Pearson Edexcel Level 3 Gce Mathematics Advanced Paper 1:Pure Mathematics VERIFIED 2024 EXAM (100% COMPLETE QUESTIONS AND MARKINGSCHEME) CHECK THE LAST PART FOR MARKING SCHEME

Pearson Edexcel Level 3 Gce Mathematics Advanced Paper 1:Pure Mathematics VERIFIED 2024 EXAM (100% COMPLETE QUESTIONS AND MARKINGSCHEME) CHECK THE LAST PART FOR MARKING SCHEME Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. Answer ALL questions. Write your answers in the spaces provided. 1. The curve C has equation y = 3x4 – 8x3 – 3 (a) Find (i) dy dx (ii) d2 y dx2 (3) (b) Verify that C has a stationary point when x = 2 (c) Determine the nature of this stationary point, giving a reason for your answer. (2) (2) Question 1 continued (Total for Question 1 is 7 marks) 2. B O 0.4 C 3 cm A O D 12 cm Figure 1 The shape ABCDOA, as shown in Figure 1, consists of a sector COD of a circle centre O joined to a sector AOB of a different circle, also centre O. Given that arc length CD = 3 cm, COD = 0.4 radians and AOD is a straight line of length 12 cm, (a) find the length of OD, (b) find the area of the shaded sector AOB. (2) (3) (Total for Question 2 is 5 marks) 3. A circle C has equation x2 + y2 – 4x + 10y = k where k is a constant. (a) Find the coordinates of the centre of C. (b) State the range of possible values for k. (2) (2) (Total for Question 3 is 4 marks) 4. Given that a is a positive constant and 2a t  1 a t dt  ln 7 show that a = ln k, where k is a constant to be found. (4) (Total for Question 4 is 4 marks) 5. A curve C has parametric equations x = 2t – 1, y = 4t –7 + 3 , t  0 t Show that the Cartesian equation of the curve C can be written in the form y  2x2  ax  b , x  1 x  –1 where a and b are integers to be found. (3) (Total for Question 5 is 3 marks) 6. A company plans to extract oil from an oil field. The daily volume of oil V, measured in barrels that the company will extract from this oil field depends upon the time, t years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions: • The initial daily volume of oil extracted from the oil field will be 16 000 barrels. • The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels. • The daily volume of oil extracted will decrease over time. The diagram below shows the graphs of two possible models. V V (0, 16 000) (0, 16 000) (4, 9000) O t Model A (4, 9000) O t Model B (a) (i) Use model A to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. (ii) Write down a limitation of using model A. (b) (i) Using an exponential model and the information given in the question, find a possible equation for model B. (ii) Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling. (2) (5) (Total for Question 6 is 7 marks) 7. B A C Figure 2 Figure 2 shows a sketch of a triangle ABC. → Given AB → = 2i + 3j + k and BC = i – 9j + 3k, show that BAC = 105.9° to one decimal place.