Power of Maths: Paper 1 – Answers
Exercise 3 1. (a) (2, 7) (b) (3, 18) (c) (17, 52) (d) (11, –5) (e) (–3, 7) (f ) (–7, –6) (g)
( 1 __
_ 2 + √
4. (a) (3, 4) (b) 4 _
√ √ (b) 1 (c) 1 (d) √
Chapter 14 Exercise 4
1. (a) 9 + 9i (b) 14 + 12i (c) 14 + 8i (d) 13 + 20i (e) 3 + 12i (f ) 8 + 18i (g) 19 + 17i (h) 13 + 3i (i) 2 + 14i ( j) (x + y) + 5i 2. (a) 10 – i (b) 2 – 6i (c) 19 + 4i (d) –12 – i (e) –10 + 2i (f ) –9 + 8i (g) 10 – i (h) 5 + 5i 3. (a) 9 + 8i (b) 4 + 5i (c) 7 – 8i (d) 23 + 6i
(e) 1 _
4. (a) 10 + 7i (b) 11 – i (c) − 21 __
2 + 7i (f ) 2 √
__ 3 + 4i (g) 2x – 2i (h) 9 + 4xi
2 − 6i (d) 12 – 2i
(e) y + 5i 5. (a) –5 – i (b) –3 + 2i (c) –2 + 7i (d) 0 + 8i (e) –8 + 9i (f ) 14 – 11i (g) 1 – 2i (h) − √
6. (a) 13 + 4i (b) (0, 0) → (–2, 8); –2 + 8i (c) (0, 0) → (7, –13); –7 + 13i (d) –9 – 4i
Exercise 5
1. (a) 12 + 8i (b) 55 + 50i (c) 42 + 14i (d) –22 – 33i (e) 24 – 6i (f ) 55 – 5i (g) 20 + 10i (h) 24 + 12i (i) –1 – 4i ( j) –2 + 3i (k) 5a + 5bi (l) k + 2ki (m) ka + kbi (n) 10 + 10i (o) 1 – 10i (p) – 4 – 5i (q) 3 – 7i (r) –11 – 8i (s) –32 – 50i 2. (a) – 6 + 0i (b) 13 + 15i (c) 2 + 12i (d) 1 – 3i (e) –28 – 24i
(f ) 17 + 12i (g) − 9 _
2 − 9 _
2 i (h) 17 + 21i (i) 1 + 6i
( j) 3 + 3i 3. P(–5, 12), 13 (a) –10 + 24i; 26 (b) 20 – 48i; 52 (c) 5
Q(4, 2); R(–8, – 4); S(–2, –1) (c) Slope of OP = 1 _
_ 2 − 6i; 13
Slope of OR = 1 _
| OP | = 2 √
OS = 1 _
Exercise 6 1. (a) 5 (b) √ (f ) √
348
__ 5 ; Slope of OQ = 1
2 , | OS | = √ 2 , | OR | = √
__ 5 (c) √
__ 2 (g) 25 (h) √
__ 2
__ 5 6. (a) 6 – 9i (b) – 6 + 2i (c) 0 – 7i
___ 80 = 4 √
_ 2 ,
___ 41 (d) √
__ 5 ; Slope of
| OQ | = 2 √
___ 29 (e) √
___ 13 (i) 2 ( j) √
___ 3 (k) 1 (l) √
__ 5
__ 5 ;
___ 40 = 2 √
___ 4
__ 5
___ 10
4. (b) P(– 4, –2); 2 ,
__ 3 − 3i (i) (x – y) – 6i ( j) (x – y) + (x + y)i
__ 2
__ 2 i (g) − 2
__ 2
2 , 3 __
_ 3 + i (h) 5 − 11
2 ) (h)
( j) 1 ___
+ 1 ___
( √
__ 2 , − 7
__ 2
__ 3 i (i) 4 + 3
i (k) ( y – 1) + 2i (l) –3 + (a – 2)i
___ √
__ 2
___ 18 = 3 √
3 (c) 5 5. P (3, 3) (a) 3 + 3i
__ 2
)
(i) (6, –5) ( j) (–8, 11) (k) (x + 1, –2) (l) (–2, y + 3) 2. (a) 1 + 0i (b) –3 + 0i (c) 0 + 5i (d) –5 + 6i (e) –7 – 2i (f ) 1
i
2. |z| = 5, |w| = 13 (a) 8 √ (c) 10; 10 (d) 2 √
(c) x2 + y2 = 25, Centre (0, 0) (d) (x + 7)2 + ( y + 5)2 = 6, Centre (–7, –5) 5. (a) ± 24 (b) ± 3 (c) ± 3 (d) ± 5
√ √
_ 3
(e) a – ci (f ) 3 + 2i (g) –5 + 2i (h) 4 – 6i (i) 3 + 0i ( j) –2 + 0i (k) 0 – 5i (l) 0 + 7i (m) –2 – 3i (n) –1 – 5i (o) √
Exercise 7 1. (a) 2 – 5i (b) 7 – 11i (c) 3 – 7i (d) √
__ 3 + 6i (p) 8 + 3i (q) (x – 7) – 2i (r) (x – 2) – (y + 3)i
(s) (x + 3) + yi (t) (4x + y – 2) + 3i 2. (a) 6i (b) 6i (c) 6 + 22i (d) 6 + 22i (e) 4i (f ) 4i (g) 6 + 14i (h) – 3 – i (i) –3 – i ( j) 6 + 16i (k) 6 + 16i (l) 3 – 11i (m) – 3 + 5i 3. (a) 15 – 21i (b) –8 + 32i (c) 7 + 11i (d) 12 – 22i (e) 4 – 3i (f ) 8 – 22i (g) 3 – 15i (h) 17 + 29i (i) 26 + 0i ( j) 13 + 29i 5. (a) –15 – 8i (b) √
__ 3 − i
Exercise 8
1. (a) – 4 + 6i (b) 5 + 5i (c) 6 + 17i (d) – 13 + 13i (e) –5 + 0i (f ) 19 – 9i (g) 2 + 8i 2. (a) 26 – 2i (b) 0 – 34i (c) –14 – 22i (d) 26 + 2i (e) 26 + 2i
(f ) 0 + 34i (g) –22 –14i (h) √ ( j) √
(b) –3 + 5i (c) – 4 + 2i (d) –3 + 2i (e) 9 + 11i (f ) 4 – 3i (g) –5 – i (h) 11 – 2i 4. (a) 2 (b) 13 (c) 29 (d) 45 (e) 5 5. (a) –11 + 60i (b) –2 – 2i (c) – 4 + 0i (d) 0 + 8i 6. (a) – 4 + 3i (c) 10; 5 (d) Halved modulus, no rotation 7. (a) – 6 – 8i (c) 10; 10 (d) Anticlockwise rotation by 90°, modulus unchanged 8. (b) 4 – 2i (d) √
____ 680 = 2 √
____ 170 (k) √
____ 680 = 2 √
modulus unchanged 11. (b) 9 + 12i 12. (b) 4 – 6i
___ 20 ; √
Exercise 9 1. (a) 0 – 3i (b) 0 + 7i (c) 0 + 5i (d) 0 − 2 _
(f ) 14 __
( j) 7 __
(b) 1 _
29 − 6 __
13 − 4 __
5 − 2 _
(f ) 11 __
(k) √
5 + 2 _
29 i (g) − 10 __
13 i (k) 11 __
5 i (c) 4 __
__ 5 (l) √
5 i (g) 11 __
Exercise 10
1. x = 2, y = 3 2. x = –3, y = 3 3. x = –4, y = 3 4. x = 1, y = –6 5. x = 2, y =1 6. x = 4, y = –2
13 − 15 __
5 − 2 _
25 − 3 __
__ 5 (m) √
13 i (h) 3 _
5 i (l) − 3 __
25 i (d) 7 __
25 + 2 __
__ 5 (n) √
5 + 1 _
25 + 24 __
5 i (i) 3 _
25 + 29 __
__ 5 (o) 1
25 i (e) 3 _
25 i (h) 1 – 2i (i) 1 _
_ 5
3 i (e) 6 _
5 − 1 _
5 − 4 _
5 i
25 i 2. (a) 1 + 2i 5 i
5 + 2 _
5 i ( j) 11 __
5 − 2 _
5 i
5 + 3 _
5 i
___ 20 (e) Anticlockwise rotation by 270°,
___ 34 (i) √
____ 170 3. (a) –3 + 12i
___ 20 = 2 √
__ 5
__ 3 − i
___ 13 ; 8 (e ) √
__ 2 ; 18 (b) 2 √
_______________ (7 + y ) 2 + (3 − x ) 2 (b)
____ 146 ; 14 (f ) √
___ 65 ; − 8
____ 401 ; −3
(g) 1; 1 (h) 1; 1 3. (b) |z| = 5, |w| = 5 (c) 5 √ 4. (a)
_______________ (x + 3 ) 2 + (y − 2 ) 2
__ 2 ; 25
__ 2
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