VJC_8865_2022_Prelim_Solutions

2022 J2 H1 Mathematics Prelim Exam Solutions with Comments Section A: Pure Mathematics [40 marks] 1 (a) Find the set of values of k for which the graphs of 22y x kx k= − + and y kx= intersect at most once. [4] Qn Solutions Comments 1(a) Equate the equations of the two graphs, 2 2 2 2 2 0 x kx k kx x kx k − + = − + = Since the graphs intersect at most once, ( ) ( )( ) ( ) 2 2 2 4 2 0 4 8 0 4 2 0 kk kk kk − −  − −   02 : 0 2 k kk     k 0 9 2 9

1 (b) MARS Café serves three types of coffee: Flat, Cappuccino and Macchiato. On 23 December 2021, a customer paid $17.30 for two cups of Flat, one cup of Cappuccino and three cups of Macchiato after using a $10 cash voucher. On the same day, the Café owner recorded a total sale of $1151.30 from selling 84 cups of Flat, 123 cups of Cappuccino and 65 cups of Macchiato. The total price of three cups of Macchiato is equivalent to the total price of five cups of Flat. Express this information as 3 linear equatio ns and hence find the price of one cup of Macchiato. [4] Qn Solutions Comments 1(b) Let $F, $C and $M represent the cost of one cup of Flat, Cappuccino and Macchiato, respectively. 2 3 27.30 (1) 84 123 65 1151.30 (2) 5 3 0 (3) F C M F C M FM + + = + + = − + = Using the GC, 3.30 4.20 5.50 F C M = = = The cost of one cup of Macchiato is $5.50.

2 The curve 1C has equation 42 xy=− . The curve 2C has equation 2 4 3xy=− . (i) Sketch 1C and 2C on the same diagram, stating the exact coordinates of any points of intersection with the axes and the equations of any asymptotes. [4] Qn Solutions Comments 2(i) (ii) Find the x-coordinates of the points of intersection of 1C and 2C , giving your answers correct to 4 decimal places. [1] Qn Solutions Comments 2 (ii) Using GC, 5.2818 or 2.4570x=− (4 decimal places) (iii) Write down as an integral an expression for the area of the region bounded by the curves C1 and C2, the line 5x=− and the y-axis. Evaluate this integral, giving your answer correct to 3 decimal places. [2] Qn Solutions Comments 2 (iii) ( ) 0 2 5 4 2 3 d 23.1864 x x x − − − − =   (3 decimal places) y x O y = 4 2C 1C

3 (i) Differentiate ( ) 2 3ln 5 e xx − . [3] Qn Solutions Comments 3(i) ( ) 22 33 2 ln 5 e ln 5 ln ln e ln 5 ln 3 xxxx xx −− = + + = + + − ( ) ( ) 2 32ddln 5 e ln 5 ln 3dd 1 2 xx x xxx xx − = + + − =+ (ii) Integrate 2 1 2 x x  + . [3] Qn Solutions Comments 3 (ii) ( ) 3 2 3 2 2 2 2 3 3 2 3 1 1 1 d 2 d 422 1 d4 1 ln43 1 2 1ln4 3 3 x x x x x xxx x x xx xxxC x x x C    + = + +       = + + = + + + = + + +     

4 A curve has equation 2353y x x x= + + − . (i) The point P is on the curve such that the x-coordinate of P is positive. The tangent to the curve at P is parallel to the line 83yx=− + . Find the equation of the tangen