CAPE Pure Mathematics Unit 2 · May/June 2016 · Paper 1

44 multiple-choice items from this paper. Open one to see it in full, then practise it on Quelpr and get it marked against the mark scheme.

1. Q11 mark · multiple choiceThe complex number z = \frac{1}{1 - i} can be represented on an Argand diagram as
2. Q21 mark · multiple choiceWhich of the following is a sketch of the locus of the point represented by the complex number z, given that |z + 5i| = 3?
3. Q31 mark · multiple choiceThe expression i [(1 + i)^2 - (1 - i)^2] is equal to
4. Q41 mark · multiple choiceIf x^2y - xy^2 = 10, then \frac{dy}{dx} is equal to
5. Q51 mark · multiple choiceThe value of \left(\cos \frac{\pi}{2} + i\sin \frac{\pi}{2}\right)^2 is
6. Q61 mark · multiple choice\int \frac{\sec^2 x}{2\tan x} \, dx =
7. Q71 mark · multiple choiceThe derivative of \ln x^{\frac{1}{3}} is
8. Q81 mark · multiple choiceIf \frac{dy}{dx} = 2xy, then the value of \frac{d^2y}{dx^2} at the point (1, 2) is
9. Q91 mark · multiple choice\int \frac{1}{x^2 + 4} \, dx =
10. Q101 mark · multiple choiceA curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The expression for \frac{dy}{dx} is given by
11. Q111 mark · multiple choice\int (\cos 5x \cos 3x) \, dx =
12. Q121 mark · multiple choiceOne square root of 3 - 4i is
13. Q131 mark · multiple choice\int \frac{2x}{(x - 1)(x + 3)} \, dx =
14. Q141 mark · multiple choiceIf f(x, y) is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] and \frac{\partial f}{\partial y} = -e^x \sin(x + y), then which of the following is TRUE?
15. Q161 mark · multiple choiceIf the terms of the sequence u_1, u_2, u_3, \dots, u_n, \dots satisfy the recurrence relation u_{n+1} = u_n + 3, n \ge 1, then the n^{\text{th}} term may be expressed as
16. Q171 mark · multiple choiceFor -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
17. Q181 mark · multiple choiceGiven that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
18. Q191 mark · multiple choiceFor the recurrence relation a_{n+1} = a_n - a_{n-1} where a_1 = 1 and a_2 = 3, the value of a_5 is
19. Q201 mark · multiple choiceThe sum to infinity of the geometric series 16 + 12 + 9 + \dots is
20. Q211 mark · multiple choiceThe expression \frac{n!}{(n - 2)!} can be simplified and written as
21. Q221 mark · multiple choice\sum_{r=1}^n \left(\frac{1}{r} - \frac{1}{r + 1}\right) =
22. Q231 mark · multiple choiceGiven that \sum_{k=1}^n k(k + 1) = S_n, then, for m < n, \sum_{k=m+1}^n k(k + 1) =
23. Q241 mark · multiple choiceThe equation e^x - x^4 = 0 has a root between
24. Q251 mark · multiple choiceBy using the Newton–Raphson method with a first approximation x_n, the second approximation x_{n+1} for a root of the equation x^5 = x^3 + 25 may be expressed as
25. Q261 mark · multiple choice^8C_3 equals
26. Q271 mark · multiple choiceIf the coefficient of x^3 in the expression of (6 - ax)^9 is -84, then the value of a is
27. Q281 mark · multiple choiceA relay team of 5 teachers is to be chosen from a group of 15 teachers. In how many ways could this relay team be chosen?
28. Q291 mark · multiple choiceThe values of x for which the expansion of \frac{1}{\sqrt{100 - 50x}} is valid are
29. Q301 mark · multiple choiceLet f be a continuous function with f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], using linear interpolation, to 3 decimal places is
30. Q311 mark · multiple choiceIn how many ways can the letters P, Q, R, S and T be arranged so that P and Q are always together, and R and S are always together?
31. Q321 mark · multiple choiceA is a 3 \times 3 matrix with determinant 14. If the matrix of cofactors of A is \begin{pmatrix} 4 & -14 & -2 \\ 3 & -7 & -5 \\ 1 & 7 & 3 \end{pmatrix}, then A^{-1} =
32. Q331 mark · multiple choiceIf P = \begin{pmatrix} 1 & -2 & 0 \\ 3 & 1 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} -7 & 6 & -10 \\ -14 & 3 & -5 \\ 7 & 0 & 7 \end{pmatrix}, then…
33. Q341 mark · multiple choiceGiven that y = 0 at x = 0, the general solution of the differential equation y'' + 6y' + 9y = 0 is
34. Q351 mark · multiple choiceA sample space X consists solely of 3 mutually exclusive events, Q, R and S. If P(Q) = 0.3 and P(R) = 0.6, then P(S) =
35. Q361 mark · multiple choiceA school debating team comprising 3 teachers, 3 boys and 3 girls is to be chosen from 5 teachers, 4 boys and 6 girls. The number of ways in which this team can be chosen is
36. Q371 mark · multiple choiceOne person is randomly selected. What is the probability that this person is female and prefers Drink B?
37. Q381 mark · multiple choiceThe FIRST ROW of the product PQ of the two 3 \times 3 matrices P = \begin{pmatrix} 2 & 3 & 1 \\ 5 & -6 & 5 \\ -1 & 2 & 3 \end{pmatrix} and Q = \begin{pmatrix} 2 & 1 & 3 \\ 5 & 0 & -1 \\ -3 & -2 & 4 \end{pmatrix}…
38. Q391 mark · multiple choiceA, B, C and D are four 3 \times 3 matrices. Given that AB = J, BC = K, CD = L, ABC = P and BCD = Q, where J, K, L, P and Q are matrices, the product of ABCD is
39. Q401 mark · multiple choiceTwo coins and a die with faces numbered 1 to 6 are thrown together once. Assuming that the die and coins are fair, the probability of obtaining 2 heads and a number less than 4 is
40. Q411 mark · multiple choiceThe general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is
41. Q421 mark · multiple choiceA suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} is
42. Q431 mark · multiple choiceThe probability that L occurs, given that K occurs, is
43. Q441 mark · multiple choiceThe number of possible values of x which satisfy the system of simultaneous equations, \[\begin{aligned} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -25 \end{aligned}\] is
44. Q451 mark · multiple choiceThe matrix A represents a system of linear equations after some elementary row operations have been performed. \[A = \begin{pmatrix} 1 & 0 & 1 & : & 3 \\ 0 & 1 & -1 & : & 2 \\ 0 & 0 & 0 & : & 2 \end{pmatrix}\] Which…