Quelpr

CAPE Pure Mathematics Unit 1 · May/June 2010 · 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 choicex - 2 is a factor of
  2. Q21 mark · multiple choiceIf p and q are positive integers such that p < q, then which of the following statement(s) is/are correct? I. -p > -q II. p^2 > pq III. p - 1 < q - 1
  3. Q31 mark · multiple choice3^{\log_3 5} =
  4. Q41 mark · multiple choiceGiven that f(x) = x^2, which of the following graphs shows y = f(x) - 1?
  5. Q51 mark · multiple choiceIf px^2 + 5x + 2 = 0 has 2 real and distinct roots, the range of possible values of p is
  6. Q61 mark · multiple choice\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x} + \sqrt{y}} may be expressed as
  7. Q71 mark · multiple choiceThe rational expression \frac{x}{x^2 + 3x - 4} + \frac{2}{x - 1} simplified as a single fraction is
  8. Q81 mark · multiple choicea^5 - b^5 =
  9. Q91 mark · multiple choiceThe value of \log_{\sqrt{6}} 36 is
  10. Q101 mark · multiple choiceThe coordinates of the point P are (4, -3). Under a one-way stretch by scale factor 2 in the y-direction with the x-axis invariant, the image of P would be
  11. Q111 mark · multiple choiceGiven \frac{8}{x^2} = \frac{\sqrt{x}}{4}, the value of x is
  12. Q121 mark · multiple choiceThe cubic equation whose roots \alpha, \beta and \gamma satisfy the following \alpha + \beta + \gamma = \frac{-2}{7}, \alpha\beta + \beta\gamma + \gamma\alpha = -1 and \alpha\beta\gamma = -\frac{3}{7} is
  13. Q131 mark · multiple choiceAn arch may be modelled by the Cartesian equation y = -2x^2 + 4x + 1, where x and y represent respectively horizontal and vertical distances. The coordinates of the HIGHEST point on the arch are
  14. Q141 mark · multiple choiceWhich of the following statements is FALSE?
  15. Q151 mark · multiple choiceIf \alpha, \beta and \gamma are the three roots of the cubic equation x^3 - 2x^2 - 4x + 5 = 0, then the value of \alpha^2\beta\gamma + \alpha\beta^2\gamma + \alpha\beta\gamma^2 is
  16. Q161 mark · multiple choiceThe centre of the circle (x - 1)^2 + (y - 2)^2 = 16 is
  17. Q171 mark · multiple choice\frac{\sin\theta (1 - \sin^2\theta)}{\cos\theta (1 - \cos^2\theta)} =
  18. Q181 mark · multiple choiceThe equation of the circle whose centre has coordinates (4, 1) and whose radius is 7 units is
  19. Q191 mark · multiple choiceThe point P has position vector \begin{pmatrix}-3 \\ 5\end{pmatrix} and Q is a point such that \vec{PQ} = \begin{pmatrix}1 \\ -7\end{pmatrix}. The position vector of Q is
  20. Q201 mark · multiple choiceThe vector \mathbf{a} is given as 5\mathbf{i} + 12\mathbf{j}. A unit vector parallel to \mathbf{a} is
  21. Q211 mark · multiple choice\sin(\alpha + 45^\circ) is equal to
  22. Q221 mark · multiple choiceIf x = r\sin\theta\cos\alpha, y = r\sin\theta\sin\alpha, z = r\cos\theta, then
  23. Q231 mark · multiple choiceThe minimum and maximum values of \frac{1}{2 + \sin\theta} respectively are
  24. Q241 mark · multiple choiceThe general solution for \sin 2\theta = \sin\frac{\pi}{6} is
  25. Q251 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
  26. Q261 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. Q271 mark · multiple choice\frac{\sin\theta - \sin\alpha}{\cos\theta + \cos\alpha} is identical to
  28. Q281 mark · multiple choice\vec{OP} and \vec{OQ} are two vectors such that \vec{OP} = \mathbf{r} - \mathbf{s} and \vec{OQ} = 2\mathbf{r} + 3\mathbf{s}. Given that \vec{OP} is perpendicular to \vec{OQ} then \mathbf{r}\cdot\mathbf{s}…
  29. Q291 mark · multiple choiceThe relationship between the curve y = x^2 - 2x + 4 and the line y = 2x is that the line
  30. Q301 mark · multiple choiceThe point (2, 3) is at one end of a diameter of the circle whose equation is x^2 + y^2 - 10x + 2y + 1 = 0. The coordinates of the other end of the diameter are
  31. Q311 mark · multiple choiceItem 31 refers to the graph below of a function y = f(x). Based on the graph above, \lim_{x \to 1^+} f(x) =
  32. Q321 mark · multiple choiceWhich of the following real functions is discontinuous?
  33. Q331 mark · multiple choice\frac{\mathrm{d}}{\mathrm{d}r}(\pi r^2) is equal to
  34. Q341 mark · multiple choiceIf \int_2^4 f(x)\,\mathrm{d}x = 12, what is the value of \int_2^3 f(x)\,\mathrm{d}x + \int_1^2 (f(x) - 1)\,\mathrm{d}x?
  35. Q351 mark · multiple choiceItem 35 refers to the diagram below. The finite region R is enclosed by the curve y = x^2, the y-axis and the line y = 3 as shown in the diagram above. This region is rotated completely about the y-axis to…
  36. Q361 mark · multiple choiceThe value of \lim_{x \to 0} \frac{\sin 3x}{x} is
  37. Q371 mark · multiple choiceA curve is defined for x > 0 by the equation y = x + \frac{2}{x}. The gradient of the curve at x = 2 is
  38. Q381 mark · multiple choice\frac{\mathrm{d}}{\mathrm{d}x}(x^3 \sin x) may be expressed as
  39. Q391 mark · multiple choiceGiven y = 3x^2 + 5\sin 2x, then \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} is equal to
  40. Q401 mark · multiple choice\lim_{x \to 2} \frac{x^3 - 2^3}{x - 2} =
  41. Q411 mark · multiple choiceIf f(x) = \begin{cases} 2 & \text{for } x \le 2 \\ -1 & \text{for } x > 2 \end{cases}, then \int_1^4 f(x)\,\mathrm{d}x equals
  42. Q421 mark · multiple choiceThe displacement, s metres, of a marble moving along a board at time t minutes is given by s(t) = 4t^3 - 30t^2 + 72t + 7 for t \ge 0. For what values of t is the displacement of the marble increasing?
  43. Q431 mark · multiple choiceAt time t years, the growth of a certain country's gross national product, G, is given by the equation \frac{\mathrm{d}G}{\mathrm{d}t} = 5 + \cos t. At the beginning of the year 1990, the gross national product…
  44. Q441 mark · multiple choiceA curve has a stationary point at (2, 4). The equation of the normal at (2, 4) on the curve is
  45. Q451 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…

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