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 ( ) ( ) 1ff xx −= for all x in the domain of f. (b) Find the value of a. [2] (c) Hence find ( ) 2025f2 . [2] Another function g is defined by 2 1 2 3g : , for , 0 5 5 5x x x x − +    . (d) Find the range of fg. [2] 9 (a) Find ( )2 1 cos dx x x− . [3] (b) Hence find the value of 2 0 2 1 cos dx x x−  . Give your answer in the form 4 cosAB− , where A and B are exact constants to be determined. [4] 10 (a) Express ( ) 1 500QQ − in partial fractions. [2] (b) A group of scientists is monitoring the population of a particular species of birds on an island. At time t years after the start of the monitoring, the number of birds on the island is Q. The scientists observe that the birth rate is proportional to the bird’s population and the birds are dying at a rate proportional to the square of the bird’s population There were 1000 birds on the island when the scientists first started monitoring the population. They discover that the population remains unchanged when there are 500 birds. (i) Show that the differential equation relating Q and t is given by ( )d 500d Q kQ Qt =− , where k is a positive constant. [2] (ii) Hence, solve the differential equation, expressing Q in terms of k and t. [5]

4 11 The points A and B have coordinates ( )15,3,0 and (5,9,5) respectively. Two lines 1l and 2l , which are perpendicular to each other, have the following equations. ( ) ( )1 : 15 3 2 4l = − + + +r i j k    , ( ) ( ) ( )2 : 5 8 9 2 5lm = + + − + +r i j k    , where  and  are parameters and m is a constant. (a) Find the value of m. [2] (b) Given that 1l and 2l intersect, find the coordinates of the point of intersection E. [2] (c) Find a cartesian equation of the plane  which contains the points A, B and E. [3] (d) The point D has coordinates ( )1 , 3,2−− . Find the position vector of the point F, the foot of perpendicular of D to  . [3] (e) Find the exact area of the circle that passes through A, D and F. [2] 12 Do not use a graphing calculator for this question. It is given that ( ) 42f6 z z z k= − + , where k is a non-zero constant. (a) If k is a purely imaginary number, determine, with justification, whether ( )f0 z = can have real roots. [1] (b) Show that ( ) ( )ff zz−= . [1] (c) Given that 2i+ is a root of the equation ( )f0 z = , determine k. Hence, or otherwise, find the remaining roots, showing your workings clearly. [6] Use the value of k found in part (c) for the rest of this question. (d) Given that the product of all the roots of ( )f0 z = is D, find the value of D, showing your workings clearly. [2] (e) A complex number w1 satisfies the equation 42 6 1 0kw w− + = . Given that w1 can be obtained from 2i+ , find w1. [2]

5 13 Frederick, a social media influencer, is starting a new online account at the start of a month. From the second month onwards: • His organic followers at the end of each month will be a times the total followers he had at the end of the previous month, where a is a positive constant. • A company he engaged in will provide him with 1000 additional followers in the middle of each month. Let ( )Fn denote the total number of organic and additional followers Frederick had at the end of n months after he started his new online account, for n + . (a) Write down a recurrence relationship between ( )Fn and ( )1Fn + for n + , giving your answer in terms of a. [2] It is found that Frederick has 3500 followers at the end of the first month. (b) Show that ( ) 23 3500 1000 1000F a a= + + . [2] (c) Find an expression for ( )Fn . Hence, find the number of months required for Frederick’s followers to exceed one million if 1.5a= . [4] (d) Given that the number of followers exceeds 20000 by the end of the first year, determine the range of values of a. [2] Frederick finds out that, instead of the additional 1000 followers in the middle of every month, the company can only provide him with additional b followers in the middle of every month. (e) Given that the number of followers remains constant since the end of the first month, find the relationship between a and b. [2]