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CAPE Pure Mathematics Unit 1 · May/June 2013 · 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 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? I. -p > -q II. p^2 > pq III. p - 1 < q - 1
  3. Q31 mark · multiple choiceTwo roots of the cubic equation 2x^3 + 3x^2 - 5x - 6 are -1 and -2. The THIRD root is
  4. Q41 mark · multiple choiceRationalising \frac{\sqrt{2}-1}{\sqrt{2}+1} gives
  5. Q51 mark · multiple choiceIf a remainder of 7 is obtained when x^3 - 3x + k is divided by x - 3, then k equals
  6. Q61 mark · multiple choiceWhich of the following are factors of 4x^4 + 8x^3 - 2x^2 - 6x - 4? I. x + 1 II. x - 1 III. x + 2 IV. x - 2
  7. Q71 mark · multiple choicea^5 - b^5 =
  8. Q81 mark · multiple choiceWhich of the following mapping diagrams does NOT represent a function?
  9. Q91 mark · multiple choiceIf g(x) is the inverse of f(x) then the correct diagram 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 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?
  12. Q121 mark · multiple choice\log 15 - \log 6 + \frac{1}{2}\log\frac{4}{25} =
  13. Q131 mark · multiple choiceThe values of x that satisfy the inequality |2x - a| > |x|, a > 0, are
  14. Q141 mark · multiple choiceThe statement p \lor \sim p is a
  15. Q151 mark · multiple choiceThe statement \sim(p \lor (\sim p \land q)) is logically equivalent to
  16. Q161 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
  17. Q171 mark · multiple choice\sin(30^\circ - A) is equal to
  18. Q181 mark · multiple choice2\sin\theta\cos\phi is equivalent 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 \cos\beta = \frac{5}{13}, then \sec\beta =
  21. Q211 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
  22. Q221 mark · multiple choiceThe value of \sin\left(\frac{\pi}{2} + p\right) is
  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 choiceWith respect to an origin O, A has coordinates (3, -2). The position vector of 3\,\vec{OA} is
  25. Q251 mark · multiple choiceThe expression \sin 6A + \sin 4A may be written as
  26. Q261 mark · multiple choice1 + \cos^4 A - \sin^4 A =
  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 choiceThe general solution for \sin 2\theta = \sin\frac{\pi}{6} is
  29. Q291 mark · multiple choiceThe cosine of the angle between the vectors -6\,\mathbf{j} and \mathbf{i} + \mathbf{j} is
  30. Q301 mark · multiple choiceIn the diagram above showing y^2 = x, y is NOT defined for
  31. Q311 mark · multiple choice\lim_{x \to 3} \frac{x^2 - 9}{x - 3} is
  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 choice\frac{d}{dx}(x^3 \sin x) may be expressed as
  34. 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
  35. Q351 mark · multiple choiceIf y = \frac{x - 6}{3 - 4x}, then \frac{dy}{dx} is
  36. Q361 mark · multiple choiceIf y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  37. Q371 mark · multiple choiceIf y = \tan 6x then \frac{dy}{dx} is
  38. Q381 mark · multiple choiceIf \frac{dy}{dx} = \cos x then
  39. Q391 mark · multiple choiceIf f''(x) = 6x, then given that f'(0) = 0, and c is a constant, f(x) =
  40. Q401 mark · multiple choiceThe path of an object is given parametrically as x = \sin t + 2, y = \cos t + 1. The slope of the tangent at t = \frac{\pi}{4} is
  41. Q411 mark · multiple choiceGiven that \int_2^5 4\,f(x)\,dx = 9, the value of \int_2^5 3\,f(x)\,dx is
  42. Q421 mark · multiple choiceThe gradient of the normal to the curve y = 3x^2 - 2x + 1 at x = 1 is
  43. Q431 mark · multiple choiceWater is leaking from a tank. The rate of change in volume of the 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…
  44. Q441 mark · multiple choiceGiven \frac{dy}{dx} = 2x, then possible sketches of the graph of y are I. A parabola opening upwards with vertex at (0, 1), passing through (-1, 1) and (1, 1)? II. A parabola opening upwards with vertex at…
  45. 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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