Power of Maths: Paper 1 – Answers
(i) x: Does not cross; y: (0, 7) ( j) x: − 7 + √
(
_________ 4
____ 137
(d) (f )
( 3 __
2 , 1 __
( 4 __
3 , 4 __
4 3
( − 7 __
x = 0 (b) (0, 1) min; x = 0 (c) 2 (e) 2 , 0
) max; x = 3 _
) max; x = 4 _
4 , − 137 ____
8
(h) (2, –9) min; x = 2 (i) (j)
( 1 __
) min; x = − 7 _
( 5 __
2 , 3 __
, 0 ; y: (0, –11) 3. (a) (0, –1) min; 4
)
( 5 __
2 , − 1 __
) min; x = 1 _
2
3 (g) (1, 4) max; x = 1 4
) min; x = 5 _
2 4 4. (a) 20 > 0, 2 points
(b) 1 > 0, 2 points (c) 676 > 0, 2 points (d) 0, 1 point (e) 16 > 0, 2 points (f ) 65 > 0, 2 points (g) 1 69 > 0, 2 points (h) 0, 1 point
Exercise 7
1. (a) –2·4, 0·9 (b) –1·6, 3·6 (c) 0·3, –3·3 (d) – 0·6, 2·3 (e) –1·5 (f ) –1, 1·5 2. (a) (i) (0, –3) (ii) (–3, 0),
( 1 __
2 , 0 ) (iii)
( − 5 __
4 , − 49 ___
8 ) 3. (b) (i) –1·8 °C
(ii) 9 °C (iii) 1·8 hours (iv) 1·6 hours, 4·4 hours (c) a = 1, b = – 6, c = 7 4. (b) (i) 375 cm (ii) 1 cm, 39 cm (iii) 400 cm 5. (b) (i) 24 m (ii) 0·9 s, 3·9 s (iii) 30 m (c) 29·8 m (d) 0·7 % 6. (b) (i) 600 m (ii) 150 m 7. (a) 2, 10 (b) 2 < x < 10, x ∈ R
8. (a) x < 25, x ∈ R (c) (i) €780 (ii) €12·50 (iii) €6, €19 (iv) 25, 100 (d) €781·25 9. (a) B(3, 0) (b) C(0, 6) (c) A(–1, 0) (d) p = –3, q = 6, r = 9 (e)
D(0, 9) (f ) P Chapter 18
Exercise 8 1. (a) x = –1·9, – 0·3, 2·1; y: (0, –1) (b) x = – 4·3, –1·7, 1·1; y: (0, 8) (c) x = –1, 2; y: (0, –2) (d) x = –3, 1; y: (0, –3) (e) x = 2·7; y: (0, –2) 2. (a) 1 root (b) 3 roots, 2 equal (c) 3 roots (d) 1 root 3. (a) x: (–2, 0), (1, 0), (2, 0) y: (0, 4) (b) x: (–2, 0), (0, 0), ( 1
_ 2 , 0); y: (0, 0) (c) x: (0, 0), (1, 0); y: (0, 0)
( 350
(d) x: (0, 0); y: (0, 0) (e) x: 3 + √
_______ 2
___ 13
y: (0, 24) (g) x: ( –1, 0 ) , −
( 3 __
(h) x: (0, 0), 3 , 0 ) ; y: (0, 0) (i) x: (−1, 0), 5 , 0 ) , 4 , 0 ) ; y: (0, 6) ( j) x: (−5, 0); y: (0, –10) 4. (a) 800
( 2 __
( √
__ 3 , 0
) ( √
__ 3 , 0
)
( 2 __
, 0 ; y: (0, 4) (f ) x: (2, 0), (3, 0), (4, 0); ,
)
__ 2
__ 2
(
_______ 2
3 − √
___ 13
, 0 , (2, 0), ) ; y: (0, –2)
( − 1 __
3 , 20 ___
3 )
(3, 0); y: (0, 3) (h) x: 2 , 0 ) , 2 , 0 ) ; y: (0, 7) −7 − √
( 1 __
( 7 __
(
_________ 4
____ 137
, 0 , )
) min; x = 5 _
2
(c) (i) 700 tonnes (ii) 2004 (iii) 2007 5. (a) –2·2, –1, 0·7 (b) 3·6 (c) –1·2, 0, 3·2 (d) – 0·9, 0·5, 4·4 (e) –2·6, – 0·4, 2 (f ) 0·8 (g) –2·6, 0·3, 2·3 6. (c) 1·5, 3, 4·5 7. (b) – 0·6, 1, 1·6 8. (b) –1·3, 0·9 (c) 3
Exercise 9
2. (a) y = (x + 2)2 (b) y = f (x – 4) (c) y = f (x – 6) (d) y = f (x) + 5
Chapter 19 Exercise 10
1. (b) 1·6 (c) 1·4 2. k = 0·5, a = 3 3. (a) 20 000 (b) (i) 80 900 (ii) 327 000 (d) 13 days 4. (a) 60 (b) 68 (d) 11·6 hrs 5. (a) 1280 (b) €40 (c) 8 yrs
Exercise 11
1. (b) (1·4, 2·6) 2. (b) – 0·9, 1·2 3. (b) 0·3 days, 5·3 days 4. (b) (0, 1) (c) (i) 1·4 (ii) –1·7 5. (a) A(0, 4), a = 8 (b) c = 2 (c) B(2, 8)
Revision Questions
1. (c) 0·4 s, 3·1 s (d) 17 m (e) 0·2 s, 4 s, 5·8 s 2. (c) (i) 96 cm3/h (ii) 72 cm3/h (iii) 11 hrs, 22 hrs (iv) 9 cm3/h 3. (a) A(–1, 0), B(6, 0) (b) D(1, 10)
(c) E
( 5 __
2 , 49 ___
4 ) (d) F 2 , 7 ) (e) | EF | = 5·25
( 5 __
4. (a) 5 A (d) (i) 1·4 A (ii) 4·5 s (e) 4 s 5. (a) 0∙06x – 400 (b) €2300 (c) €25 000 6. (a) 0·2 (b) 1200 (e) (i) 2300 (ii) 6·5 days (f ) P = 1000(1·2)t 7. (b) (i) 219 (ii) 263 (c) N = 2R – 1 (d) 9·2 × 1015 m (e) 1·5 × 1011 m (f ) 61 333
Section 8 Chapter 20
Exercise 1
1. (a) 5 m/s (b) 5 m/s (c) Straight line – Constant slope 2. (a) 27 (b) 30 3. (a) (i) –2, (ii) –1 (b) –2⋅5; V is decreasing (c) –2⋅5 4. (a) 2
(c) 1 (d) 1 5. (a) 200 mA/V (b) 333· ˙
3 mA/V; dI ___
_ 3 (b) 1
dV
Exercise 2 1. (a) 2x (b) 6x 2 (c) 4 (d) 12x (e) –3x 2 (f ) –15x 2
(g) 2x ___
(d) 2ax + b (e) 12x 2 – 10x + 6 (f ) 3ax 2 + 2bx + c
5 (h) x2 (i) 4x ___
5 ( j) − 9x 2 ___
4 2. (a) 2 (b) a (c) 6x – 5
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