OCR A-Level Maths 2027 Predicted Practice Paper — Pure and Statistics

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. Answer ALL questions.

• 1

  Question SECTION-A-PURE-MATHEMATICS1 (6 marks)

  Solve the equation 3 sin(2x) = 2 cos(x) for 0 <= x < 360 degrees. Give each solution to one decimal place where appropriate.

  (Total for Question SECTION-A-PURE-MATHEMATICS1 is 6 marks)
• 2

  Question SECTION-A-PURE-MATHEMATICS2 (7 marks)

  A geometric sequence has first term 200 and common ratio 0.85. (a) Show that the sum to infinity is 4000/3. (b) Find the value of the first term of the sequence that is less than 20, stating which term number this is.

  (Total for Question SECTION-A-PURE-MATHEMATICS2 is 7 marks)
• 3

  Question SECTION-A-PURE-MATHEMATICS3 (7 marks)

  A curve is defined by the parametric equations x = 2t^2, y = t^3 - 3t. (a) Show that dy/dx = (3t^2 - 3)/(4t). (b) Find the coordinates of the two stationary points of the curve.

  (Total for Question SECTION-A-PURE-MATHEMATICS3 is 7 marks)
• 4

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

  The function f(x) = (1 + 4x)^(1/2). (a) Find the binomial series expansion of f(x) in ascending powers of x up to and including the term in x^3. (b) State the range of values of x for which the expansion is valid.

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

  Question SECTION-A-PURE-MATHEMATICS5 (7 marks)

  The curve C has equation x^2 + 3xy + y^2 = 11. The point P(1, 2) lies on C. Find the equation of the tangent to C at P, giving your answer in the form ax + by + c = 0 where a, b and c are integers.

  (Total for Question SECTION-A-PURE-MATHEMATICS5 is 7 marks)
• 6

  Question SECTION-A-PURE-MATHEMATICS6 (8 marks)

  (a) Use integration by parts to find the exact value of the integral of x e^(-2x) with respect to x from x = 0 to x = 1. (b) Hence state the value to 4 decimal places.

  (Total for Question SECTION-A-PURE-MATHEMATICS6 is 8 marks)
• 7

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

  The variables x and y satisfy the differential equation dy/dx = xy, where y = 2 when x = 0. (a) Solve the differential equation to find y in terms of x. (b) Find the exact value of y when x = 2.

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

  Question SECTION-A-PURE-MATHEMATICS8 (7 marks)

  The functions f and g are defined by f(x) = ln(x - 1) for x > 1 and g(x) = e^(3x) for all real x. (a) Find an expression for the composite function fg(x) and state its range. (b) Find f^(-1)(x) and state its domain.

  (Total for Question SECTION-A-PURE-MATHEMATICS8 is 7 marks)

Section B: Statistics

Statistics, probability and data interpretation questions. Answer ALL questions.

• 1

  Question SECTION-B-STATISTICS1 (9 marks)

  The mass, in grams, of apples from an orchard is modelled by a normal distribution with mean 62 g and standard deviation 8 g. (a) Find the probability that a randomly chosen apple has mass greater than 75 g. (b) Find the probability that a randomly chosen apple has mass between 55 g and 70 g. (c) Apples with mass in the heaviest 10 percent are graded 'Premium'. Find, to the nearest gram, the minimum mass for a Premium grading.

  (Total for Question SECTION-B-STATISTICS1 is 9 marks)
• 2

  Question SECTION-B-STATISTICS2 (9 marks)

  In a large batch of manufactured components, 25 percent are known to be substandard. A quality inspector selects a random sample of 18 components. Let X be the number of substandard components in the sample. (a) State a suitable distribution for X, giving its parameters. (b) Find P(X = 5). (c) Find P(X <= 3). (d) Find the mean and standard deviation of X.

  (Total for Question SECTION-B-STATISTICS2 is 9 marks)
• 3

  Question SECTION-B-STATISTICS3 (9 marks)

  The proportion of adults in a town who cycle to work is claimed to be 0.25. A local councillor believes the true proportion is higher. She takes a random sample of 18 adults and finds that 9 of them cycle to work. Test, at the 5 percent significance level, whether there is evidence to support the councillor's belief. State your hypotheses and conclusion clearly.

  (Total for Question SECTION-B-STATISTICS3 is 9 marks)
• 4

  Question SECTION-B-STATISTICS4 (9 marks)

  A researcher records the number of hours of study (x) and the exam score (y) for a random sample of students. The product moment correlation coefficient for the sample is calculated to be r = 0.68. (a) Describe the correlation shown by this value. (b) The researcher fits the least squares regression line y = 12.4 + 3.7x. Use the model to estimate the exam score for a student who studies for 15 hours, and comment on the reliability of this estimate if the sample data covered study times from 2 to 12 hours. (c) Give one reason why it would be inappropriate to use this regression line to estimate the number of hours studied by a student who scored 80.

  (Total for Question SECTION-B-STATISTICS4 is 9 marks)
• 5

  Question SECTION-B-STATISTICS5 (9 marks)

  A spinner has four sectors labelled 1, 2, 3 and 4. The probability of landing on each number is given in the table: P(1) = 0.4, P(2) = 0.3, P(3) = 0.2, P(4) = 0.1. The spinner is spun twice, and the outcomes are independent. Let T be the total of the two scores. (a) Find P(T = 4). (b) Find P(T = 4 given that the first spin landed on 2). (c) Determine whether the event 'T = 4' and the event 'the first spin landed on 2' are independent, justifying your answer.

  (Total for Question SECTION-B-STATISTICS5 is 9 marks)