RVHS 2022 J1 H2MA Test 2 Std Soln

RVHS 2022 JC1 H2 Maths Lecture Test 2 1 2022 H2 Maths JC1 Lecture Test 2 Solution (Student Copy) Question 1 [6 marks] AP/GP (a) Let an be the amount of gold received for logging in on the nth day. Then 1 2 3100, 130, 160, ...a a a= = = Which forms an arithmetic progression. Thus, ( ) ( ) ( )( ) 2 212 900 [2 100 1 30 ]2 0 15 85 900 5.41 or 11.1 (rej) n nS a n d n n nn nn = + −  + −  + −  − Therefore, it will take 6 days. (b)i. Let bn be the amount of XP received for completing the quest on the nth time. Then ( ) ( ) 2 1 2 31400, 1400 0.9 , b 1400 0.9 , ...bb= = = Which forms a geometric progression. Thus ( ) ( ) 10 1 1 1400 1 0.9 1 0.9 9118.501839 n n ar S r − = − − = − = Therefore, 9120 XP (3 s.f.) would be earned (b)ii. Maximum XP is the amount earned if played infinitely many times. 1 1400 1 0.9 14 000 aS r  = − = − = The maximum theoretical amount of XP to be earned is 14 000 XP

RVHS 2022 JC1 H2 Maths Lecture Test 2 2 Question 2 [8 marks] MOD and Sigma Notation (i) ( ) ( )f 1 f 11ln ln 1 1 1ln 1 ln 1 nn nn n n n n +−    =−    +     +=  =  + (ii) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ln 1 f 1 f f 2 f 1 1 f 3 f 2 2 ... ... f f 1 1 f 1 f f 1 f 1 1ln 1 ln 1 ln 1 N n N n n n nn n n NN nN NN nN N N N = =   + = + − − =  +− = = + − − =− + + − = =− + + =− +  + =− +   (iii) ( ) 11 ln lim ln11 lim ln 1 N Nnn N nn nn N  →== → =++ = − + =−  Hence, the series diverges (iv) ( ) 2 2 4 5 1 1 ln ln ln1 1 1 ln 2 1 ln 5 5ln 21 NN n n n n n n n n n N N = = =      =−     + + +      =− + + =  +   

RVHS 2022 JC1 H2 Maths Lecture Test 2 3 Question 3 [8 marks] Graphing Techniques (i) 2 47ax xy bx −+= − with asymptotes 3yx=− + and 1x=− By observation, 1b= , 1a= 2 4 7 4 311 xxyx xx −+= =− + +−− (ii) Consider the points of intersection between y = k and 2 47 1 xxy x −+= − . ( ) ( ) 2 2 47 1 0 4 7 xxk x x k x k −+= − = + − + − For no points of intersection, ( ) ( )( ) ( )( ) 2 2 2 4 4 1 7 0 8 16 4 28 0 4 12 0 6 2 0 kk k k k kk kk − − −  − + + −  − −  − +  26 k−   Therefore, there are no y values between –2 and 6. (iii) y x 2 47 1 xxy x −+= − ( ) ( ) ( ) 0,7 1,6 3, 2 A B C − −

RVHS 2022 JC1 H2 Maths Lecture Test 2 4 Question 4 [8 marks] Graphing Transformations (a) ( ) 22 2 2 2 2 1 Replace with 3 Scale parallel to -axis, with factor 3 13 Replace with 1 Translate 1 unit in the negative -direct ion 1 19 xy yy y yx yy y yx +=  += + ++= Alternatively: ( ) 22 2 2 2 2 2 2 1 1Replace with 3 1Translate units in the negative directi on3 1 13 yReplace with 3 Scale parallel to -axis, s.f. 3 1 33 1 19 xy yy y xy y y yx yx += +  − + + =  ++ ++= (b)i. ( ) ( ) f Replace with 1 Translate 1 unit in the positive directi on f1 Replace with 2 Scale parallel to -axis, with factor 2 f1 2 yx xx x yx xx x xy = − − =−  =−

RVHS 2022 JC1 H2 Maths Lecture Test 2 5 (b)ii. y x ( ) 1 fy x= f1 2 xy =− y x A’(-2,0) B’(2,0) C’(4,-1) D’(8,0) x = 0