HCI_9758_2025_Prelim Paper 2_Solutions

2025 C2 H2 Mathematics Preliminary Examination Paper 2 Suggested Solutions Section A: Pure Mathematics (40 marks) 1 The region A is bounded by the curves 1yx=+ , 72yx=− , the x–axis and the y–axis. (a) Find the exact area of A. [4] (b) Find the volume of the solid obtained when A is rotated through 2 radians about the y–axis. [3] Qn Suggested Solutions (a) Coordinates of point of intersection: ( )2, 3 Method 1: Using x-axis. ( ) ( ) 2 3.5 02 2 3.5 02 2 3.533 22 02 2 Area d d 1 d 7 2 d 2 1 7 2 33 2 27 2 27 3 3 3 22 3 3 3 23 3 units3 y x y x x x x x xx =+ = + + −    +−   =−             = − − −          = − + =−   y (2, ) (3.5,0 ) x A (0,1) O

Method 2: Using y-axis. 33 01 233 2 01 3333 01 2 Area d dy 7 d 1 d2 7 2 6 3 7 3 3 3 3 3 1 312 6 3 3 23 3 units3 x y x y y y y y y y y =− −= − −    = − − −           = − − − − −              =−   (b) 2 72 7 2 yx yx =− −= ( ) 2233 22 01 3 7Vol d 1 d 2 47.38326 47.4 units (3 s.f.) y y y y  −= − − =  

2 It is given that ln sin 4yx  =+   , where 3 44 x−   . (a) Show that 22 2 dd 10dd yy xx + + = . Hence find the first four non-zero terms of the Maclaurin expansion of y, leaving your answer in exact form. [6] (b) Verify the result obtained in part (a) is obtained using standard series from the List of Formulae (MF27). [5] Qn Suggested Solutions (a) e sin 4 y x =+  de cosd4 y y xx =+  2 2 2 dde e sind d 4 yy yy xxx    + =− +       2 2 2 dde e edd y y y yy xx  + =− 2 2 2 dd 10dd yy xx  + + = Alternative Solution cosd 4 cotd4 sin 4 xy xx x     + = = +  + 2 2 2 d cosecd4 y xx =− +  22 22 2 dd 1 cot cosec 1 0d d 4 4 yy xxxx      + + = + − + + =           Applying further implicit differentiation 23 23 d d d20 d d d y y y x x x  +=  When x = 0, 23 23 1 d d dln , 1, 2, 4d d d2 y y yy x x x = = =− =

Maclaurin expansion of y is 2312ln ... 32 y x x x= + − + + (b) ln sin 4yx  =+   ln sin cos sin cos44xx =+  ( ) 11ln sin cos 22 1ln sin cos 2 xx xx =+  =+  231ln ln 1 262 xxx + + − − 22 3 2 3 323 11ln 2 6 2 2 62 1 ... 3 2 6 x x x xxx xxx    = + − − − − −       + − − + ( ) 22 3 2 3 23 2 3 3 1 1 1ln ... 2 6 2 2 32 1 1 1ln ... 2 6 2 32 x x xx x x xxx x x x    = + − − − − + +       = + − − − − + + 2312ln ... 32 x x x= + − + + (verified)

3 The parametric equations of the curve C are 1 3cosecx=−  and 2cot 3y=−  , where 0 . (a) Show d2 secd3 y x =− . Hence find the equation of the normal to C at the point where 4  = . Give the equation in the form ,=+y Ax B where A and B are exact constants to b