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 ( )fyx= correspond to the asymptotes 0x= and yx   = + + respectively on ( )fyx =+ . (b) Find the value of  and of . [2] (c) On a separate diagram, sketch the graph of ( )f'yx= , giving the coordinates of the points where the graph cross es the axes and the equations of the asymptotes, if any. [3] Diagram 1 Diagram 2

4 8 The function f is defined by ( ) ( )2 ln e for e 0, f cos for 0 .12 xx x xx  − + −   =    (a) Given that the inverse function 1f − exists, find the exact value of m if ( ) 1f1 m− =− . [2] Another function g is defined by ( )g sin , for 0 4x x x =   . (b) Explain why the composite function fg exists. [2] (c) Find the exact value of fg 6   . [2] The domain of f is further restricted to 01 x . (d) By considering ( ) ( ) 23f 1 and f 1 , deduce the value(s) of ( )f1n where n is a positive integer. [2] 9 Referred to the origin O, a variable point R has position vector r and the fixed points A and B have position vectors a and b respectively, where a and b are non-zero vectors. (a) If a and b are parallel, find ab . [1] (b) If a and b are non-parallel, interpret geometrically the following vector equations: (i) ( ) − =r b a 0 , [2] (ii) ( ) 0 −=r b a . [2] It is now given that the coordinates of A and B are (−1, 2, 2) and (6, −2, −4) respectively. (c) Use a vector product to find the area of the triangle OAB. [3]

5 10 It is given that 4 d 19 n n xIx x = − . (a) Find 3I . [2] (b) By using the substitution 2ux= , find 1I . [4] (c) Using integration by parts and the result in part (a), find the exact value of ( ) 1 123 0 sin 3 dx x x−  . [4] 11 A construction company is installing cables on the side of a building. Points ( ),,x y z are defined relative to a main anchor point O located at ( )0,0,0 , where units are in metres. Cables are laid in straight lines and the widths of the cables can be neglected. An existing cable L starts at the main anchor point O and goes in the direction of 1 2 4    . A new cable M passing through points ( )1, 1, 1P − and ( )3, 0, 0Q is installed. (a) Using a non-calculator method, show that the cables L and M do not meet. [4] The construction company wishes to connect the point P to an inclined rectangular roof extension that is modelled by the plane with equation 2 3 28x y z+ + = . It was found that the best way is to use a cable to connect the point P to the point F on the rectangular roof extension that is closest to P. (b) Find the coordinates of the point F. [3] (c) Hence find the exact length, in metres, of the cable needed to connect point P to the rectangular roof extension. [2]

6 12 The parametric equations of the curve C are tan 1 and 2secxy= − = , where ππ .22−   (a) Find d d y x in terms of  , leaving your answer in a single trigonometric function. [2] (b) Find the equation of the tangent to C at the point where it cuts the y-axis, leaving your answer in exact form. [3] (c) Find the Cartesian equation of C. [1] (d) Find the equations of the asymptotes and state the coordinates of any turning point(s). [3] (e) The region bounded by the tangent found in part (b), the curve C and the line 1x=− is rotated completely about the x-axis. Calculate the volume obtained. [3]

7 13 At the start of January 2014, Leon took a home loan of $500 000 from a bank that offers a fixed monthly interest rate of 0.3%, to buy a n apartment unit in a condominium. The interest was added to the amount ow ed on the 15th of every month, with the first interest amount added on 15 January 2014. Leon made a monthly repayment of $ x on the 20th of every month, with the first repayment on 20 January 2014. (a) Show that the amount (in dollars) that Leon owed at the end of n months was ( ) ( )1000500000 1.003 1.003 13 nn x−− . [3] Leon planned to repay the loan fully in 360 monthly repayments. (b) Find the value of x, giving your answer to the nearest cents. [2] The estate management collects a monthly maintenance fee from each apartment unit within the condominium at the start of every month. The monthly maintenance fee on 1 January 2014 was $400. To cope with the rising cost of maintenance over time, the estate management increased the monthly maintenance fee by $k every 2 years such that the monthly maintenance fe e increased to ( )$ 400 k+ on 1 January 2016 and to ( )$ 400 2 k+ on 1 January 2018 and so on. (c) Show that the total maintenance fee (in dollars) that Leon paid at the end of 10 years was of the form s tk+ , where s and t are positive constants to be determined. [3] The value of Leon’s apartment unit increased to $750 000 at the end of December 2023. It was also known that 50k= . (d) By considering the amount Leon owed, along with t