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