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CAPE Pure Mathematics Unit 1 · May/June 2015 · 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 choicea^5 - b^5 =
  2. Q21 mark · multiple choiceIf a remainder of 3 is obtained when 8x^3 + 4x + k is divided by x - 1, then k equals
  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 choice4\sqrt{x} + \frac{4\sqrt{3x}}{\sqrt{48}} =
  5. Q51 mark · multiple choice3\log_2 2q - 2\log_2 3q + 1 expressed as a single logarithm in its SIMPLEST form is
  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 choiceThe annual growth, g(x), (in thousands) of the population in a country for x years is represented by g(x) = 2^x. In how many years will a growth of 32 thousand be achieved?
  8. Q81 mark · multiple choiceThe general quadratic equation with roots \alpha and \beta may be written as
  9. Q91 mark · multiple choiceIf g(x) is the inverse of f(x) then the correct diagram is
  10. Q101 mark · multiple choiceThe statement \sim(p \vee (\sim p \wedge q)) is logically equivalent to
  11. Q111 mark · multiple choiceWhich of the following statements is true?
  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 choice3^{\log_3 5} =
  15. Q151 mark · multiple choiceIf f(x) = 3x - 4 and f(g(x)) = x, then g(x) is
  16. Q161 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
  17. Q171 mark · multiple choice\sin(30^\circ - A) is equal to
  18. Q181 mark · multiple choiceThe centre of the circle (x - 1)^2 + (y - 2)^2 = 16 is
  19. Q191 mark · multiple choiceThe relationship between the curve y = x^2 - 2x + 4 and the line y = 2x is that the line
  20. Q201 mark · multiple choiceIf \beta is an acute angle and \cos \beta = \frac{5}{13}, then \sec \beta =
  21. Q211 mark · multiple choiceWith respect to an origin O, A has coordinates (3, -2). The position vector of 3\,\vec{OA} is
  22. Q221 mark · multiple choiceThe curves y^2 = x + 7 and xy = 6 intersect at three points. The y coordinates of these points are
  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 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
  25. Q251 mark · multiple choiceA circle has centre (-1, -1). The equation of the tangent to the circle at the point (0, -3) on the circle is
  26. Q261 mark · multiple choice1 + \cos^4 A - \sin^4 A \equiv
  27. Q271 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
  28. Q281 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
  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 choice\frac{1}{\text{cosec}^2 x} =
  31. Q311 mark · multiple choiceGiven that f(x) = (2x + 1)^3 then f''(2) equals
  32. Q321 mark · multiple choice\lim_{x \to -1} \frac{x^2 - x - 2}{x - 1} =
  33. Q331 mark · multiple choice\frac{d}{dx}(x^3 \sin x) may be expressed as
  34. Q341 mark · multiple choiceA dial on a plane preparing for landing registers the number 200 + 5\left(\frac{\sin h}{h}\right), where h is the height above the ground. Just as the plane lands the dial reads
  35. Q351 mark · multiple choiceIf \int_2^4 f(x)\,dx = 12, what is the value of \int_2^3 f(x)\,dx + \int_3^4 (f(x) - 1)\,dx?
  36. Q361 mark · multiple choiceIf y = \sqrt{2x + 1} then \frac{d^2y}{dx^2} is
  37. Q371 mark · multiple choiceThe gradient of the normal to the curve y = \ln x at x = 2 is
  38. Q381 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
  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 choiceIf f'(x) = \sin x, then f(x) =
  41. Q411 mark · multiple choiceAt time t years, the growth of a certain country's gross national product, G, is given by the equation \frac{dG}{dt} = 5 + \cos t. At the beginning of the year 1990, the gross national product was recorded as…
  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 choiceThe TOTAL shaded area in the diagram below is given by
  44. Q441 mark · multiple choiceGiven \frac{dy}{dx} = 2x then a sketch graph of y may be represented by I. II. III. IV.
  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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