CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 1

43 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 choiceExpressed in the form a + bi, where a, b, c \in \mathbb{R}, \frac{-2 + 2i}{1 + i} =
2. Q21 mark · multiple choiceThe quadratic equation with roots 2 \pm i\sqrt{3} is
3. Q31 mark · multiple choiceThe modulus of the complex number \frac{1}{2} - \frac{1}{2}i is
4. Q41 mark · multiple choiceThe value of \left[\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right]^4 is
5. Q51 mark · multiple choiceIf |z - 5 + 2i| = 3, the locus of the point (x, y) is
6. Q61 mark · multiple choiceGiven that y = \tan^{-1}(2x), then \frac{dx}{dy} equals
7. Q71 mark · multiple choiceThe complex number z = \sqrt{3} + i can be expressed as
8. Q81 mark · multiple choiceGiven that z = -1 + \sqrt{3}\,i, then the exponential form of the complex number z is
9. Q91 mark · multiple choiceThe derivative of the function y = \ln\left(\frac{\cos x}{\sin x}\right) with respect to x is
10. Q101 mark · multiple choiceIf f(x) = \ln 2x, then f''(x) =
11. Q111 mark · multiple choiceThe integral of \frac{1}{1 - \sin^2 x} with respect to x is
12. Q121 mark · multiple choice\int 2x e^{-x}\,dx =
13. Q131 mark · multiple choiceGiven that a, b, c and k are constants, then \int \frac{3}{x^2(x - 1)}\,dx can be expressed as
14. Q141 mark · multiple choiceIf f 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 choiceThe value of the term that is independent of x in the binomial expansion of \left[x^2 + \frac{1}{x}\right]^{12} is
16. Q171 mark · multiple choiceGiven that u_n represents the n^{\text{th}} term of a sequence, which of the following converges?
17. Q181 mark · multiple choiceA sequence is defined as u_{n+1} = 1 - \frac{1}{1 + u_n} where u_1 = 1 and n \in \mathbb{N}. The 20th term of the sequence is
18. Q191 mark · multiple choiceThe Maclaurin series for \sin x, up to the term in x^3, is
19. Q201 mark · multiple choice\sum_{r=1}^n \left[\frac{1}{r} - \frac{1}{r+1}\right] =
20. Q211 mark · multiple choiceWhich of the following series are arithmetic series? \nI. \sum_{r=1}^n (7 + 3r) \nII. \sum_{r=1}^n 2(3^r) \nIII. \sum_{r=1}^n \log_{10} 3^{(r+1)}
21. Q221 mark · multiple choiceThe equation e^x - x^4 = 0 has a root between
22. Q231 mark · multiple choiceThe sum of the infinite geometric series 180 - 60 + 20 - \dots is
23. Q241 mark · multiple choiceGiven that S_n = \sum_{i=1}^n \left[\frac{1}{i} - \frac{1}{i+1}\right], \lim_{n\to\infty} S_n is
24. Q251 mark · multiple choiceThe equation x^3 - x - 3 = 0 has one real positive root in the interval (n, n + 1). The value of n is
25. Q261 mark · multiple choiceThe value of \sum_{r=1}^\infty 2\left[\frac{1}{4}\right]^{r-1} is
26. Q271 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) =
27. Q281 mark · multiple choiceA continuous function is defined by f(0) = 1 and f(0.8) = -0.76. \nThe first approximation to the root in [0, 0.8], to 3 decimal places, using linear interpolation is
28. Q291 mark · multiple choiceThe binomial coefficient \binom{n}{4} is equivalent to
29. Q301 mark · multiple choiceIt is given that the equation e^{0.5x} + x^2 - 3.5x = 0 has exactly one root in the interval [0, 1]. \nApplying the interval bisection twice, a more accurate determination of the interval containing the root is
30. Q311 mark · multiple choiceWhich of the following statements is true?
31. Q321 mark · multiple choiceFrom the letters A, B, C, D and E, the number of three-letter words that can be made if no letter is repeated is
32. Q331 mark · multiple choiceItem 33 refers to the following Venn diagram which shows the probabilities associated with events K and L in a sample space S. \nThe probability that L occurs, given that K occurs, is
33. Q341 mark · multiple choiceA committee of 3 teachers, 3 doctors and 3 lawyers is to be chosen from 5 teachers, 4 doctors and 6 lawyers. The number of ways in which this committee can be chosen is
34. Q351 mark · multiple choiceGiven that y = 0 at x = 0, the general solution of the differential equation y'' + 6y' + 9y = 0 is
35. Q361 mark · multiple choiceGiven that \mathbf{H} is a non-singular, square matrix, the determinant |\mathbf{H}^2| of \mathbf{H}^2 is
36. Q371 mark · multiple choiceA relay team of 5 teachers is to be chosen from a group of 15 teachers. \nIn how many ways could this relay team be chosen?
37. Q391 mark · multiple choiceThe planes represented by the equations \begin{aligned} 2x + y - z &= 4 \\ x + y + z &= 1 \\ 3x - 2y - z &= 2 \end{aligned} \nI. are inconsistent\nII. are not parallel\nIII. have a unique solution
38. Q401 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 &= -16 \end{aligned}\nis
39. Q411 mark · multiple choiceThe matrix \mathbf{M} = \begin{pmatrix} 2 & -5 & 6 \\ 1 & 1 & -2 \\ a & 2 & 2 \end{pmatrix}. \nIf \det \mathbf{M} = 22, then a =
40. Q421 mark · multiple choiceIf the auxiliary equation for a homogeneous second order differential equation with real, constant coefficients is given by \lambda^2 + 6\lambda + 50 = 0, then the general solution of the differential equation may be…
41. Q431 mark · multiple choiceA suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x}\nis
42. Q441 mark · multiple choiceGiven that y = \frac{\pi}{4} and x = \frac{1}{2}, then the particular solution of \frac{dy}{dx} = 2x\cos^2 y is
43. Q451 mark · multiple choiceItem 45 refers to the matrix \mathbf{M} below. \mathbf{M} = \begin{pmatrix} 1 & 2 & 4 \\ -1 & 3 & 0 \\ 0 & 1 & 5 \end{pmatrix} \nThe co-factor of the element 3 in \mathbf{M} may be written as