Power of Maths: Paper 1 – Answers
(ii) 6t – 12 (c) Maximum distance = 4 m, minimum distance = 0 m 12. (a) (i) 240 m (ii) 36 m/s (b) 294 m
Revision Questions
Concepts and skills 1. (a) 2x2 + 6x + 5 (b) f ′(x) = 2; g ′(x) = 2x + 3; h ′(x) = 4x + 6 (c) x > − 3
2. (a) 6x (b) 1 (c) 3x3 – 6x2 + x – 2 (d) 9x2 – 12x + 1 (e) 3x2 + 1 (f ) 6x2 – 12x 3. (a) 3x2 – 2x (b) x – y – 3 = 0 (c) 5x – y + 1 = 0 (d) (–1, – 4)
_ 2 , x ∈ R (d) 2x + y + 3 = 0
(e) (0, –2);
( 2 __
3 , − 58 ___
27 ) 4. (b) x – y – 8 = 0 (c) x + y + 4 = 0 (e) √
(c) 2πh – πh2 (d) (i) 3 _
(b) 200x2 (c) 20 000x – 150x2 (d) 20 000 – 300x (e) 66·67 (f ) €6667 6. (a) 0 m3 (b) 2π
__ 8 = 2 √
4 π (ii) π (e) 2π – 2πh (f) 3 _
___ 3 m3
4 m
7. (a) 6x2 – 12x + 6 = 6(x2 – 2x + 1) = 6(x – 1)2 for all x ∈ R (b) (i) 24 (ii) –18 (c) –34 8. (a) –2x (b) 2
(c) (i) 12x2 + 16x – 1 (ii) 88
Contexts and applications 1. (a) t (Hours) 02468 10
N
_ 3
__ 2 5. (a) 50x2 + 20 000x
(b) 60 – 10t (c) 14:00 or 2 p.m. (d) 180 (e) 1400 2. (a) 125x − 5x 2 (b)
___ 3
x 015 30 45 60 75
R (€) 0 1500 2250 2250 1500 0 C (€) 0 450 1050 1800 2700 3750
(c) 50 covers/week; €2100 (e) 100 – 4x; €1250 3. (a) (i) 250 m (ii) 385 m (c) 378 m (d) 410 m (e) 250 m (f) 410 m 4. (a) 3t 2 – 40t + 100 (b) 6t – 40 (c) 10 s (d) 0 m/s
(e) t (s) 46 8
5. (a) 1 _
x = 5 (b)
(e) 18 m 6. (a) 9 (b) − 4 __
_______ 3
7 − √
___ 13
_______ 3
___ 13
10 12
v (m/s) 144 96 32 0 48 (f) 400 m (g) (i) –25 m/s2 (ii) 0 m/s2 7 ± √
(c) d 2y , L Min at x =
___ dx 2 = 6x − 14 7 + √
_______ 3
___ 13
_______ 5
(d) 121 m (d ) t ( years) 0369 12
81 ; L Max (c) 23 years 15
18 21 E ( years) 21·0 22·1 22·8 23·0 19·7 16·4 13·1 9·9 12 0 100 160 180 160 100 0
(e) 30 years 7. (a) 6 – x (b) 6x2 – x3 (c) 12x – 3x 2; (0, 0); (4, 32) (d) 12 – 6x; Vmax at x = 4 (e) 32 cm3 8. (c) 10 m (d) 95 m (e) – 0·40 9. (a)
( 1, − 9 __
4
) L Min (c) (0, –1) (d) − 4 _
( − 4 __
3 , 25 ___
3 < x < 1, x ∈ R 27 ) L Max;
; L Max at
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