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CAPE Pure Mathematics Unit 2 · May/June 2017 · 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.

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  1. Q11 mark · multiple choiceExpressed in the form a + bi, where a, b, \in \mathbb{R}, \frac{-2 + 2i}{1 + i} =
  2. Q21 mark · multiple choiceThe complex number z = \sqrt{3} + i can be expressed as
  3. Q31 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?
  4. Q41 mark · multiple choiceOne square root of 3 - 4i is
  5. Q51 mark · multiple choice\bar{z} is the conjugate of z. Which of the following are always true? I. |\bar{z}| = |z| II. \arg z = \arg \bar{z} III. z\bar{z} is real
  6. Q61 mark · multiple choiceGiven that y = \tan^{-1}(2x), then \frac{dx}{dy} equals
  7. Q71 mark · multiple choiceGiven e^{x+y} - x = 0, then \frac{dy}{dx} is equal to
  8. Q81 mark · multiple choiceThe number of bacteria present in a culture is modelled by y = y_0 e^{kt}, where k > 0, y is the population after t hours, and y_0 is the initial population. The rate of growth, c, when t = 5 is given by
  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 choiceGiven that f(x) = \ln 3x^2, then f'(-2) equals
  11. Q111 mark · multiple choiceThe integral of \frac{1}{1 - \sin^2 x} with respect to x is
  12. Q121 mark · multiple choice\int 4e^{2x - 1} dx =
  13. Q131 mark · multiple choiceThe argument of the complex number z = -\frac{1}{2} + i\frac{\sqrt{3}}{2} is
  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 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
  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 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
  19. Q201 mark · multiple choiceWhich of the following series are arithmetic series? I. \sum_{r=1}^n (7 + 3r) II. \sum_{r=1}^n 2(3^r) III. \sum_{r=1}^n \log_{10}(r + 1) IV. \sum_{r=1}^n \log_{10} 3^{(r+1)}
  20. Q211 mark · multiple choiceThe sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
  21. Q221 mark · multiple choiceThe coefficient of a^2b^5 in the expansion of (a + b)^7 is
  22. Q231 mark · multiple choiceThe sum of the first n terms of a geometric series is \left[1 - \left(\frac{1}{2}\right)^n\right]. The value of the SECOND term 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 the term that is independent of x in the binomial expansion of \left(x^2 + \frac{1}{x}\right)^{12} 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 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], to three decimal places, using linear interpolation is
  28. Q291 mark · multiple choiceIf the coefficient of x^3 in the expansion of (6 - ax)^9 is -84, then the value of a is
  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]. Applying the interval bisection twice, a more accurate determination of the interval containing the root is
  30. Q311 mark · multiple choiceIn how many ways can the letters ABCDE be arranged so that the A and B are always together?
  31. Q321 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…
  32. Q331 mark · multiple choiceThe probability that L occurs, given that K occurs, is
  33. Q341 mark · multiple choiceThe determinant of the matrix M = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
  34. Q351 mark · multiple choiceA sample space S consists of 3 mutually exclusive and exhaustive events Q, R and S. If P(Q) = 0.3 and P(R) = 0.6, then P(S) =
  35. Q361 mark · multiple choiceGiven that H is a non-singular, square matrix, the determinant |H^2| of H^2 is
  36. Q371 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} is
  37. Q381 mark · multiple choiceThe letters of the word IRREGULAR are to be arranged in a line. The number of possible arrangements in which the 3 Rs are NOT together is
  38. 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} I. are inconsistent II. are not parallel III. have a unique solution
  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 choiceA bag contains 6 blue balls and 4 red balls. Terry chooses 2 balls at random from the bag without replacement. The probability that BOTH balls are red is
  41. 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…
  42. Q431 mark · multiple choiceA suitable integrating factor for the solution of the differential equation \frac{dy}{dx} + \frac{2y}{x} = \frac{1}{x} is
  43. 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
  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…

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