RI 9758 2023 Prelim P1 Solution

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ 2023 Year 6 H2 Mathematics Preliminary Examination Paper 1: Solutions 1 Solution [4] ( )( ) 95 ,111 1 x xx xx +< ≠±−+ + ( )( ) ( )( ) 9 51 011 xx xx −+ − <−+ ( )( ) 2 44 011 xx xx ++ <−+ ( ) ( )( ) ( ) 2 21 1 0 *x xx+ − +<  Method 1: 2 or 2 1 or 1x xx∴ <− − < <− > Method 2: Since ( ) 2 2 0 for ,xx+≥ ∈  (*) is equivalent to ( )( )11 0xx− +< and 2x≠− 1 or 1 and 2xxx∴ <− > ≠− 2 Solution [4] Let $x, $y and $z denote the usual selling price of a small, medium and large bag of Griffles popcorn respectively. To receive a total of 3 small, 7 medium and 1 large bag of Griffles popcorn, Beatrice bought 2 small, 5 medium and 1 large bag of popcorn. ( )0.95 3 7 85.5 3 7 90 (1)x yz x yz++= ⇒++ =  2 5 85.5 18.50 2 5 67 (2)x yz x yz+ += − ⇒ + +=  2.4 2.4 0 (3)z x xz= ⇒ −=  On solving, 5, 9 and 12xy z= = = The usual selling price of a small, medium and large bag of Griffles popcorn is $5, $9 and $12 respectively. x

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ 3 Solution (a) [2] We have 2 dd48 dd ArAr r ttππ= ⇒= . Hence dd100 8 (5) 0.796 cm/sdd rr ttπ−= ⇒= − Alternative 2 d48 dr AAr r ππ= ⇒= ddd d dd 1 100 8 (5) 5 2 r Ar t tA π π = × = −× =− Hence the radius is decreasing at 5 cm/s2π . (b) [2] 324d dVolume of meteorite, 43d d VrVr r ttππ= ⇒= Since V decreases with t, we have d d V kAt =− for proportionality constant 0k > . This means that ( ) 22dd44 dd rrr kr kttππ = − ⇒= − , which is a negative constant. Hence the radius is decreasing at a constant rate. Alternative We have 2d (4 )d V kA k rt π= −= − for proportionality constant 0k > . 324d 43d VVr r rππ= ⇒= 2 2 dd d d dd 1 (4 ) 4 0 rVr t tV kr r k π π = × = −× = −< Since k is a negative constant, thus the radius is decreasing at a constant rate.

Raffles Institution H2 Mathematics 2023 Year 6 _____________________________________________________________________________________________ 4 Solution (a) [3] Method 1 Reflect the graph of ( )fyx= about the x-axis, followed by scaling of the resulting graph by a factor of 2 parallel to the y -axis, and translating the resulting graph by 1 unit in the positive y-direction. Method 2 Translate the graph of ( )fyx= by 1 2 unit in the negative y-direction, followed by reflecting the resulting graph about the x-axis, and scaling by a factor of 2 parallel to the y-axis. Method 3: Reflect the graph of ( )fyx= about the x-axis, followed by translating the resulting graph by 1 2 unit in the positive y-direction, and scaling by a factor of 2 parallel to the y-axis. Method 4: Scale the graph of ( )fyx= by a factor of 2 parallel to the y-axis, followed by transl