OCR GCSE Maths 2027 Predicted Practice Paper — Paper 5 Higher

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

Section A

Short questions — fluency and direct application. Answer ALL the questions.

• 1

  Question SECTION-A1 (3 marks)

  Write the number 0.272727... (with the digits 27 recurring) as a fraction in its simplest form. You must show your working.

  (Total for Question SECTION-A1 is 3 marks)
• 2

  Question SECTION-A2 (3 marks)

  (a) Simplify fully (3 + sqrt(5))(3 - sqrt(5)). (b) Hence write 1/(3 + sqrt(5)) in the form (a + b*sqrt(5))/c, where a, b and c are integers.

  (Total for Question SECTION-A2 is 3 marks)
• 3

  Question SECTION-A3 (3 marks)

  A quantity increases by 20% and then the new amount decreases by 20%. The final amount is 240. Work out the original amount.

  (Total for Question SECTION-A3 is 3 marks)
• 4

  Question SECTION-A4 (3 marks)

  The nth term of a sequence is given by the quadratic rule T(n) = n^2 + 2n. Write down the first three terms, and state the value of n for which T(n) = 80.

  (Total for Question SECTION-A4 is 3 marks)
• 5

  Question SECTION-A5 (3 marks)

  Solve the inequality 5 - 2x < 11, and represent the solution on a number line description (state clearly the direction and whether the endpoint is included).

  (Total for Question SECTION-A5 is 3 marks)
• 6

  Question SECTION-A6 (3 marks)

  A bag contains only red, blue and green counters. The probability of taking a red counter at random is 0.35 and the probability of taking a blue counter is 0.4. There are 30 green counters. Work out the total number of counters in the bag.

  (Total for Question SECTION-A6 is 3 marks)
• 7

  Question SECTION-A7 (3 marks)

  Work out the value of 27^(2/3) x 2^(-2) x 5^0, giving your answer as an exact fraction.

  (Total for Question SECTION-A7 is 3 marks)

Section B

Standard questions — reasoning and linked methods. Answer ALL the questions.

• 1

  Question SECTION-B1 (5 marks)

  Solve the simultaneous equations 3x + 2y = 16 and 5x - 2y = 8. Show all your working.

  (Total for Question SECTION-B1 is 5 marks)
• 2

  Question SECTION-B2 (4 marks)

  The point A has coordinates (1, 2) and the point B has coordinates (5, 10). Find the equation of the straight line that passes through A and B, giving your answer in the form y = mx + c.

  (Total for Question SECTION-B2 is 4 marks)
• 3

  Question SECTION-B3 (5 marks)

  Factorise fully 6x^2 + x - 12, and hence solve 6x^2 + x - 12 = 0.

  (Total for Question SECTION-B3 is 5 marks)
• 4

  Question SECTION-B4 (5 marks)

  A cone has a base radius of 6 cm and a perpendicular height of 8 cm. Work out the exact total surface area of the cone (curved surface plus base), giving your answer in terms of pi. [Use slant height l where l^2 = r^2 + h^2; curved surface area = pi r l.]

  (Total for Question SECTION-B4 is 5 marks)
• 5

  Question SECTION-B5 (5 marks)

  A car travels the first 90 km of a journey at an average speed of 60 km/h and the remaining 60 km at an average speed of 40 km/h. Work out the average speed for the whole journey, in km/h.

  (Total for Question SECTION-B5 is 5 marks)
• 6

  Question SECTION-B6 (4 marks)

  In a survey, the ratio of people who preferred tea to coffee to neither was 5 : 3 : 2. Altogether 140 more people preferred tea than preferred coffee. Work out the total number of people surveyed.

  (Total for Question SECTION-B6 is 4 marks)
• 7

  Question SECTION-B7 (5 marks)

  A stone is thrown and its height h metres above the ground after t seconds is modelled by h = 20t - 5t^2. Find the time(s) when the stone is at a height of 15 m above the ground.

  (Total for Question SECTION-B7 is 5 marks)
• 8

  Question SECTION-B8 (6 marks)

  Prove that the sum of the squares of any two consecutive odd numbers is even but is not a multiple of 4. Use algebra to justify your answer.

  (Total for Question SECTION-B8 is 6 marks)

Section C

Extended problem-solving questions. Answer ALL the questions.

• 1

  Question SECTION-C1 (7 marks)

  OABC is a parallelogram. Vector OA = a and vector OC = c. The point M is the midpoint of AB, and the point N divides OC such that ON : NC = 2 : 1. (a) Express the vectors AB, OM and MN in terms of a and c. (b) Hence determine whether the points O, and the midpoint of MN, and B are collinear, justifying your answer.

  (Total for Question SECTION-C1 is 7 marks)
• 2

  Question SECTION-C2 (7 marks)

  A, B, C and D are points on the circumference of a circle with centre O. BD is a diameter. Angle BAC = 40 degrees and angle ADB = 25 degrees. Using circle theorems, work out the sizes of (a) angle BDC, (b) angle DBC, and (c) angle ABD. Give a reason for each step.

  (Total for Question SECTION-C2 is 7 marks)
• 3

  Question SECTION-C3 (7 marks)

  Two solid statues are mathematically similar. The smaller statue has a height of 30 cm and a surface area of 900 cm^2, and it has a mass of 5.4 kg. The larger statue has a height of 50 cm. (a) Work out the surface area of the larger statue. (b) Work out the mass of the larger statue, assuming they are made of the same material. Give your answers to a suitable degree of accuracy.

  (Total for Question SECTION-C3 is 7 marks)
• 4

  Question SECTION-C4 (7 marks)

  A number x satisfies the equation x^3 + 4x - 9 = 0. (a) Show that this equation can be rearranged into the iterative formula x_(n+1) = cube_root(9 - 4 x_n). (b) Using a starting value x_0 = 1.3, apply the iteration to find x_1, x_2 and x_3, giving each to 4 decimal places. (c) Explain what the sequence of values suggests about a root of the equation. (Non-calculator: you may leave cube roots evaluated to the stated accuracy assuming their decimal values; show the substitution clearly.)

  (Total for Question SECTION-C4 is 7 marks)
• 5

  Question SECTION-C5 (6 marks)

  A field is in the shape of a triangle ABC. AB = 80 m, BC = 65 m and the angle ABC = 68 degrees. (a) Work out the area of the field. (b) Work out the length AC. (c) A fence runs from B perpendicular to AC. Work out the length of this perpendicular fence. Give each answer to 3 significant figures. [You may use sin 68 = 0.927, cos 68 = 0.375.]

  (Total for Question SECTION-C5 is 6 marks)
• 6

  Question SECTION-C6 (6 marks)

  A cumulative frequency analysis is carried out on the times, in minutes, taken by 200 runners to finish a race. The times are grouped and the cumulative frequencies build up so that the median is 46 minutes, the lower quartile is 39 minutes and the upper quartile is 55 minutes. It is also known that 30 runners took longer than 62 minutes. (a) Work out the interquartile range and explain what it represents. (b) Estimate the number of runners who took between 39 and 55 minutes. (c) Estimate the percentage of runners who took longer than 62 minutes, and comment on whether 62 minutes could reasonably be described as an outlier boundary using the 1.5 x IQR rule.

  (Total for Question SECTION-C6 is 6 marks)