OCR A-Level Maths 2027 Predicted Practice Paper — Pure Mathematics

Answer ALL questions. Write your answers in the spaces provided. You must write down all the stages in your working.

Section A: Pure Mathematics

Pure mathematics questions with a gradient of difficulty. Answer ALL questions.

• 1

  Question SECTION-A-PURE-MATHEMATICS1 (3 marks)

  The line L passes through the points A(-2, 5) and B(4, -7). (a) Find the gradient of L. (b) Find an equation of the line perpendicular to L that passes through the midpoint of AB, giving your answer in the form ax + by + c = 0 where a, b and c are integers.

  (Total for Question SECTION-A-PURE-MATHEMATICS1 is 3 marks)
• 2

  Question SECTION-A-PURE-MATHEMATICS2 (4 marks)

  Given that f(x) = 2x^3 - 5x^2 - 4x + 3, (a) show that (x - 3) is a factor of f(x), and (b) hence factorise f(x) completely.

  (Total for Question SECTION-A-PURE-MATHEMATICS2 is 4 marks)
• 3

  Question SECTION-A-PURE-MATHEMATICS3 (5 marks)

  Find the first four terms, in ascending powers of x, of the binomial expansion of (1 + 3x)^(-2), and state the range of values of x for which the expansion is valid.

  (Total for Question SECTION-A-PURE-MATHEMATICS3 is 5 marks)
• 4

  Question SECTION-A-PURE-MATHEMATICS4 (6 marks)

  A geometric series has first term a and common ratio r. The second term is 6 and the sum to infinity is 32. (a) Show that 16r^2 - 16r + 3 = 0. (b) Given that the series converges, find the two possible values of r and the corresponding values of a.

  (Total for Question SECTION-A-PURE-MATHEMATICS4 is 6 marks)
• 5

  Question SECTION-A-PURE-MATHEMATICS5 (6 marks)

  The curve C has equation y = x^3 - 6x^2 + 9x + 2. (a) Find dy/dx and hence the coordinates of the two stationary points of C. (b) Use the second derivative to determine the nature of each stationary point.

  (Total for Question SECTION-A-PURE-MATHEMATICS5 is 6 marks)
• 6

  Question SECTION-A-PURE-MATHEMATICS6 (7 marks)

  (a) Express 3sin(x) - 4cos(x) in the form Rsin(x - alpha), where R > 0 and 0 < alpha < 90 degrees, giving alpha to 1 decimal place. (b) Hence solve the equation 3sin(x) - 4cos(x) = 2.5 for 0 <= x <= 360 degrees, giving your answers to 1 decimal place.

  (Total for Question SECTION-A-PURE-MATHEMATICS6 is 7 marks)
• 7

  Question SECTION-A-PURE-MATHEMATICS7 (7 marks)

  A curve is defined by the parametric equations x = t^2 - 1, y = t^3 - 3t, for t in the real numbers. (a) Find dy/dx in terms of t. (b) Find the coordinates of the point(s) on the curve where the tangent is parallel to the x-axis.

  (Total for Question SECTION-A-PURE-MATHEMATICS7 is 7 marks)
• 8

  Question SECTION-A-PURE-MATHEMATICS8 (8 marks)

  (a) Use the substitution u = 1 + x^2 to find the integral of 2x/(1 + x^2)^3 with respect to x. (b) Hence evaluate the definite integral of 2x/(1 + x^2)^3 from x = 0 to x = 1, giving your answer as an exact fraction.

  (Total for Question SECTION-A-PURE-MATHEMATICS8 is 8 marks)
• 9

  Question SECTION-A-PURE-MATHEMATICS9 (8 marks)

  The functions f and g are defined for all real x by f(x) = 3x - 1 and g(x) = x^2 + 2. (a) Find the composite function fg(x) and gf(x), simplifying each. (b) Solve the equation fg(x) = gf(x). (c) Explain why the function g does not have an inverse over its given domain.

  (Total for Question SECTION-A-PURE-MATHEMATICS9 is 8 marks)
• 10

  Question SECTION-A-PURE-MATHEMATICS10 (9 marks)

  Relative to a fixed origin O, the points P and Q have position vectors p = 2i + 3j - k and q = 5i - j + 2k. (a) Find the vector PQ and its exact magnitude. (b) Find a unit vector in the direction of PQ. (c) The point R lies on the line PQ such that PR:RQ = 2:1. Find the position vector of R.

  (Total for Question SECTION-A-PURE-MATHEMATICS10 is 9 marks)
• 11

  Question SECTION-A-PURE-MATHEMATICS11 (9 marks)

  (a) Prove that (cos(2x))/(sin(x) + cos(x)) = cos(x) - sin(x) for all values of x for which sin(x) + cos(x) is not zero. (b) Hence, or otherwise, solve cos(2x) = 2(sin(x) + cos(x))(cos(x)) - 1 for 0 <= x <= 2pi, giving answers in radians as exact multiples of pi where appropriate.

  (Total for Question SECTION-A-PURE-MATHEMATICS11 is 9 marks)
• 12

  Question SECTION-A-PURE-MATHEMATICS12 (10 marks)

  The population P (in thousands) of a colony of insects is modelled by the differential equation dP/dt = 0.5P(1 - P/8), where t is the time in weeks and t >= 0. Initially P = 2. (a) By using partial fractions, show that ln(P/(8 - P)) = 0.5t + c for some constant c, and find c. (b) Hence show that P = 8/(1 + 3e^(-0.5t)). (c) State the value that P approaches as t becomes large, and interpret this in context.

  (Total for Question SECTION-A-PURE-MATHEMATICS12 is 10 marks)
• 13

  Question SECTION-A-PURE-MATHEMATICS13 (10 marks)

  The equation x^3 + 2x - 7 = 0 has a single real root, alpha. (a) Show that alpha lies between x = 1 and x = 2. (b) Show that the equation can be rearranged into the iterative form x_(n+1) = cbrt(7 - 2x_n). (c) Using x_0 = 1.5, apply the iteration to find x_1, x_2 and x_3, giving each to 4 decimal places. (d) Use the Newton-Raphson method once, starting from x_0 = 1.5, to obtain an alternative estimate of alpha, giving your answer to 4 decimal places.

  (Total for Question SECTION-A-PURE-MATHEMATICS13 is 10 marks)
• 14

  Question SECTION-A-PURE-MATHEMATICS14 (8 marks)

  A closed cylindrical can is to be made to hold a fixed volume of 500 cubic centimetres. The can has base radius r cm and height h cm. (a) Show that the total surface area A (including both circular ends) is given by A = 2pi r^2 + 1000/r. (b) Find the value of r that minimises A, giving your answer to 3 significant figures, and justify that this value gives a minimum.

  (Total for Question SECTION-A-PURE-MATHEMATICS14 is 8 marks)