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CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 1

45 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.

Practise this paper
  1. Q11 mark · multiple choiceThe conjugate of the complex number 7 + \frac{1}{2}i is
  2. Q21 mark · multiple choiceThe complex number z = \sqrt{3} + i can be expressed as
  3. Q31 mark · multiple choiceGiven that \cos 2x = 1 - 2\sin^2 x, then \int_0^{\pi} \sin^2\left(\frac{x}{4}\right) dx is
  4. Q41 mark · multiple choiceOne square root of 3 - 4i is
  5. Q51 mark · multiple choiceThe complex number z = \frac{1}{1-i} can be represented on an Argand diagram as
  6. Q61 mark · multiple choiceIf f(x) = \ln 2x, then f'(x) =
  7. Q71 mark · multiple choiceThe equation e^x - x^4 = 0 has a root between
  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 choice\frac{d}{dx}(\ln x)^3 =
  10. Q101 mark · multiple choiceGiven that a, b, c and k are constants, then \int \frac{3}{x^2(x - 1)} dx can be expressed as
  11. Q111 mark · multiple choiceThe integral of \frac{1}{1 - \sin^2 x} with respect to x is
  12. Q121 mark · multiple choiceThe partial fractions of \frac{x + 3}{(2x + 5)(x - 1)^2} may be expressed in the form
  13. Q131 mark · multiple choiceA curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The simplest expression for the gradient of the tangent in terms of t is
  14. Q141 mark · multiple choice\int \frac{dx}{\sqrt{1 - 9x^2}} =
  15. Q151 mark · multiple choiceThe argument of the complex number z = -\frac{1}{2} + i\frac{\sqrt{3}}{2} is
  16. Q161 mark · multiple choiceFor -1 < 2n < 1, \sum_{r=0}^{\infty} (2n)^r =
  17. Q171 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
  18. Q181 mark · multiple choiceThe 5^{\text{th}} term in the sequence that is defined by the relation u_n = (-1)^{n+1}\frac{n}{3n - 1}, n \ge 1, is
  19. Q191 mark · multiple choiceWhich of the following sequences, \{u_n\}, converges?
  20. Q201 mark · multiple choiceWhich of the following series are arithmetic series? \begin{align*} \text{I.} & \quad \sum_{r=1}^n (7 + 3r) \\ \text{II.} & \quad \sum_{r=1}^n 2(3^r) \\ \text{III.} & \quad \sum_{r=1}^n \log_{10}(r + 1) \\ \text{IV.} &…
  21. Q211 mark · multiple choiceThe sum to infinity of a geometric series is \frac{1}{1 - 2x}. The range of x is
  22. Q221 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
  23. Q231 mark · multiple choiceThe binomial coefficient \binom{n}{2} is equivalent to
  24. 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
  25. Q251 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. Q261 mark · multiple choiceGiven that the coefficient of the term in b^3 in the binomial expansion of (a + b)^5 is 40, then a =
  27. Q271 mark · multiple choiceIf \sum_{n=2}^\infty 2^{-n} = a, then a is
  28. Q281 mark · multiple choiceThe Maclaurin series for \sin x, up to the term in x^3, is
  29. Q291 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
  30. Q301 mark · multiple choiceGiven that the n^{\text{th}} approximation of the root of the equation x^5 = x^3 + 25 based on the Newton-Raphson method is x_n, then x_{n+1} may be expressed as
  31. Q311 mark · multiple choiceIn how many ways can the letters ABCDE be arranged so that the A and B are always together?
  32. Q321 mark · multiple choiceThe number of distinct permutations of the letters of the word POSSIBILITY is
  33. Q331 mark · multiple choiceA relay team of five teachers is to be chosen from a group of 15 teachers. In how many ways could this relay team be chosen?
  34. Q341 mark · multiple choiceX and Y are mutually exclusive events. If P(X) = \frac{1}{4} and P(Y) = \frac{1}{5}, then P(X \cup Y) =
  35. Q351 mark · multiple choiceThe matrix A 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} =
  36. Q361 mark · multiple choiceThe number of possible values of x which satisfy the system of simultaneous equations, \begin{align*} 2x + 3y + 2z &= -5 \\ 4x + 6y + 4z &= -10 \\ 6x + 9y + 6z &= -16 \end{align*} is
  37. Q371 mark · multiple choiceIf M = \begin{pmatrix} 1 & 1 & 4 \\ 3 & 2 & -1 \\ 6 & 0 & 5 \end{pmatrix}, then the cofactor of the element 3 in M above may be written as
  38. Q381 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…
  39. Q391 mark · multiple choiceThe letters of the word I R R E G U L A R are to be arranged in a line. The number of possible arrangements in which the 3 Rs are NOT together is
  40. 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
  41. Q411 mark · multiple choiceThe determinant of the matrix M = \begin{pmatrix} 3 & -1 & 5 \\ 2 & 3 & -2 \\ 0 & 5 & 4 \end{pmatrix} is
  42. 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…
  43. Q431 mark · multiple choiceThe general solution of the differential equation (x - 2)\frac{dy}{dx} = y is
  44. 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
  45. Q451 mark · multiple choiceThe general solution of the differential equation \frac{dy}{dx} = \frac{y}{x} is

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