equations.
45. 6(c)(iv) a)1 markShow that (x, y, z) = (1, 1, 1) is a solution of the system of equations: x + y + z = 3 2x + 2y + 2z = 6 3x + 3y + 3z = 9.
46. 6(c)(iv) b)5 marksHence, find the general solution of the system.