ACSI 2016 Promo Paper 1 Ans

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 1 FINAL EXAMINATION 2016 YEAR 5 IB DIPLOMA PROGRAMME PAPER 1 MATHEMATICS HIGHER LEVEL – SOLUTIONS Qn Solution 1. General Term for  10 1 x :   10 210 1011 r rrr x xrr          101 11x x  r = 2:  22 1210 10.914 52 2.1x xx  r = 0:  00 210 110 x  1 114 5 . . 4 514 4xx      Comment: Mostly well done although a small number do not know how to evaluate 10C2 and a handful do not remember how to do Binomial Expansion. 2. (i)   18 27 ... 99 10 18 992 5 117 585      Comment: A number of students did by pure arithmetic calculation and some missed out on 99 as the last term. (ii) 10 1 99 r r   (divisible by 9) Total sum – sum of all divisible by 9 =   90 10 11 99 9 rr rr    Comment: Well done.

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 2 Qn Solution 3.  22 2 22 2 22 2 22 A : B 11 1 : 22 2 : 3 4 20 Area R rr Rr r Rr r Rr Rr R        Comment: Very well done except some forgot the formula for area of sector. 4. (i) 22 12 2 1 21 12 1 21 21 21 21 21 x x x x x S           Comment: Some did not manipulate answer to the required form. (ii)    23 1 1 123. .1 1 11 11 12 0.5 66 33 2 2 2 .... 2 2 2 2 1 2 x xx x x x                Comment: A number forgot to substitute the value of x in their final answer.

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 3 Qn Solution 5.      1 2 3 1 1 1 1 Let P be statement ! 2 3, 3! 6 24 is true. Assume P is true, ! 2 To show that P is true: 1 ! 2 !2 1! 2 1 11! 2 2 1 2, 1, 2 11! 2 2 n n k k k k k k k k n nL H S RHS LHS P k k k kk kk kif k kk                         31 21 1! 2 Since P and P is true, by Mathematical Induction, P is true for all 2, k k kn k nn        Comment: Some vague argument on the validity of P(k+1). 6.  32 32 32 16 28 2 2 81 68 1 6 022 2 81 6 4 1 6 084 4 4 16 0 or equivalent let y x yx yy y yy y yyy                        Comment: A number of the students did by finding out the coefficients of the function in relation to the sum and product of roots. A small group of students did by solving for the original roots.

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 4 Qn Solution 7.  1sin arctan44 6 3 sin 46 sin cos cos sin46 4 6 13 1 1 2222 31 22 arc k k                           Comment: Well done although some mistakenly gave the value of 3 1arctan as 3  . 8. (i)        11 0, 0 2 0 0 0 x aaax ax ax ax axax x axax xx a x a ax x a             or Comment: Badly done with many stopping at   02   xaxa x . Some continued with incoherent arguments. (ii)  a = 2,   ysubst x e   20 2l n 2yyee y    (rej) or Some have notion that 2a and yex but could not obtain the right solution.

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 5 Qn Solution 9. (i)      3 2 30 2 11 11 1 1 1lim or or 2 22 h fx h fx xh x hh xx h hx hx xx hxx h hx hx x x h xxh h x xh xh x x x xh xh x x x x x x x x                       Comment: Several students did not do the key step of multiplying by the conjugate of the surds in order to arrive at the correct answer. (ii)   2 2c o s ln 2 ln cos ln 2 ln cos : sinln 2 ln 2 1 cos ln 2 1 tan ln 2 1 tan ln 2 tan ln cos 1 tan ln 2 ln 2 ln 2 ln 2 xy x x ex xy e x xy x x Diff dy xyx dx x dyxx y dx dy x y x x x y or ordx x x x x x                  Alternatively:     2c o s : 2l n 2 c o s s i n 2 l n2 c o s s i n 2 l n2 c o ss i n 2l n 2 c o ss i n 1s i n or 2l n 2 2l n 2 l n 2 2l n 2 xy x xy x x xy x xy xx y x x xy xy xy ex Diff dyyx e xe xdx dyxe x x ydx ex x y ex xdy y y e x dx x x x x x x              Comment: Badly done. There were mistakes in implicit differentiation and algebraic manipulation. Most common mistake is with differentiating 2xy . Some wrote wrongly that 22 . 2xy x y

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 6 Qn Solution 10. (a)      1 2 2 2 2 1 2 21 1 11 11 11 11 11 1 ], 1 [ff yx x yx x yx xy xy xy fx x x DR                        Comment: Several students interpreted the question wrongly into testing for whether the inverse function exist and did the horizontal line test. This is not required. Common mistake in forgetting the  and in not choosing the correct rule for the function. (b) (i) ]0, [ \{0} exists. hgRD gh      Comment: need to state clearly both the range of h and the domain of g in correct notations. (b) (ii)      12 2 ], 0 [gh h gh x g x x xx DD       (b) (iii)      2 2016 (for even numbers) 1 gx x gg x gx ghx hx    Comment: the question requires the answer in terms of h(x). (c)     11 2 1fx fx f x f x      Comment: Badly done. Usually the wrong sequence of transformations or wrong transformations given. 1   1 2 4

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 7 Qn Solution (d) Comment: some mistakes with the shape and symmetry of the curve. (e)                   Comment: badly done. Especially the LHS has a horizontal asymptote to y = 0.5. 11. (a) 11 1. 1 41cos 18 3 114 36 22 6 or equivalent336 or or            Comment: some wrongly took the angle to be between vectors OA and OB. (b) ~ 1 ~ 11 21 :1 1 , 2 1 41 51 rO A A B Lr o r                                 Comment: well done. 11 1 2 0  0.5  (1, ‐1)

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 8 Qn Solution (c)  2 ~ 22 :4 1 , 73 11 2 2 11 4 1 41 7 3 21 , 5 , 33 2, 3 112 13 1 2 o r 2 , 2 , 1 41 1 Lr OP                                                         Comment: Generally well done. (d)     ~ 22 2 2 41 4 73 7 3 .0 22 2 2 41 0 . 1 0 73 7 3 22 1 4 33 0 14 14 0 1 22 0 41 3 73 4 OF CF d                                                             Comment: Generally well done. (e) 1 '2 '2 02 2 2 3 10 16 47 1 OF OC OC OC OF OC                        Comment: Generally well done.

ACS (Independent) Mathematics Department / Mathematics HL / Year 5 / 2016 Final Examinations / Paper 1 / Solutions 9 Qn Solution (f) (i) Area of first parallelogram ~~ ~ ~ ~~ ~ ~ ~ .u v vu v v vu v    Area of second parallelogram ~~ ~ ~ ~~ ~ ~ ~ .u v vu v v vu v       Hence the area