SUCCEEDING IN THE VCE 2017 UNIT 3 SPECIALIST MATHEMATICS STUDENT SOLUTIONS
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1 SUCCEEDING IN THE VCE 07 UNIT SPECIALIST MATHEMATICS STUDENT SOLUTIONS FOR ERRORS AND UPDATES, PLEASE VISIT QUESTION (a) (b) i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
2 (c) i ( 7i) i QUESTION (a) ()(0) i (b) 7 0 ()(7) 8 9i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
3 QUESTION (a) (b) w w ( ) ( 7) i i w w ( ) ( 7) i i QUESTION 8 (a) (b) (c) a i (d) (e) 9 7 i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
4 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
5 QUESTION 0 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
6 QUESTION (a) i (b) w i (c) (e) w i (d) 9 i wi. w i i (f) i i i (g) i i (h) i i i i QUESTION A (a) i (b) i (c) i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
7 QUESTION B (a) i (b) i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 7
8 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 8 QUESTION (a) () ) ()( 0 ) ( ) ( i (b) Arg() = tan
9 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 9 QUESTION 8 (a) cis sin cos i sin cos i i i (b) cis sin cos i sin cos i i i i
10 QUESTION If (a) cis and w cis find:. w (b) w The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 0
11 QUESTION (a) Find in eact Cartesian form. (b) Epress in polar form. cis (c) Given that cos isin, find the polar form of. (d) Hence find the eact value of cos. From (a): From (c): cis cos i i csin. Equate real parts of each epression for : cos cos The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
12 QUESTION * QUESTION (b) (c) The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
13 QUESTION i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
14 Using technology to check your answer: QUESTION 7 (a) Simplify the following epressions, giving your answer in Cartesian form. (i) ( ) i = 0 i 8 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
15 (ii) cis ( i) i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
16 (iii) ( i) ( + i) = cis cis = ( ) cis cis = cis ( ) cis = 8 cis = 8 i = i (b) If acis and i find the value of a. The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
17 b (c) If acis and w cis and cis w find the value of a and b. b (d) Given that acis and w cis find the value of a and b if w 08cis. The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 7
18 QUESTION 8* (a) Find in eact Cartesian form. ( i)(7 i) 8 iii i (b) Hence find in eact Polar form. Arg tan tan () cis (c) Write both and in eact Polar form. (d) 7 0 Arg tan tan Arg tan tan 7 7 Hence find the eact value of tan tan 7. From (b) cis From (c) cis tan tan 7 Therefore cis tan tan cis 7 Therefore tan tan 7 cis tan cis tan 7 The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 8
19 QUESTION 9* cis m m cis m i i cis m m cis m i i m m m m cis cis 0 m cos m isin m cos m isin m 0 m cos m isin m cos m isin m 0 m m i sin 0 m m i sin 0 as 0 m sin 0 m n n Z m n n Z The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 9
20 QUESTION 0* (a) w r cis n n w r cis and n n w a. Equating these equations gives: n 0 as required r n cis n a n r cos n isin n a0i n r cos n a and r sin n 0 sin n (b) (c) wr cis n n w r cis n n n n r cos n r isin n n r cos n r isin n n a r i (0) from (a) a n n n n w w w a n n n n w ( w) ( w) a Therefore w and w satisfy n a The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 0
21 QUESTION * (b) Hence show that a solution to 0 0 is sin. Substitute sin into the result given in (a) (i): sin( ) 0. Then the equation 0 0 given to solve becomes sin( ) 0 : sin( ) 0 m, where m Z m Therefore Let m : m sin sin. sin is a solution. (c) Hence show that the eact value of sin is equal to or 0 0 Therefore from the quadratic formula. 8 Which gives 0. 8 Now as sin sin sin then sin 0 sin then The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
22 (d) Find an eact solution to 0 of the form a( b c), where a, b and c are integers. Substitute cos into the result given in (a) (ii): cos( ) 0. Then the equation 0 given to solve becomes cos( ) : cos( ) (m ), where m Z ( m ). (m ) Therefore cos cos. Let m 0 : cos is a solution. Substitute sin 0 for cos : cos from part (c) into cos sin and solve The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
23 QUESTION (b) 8 (c) i The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
24 Solution QUESTION Im() Re() (b) Im() Re() The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
25 QUESTION A The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
26 (e) cis Im() cis Re() QUESTION B (a) cis (b) The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page
27 QUESTION Use technology to check your answers: The School For Ecellence 07 Succeeding in the VCE Unit Specialist Maths Page 7
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