HCI_9758_2025_Prelim Paper 1

1 A curve has equation 31 1yx x= + + − . Using differentiation, find the set of values of x where the curve is strictly decreasing. Give your answers in exact form. [4] 2 A sequence of real numbers, nx , satisfies the recurrence relation ( ) 2 1 3 n nnx a bn cx+ = − + + , for , , ,n a b c and 1n . Given that 1 2 31, 8, 70,x x x= =− = and 4 377x =− , find the values of a, b and c. [3] Hence, find the values of 10x and 11x . [2] 3 It is given that 2 2 d12 cosd yxy y xx xy  += , where 0 2x  and 0y . (a) Using the substitution 2u xy= , show that the differential equation can be reduced to d sec d ux xu= . [3] (b) Hence find the general solution to the differential equation. [3] 4 The diagram shows a sketch of the function 3f ( ) e 1, xx =+ for 01 x The region bounded by the curve and the lines 0y= , 0x= and 1x = is A The region A is split into 5 vertical strips of equal width h, as shown in the diagram (a) State the value of h and using a suitable sketch, explain whether ( )( ) 5 1 f k h kh =  is less or more than the area of A. [3] (b) A is now split into n vertical strips of equal width . Using calculus, f ind the exact value of 3 6 9 3 3 31lim e e e ... e e n n n n n n nn − →  + + + + + +  . [3] O y x h

2 5 The tourism board plans to construct a tram track around a tourist attraction. The diagram above shows part of the blueprint of the track, where the tram will run along a circular path with centre O and radius 450 m, from a fixed point A to a variable point B, and then straight across to a fixed point C. The angle AOB is denoted as  , where 0 . The speed of the tram along the circular path will be maintained at 8 m/s and the speed along the straight path BC will be maintained at 6 m/s. (a) Show that the time taken for the tram to travel from point A to C is 225 150 6 cos42 +  . [3] (b) Use calculus to find the maximum time taken for the tram to travel from point A to point C. (You need not show that your answer gives a maximum.) [3] 6 Given that b is a real constant such that 04 b , describe fully a sequence of transformations that transforms the curve 2yx= to the curve 241y x bx= + + [4] Sketch the curve 2 1 41y x bx= ++ Give the equation of any asymptotes and the coordinates of any axial intercepts and turning points, in terms of b where appropriate [3] 7 (a) Use the formula listed in the List of Formulae (MF27) to explain why 0 1 e!r r  = = . [1] (b) Show that 0 1 2e!r r r  = + = . [3] (c) Find the value, in terms of e, of 6 1 !r r r  = +  . [3] O  C A B

3 8 The function f is defined by 3f : , 54 axx x − − for x , 4 5x , and a . (a) Find the range of f in terms of a. [1] The function f is such that ( ) (