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CAPE Pure Mathematics Unit 1 · May/June 2009 · 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 choiceGiven that x and y are negative integers, and that x > y, which of the following is true?
  2. Q21 mark · multiple choice\log_5 5\sqrt{5} is equal to
  3. Q31 mark · multiple choiceWhich of the following BEST represents f(x) = x(1 - x)?
  4. Q41 mark · multiple choiceThe tables below show the values of the functions f and g. \begin{tabular}{|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{tabular}…
  5. Q51 mark · multiple choiceThe range of values of x that satisfies the inequality |x - b| < a is
  6. Q61 mark · multiple choice4\sqrt{x} + \frac{4\sqrt{3x}}{\sqrt{48}} =
  7. Q71 mark · multiple choiceGiven (x - p)^3 - q^2(x - p) = 0, the values of x are
  8. Q81 mark · multiple choiceThe range of values of x for which \frac{2x + 3}{x} \ge 8 is
  9. Q91 mark · multiple choiceIf \log_a 4 + \log_a x - \log_a 7 = 2, then the value of x is
  10. Q101 mark · multiple choiceThe point (1, 1) lies on the curve y = f(x). Under which of the following transformations is (1, 1) invariant? \begin{align*} \text{I.} & \quad y = f^{-1}(x) \\ \text{II.} & \quad y = f(-x) \\ \text{III.} & \quad…
  11. Q111 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?
  12. Q121 mark · multiple choiceThe values of x that satisfy the inequality |2x - a| > |x|, a > 0, are
  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 choiceTwo functions, f and g, are defined as f: x \to 2x + 1, 0 < x < 10, and g: x \to x^2, 0 < x < 3. The composite function gf can only be formed if the domain of f is restricted to
  15. Q151 mark · multiple choiceThe number of visas V(x) issued by an embassy annually, is given by V(x) = 7x^2 - 42x + 72. The LEAST number of visas issued in a particular year, x, is
  16. Q161 mark · multiple choiceThe graph of y = \sin 2x is
  17. Q171 mark · multiple choiceThe value of \sin\left(\frac{\pi}{2} + p\right) is
  18. Q181 mark · multiple choice\frac{1}{\csc^2 x} \equiv
  19. Q191 mark · multiple choiceIf P = (2\sin^2\theta + 2\cos^2\theta)(\sec^2\theta - \tan^2\theta) then P is equal to
  20. Q201 mark · multiple choiceThe vector \begin{pmatrix} p \\ q \end{pmatrix} is perpendicular to the vector \begin{pmatrix} 3 \\ -1 \end{pmatrix}. The relation between p and q is
  21. Q211 mark · multiple choice1 + \cos^4 A - \sin^4 A =
  22. Q221 mark · multiple choiceThe expression \cot x + \tan x can be written as
  23. Q231 mark · multiple choiceThe normal at P(4, 3) to the circle (x - 2)^2 + y^2 = 25 has gradient \frac{3}{4}. The equation of the tangent at P to (x - 2)^2 + y^2 = 25 is
  24. Q241 mark · multiple choiceThe line through the points P(k, 2) and Q(6, 8) is parallel to the line with equation 3x + y - 21 = 0. The value of k is
  25. Q251 mark · multiple choiceA curve C is given by the parametric equations x = 2\sin\theta, y = \cos\theta. The Cartesian equation of C is
  26. Q261 mark · multiple choiceA vector equation is given as s\begin{pmatrix} -2 \\ 1 \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 p = 2\mathbf{i} + \mathbf{j} and q = \lambda\mathbf{i} + 6\mathbf{j} are perpendicular vectors, then the value of \lambda is
  28. Q281 mark · multiple choiceThe distance (d) metres of a reciprocating 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…
  29. Q291 mark · multiple choiceFor n \in \mathbb{Z}, the general solution of \sqrt{3}\sin\theta - \cos\theta = 0 is \theta =
  30. Q301 mark · multiple choiceThe vector \mathbf{u} has magnitude 4\sqrt{5} units and is parallel to the vector \mathbf{v} = \mathbf{i} - 2\mathbf{j}. A unit vector parallel to \mathbf{u} is
  31. Q311 mark · multiple choiceBased on the diagram above, which of the following statements is NOT correct?
  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. Q331 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. Q341 mark · multiple choiceGiven that \int_2^4 4f(x)\,dx = 9, the value of \int_4^2 3f(x)\,dx is
  35. Q351 mark · multiple choiceThe portion of the curve y = 2x between x = a and x = b is rotated 360^\circ (2\pi radians) about the x-axis. The volume, v, of the solid generated is BEST represented as
  36. Q361 mark · multiple choiceIf y = \tan 6x then \frac{dy}{dx} is
  37. Q371 mark · multiple choiceIf y = \frac{x - 6}{3 - 4x} then \frac{dy}{dx} is
  38. Q381 mark · multiple choiceIf y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  39. Q391 mark · multiple choiceThe gradient at x = \frac{\pi}{6} on the curve y = \sin x is
  40. Q401 mark · multiple choiceA curve is defined by the equation y = -5(1 - 2x)^2. Given that x increases at a rate of 1 unit per second when x = 1, what is the corresponding rate of change for y?
  41. Q411 mark · multiple choice\int_0^\frac{\pi}{4} \sec^2 x\,dx =
  42. Q421 mark · multiple choiceThe area of R is
  43. Q431 mark · multiple choiceIf the rate of change of y with respect to x is 3x^2 + \frac{4}{x^3} then y is equal to
  44. Q441 mark · multiple choiceA rod is heated and its length at time t seconds is given by L = 5t^2 + 100\text{ centimetres}. When t = 3, the rate of increase of L, in \text{cm s}^{-1}, is
  45. Q451 mark · multiple choiceIf a rope of length k\text{ metres} forms three sides of a rectangle of width x\text{ metres} then the area, R, in square metres, of the rectangle is given by R = x(k - 2x). The MAXIMUM value of R is

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