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CAPE Pure Mathematics Unit 1 · May/June 2011 · 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 choice\sqrt{8} + \sqrt{32} - \sqrt{162} can be simplified as
  2. Q21 mark · multiple choiceIf p and q are positive integers such that p < q, then which of the following statements is/are correct? \begin{align*} \text{I.} & \quad -p > -q \\ \text{II.} & \quad p^2 > pq \\ \text{III.} & \quad p - 1 < q - 1…
  3. Q31 mark · multiple choiceWhich of the following diagrams BEST represents f(x) = x(1 - x)?
  4. Q41 mark · multiple choiceThe range of values of x that satisfies the inequality |x - b| < a is
  5. Q51 mark · multiple choiceThe (k + 1) term in \sum_{r=1}^n r(r - 1) is
  6. Q61 mark · multiple choiceIf a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  7. Q71 mark · multiple choiceIf \alpha and \beta represent roots of the equation x^2 - px + q = 0, then the value of \alpha^2 + \beta^2 is
  8. Q81 mark · multiple choiceThe range of values of x, (x > 0), for which \frac{2x + 3}{x} \ge 8 is
  9. Q91 mark · multiple choiceThe exact value of \left(\frac{25}{16}\right)^{-\frac{1}{2}} is
  10. Q101 mark · multiple choiceWhich of the following is true if \alpha, \beta and \gamma are roots of the cubic equation 3x^3 - 4x^2 - 7x - 10 = 0?
  11. Q111 mark · multiple choiceThe value of \log_{\sqrt{6}} 36 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 \alpha\beta\gamma = -\frac{3}{7} is
  13. Q131 mark · multiple choiceGiven that \alpha, 2\alpha and 3\alpha are the roots of the equation x^3 + kx^2 + 48 = 0, the value of the constant k is
  14. Q141 mark · multiple choiceThe annual growth, g(x), (in thousands) of the population over x years is represented by g(x) = 2^x. Over how many years will an annual growth of 32 thousand be achieved?
  15. Q151 mark · multiple choiceThe range of real values of x for which x^2 - 3x - 28 < 0 is
  16. Q161 mark · multiple choiceThe cosine of the angle between the vectors -6\,\mathbf{j} and \mathbf{i} + \mathbf{j} is
  17. Q171 mark · multiple choiceThe value of \sin\left(\frac{\pi}{2} + p\right) is
  18. Q181 mark · multiple choiceThe circle with equation x^2 + y^2 - 2x - 4y - 11 = 0 has radius equal to
  19. Q191 mark · multiple choiceThe equation of the circle whose centre has coordinates (4, 1) and whose radius is 7 units is
  20. Q201 mark · multiple choiceIf \beta is an acute angle and \sin\beta = \frac{12}{13}, then \sec\beta =
  21. Q211 mark · multiple choice\frac{1}{\text{cosec}^2 x} \equiv
  22. Q221 mark · multiple choiceThe expression \cot x + \tan x may be written as
  23. Q231 mark · multiple choiceWhat value of \theta, 0 \le \theta \le \pi, satisfies the equation 2\cos^2\theta + 3\cos\theta - 2 = 0?
  24. Q241 mark · multiple choiceThe curve with parametric representation x = 2t, y = t^2 has Cartesian equation
  25. Q251 mark · multiple choiceThe expression \sin 6A + \sin 4A may be written as
  26. Q261 mark · multiple choiceA vector equation is given as s\begin{pmatrix} 1 \\ -2 \end{pmatrix} + t\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} -5 \\ -1 \end{pmatrix}. The values of s and t are, respectively
  27. Q271 mark · multiple choiceIf \mathbf{p} = 2\,\mathbf{i} + \mathbf{j} and \mathbf{q} = \lambda\,\mathbf{i} + 6\,\mathbf{j} are perpendicular vectors, then the value of \lambda is
  28. Q281 mark · multiple choiceWhich of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, (0 \le x \le 2\pi)?
  29. Q291 mark · multiple choiceWhich of the following equations BEST represents the graph shown above?
  30. Q301 mark · multiple choiceIf 2\cos\theta + 9\sin\theta \equiv r\cos(\theta - \alpha) where r > 0 and 0 < \alpha < \frac{\pi}{2}, then the maximum value of the expression is
  31. Q311 mark · multiple choiceIn the diagram above showing y^2 = x, y is NOT defined for
  32. Q321 mark · multiple choiceGiven that \lim_{x \to 0} \frac{\sin x}{x} = 1, where x is measured in radians, then \lim_{x \to 0} \frac{\sin 3x}{2x} is
  33. Q341 mark · multiple choiceThe function g is defined as g(x) = \begin{cases} 3x + 5 & \text{for } x < 3 \\ px + 2 & \text{for } x \ge 3 \end{cases} For the function to be continuous at x = 3, the value of 'p' should be
  34. Q351 mark · multiple choiceThe volume generated by rotating the bounded, shaded region through 360^\circ about the x-axis is
  35. Q361 mark · multiple choiceIf y = 3x^2 + 5\sin 2x, then \frac{d^2y}{dx^2} is equal to
  36. Q371 mark · multiple choiceIf y = \tan 6x then \frac{dy}{dx} is
  37. Q381 mark · multiple choiceIf \frac{dy}{dx} = \cos x then
  38. Q391 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*}
  39. Q401 mark · multiple choiceBased on the graph above, \lim_{x \to 1^+} f(x) =
  40. Q411 mark · multiple choiceGiven that \int_2^5 4 f(x)\,dx = 9, the value of \int_2^5 3 f(x)\,dx is
  41. Q421 mark · multiple choiceA curve has a stationary point at (2, 4). The equation of the normal to the curve at (2, 4) is
  42. Q431 mark · multiple choiceGiven \frac{dy}{dx} = 2x, then a sketch of the graph of y is
  43. Q441 mark · multiple choiceA rod is heated and its length at time t seconds is given by L = 5t^2 + 100 centimetres. When t = 3, the rate of increase of L, in \text{cm s}^{-1}, is
  44. Q451 mark · multiple choiceThe radius of a circle is increasing at a rate of 0.1\,\text{cm s}^{-1}. At the instant when the radius is 3\text{ cm}, the rate of increase of the area in \text{cm}^2\text{ s}^{-1} is

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