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 tangent at P, giving your answer in the form ax + by + c = 0 where a, b and c are integers. [4] Qn Solutions Comments 4(i) ( )( ) 2 2 2 2 d 1 6 3d d 8d 1 6 3 8 3 6 9 0 2 3 0 1 3 0 1 or 3 (N.A. 0) y xxx y x xx xx xx xx xx x = + − =− + − =− − − = − − = + − = =− =  When 3x= , ( ) 235 3 3 3 3 8 y= + + − = Equation of tangent at the point P is ( )8 8 3 8 32 8 32 0 yx yx xy − =− − =− + + − = (ii) The tangent at P meets the x-axis at point T. Find the area of triangle PTO, where O is the origin. [2] Qn Solutions Comments 4 (ii) When y = 0, 0 8 32 8 32 4 x x x =− + = = Area of triangle POT = 1 4 8 162  =

5 A fish farm which supplies sea bass to restaurants in Singapore started operations in January 2001. Based on an expert model, the population of sea bass in the fish farm, P thousands, at time t years is given by 0.215e , tPN −=− where N is a positive real number. The fish farm had a population of 15000 sea bass when it started operations. (i) Show that 30N = . [1] Qn Solutions Comments 5(i) ( ) 0.2(0) When 0, 15, 15e 15 15 1 15 30 tP N N N − == −= −= = (ii) Find d .d P t [2] Qn Solutions Comments 5 (ii) 0.2 0.2d 15( 0.2)e 3ed ttP t −−=− − = (iii) Using the expression for d ,d P t explain why the sea bass population in the fish farm will keep increasing. [1] Qn Solutions Comments 5 (iii) 0.2d 3ed tP t −= Since 0.2e > 0t− for all values of t. d d P t is always positive. Therefore, population of sea bass in the fish farm is always increasing.

(iv) Sketch the graph of P against t. [2] Qn Solutions Comments 5 (iv) (v) The fish farm is designed to have a maximum capacity of 25000 fishes. Determine the year in which the fish farm is estimated to reach its maximum capacity. [1] Qn Solutions Comments 5(v) 0.2Using G.C to solve 30 15e 25,t−−= t = 5.4930 The fish farm will reach maximum capacity in the year 2006 Alternative, 0.230 15 25 5ln 3 5.49306 te t t −−   The fish farm will reach maximum capacity in the year 2006. (0,15) t P P = 30

The owner of the fish farm is also interested in modelling his revenue, R thousand dollars per year, at any point in time. The model he uses is 32 9 14 15, for 0 532 ttR t t= − + +   . (vi) Use differentiation to find the stationary value of R and justify whether this value is a minimum or maximum. [4] Qn Solutions Comments 5 (vi) 32 9 14 1532 ttRt= − + + 2d 9 14d ( 2)( 7) R ttt tt = − + = − − dFor stationary points, 0,d ( 2) n ( 7) 0 2 or rejected si ce 0 57( ) R t tt tt −=  = − = By first derivative test, x 1.99 2 2.01 d d R t 0.0501 0 0.0499− The stationary value of R is maximum at t = 2. Alternative By second derivative test, 2 2 2 2 d 29d dWhen 2, 5 0 d R tt Rt t =− = =−  The stationary value of R is maximum at t = 2. When t = 2, ( ) ( ) 23 922 83 14 2 15 27.6667 27.7 (3 s.f.)3 2 3R= − + + = =  Therefore, the stationary value of R = 27.7 is maximum when t = 2.

(vii) Use your calculator to find the value of 5 32 0 9 14 15 d .32 tt tt − + +   In the context of the question, what does this value represent? [2] Qn Solutions Comments 5 (iv) 5 32 0 9 14 15 d 114.58 11532 tt tt − + + =    The total revenue of the fish farm owner at the end of 5 years after the fish farm started operations in January 2001 is 115 thousand dollars.

Section B: Probability and Statistics [60 marks] 6 A and B are events such that P( | ) 0.3AB = , P( | ) 0.6BA = and P( ) 0.72AB= . (i) Find P( )AB . [3] Qn Solutions Comments 6(i) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) P | P P 0.3P P PP (1) 0.3 P | P P 0.6P P PP (2) 0.6 P =P P P 0.72 P P P (3) Sub (1) and (2) into (3), PP0.72 P 0.6 0.3 0.432 P 2P 0.6P 0.432 2.4P P 0.18 A B B A B B A B ABB B A A A B A A B ABA A B A B A B A B A B A B A B AB A B A B A B AB AB = = = = = =  + −  = + −  = + −  =  +  −  = = (ii) Determine whether the events A and B are independent. [2] Qn Solutions Comments 6 (ii) P( ) 0.6P( ) 0.18 3P( ) 0.6 10 AB A A  = == Since P(A) = P(A|B), events A and B are independent. Alternative