ASRJC_8865_2022_Prelim_Solutions_vetted

ASRJC H1 Math Prelims Solutions 1 (i) ( )( ) 22 3 2 0 2 1 2 0 xx xx − −  + −  Sketch the graph: 12 or 2xx − (ii) 1 < e 22 x− ln 2x 2 (a) 2 2 2Let 3yx x =− 3 4 2 49 12y x x x= − + 3 2 d4 36 18d y xxxx= − − (b) 3 2 d 54 x x− = 1 32 (5 4 ) dxx − − ( ) 2 23 354 32 C (5 4 ) C2 4( 4)3 x x−= + =− − + − where C is an arbitrary constant. 1 2− x 2

ASRJC H1 Math Prelims Solutions 3 (a) 14d 8ed xy x −= When x = 1, y = 8 – 2e– 3 1 4(1) 3d 8e 8ed y x −−== Equation of tangent at the point x = 1: y – (8 – 2e– 3) = 38e ( 1)x− − y = 3 3 38e 8e 8 2ex− − −− + − y = 338e 10e 8x−− −+ where m = 38e− and c = 38 10e −− (b) A = 1 14 1 4 8 2e d x x−− 1 14 1 4 1 8 e 2 xx −=+ 311 8 e 222 −   = + − +       3 115 2 2e=+ 115 and 22pq== 4 (i) y = x4 – 10x3 + 36x2 – 54x + 17 32d 4 30 72 54d y x x xx = − + − d 0d y x = 324 30 72 54 0x x x− + − = Using GC, x =1.5 or x = 3 When x = 1.5, y = – 11.6875 When x = 3, y = – 10 Coordinates of stationary points are (1.5, – 11.6875) and (3, – 10) (ii) x 1.4 1.5 1.6 d d y x – 1.0240044 0 0.7839964 Slope (1.5, – 11.6875) is a minimum point. x 2.9 3 3.1 d d y x 0.0560016 0 0.0640024 Slope

ASRJC H1 Math Prelims Solutions (3, – 10) is an inflexion point. (iii) (iv) x4 – 10x3 + 37x2 – 52x + 18 = 0 x4 – 10x3 + 36x2 – 54x + 17 + x2 + 2x +1 = 0 x4 – 10x3 + 36x2 – 54x + 17 + x2 + 2x +1 = 0 x4 – 10x3 + 36x2 – 54x + 17 = – x2 – 2x – 1 (v) Using GC, x = 0.503 or x = 2.246 (to 3 dp) 5 (a) Let x, y and z be the price of a Brand A, Brand B and Brand C laptop respectively. 16x + 23y + 20z = 109 741 ------------ (1) 20z = 16x + 23 596 – 16x + 20z = 23 596 ------------ (2) 2x + 2y + 2z = 10 994 ------------ (3) Using GC, x = 1399, y = 1799 and z = 2299. 2 equations correct – 1 mark 3 equations correct – 2 marks The total price of two Brand A laptops and three Brand C laptops = $[2(1399) + 3(2299)] y = x4 – 10x3 + 36x2 – 54x + 17 y = – x2 – 2x – 1

ASRJC H1 Math Prelims Solutions = $9695 (b) (i) (ii) P = 1.8 + 0.6ln (9 – 2) = 2.9675 = 2.97 million dollars The value 2.97 million dollars represents the annual profits in the year up to time t = 3 years. (iii) P = 1.8 + 0.6ln (3t – 2) d 1.8 9 ord 3 2 5(3 2) P t t t= −− (iv) For t  1, d 1.8 9 0d 3 2 5(3 2) P t t t= = −− As the years t increases, the annual profits are expected to increase at a decreasing rate. (v) When t = 5, d 1.8 0.13846 0.138d 3(5) 2 P t = = =− million dollars/year P/ million dollars t /years

ASRJC H1 Math Prelims Solutions 6 (i) The customers can be numbered from 1 to 10 000. Using a random number generator, 1000 numbers can be generated.