HCI_9758_2025_Prelim Paper 1_Solutions

HCI 2025 C2 H2 Mathematics Preliminary Examination Paper 1 Suggested Solutions 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] Suggested Solutions ( ) 1 31 1 1 3 1 yx x xx − = + + − = + + − Hence ( ) 2 2 d 1 0 3( 1)( 1)d 31 1 y xx x −= + + − − =− − For strictly decreasing, ( ) 2 d 0d 310 1 y x x   −  − Method 1 ( ) ( ) 2 2 2 2 2 1 3 0 1 22 0 1 xx x xx x − + −  − −−  − Critical values: 1x = and 2 2 2 0xx− − = ( )2 4 4 2 132x  − − = =  + - - + 13− 1 13+  |1 3 1 or 1 1 3x x x  −     +

2 Method 2 ( ) 2 31 1x  − Since ( ) 2 10x− , ( ) 2 13x− for 1x  (IMPT!!) 2 2 2 1 3 0 2 2 0 xx xx  − + −  − −  Consider 2 2 2 0xx− − = ( )2 4 4 2 132x  − − = =   |1 3 1 3, 1x x x  −   + 

3 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] Suggested Solutions ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 22 3 32 4 3 8 3 (1) 3 2 8 70 9 4 8 (2) 3 3 70 377 27 9 70 (3) x a b c abc x a b c a b c x a b c a b c = − + + − =− + + −−−−−−−−− = − + + − = + − −−−−−−−−− = − + + − =− + + −−−−−−−− By GC, a = 2, b = 3, c = –5 From GC, 10 11 7210868 36172738 x x =− =

4 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] Suggested Solutions (a) 2 2dd 2dd u xy uy y x yxx = =+  2 2 d1cos 2 d d1cos d d sec d yx y xy x xy ux xu ux xu  +=  = = (b) ( ) ( ) 2 24 d sec d 1 ln sec tan 2 sec , tan 0 for 0 2 2ln sec tan u u x x u x x C x x x x y x x C  = = + +     = + + 

5 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] (a) Suggested Solutions (a) 1 5h= ( ) 5 1 f k h kh =  = 1 1 2 3 4 5f f f