HCI_9758_2024_Promo

1 An athlete on a special diet would like to have battered fish fillet, coleslaw and fries for lunch. He must ensure that his intake (in grams) of protein, carbohydrates and fat per meal is 55g, 130g, and 70g respectively. The table below shows the nutritional breakdown for one serving of each item. Calculate the number of servings of battered fish fillet, coleslaw and fries that the athlete should take for his lunch. [3] 2 (a) Given that ( )( ) 2 1 1 2 16 n r nr n n = = + + , find 1 11 23 rn r rr =    −+        , in terms of n. [4] (b) Explain why 1 11 123 rn r rr =    − +         for all positive integers n. [1] 3 The diagram below shows a curve with equation 1yx=− . Let A be the area bounded by the curve and the axes. A total of n rectangles, each of equal width, are constructed to approximate the value of A. (a) Show that the total area of n rectangles is given by 1 0 1 n r nr nn − = − . [2] (b) Without using a calculator, evaluate 1 0 1lim n n r nr nn − → = − . [3] Protein (in grams) Carbohydrates (in grams) Fat (in grams) Battered fish fillet 20 20 15 Coleslaw 4 15 8 Fries 3 45 16 1 0 ⋯

2 4 (a) Sketch the curve with equation 2 1 xqy x −= − , where 2q  , stating the equations of the asymptotes and the coordinates of the points where the curve meets the axes. [3] (b) Hence, by adding a suitable line on the same diagram in part (a), solve the inequality 2 201 xq xx − −− , giving your answer in terms of q. [3] 5 Show that the distance L between any point ( ),xy on the curve 2e3xy=− and the fixed point ( )2, 1A − satisfies the equation ( ) ( ) 2222 2 e 2 xLx= − + − . [1] Using differentiation, find the x-coordinate of the point P on the curve that is closest to the point A, leaving your answer in 4 decimal places. You do not need to show that P is closest to A. [3] A variable point Q moves along the curve 2e3xy=− such that its x-coordinate is increasing at a rate of 0.5 units per second. Find the exact rate of change of L at the instant when Q is on the y-axis. [3] 6 The curve C has equation 22ln y x y= , where , 1 0xy −   . It is given that C has only one turning point. (a) Show that ( ) 22d 22d y x y xyx −= . [3] (b) Find the coordinates of the turning point. [2] (c) Calculate the value of 2 2 d d y x at the turning point and determine whether the turning point is a maximum or a minimum. [3]

3 7 (a) Describe a sequence of two transformations that will transform the graph of ( )fyx= to the graph of ( )fyx =+ , where  and  are positive constants. [2] Diagram 1 and Diagram 2 show the graphs of ( )fyx= and ( )fyx =+ respectively. The turning points with coordinates (0,0) and (2,4) on ( )fyx= correspond to the points with coordinates ( 2,0)− and (2,4) respectively on ( )fyx =+ . The asymptotes 1x= and yx =+ on