HCI_9758_2024_Promo_Solutions

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] Suggested Solutions Let x, y and z be the number of servings of battered fish fillet, coleslaw and fries respectively. 20 4 3 55x y z+ + = 20 15 45 130x y z+ + = 15 8 16 70x y z+ + = By GC, x = 2, y = 3, z = 1 Therefore, the runner should take 2 servings of battered fish fillet, 3 servings of coleslaw and 1 serving of fries for his lunch. Protein (in grams) Carbohydrates (in grams) Fat (in grams) Battered fish fillet 20 20 15 Coleslaw 4 15 8 Fries 3 45 16

2 © HCI 2024 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] Suggested Solutions (a) ( )( ) ( ) ( )( ) ( ) 1 2 1 1 1 2 11 23 11 23 11122 1 2 1 11 661 2 11 1 2 1 126 111 23 rn r rn n n r r r n n n rr rr nnn n n n nn n n = = = =    −+       = − −  − = − + + − + − = − − + + + = − − +     (b) Explain why 1 11 123 rn r rr =    − +         for all positive integers n. [1] Suggested Solutions (b) Since n + , ( ) 21 0 and 1 023 n n n  +  . Thus 1 11 123 rn r rr =    − +         . 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. 1 0 ⋯

3 © HCI 2024 [Turn over (a) Show that the total area of n rectangles is given by 1 0 1 n r nr nn − = − . [2] Suggested Solutions (a) Area of n rectangles ( ) 1 0 1 0 1 0 1 1 1 11 1 ... 1 11 0 1 ... 1 1 n r n r n n n n n n n nnnn n n n n nr nn nr nn − = − =       −= − + − + + −                 −−−−= + + +  −= −  

4 © HCI 2024 (b) Without using a calculator, evaluate 1 0 1lim n n r nr nn − → = − . [3] Suggested Solutions (b) 1 0 1 0 1 0 1lim Area of infinitely many rectangles Area under the curve from 0 to 1 1 d 1 d n n r nr nn xx xx xx − → = −= = = = =− =− − −    ( ) 1 3 0 2 1 3 2 3 x= − − = 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] Suggested Solutions (a)