561 Answers Chapter 3
561 Answers Chapter 3
561 Answers Chapter 3
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Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.<br />
Exercise 3F<br />
2π<br />
1 a 8cisπ b 16cis ⎛–-----⎞<br />
⎝ 3 ⎠<br />
1 π<br />
π<br />
c ----- cis -- d 2 cis --<br />
32 3<br />
⎝ ⎛ 6⎠<br />
⎞<br />
2 a −2 + 2 3i b 128 + 128i<br />
c −128 − 128 3i d – 31 3 + 118i<br />
3 a −2 + 2i b<br />
c<br />
–4 2 – 4 2i d ( 2 3 – 2i) 3<br />
16<br />
4 a –-----<br />
i<br />
b<br />
27<br />
c<br />
d<br />
e<br />
–3 2 – 3 6<br />
f ------------------------------ + i⎛--------------------------<br />
3 2– 3 6 ⎞<br />
16 ⎝ 16 ⎠<br />
1<br />
5 a --i<br />
2<br />
π<br />
6 a modulus = 16, argument = -- + 2πk,<br />
6<br />
k = 0, ±1, ±2, ...<br />
b modulus = 1, argument =<br />
k = 0, ±1, ±2, ...<br />
24 3i<br />
1<br />
c modulus = --, argument = – 2π ----- + 2πk,<br />
8<br />
3<br />
k = 0, ±1, ±2, ...<br />
2<br />
d modulus = ------, argument = – π -- + 2πk,<br />
32<br />
2<br />
k = 0, ±1, ±2, ...<br />
3π<br />
7 a w = 2 cis ----- z = cis<br />
⎝<br />
⎛ 4 ⎠<br />
⎞ π<br />
, 2 2 ⎛–--⎞<br />
⎝ 4⎠<br />
8 a i 8cis ⎛ π<br />
–--⎞<br />
ii –8i<br />
⎝ 2⎠<br />
2187 3<br />
------------------<br />
512<br />
6+<br />
2 6–<br />
2<br />
– 78 125⎛-------------------- + i⎛-------------------⎞⎞<br />
⎝ 4 ⎝ 4 ⎠⎠<br />
27<br />
-----<br />
4<br />
27 3<br />
+ ------------ i<br />
4<br />
2( 3+<br />
1) + i 21 ( – 3)<br />
b i 16cis ⎛2π<br />
-----⎞ ii −8 + 8 3i<br />
⎝ 3 ⎠<br />
–<br />
– π -- + 2πk,<br />
3<br />
2187<br />
-----------i<br />
512<br />
2<br />
3π<br />
b modulus = ------, argument = ----- + 2 πk,<br />
4<br />
4<br />
k = 0, ±1, ±2, ...<br />
1 π<br />
c i -- cis -- ii<br />
8 ⎝ ⎛ 2⎠<br />
⎞ 1 --i<br />
8<br />
2<br />
d i ------<br />
π<br />
cis –-- ii<br />
9 ⎝<br />
⎛ 4⎠<br />
⎞ 1 1<br />
-- – --i<br />
9 9<br />
e i 108 2 cis ⎛ 3π<br />
–-----⎞<br />
ii −108 − 108i<br />
⎝ 4 ⎠<br />
9 2 π<br />
f i --------- cis -- ii<br />
2 ⎝ ⎛ 4⎠<br />
⎞ 9 9<br />
-- + --i<br />
2 2<br />
g i – 1 1<br />
----- – -----i ii – 1 1<br />
----- – -----i<br />
12 12<br />
12 12<br />
h i<br />
1<br />
1<br />
-------- cis0 ii --------<br />
216<br />
216<br />
11 a – 1 = cis = cis<br />
2 -- + ------i ----- -- – ------i<br />
12 2i<br />
13<br />
1<br />
d i ----- cis ⎛ 2π<br />
–-----⎞ ii –-----<br />
1 i 3<br />
– --------<br />
16 ⎝ 3 ⎠ 32 32<br />
2<br />
9 a i ------<br />
3π<br />
cis –----- ii −<br />
2 ⎝<br />
⎛ 4 ⎠<br />
⎞ 1 1<br />
–-- --i<br />
2 2<br />
b i<br />
2 π<br />
-------- cis -- ii +<br />
108 ⎝ 4⎠<br />
1<br />
-------- --------i<br />
108 108<br />
c i<br />
1<br />
--<br />
1<br />
cisπ ii –--<br />
4<br />
4<br />
b 0<br />
3<br />
2<br />
– 3 1<br />
-------- – --------i<br />
256 256<br />
Exercise 3G<br />
1 a (z + 3i )(z − 3i )<br />
b (z + 7i )(z − 7i )<br />
c (z + 2 2i )(z − 2 2i )<br />
d (z 2 − 8i )(z 2 + 8i )<br />
e (3z − 4i)(3z + 4i)<br />
2π<br />
⎝<br />
⎛ 3 ⎠<br />
⎞ 1 , 2<br />
f (z 3 − i)(z 3 i – 3 i + 3<br />
+ i) = (z + i)(z − -------------- )(z − -------------- )<br />
2 2<br />
i – 3 i + 3<br />
(z − i)(z + -------------- )(z + -------------- )<br />
2 2<br />
1 11<br />
d -- ± ---------i e ± 3i<br />
2 2<br />
3<br />
2<br />
⎛ π<br />
–--⎞<br />
⎝ 3⎠<br />
7 3i<br />
2 a 2 ± 2i b −3 ± i c -- ± --------<br />
2 2<br />
<strong>Answers</strong><br />
565
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