ASRJC 9758 2023 Prelim P1

[Turn over P Q R S d d 1 In a particular soccer league of 20 teams, each team plays a total of 38 games over the course of a season. Teams are awarded 3 points for a win, 1 point for a draw, and no point for a loss. Lucy’s favorite team scored a total of 54 points this season and she is especially happy because the sum of games won or drawn this season exceeded twice the number of games lost by 8 games. (a) Find the number of games won by Lucy’s favorite team this season. [3] Mark’s favorite team won 2 games fewer and drew 5 games more than Lucy’s favorite team this season. (b) Explain, with clear workings, which team performed better this season. [1] 2 (a) Show that 2d (tan ) secd xxx = [2] (b) Show that 2sin 2 cot 2cosxx x = [1] (c) Hence, find the exact value of ( ) ( )15 30 cosec 10 tan 5 dx xx π π∫ . [4] 3 (a) The cross section PQRS of a water passageway of depth d is as shown in the figure below. It is given that 2PQ QR RS a++= and with PQ and RS each inclined to the line QR at an angle θ. Show that the area of the cross-section PQRS is ( ) 22 cot 2cosecA ad d θθ= +− . [3] (b) Find, in terms of a and d, the maximum value of A, as θ varies. [4]

2 © ASRJC 2023 4 The equations of planes p1 and p2 are respectively given by 03 1 1 , 12 s t s,t      = +− ∈       r  and 37 α β   = r. , where α and β are real constants. (a) Given that the line 1l : 12 43 xy z+−= = lies in p2, find α and β . [3] It is now given that 3α =− and 2β = . (b) Find the equation of the line of intersection of p1 and p2. [3] (c) The plane p3 with equation 3 40yz+ += is parallel to 1l . Find the distance between 1l and p3. [2] 5(a) Show that 3 4 1 8 20 2 1 2 1 2 3 (2 1)(2 1)(2 3) r r r r rrr +−+=− + + −++ . [2] (b) Hence show that 1 25 (2 1)(2 1)(2 3) n r r rrr= + −++∑ 22 3 (2 1)(2 3) n nn += − ++ . [4] (c) Find the smallest possible integer k such that 1 25 (2 1)(2 1)(2 3) k r r rrr= + −++∑ is within 0.004 of the sum to infinity. [3] 6 (a) Find 2cot 3 dxx∫ . [2] (b) Find the exact value of 24 22 43 d48 xx xxx −+ −+∫ . [3] (c) Use the substitution 2cotx θ= to find the exact value of 22 23 22 3 4 d(4 ) x xx − +∫ . [5]

3 © ASRJC 2023 [Turn over 7 (a) The parametric equations of a curve C are 3secxt= and 2tanyt=− , where 0 2t π<< . (i) Find the cartesian equation of the curve, stating any restriction on the values of x and y. [2] (ii) Sketch the curve C, indicating the equation of the asymptote if any. [2] (b) A curve C1 has parametric equations 2sin 1, 2cos3 4sinx ty t t= += + , 33 tππ− ≤≤ . The tangent to the curve C1 at the point (1, 2) is given by L as shown in the diagram below. The line L cuts the line 1C at ( )2, 4 . Show that the area bounded by the curve C1, the line L and the line x = 2 is given by 0 4cos3 cos 8sin cos da tt tt t θ −+∫ , where a and θ are constants to be determined in exact form. Hence find the exact value of this area. [6] 8 It is given that 1 3 sin 2y x= + . (a) Show that . Hence find the Maclaurin series for y, up to and including the term in 2x . [4] (b) Given further that x is small so that 3x and higher powers of x can be neglected, use appropriate expansions from the List of Formulae ( MF26) to verify the correctness of the series of y in part (a). [3] (c) Use the series expansion found in part ( a) and integration to find an approximate value for 1.5 0 1 d3 sin 2 xx+∫ . [2] 2 2 2 dd 4 sin 2 4 cos 2 0dd yy y xy xxx −+ =

4 © ASRJC 2023 9 Do not use a calculator in answering this question. (a) Three complex numbers are 1 1iz = + , 2 3iz = − and 3 cos isin33z ππ= + . Find 12 2 3 zz z in the form ( )cos isinr θθ+ , where 0r > and πθπ−<≤ . [4] (b) A fourth complex number 4z is given by cos isinθθ+ . (i) Show that 4 4 1 1 z z + − can be expressed as cot2k θ , where k is a complex number to be determined. [4] (ii) For the case 4 πθ = , show that ( ) 4 4 1 1 2i1 z z + = +− . [2] (iii) Hence or otherwise, find the exact value of tan 8 π . [2] 10 Glucose is a type of sugar in our blood and our body uses it for energy. Having low sugar in the blood for long periods of time can cause health problems if it is not treated. At 10:00 am, a patient is given glucose via an intravenous drip at a constant rate of r units of glucose per hour. It is known that the rate at which the glucose in the blood stream is absorbed by the body, is proportional to the amount of glucose present in the blood stream. At the start of the treatment, the patient has 0x units of glucose in his blood stream. It is given that x denotes the amount of glucose present in the blood stream of the patient at time t hours after he starts the treatment. (a) Form a differential equation in x and show that 0 e ktrrxx kk −= −+ where k is a positive constant. [4] (b) State the theoretical limiting value of x if the patient is given the treatment over a long period. [1] (c) Sketch the graph of x for t ≥ 0. [2] (d) At what time (to the nearest minute), would the glucose in his blood stream first exceed 90% of the value found in part (b)? [5]

5 © ASRJC 2023 [Turn over 11 On 1st Jan 2023, Mr Kim borrows $40 000 for his business from a bank. The bank provides two options for repaying the loan. Under Plan A, the bank imposes a one -time administrative fee of $ 4660, which is added to the loan amount. Mr Kim needs to pay a monthly installment of $700 at the start of each month, and the installment amount increases by $60 for each subsequent month, until the loan amount is fully paid. (i) (a) Show that he would have paid a total amount of $( ) 230 670kk+ after the kth payment. [2] (b) How many payments will it take for Mr Kim to fully repay his loan? [2] Under Plan B, an interest rate of 1.5% per month will be charged to the outstanding amount at the end of every month, starting from 31st Jan 2023. Mr Kim needs to pay a fixed monthly instalment of $p at the end of every month after the interest has been charged. (ii) (a) Find the amount he owes at on 1st Mar 2023. [1] (b) Show that the amount he owes at the end of nth month after payment of instalment is given by ( )40000 1nn pαβα−− , where α and β are constants to be determined. [3] (c) Mr Kim intends to pay off his loan in k months. He decides to pay $1585 per month for the first ( k – 1) months and to pay the outstanding amount $m (where 0 1585m<≤ ) left in the loan at the end of the k th month after interest has been charged. Find the values of k and m. [4]