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

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 (5 marks)

  Find the first three terms, in ascending powers of x, in the binomial expansion of sqrt(1 + 3x). State the range of values of x for which the expansion is valid.

  (Total for Question SECTION-A-PURE-MATHEMATICS1 is 5 marks)
• 2

  Question SECTION-A-PURE-MATHEMATICS2 (6 marks)

  The curve C has equation y = x^2 ln x for x > 0. Find the exact x-coordinate of the stationary point of C, and determine whether it is a maximum or a minimum.

  (Total for Question SECTION-A-PURE-MATHEMATICS2 is 6 marks)
• 3

  Question SECTION-A-PURE-MATHEMATICS3 (6 marks)

  Use the substitution u = 1 + x^2 to evaluate the integral of (x / (1 + x^2)) dx between the limits x = 0 and x = 2. Give your answer in an exact form involving a logarithm.

  (Total for Question SECTION-A-PURE-MATHEMATICS3 is 6 marks)
• 4

  Question SECTION-A-PURE-MATHEMATICS4 (7 marks)

  The equation x^3 - 5x - 3 = 0 has a root alpha lying between 2 and 3. (a) Show that the equation may be rearranged into the form x = (5x + 3)^(1/3). (b) Using the iteration x_(n+1) = (5x_n + 3)^(1/3) with x_0 = 2.5, find x_1, x_2 and x_3, giving each to 4 decimal places. (c) By choosing a suitable interval, show that alpha = 2.4909 correct to 4 decimal places.

  (Total for Question SECTION-A-PURE-MATHEMATICS4 is 7 marks)
• 5

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

  A curve is defined parametrically by x = t^2 + 1 and y = t^3 - 3t, where t is a real parameter. (a) Find dy/dx in terms of t. (b) Find the equation of the tangent to the curve at the point where t = 2, giving your answer in the form y = mx + c.

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

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

  (a) Express 5 cos(theta) - 12 sin(theta) in the form R cos(theta + alpha), where R > 0 and 0 < alpha < 90 degrees, giving alpha to 1 decimal place. (b) Hence solve the equation 5 cos(theta) - 12 sin(theta) = 6 for 0 <= theta <= 360 degrees, giving your answers to 1 decimal place.

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

  Question SECTION-A-PURE-MATHEMATICS7 (8 marks)

  Use integration by parts to evaluate the exact value of the integral of x e^(2x) dx between x = 0 and x = 1.

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

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

  The functions f and g are defined for all real x by f(x) = 2x - 1 and g(x) = x^2 + 4. (a) Find an expression for the composite function gf(x), simplifying your answer. (b) Explain why f has an inverse but g does not, given their stated domains, and find f^(-1)(x). (c) Solve the equation gf(x) = 8.

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

Section B: Mechanics

Mechanics modelling and problem-solving questions. Answer ALL questions.

• 1

  Question SECTION-B-MECHANICS1 (8 marks)

  A ball is projected from ground level with speed 24.5 m s^-1 at an angle of 30 degrees above the horizontal. The ball is modelled as a particle moving freely under gravity, with g = 9.8 m s^-2 and air resistance neglected. (a) Find the time of flight until the ball returns to ground level. (b) Find the horizontal range. (c) Find the greatest height reached.

  (Total for Question SECTION-B-MECHANICS1 is 8 marks)
• 2

  Question SECTION-B-MECHANICS2 (9 marks)

  Two particles A of mass 5 kg and B of mass 3 kg are connected by a light inextensible string passing over a smooth fixed pulley. The particles hang vertically and are released from rest with the string taut. Taking g = 9.8 m s^-2, (a) find the acceleration of the system, (b) find the tension in the string, and (c) state one modelling assumption implied by describing the string as 'light and inextensible' and explain its effect.

  (Total for Question SECTION-B-MECHANICS2 is 9 marks)
• 3

  Question SECTION-B-MECHANICS3 (9 marks)

  A box of mass 8 kg is released from rest on a rough plane inclined at 20 degrees to the horizontal. The coefficient of friction between the box and the plane is 0.3. Taking g = 9.8 m s^-2, (a) show that the box moves, (b) find the acceleration of the box down the plane, and (c) find the speed of the box after it has travelled 4 m down the plane.

  (Total for Question SECTION-B-MECHANICS3 is 9 marks)
• 4

  Question SECTION-B-MECHANICS4 (9 marks)

  A particle P moves in a straight line so that at time t seconds (t >= 0) its velocity v m s^-1 is given by v = 3t^2 - 12t + 9. (a) Find the times at which P is instantaneously at rest. (b) Find the acceleration of P when t = 1. (c) Given that P is at the origin when t = 0, find the total distance travelled by P in the interval 0 <= t <= 3.

  (Total for Question SECTION-B-MECHANICS4 is 9 marks)
• 5

  Question SECTION-B-MECHANICS5 (10 marks)

  A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a light rigid tow-bar. The engine produces a constant driving force of 3200 N. Resistances to motion of 600 N on the car and 200 N on the trailer act throughout. (a) Find the acceleration of the car and trailer. (b) Find the tension in the tow-bar. (c) When travelling at 20 m s^-1 the tow-bar suddenly breaks. Assuming the same resistance continues to act on the trailer, find how far the trailer travels before coming to rest.

  (Total for Question SECTION-B-MECHANICS5 is 10 marks)