561 Answers Chapter 3
561 Answers Chapter 3
561 Answers Chapter 3
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<strong>Answers</strong><br />
Selected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.<br />
d – 5 3 5<br />
--------- – --i e −4 f 3 3 – 3i<br />
2 2<br />
c z = 2 7 cis ⎛–π<br />
----- ⎞ d z = 2 5 cis ⎛ 5π<br />
–-----⎞<br />
⎝ 3 ⎠<br />
⎝ 6 ⎠<br />
564<br />
2π π<br />
3 a –----- b –-- c π d 0<br />
3 2<br />
– 3<br />
4 2 – 3i = 7 ; tanθ = --------- ⇒ θ = tan −1 ⎛ 3<br />
------ ⎞ ⇒<br />
2<br />
⎝ 2 ⎠<br />
θ = −0.7137 c<br />
6 a i<br />
3π<br />
–-----<br />
ii – 3π ----- + 2kπ, k = 0, ±1, ±2<br />
4<br />
4<br />
b i<br />
π<br />
–--<br />
6<br />
ii – π -- + 2kπ, k = 0, ±1, ±2, ...<br />
6<br />
c i<br />
π<br />
–--<br />
3<br />
ii – π -- + 2kπ, k = 0, ±1, ±2, ...<br />
3<br />
d i<br />
π<br />
--<br />
π<br />
ii -- + 2kπ, k = 0, ±1, ±2, ...<br />
6<br />
6<br />
e i<br />
2π<br />
–-----<br />
ii – 2π ----- + 2kπ, k = 0, ±1, ±2, ...<br />
3<br />
3<br />
f i<br />
3π<br />
–-----<br />
ii – 3π ----- + 2kπ, k = 0, ±1, ±2, ...<br />
4<br />
4<br />
g i<br />
π<br />
–--<br />
2<br />
ii – π -- + 2kπ, k = 0, ±1, ±2, ...<br />
2<br />
h i<br />
π<br />
--<br />
π<br />
ii -- + 2kπ, k = 0, ±1, ±2, ...<br />
2<br />
2<br />
i i π ii π + 2kπ, k = 0, ±1, ±2, ...<br />
j i 0 ii 2kπ, k = 0, ±1, ±2, ...<br />
π<br />
k i –--<br />
ii – π -- + 2kπ, k = 0, ±1, ±2, ...<br />
2<br />
2<br />
5π<br />
l i –-----<br />
ii – 5π ----- + 2kπ, k = 0, ±1, ±2, ...<br />
6<br />
6<br />
3π<br />
7 a 2 2 cis –----- b 4cis<br />
⎝<br />
⎛ 4 ⎠<br />
⎞ ⎛–--<br />
π ⎞<br />
⎝ 6⎠<br />
c cis ⎛ π<br />
–--⎞ π<br />
d 2 2 cis --<br />
⎝ 3⎠<br />
⎝ ⎛ 6⎠<br />
⎞<br />
e 4 2 cis ⎛ 2π<br />
–-----⎞ f 3 2 cis ⎛ 3π<br />
–-----⎞<br />
⎝ 3 ⎠<br />
⎝ 4 ⎠<br />
π<br />
g 2cis –--<br />
h 4cis<br />
⎝<br />
⎛ 2⎠<br />
⎞ π --<br />
⎝ ⎛ 2⎠<br />
⎞<br />
i 3cisπ j 5cis0<br />
1<br />
k ------<br />
π<br />
cis –-- l cis<br />
3<br />
⎝<br />
⎛ 2⎠<br />
⎞ 2 ---------<br />
3 ⎛ 5π<br />
–-----⎞<br />
3 ⎝ 6 ⎠<br />
3π<br />
8 a z = 3 2 cis ----- b z = cis<br />
⎝<br />
⎛ 4 ⎠<br />
⎞ ------<br />
2 ⎛– 3π<br />
---------⎞<br />
3 ⎝ 4 ⎠<br />
Specialist Maths Dimensions Units 3 & 4<br />
π<br />
e z = 5 2 cis -- f z = cis<br />
⎝ ⎛ 4⎠<br />
⎞ ---------<br />
10 ⎛3π<br />
-----⎞<br />
5 ⎝ 4 ⎠<br />
π<br />
g z = cis -- h z = cis<br />
⎝ ⎛ 4⎠<br />
⎞<br />
⎛ 5π<br />
–-----⎞<br />
⎝ 6 ⎠<br />
9 a 1 + 3i b 1 − i c – 3 2 -- 3 3<br />
– --------- i<br />
2<br />
d – 3 e −2 f 3i<br />
2 -- 3<br />
+ ------i<br />
2<br />
g −5i h 1 −<br />
11 a −33 − 56i c<br />
Exercise 3E<br />
3π<br />
1 a i 6cis ----- ii cis<br />
⎝<br />
⎛ 4 ⎠<br />
⎞ 3 π<br />
-- --<br />
2 ⎝ ⎛ 4⎠<br />
⎞<br />
–5π<br />
b i 2cis --------- ii cis<br />
⎝<br />
⎛ 6 ⎠<br />
⎞ 1 -- ⎛–5π<br />
---------⎞<br />
2 ⎝ 6 ⎠<br />
7π<br />
c i 20cis ----- ii cis<br />
⎝<br />
⎛ 12⎠<br />
⎞ 5 -- ⎛ π<br />
----- ⎞<br />
4 ⎝12⎠<br />
11π<br />
d i 3 2 cis -------- ii cis<br />
⎝<br />
⎛ 12 ⎠<br />
⎞ 3 ---------<br />
2 ⎛5π<br />
-----⎞<br />
2 ⎝12⎠<br />
3 π<br />
e i 5 3 cisπ ii ------ cis --<br />
5 ⎝ ⎛ 2⎠<br />
⎞<br />
π<br />
f i 2 2 cis –-- ii cis<br />
⎝<br />
⎛ 3⎠<br />
⎞ ------<br />
2 ⎛ 2π<br />
–-----⎞<br />
2 ⎝ 3 ⎠<br />
3<br />
2 a 2 3i b −1 − ------i c – 3 3<br />
4 -- 3<br />
+ ------i<br />
4<br />
3 3 5<br />
d --------- – --i e – 3 + 3i f<br />
3 2<br />
g<br />
65<br />
---------<br />
25<br />
– 3 1<br />
------ – --i h –--------- 9 + i . 3 ---------<br />
3<br />
6 2 4 2 4 2<br />
11π<br />
3 a –1 – 3 + ( 1–<br />
3)i<br />
= 2 2 cis ⎛–--------⎞<br />
⎝ 12 ⎠<br />
–1 + 3 ( 1+<br />
3)i<br />
2<br />
b ------------------- – ---------------------- = ------ cis ⎛5π<br />
-----⎞<br />
8 8 4 ⎝12⎠<br />
5π<br />
c 2 3 – 2 + i( 2 3 + 2) = 4 2 cis ⎛-----⎞<br />
⎝12⎠<br />
3 3 3 2 π<br />
d -- + --i = --------- cis --<br />
4 4 4 ⎝ ⎛ 4⎠<br />
⎞<br />
3i<br />
3<br />
------<br />
2<br />
1<br />
+ --i<br />
2
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