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CAPE Pure Mathematics Unit 1 · May/June 2012 · 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 choiceIn the real number system the inverse of addition is represented by
  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 choiceWhich of the following statements is true?
  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. Q51 mark · multiple choicex - 2 is a factor of
  6. Q61 mark · multiple choiceThe range of values of x that satisfies the inequality |x - b| < a is
  7. Q71 mark · multiple choicea^5 - b^5 =
  8. Q81 mark · multiple choiceThe graph of f(x) = |x - 2| + 1 is BEST illustrated by
  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 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 that the roots of x^2 - 5x + a = 0 are equal, then a =
  12. Q121 mark · multiple choiceThe function f(x) is decreasing for the range
  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 choiceThe general quadratic equation with roots \alpha and \beta may be written as
  15. Q151 mark · multiple choiceThe expression 2 - 4x + 3x^2 can be written as
  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 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
  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 choiceWhich of the following sketches BEST represents the curve y = \cos\frac{1}{2}x, (0 \le x \le 2\pi)?
  21. Q211 mark · multiple choice\sin(\alpha + 45^\circ) is equal to
  22. Q221 mark · multiple choiceWhich of the following equations best represents the graph?
  23. 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
  24. Q241 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
  25. Q251 mark · multiple choiceGiven that \alpha is an acute angle and \tan\alpha = \frac{3}{4}, then \sin(90^\circ - \alpha) =
  26. Q261 mark · multiple choice\sin(30^\circ - A) is equal to
  27. Q271 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
  28. Q281 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
  29. 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
  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 choiceThe first derivative of \frac{-1}{x^2 - 1} with respects to x is
  32. Q321 mark · multiple choiceThe function g is defined as g(x) = \begin{cases} 3x + 5, & x < 3 \\ px + 2, & x \ge 3 \end{cases}. For the function to be continuous at x = 3 the value of p should be
  33. Q331 mark · multiple choice\frac{d}{dr}(\pi r^2) is equal to
  34. Q341 mark · multiple choiceGiven that \int_3^5 4f(x)\,dx = 9, the value of \int_3^5 f(x)\,dx is
  35. Q351 mark · multiple choiceAn expression for obtaining the volume generated by rotating the bounded, shaded region through 360^\circ about the x-axis is
  36. Q361 mark · multiple choiceThe value of \lim_{x \to 0} \frac{\sin 3x}{x} is
  37. Q371 mark · multiple choiceIf y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  38. Q381 mark · multiple choiceIn the graph showing y^2 = x, y is NOT defined for
  39. Q391 mark · multiple choiceGiven y = 3x^2 + 5\sin 2x, then \frac{d^2y}{dx^2} is equal to
  40. Q401 mark · multiple choice\int_0^{\frac{\pi}{4}} \sec^2 x\,dx =
  41. Q411 mark · multiple choiceIf y = \tan 6x then \frac{dy}{dx} is
  42. Q421 mark · multiple choice\int_0^{\frac{\pi}{2}} \cos 5x\,dx is
  43. Q431 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?
  44. Q441 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
  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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