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CAPE Pure Mathematics Unit 1 · May/June 2014 · 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.

Practise this paper
  1. Q11 mark · multiple choiceIn the real number system the inverse of addition is represented by
  2. Q21 mark · multiple choiceRationalising \frac{\sqrt{2}-1}{\sqrt{2}+1} gives
  3. Q31 mark · multiple choiceThe inverse of p \rightarrow q is
  4. Q41 mark · multiple choiceIf a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  5. Q61 mark · multiple choice\ln x^y =
  6. Q71 mark · multiple choiceIf \alpha and \beta represent the roots of the equation x^2 - px + q = 0, then the value of \alpha^2 + \beta^2 is
  7. Q81 mark · multiple choiceThe graph of f(x) = |x - 2| + 1 is BEST illustrated by
  8. Q91 mark · multiple choiceIf \log_a 4 + \log_a x - \log_a 7 = 2, then the value of x is
  9. Q101 mark · multiple choiceThe cubic equation whose roots \alpha, \beta and \gamma satisfy the following conditions \alpha + \beta + \gamma = \frac{-2}{7},\ \alpha\beta + \beta\gamma + \gamma\alpha = -1\ \text{and}\ \alpha\beta\gamma =…
  10. Q111 mark · multiple choiceWhich of the following sets of ordered pairs represent functions? \begin{align*} \text{I.} & \quad \{(-1, 1), (0, 2), (1, 3), (4, 6)\} \\ \text{II.} & \quad \{(-2, 4), (1, 1), (1, 4), (2, 4)\} \\ \text{III.} & \quad…
  11. Q121 mark · multiple choicea^4 - b^4 =
  12. Q131 mark · multiple choice3^{\log_3 5} =
  13. Q141 mark · multiple choiceGiven (x - p)^3 - q^2(x - p) = 0, the values of x are
  14. Q151 mark · multiple choiceThe tables below show the values for two functions, f and g. \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 7 & 5 & 3 & 2 & -7 & -5 \\ \hline \end{array}…
  15. Q161 mark · multiple choiceThe centre of the circle (x - 1)^2 + (y - 2)^2 = 16 is
  16. Q171 mark · multiple choiceThe value of \cos\left(\frac{\pi}{2} - p\right) is
  17. Q181 mark · multiple choiceThe vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relationship between p and q is
  18. Q191 mark · multiple choiceThe expression \sin 6\theta + \sin 4\theta may be expressed as
  19. Q201 mark · multiple choiceWhich of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, \quad (0 \le x \le 2\pi)?
  20. Q211 mark · multiple choiceWhat value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2\theta + 3\cos\theta - 2 = 0?
  21. Q221 mark · multiple choiceThe equation of the circle with centre (-3, 5) and radius 6 is
  22. Q231 mark · multiple choiceA curve is defined by the parametric equations x = 3 + 2t and y = \frac{1}{t}. The Cartesian equation of the curve is
  23. Q241 mark · multiple choiceIf P = (2\sin^2\theta + 2\cos^2\theta)(\sec^2\theta - \tan^2\theta), then P is equal to
  24. Q251 mark · multiple choiceThe expression \sin\left(\alpha + \frac{\pi}{4}\right) is equivalent to
  25. Q261 mark · multiple choiceThe variable point P(x, y) moves so that it is the same distance from the points (1, 6) and (3, 2). The equation of the locus of P may be obtained from
  26. Q271 mark · multiple choiceGiven that \mathbf{a} = \begin{pmatrix} -3 \\ 7 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}, then |3\mathbf{a} + 2\mathbf{b}| is equal to
  27. Q281 mark · multiple choiceA circle has centre (-1, -1). The equation of the tangent to the circle at the point (0, -3) on the circle is
  28. Q291 mark · multiple choiceThe distance, d metres, of an arm of a shaping machine from its starting position can be modelled by the equation d = 12\cos\theta + 5\sin\theta. The MAXIMUM distance, in metres, from the starting point is
  29. Q301 mark · multiple choiceThe vector \mathbf{a} is given as 5\mathbf{i} + 12\mathbf{j}. A unit vector parallel to \mathbf{a} is
  30. Q311 mark · multiple choiceFrom the diagram above, which of the following statements are true? \begin{align*} \text{I.} & \quad f'(1) < 0 \\ \text{II.} & \quad f(1) > k \\ \text{III.} & \quad f(2) = 0 \\ \text{IV.} & \quad f'(2) = k \end{align*}
  31. Q321 mark · multiple choice\lim_{x \to -5} \frac{x + 5}{x^2 - 25} =
  32. Q331 mark · multiple choice\frac{d}{dr}(\pi r^2) is equal to
  33. Q341 mark · multiple choiceA curve is given parametrically by the equations x = t^2 - 2t, y = t^2 + 2t. The simplest expression for \frac{dy}{dx} is given by
  34. Q351 mark · multiple choiceGiven that \cos 2x = 2\cos^2 x - 1, then \int \cos^2\left(\frac{x}{4}\right) dx is
  35. Q361 mark · multiple choiceThe value of \lim_{x \to 0} \frac{\sin 3x}{x} is
  36. Q371 mark · multiple choiceIf y = \sqrt{2x + 1}, then \frac{d^2y}{dx^2} is
  37. Q381 mark · multiple choiceIf \frac{dy}{dx} = 2xy, then the value of \frac{d^2y}{dx^2} at the point (1, 2) is
  38. Q391 mark · multiple choiceThe gradient at x = \frac{\pi}{6} on the curve y = \sin x is
  39. Q401 mark · multiple choiceWater is leaking from a tank. The rate of change in volume of water in the tank with respect to time, t, is inversely proportional to the volume, V, of water in the tank. If k is a positive constant of…
  40. Q411 mark · multiple choiceAn expression for obtaining the volume generated by rotating the shaded region through 360^\circ about the x-axis is
  41. Q421 mark · multiple choiceBased on the diagram above, which of the following statements is NOT correct?
  42. Q431 mark · multiple choiceGiven that f(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ ax + 2 & \text{for } x \ge 3 \end{cases} For the function to be continuous at x = 3, the value of a should be
  43. Q441 mark · multiple choice\lim_{x \to 2} \frac{x^3 - 2^3}{x - 2} =
  44. Q451 mark · multiple choice\frac{d}{dx}(x^3 \sin x) may be expressed as

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