(3) Given that the equation f(x) = where is a constant, has two distinct roots, k, k (c) state the set of possible values for k. (2) (Total for Question 11 is 6 marks) ___________________________________________________________________________ 12. (a) Solve, for < 180°, the equation x –180° 3 sin2 + sin + 8 9 cos2 x x x giving your answers to 2 decimal places. (6) (b) Hence find the smallest positive solution of the equation 3 sin2 (2 30) + sin (2 30) + 8 = 9 cos2 (2 30) – – – giving your answer to 2 decimal places. (2) (Total for Question 12 is 8 marks) ___________________________________________________________________________ (a) Express 10 cos 3 sin in the form cos ( + where > 0 and 0 < < 90. 13. R R – 𝛼), 𝛼 Give the exact value of and give the value of in degrees, to 2 decimal places. R 𝛼, (3) Figure 3 The height above the ground, metres, of a passenger on a Ferris wheel minutes after the H t wheel starts turning, is modelled by the equation = 10 cos (80t)° + 3 sin (80t)° H a – where is a constant. a Figure 3 shows the graph of against for two complete cycles of the wheel. H t Given that the initial height of the passenger above the ground is 1 metre, (b) (i) find a complete equation for the model. (ii) Hence find the maximum height of the passenger above the ground. (2) (c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle.
(3) Given that the equation f(x) = k, where k is a constant, has two distinct roots, (c) state the set of possible values for k. (2) (Total for Question 11 is 6 marks) ___________________________________________________________________________ 12. (a) Solve, for –180° x < 180°, the equation 3 sin2 x + sin x + 8 9 cos2 x giving your answers to 2 decimal places. (6) (b) Hence find the smallest positive solution of the equation 3 sin2 (2 – 30) + sin (2 – 30) + 8 = 9 cos2 (2 – 30) giving your answer to 2 decimal places. (2) (Total for Question 12 is 8 marks) ___________________________________________________________________________ 13. (a) Express 10 cos – 3 sin in the form R cos ( + 𝛼), where R > 0 and 0 < 𝛼 < 90. Give the exact value of R and give the value of 𝛼, in degrees, to 2 decimal places. (3) Figure 3 The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the equation H = a – 10 cos (80t)° + 3 sin (80t)° where a is a constant. Figure 3 shows the graph of H against t for two complete cycles of the wheel. Given that the initial height of the passenger above the ground is 1 metre, (b) (i) find a complete equation for the model. (ii) Hence find the maximum height of the passenger above the ground. (2) (c) Find the time taken, to the nearest second, for the passenger to reach the maximum height on the second cycle.
02a_a_level_mathematics_sams_-_paper_2_pure_mathematics.docx