2024 RI H2 Math Prelim P1 (Soln) - new

RAFFLES INSTITUTION 2024 Year 6 H2 Mathematics Prelim Exam Paper 1 Questions and Solutions with comments       ______________________  2024 Yr 6 H2 Math Prelim Exam Paper 1 Solution with Comments 1 A function f is defined by 32f( ) .xa xb xc x d  The graph of f( )yx passes through the points (3 , 4 )  and (1, 8). Given that the graph of 1 f( )y x has a turning point at 1 4(2, ) find the values of ,, a n d abc d . [4] [4] 32f( ) f ( 3) 4 27 9 3 4 ----- (1) f (1) 8 8 ----- (2) xa xb xc x d abc d abcd        Since the graph of 1 f( )y x has a turning point at 1 4(2, ) , the graph of f( )yx has a turning point at (2,4) . f( 2 ) 4  and f( 2 ) 0  .  f (2) 4 8 4 2 4 ------ (3)abc d    2f( ) 3 2 f (2) 0 12 4 0 ------ (4) xa xb x c ab c          From the GC, 1, 1, 8 and 16ab c d    . Question was well done by most students. Some students had difficulty with understanding the statement: “ 1 f( )y x has a turning point at 1 4(2, ) ” with some working out  2 f()d 0d f( ) yx x x  .

2 ______________________  2024 Yr 6 H2 Math Prelim Exam Paper 1 Solution with Comments 2 [The volume of a sphere with radius r is given by 34 3 r and the surface area of a sphere with radius r is given by 24 r .] (a) The volume of an expanding sphere is increasing at a constant rate of 315 cm s .   Show that, at any instant, the rate of increase of the surface area is 21cm s ,k r  where r is the radius of the sphere and k is a constant to be determined. [3] (b) Find the exact rate of change of surface area of the expanding sphere when the surface area is 220 cm . [2] (a) [3] Volume, 34 3Vr         and Surface area,  24A r                 2d 4d V rr  d 8d A rr  2 dd d dd d dd d ddd 1 8 5 4 10 , where 10. (shown) AA r tr t Ar V rV t r r kr              Question was well done by most students. Common error: Some students related but then treated A or V as constants. So, by writing 3or3 Ar VVA r , one should actually get d1 d d3 d VA Arrr   or 2 1dd 3 dd A VV rr r r  and not d d3 VA r  or 2 3d d A V rr . (b) [2] When 20,A 242 0r                                       5r   (since  0r  ) d 10 2 5d5 A t    21cm s .

3 ______________________  2024 Yr 6 H2 Math Prelim Exam Paper 1 Solution with Comments 3 (a) Without using a calculator, solve exactly 2 1 1.1 x x x     [4] (b) Hence solve exactly 2 1 11    xx x . [4] (a) [4] 2 1 11 x x x     2 1 101 x x x      2 1 01 1 xxx x       2 22 01 x x x    Consider 2 22 0 :xx    2 22 4 ( 1 ) ( 2 ) 12(1) 3x     So  11 33 01 x x x       011 ) 3( 31xxx       1x or 13 13 x Most students are able to manipulate the inequa