RVHS 2023 J2 H2MA Test 1 4b Soln (Std)

RVHS 2023 JC2 H2 Maths Lecture Test 1 1 Name : Index Number : Class : Date : 3 Feb 2023 Duration : 50 mins Max. No. of Marks : 30 List of Formulae Derivatives f ( )x f '( )x 1sin x 2 1 1 x 1cos x 2 1 1 x   1tan x 2 1 1 x cosec x cos ec cotxx sec x sec tanxx Integrals (Arbitrary constants are omitted; a denotes a positive constant.) f ( )x f ( )dxx 22 1 xa 11 tan x aa    22 1 ax 1sin x a    ()xa 22 1 xa 1 ln2 xa a x a    ()xa 22 1 ax 1 ln2 ax a a x    ()xa tan x ln(sec )x 1( π)2x  cot x ln(sin )x (0 π)x cosec x ln(cosec cot )xx (0 π)x sec x ln(sec tan )xx 1( π)2x  [Answer all the questions on writing papers. Up to 1 mark will be deducted for poor presentation.] 1 (i) Find e dxxx  . [2] (ii) Find 2 1 d14 xx . [2] (iii) Hence find the general solution to the differential equation 2d e 4 ed xxy x xyx  . [2] River Valley High School Integrated Programme 2023 JC2 H2 Mathematics (9758) Lecture Test 1 (Term 1)

RVHS 2023 JC2 H2 Maths Lecture Test 1 2 ~ The End ~ 2 The curve C has equation   3 2 3 2 449 xy x   . (i) The finite region bounded by the curve C, the line 33 2x and the x-axis is rotated through 2π radians about the x-axis. Find the volume of the solid obtained, giving your answer correct to 3 significant figures [2] (ii) The finite region bounded by the curve C, the lines 33 2x , 1y and the y-axis is rotated through 2π radians about the x-axis. Find the volume of the solid obtained, giving your answer correct to 3 significant figures. [3] 3 A 1500-litre tank contains 600 litres of water with 5 kg of sugar dissolved in it initially. Sugar solution of concentration 1 5 kg per litre enters the tan k at a constant rate of 9 litres per hour. A well-mixed solution leaves the tank at a constant rate of 6 litres per hour. (i) Denoting the amount of sugar in the tank as Q kg, show that d 2 9 d 200 5 Q t Q t  . [2] (ii) Using the substitution   2 200u t Q , show that the differential equation reduces to   2d9 200d5 u tt  . [3] (iii) Hence, find the amount of sugar in the tank at the time when it is full. [6] 4 Alice has 3 red, 3 blue and 3 yellow dominoes. (a) Find the number of ways to arrange the dominoes in a line if (i) there are no restrictions, [2] (ii) the red dominoes cannot be all together. [3] (b) Find the number of ways to arrange the dominoes in a circle if the blue dominoes are to be separated. [3]

RVHS 2023 JC2 H2 Maths Lecture Test 1 3 RVHS 2023 JC2 H2 Lecture Test 1 Marking Scheme 1 Solution [10] Integration Techniques (a) e d e e d ee x x x xx x x x x xc      d ed d 1ed x x vux x u vx   (b)     2 2 1 d14 12 d2 12 1 1 1 2 ln2 2 1 1 2 1 1 2ln4 1 2 xx x x x cx x cx            Alternatively    2 1 d14 1 d1 2 1 2 1122 + d1 2 1 2 1 1 1ln 1 2 ln 1 22 2 2 1 1 2ln4 1 2 xx xxx xxx x x c x cx                (iii) 2 2 2 d e 4 ed 1d e1 4 d 1 d e d14 1 1 2ln e e4 1 2 xx x x xx y x xyx y xyx y x xy y xcy         

RVHS 2023 JC2 H2 Maths Lecture Test 1 4 2 Solution [12] Differential Equations (i) Volume of solid formed     33 22 3 2 3 2 4 0 3 πd 49 0.09375π =0.295 units to 3sf x x x        (ii) Volume of solid formed =Volume of cylinder – volume of the solid in (i)     2 3 33π 1 0.09375π2 =7.86757 =7.87 units to 3sf  

RVHS 2023 JC2 H2 Maths Lecture Test 1 5 3 Solution [12] Differential Equations (i)     d rate in rate outd 196 5 600 3 92 5 200 Q t Q t Q t                d 2 9 (shown)d 200 5 Q tt Q    (ii)         2 2 2 200 dd 200 2 200dd 1 d d 2 d d 200200 u t Q uQ t Q ttt uQ t t tt Q        Sub into differential equation       2 2 d 2 9 d 200 5 1 d 9 d5200 d9 200 shownd5 Q tt u tt u tt Q     Alternatively,             2 2 23 200 1 200 d 1 d 2 2dd 200 200 u t Q uQ t u utt tt Q      Sub (1) and (2) into d 2 9 d 200 5 Q t Q t              2 3 3 2 2 1 d 2 2 9 d5200 200 200 1 d 9 d5200 d9 200 shownd5 u uutt t t u tt u tt         

RVHS 2023 JC2 H2 Maths Lecture Test 1 6 (iii)       2 3 3 d9 200d5 2009 53 3 2005 u tt tuc tc              23 2 3200 200 5 3 2005 200 t Q t c cQt t         When t = 0, Q = 5.     2 35 2005 200 4800000 c c       2 3 48000002005 200 Qt t     Tank is full when t = 300     2 3 4800000500 280.8 kg5 500 Q  

RVHS 2023 JC2 H2 Maths Lecture Test 1 7 4 Solution [8] P & C (a) (i) Number of ways 9! 16803!3!3! (ii) Number of ways 7!1680 15403!3!   , Alternatively, Case1: red dominoes are all separated Number of ways 76! 70033!3!     , Case2: 2 red dominoes are together with the other one separated from the group of 2. Number of ways 76! 2! 84023!3!     , Total number of ways 700 840 1540  