CAPE Pure Mathematics Unit 2 · May/June 2018 · 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 choiceWhich of the following is a sketch of the locus of the point represented by the complex number z, given that |z - 5i| = 3?
2. Q21 mark · multiple choiceThe roots of the equation x^2 + 1 = 0 are
3. Q31 mark · multiple choiceGiven that z = -1 + \sqrt{3}i, then the exponential form of the complex number z is
4. Q41 mark · multiple choiceThe derivative of \ln\left(\frac{1}{\sqrt[3]{x}}\right) is
5. Q51 mark · multiple choiceIf f(x, y) is such that \frac{\partial f}{\partial x} = e^x [-\sin(x + y) + \cos(x + y)] \text{ and} \frac{\partial f}{\partial y} = -e^x \sin(x + y), \text{ then which of the} following is true?
6. Q61 mark · multiple choiceThe locus of the points described by a complex number z is given by |z - 1 - 2i| = 3. The locus describes a circle with
7. Q71 mark · multiple choiceThe complex number z = \frac{1}{1 - i} can be represented on an Argand diagram as
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 choiceThe expression i[(1 + i)^2 - (1 - i)^2] is equal to
10. Q101 mark · multiple choice\int \frac{1}{1 + 9x^2}\,dx is
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{\sec^2 x}{2\tan x}\,dx =
14. Q141 mark · multiple choiceThe principal value of the argument of the complex number -2 + 2i is
15. Q161 mark · multiple choiceFor -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
16. Q171 mark · multiple choiceThe binomial coefficient \begin{pmatrix} n \\ 4 \end{pmatrix} is equivalent to
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 choiceThe equation e^x - x^4 = 0 has a root between
19. Q201 mark · multiple choiceThe sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
20. Q211 mark · multiple choiceLet a_n and S_n denote, respectively, the value of the n^{\text{th}} term and the n^{\text{th}} partial sum of a series. The value of S_{n+2} - S_n when calculated on the series is
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 coefficient of x^4 in the Taylor series expansion of f(x) = \cos x about x = 0 is
24. Q251 mark · multiple choiceThe value of the term independent of x in the binomial expansion of \left(x^2 + \frac{1}{x}\right)^{12} is
25. Q261 mark · multiple choiceWhich of the following statements is true?
26. Q271 mark · multiple choiceIf the coefficient of x^3 in the expansion of (6 - ax)^9 is -84, then the value of a is
27. Q281 mark · multiple choiceThe sum of the infinite geometric series 180 - 60 + 20 - \dots is
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 choiceA continuous function is defined by f(0) = 1 and f(0.8) = -0.76. The first approximation to the root in [0, 0.8], to 3 decimal places, using linear interpolation is
30. Q311 mark · multiple choiceIf \mathbf{M} = \begin{pmatrix} 1 & 3 & 3 \\ -2 & 4 & 1 \\ 0 & 5 & 6 \end{pmatrix}, the FIRST ROW of the co-factor matrix of \mathbf{M} is
31. Q321 mark · multiple choiceThe determinant of the matrix \mathbf{M} = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
32. Q331 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
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 choiceIf marbles are chosen, without replacement, from a bag of 10 blue and 5 red marbles, then the probability of getting a red marble followed by 2 blue marbles is
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 choiceIf \mathbf{M} = \begin{pmatrix} 1 & 1 & 4 \\ 3 & 2 & -1 \\ 6 & 0 & 5 \end{pmatrix}, then the co-factor of the element 3 in \mathbf{M} above may be written as
37. Q381 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
38. Q391 mark · multiple choice\mathbf{A}, \mathbf{B}, \mathbf{C} and \mathbf{D} are four 3 \times 3 matrices. Given that \mathbf{AB} = \mathbf{J}, \mathbf{BC} = \mathbf{K}, \mathbf{CD} = \mathbf{L}, \mathbf{ABC} = \mathbf{P} and…
39. Q401 mark · multiple choiceA suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} \text{ is}
40. Q411 mark · multiple choiceThe general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is
41. Q421 mark · multiple choiceThe probability that L occurs, given that K occurs, is
42. Q431 mark · multiple choiceChad, Matthew, Josh, Paul and Tifanny are travelling in a five-seater car with 3 persons in the back and 2 persons in the front. Each person occupies a seat. The number of different ways they can sit in the car if…
43. Q441 mark · multiple choiceThe general solution of the differential equation \frac{d^2y}{dx^2} - 3\frac{dy}{dx} = 0 is
44. Q451 mark · multiple choiceThe matrix \mathbf{A} represents a system of linear equations after some elementary row operations have been performed. \mathbf{A} = \begin{pmatrix} 1 & 0 & 1 & : & 3 \\ 0 & 1 & -1 & : & 2 \\ 0 & 0 & 0 & : & 2…