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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
1
Answers
Skills check
a y= 2.5x2+x 1
whenx= 3,y= 18.5
b h= 3 2t 1 when t= 0, h= 2
c d= 2 t3 5t1+ 2 when t=1
2, d= -31
4
a x2+x 3 = 0 x= 1.30, 2.30
b 2t2 t= 2
2t 2 t 2 = 0 t= 0.781, 1.28
c x 2y= 3
3x 5y= 2 x= 19,y= 11
a A(7, 2) B(1, 4) m = = =+
4 2
1 7
6
8
3
4
b A(3, 2) B(1, 8) m = = =++
8 2
1 3
10
4
5
2
Exercise 4A a B
Mrs. Urquiza
Mr. Genzer
A
MickLucyLidia
Diana
This is a function since each student is in only
one mathematics class.
b B
Mrs. Urquiza
Mr. Genzer
A
MickLucyLidiaDiana
This is not a function since each teachers
teaches two of the student.
a A3
750
B12
1649100
This is a function since each element of A is
related to one and only one element of B.
b B121649
100
A3750
This is not a function since one element of
B(16) is not related to any element of A.
c C49
100
A3750
This is a function since each element of C is
related to one and only one element of A.
Mathematical models4 a i A
1234
B246
This is not a function since one element of
A(4) is not related to any element of B.
ii A1234
C1246
This is not a function since one element of
A(4) is not related to any element of C.
iii C1246
A1234
This is not a function since one element of
C(1) is not related to any element of A.
iv B246
C1246
This is a function since each element of Bis related to one and only one element of C.
v C1246
A1234
This is a not a function since one element
of C(6) is not related to any element of A.
a y= 2x b y=x
2
c y=x
3 dy
x=
3
2 a Function
b Function
c not a function since negative elements in the
first set are not related to any element in the
second set
d Function
Exercise 4B
a i
ii domain is the set of all real numbers
iii yes, sincey= 0 is the image of x= 0
x1
2
0 1 3.5 6
y = 2x 1 0 2 7 12
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
2
b i
ii domain is the set of all real numbers
iii no, since there is no solution to the
equation
0 =x 2+ 1
c i
ii domain is the set of all real numbers except
x= 1
iii no, since there is no solution to the
equation
0 =
1
1x-
d ix 3 0
1
4 1 9 100
y x x 0 1
2 1 3 10
ii domain is the set of all non-negative real
numbers
iii yes,y= 0 is the image of x= 0
a False, there is no solution to the equation
0 =2
x
b true,y=x 2 0 for all values ofx
c true,y=x 2 + 3 3 for all values ofx
d true,y= 3 whenx= 2
e true,y= -33
1 = 2
f false, the image ofx= 1 isy= 4
Exercise 4C
a y y= 2x 4
x2 04 2 4 6
2
46
8
6
8
4
26
b i (2, 0)
ii (0, 4)
c A (250, 490) 490 2 250 4 no
d B (3,y) y= 2 3 4 = 10 a i {x: 4 x 6} ii {y: 4 y 1}
iii (4, 0) iv (0, 2)
x 3 0 2 1
4 2 x
y = x 2+1 10 1 5 17
16 5 5
x 2 1 0 1
2 3 5
yx
1
11 x 1
2
3
1
4
1
6
b i {x:x} ii {y:y 8}
iii (4, 0), (0, 0) iv (0, 0)
c i {x: 1 x 1} ii {y: 0 y 1}
iii (1, 0), (1, 0) iv (0, 1)
d i {x:x 1} ii {y:y 4}
iii no points iv (0, 8)
a i F ii F iii T b i F ii T iii F
c i F ii T iii F
d i F ii F iii T
a yy= 2x+ 3
x2 04 2 4 6
2
4
6
8
6
8
4
26
b y
y= x+ 2
x2 04 2 4 6
2
4
6
8
6
8
4
26
c yy= 3x 4
x2 04 2 4 6
2
4
6
8
6
8
4
26
Exercise 4D
f (x) =x (x 1) (x+3)
a f (2) = 2(1) (5) = 10
b f1
2
=
=
12
1
2
7
2
7
8
c f (3) = 3(4)(0) = 0
d f (1) = 1(2)(2) = 4
(1, 4) lies on the graph of f
d(t) = 5tt 2
a t
b d (2.5) = 5(2.5) 2.52= 6.25
c d (1) = 51 = 4
d d (1) = 4 d(4) = 2016 = 4 d (1) = d(4)
C (n) = 100 10n
a C (2) = 100 20 = 80
b b= C (3) = 100 30 = 70
c C (a) = 0 100 10a= 0 a= 10
a i v (1) = 3 ii v (3) = 3
b 3m+ 6 = 9 m= 1 c t= 2
d v (t) < 0 for t> 2
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
3
f (x) = 0.5 (3 x)
a yy= 0.5(3 x)
x2 04 2 4 6
2
4
6
8
6
8
4
26
b (3, 0)
c (0, 1.5)
d 0.5 (3 x) = 2 3 x= 4 x= 1
h (x) = 3 22x
a i h(0) = 3 ii h(1) = 3 21= 6
b 3 2x= 24 2x= 8 x= 3
Exercise 4E
a i l= 30 2x ii w= 15 2x
b V= (30 2x)(15 2x)x
i V (3) is the volume of the box when the
squares cut from each corner have side
length 3 cm.
ii V (3) = (24) (9) (3) = 643 cm 3
iii V (3.4) = (23.2) (8.2) 3.4 = 646.816 cm 2
iv No,x< 7.5 since the width of the card is
only 15 cm a width = 12 x
b A=x(12 x)
c i A (2) is the area of the rectangle when the
length is 2 cm
ii A (2) = 2(10) = 20 cm2
d No, ifx= 12 the width would be 0.
a C= 300 + 150n
b C (30) = 300 + 150(30) = 4800USD c i 300 + 150n< 2300
ii 300 + 150(14) = 2400, no
iii 150n< 2000, n< 13.3 13 days.
C (x) = 0.4x 2+ 1500 I(x) = 0.6x 2+ 160x
a P(x) = I(x) C(x) = 0.6x 2+ 160x
(0.4x 2+ 1500) = x 2+ 160x 1500
b P(6) = 62+ 160(6) 1500 = 576 AUD, a
loss of 576 AUD
c i P(40) = 402+ 160(40) 1500 = 3300 AUD
ii I(40) = 0.6(40)2+ 160(40)
= 5540
Assuming all books have same price, one
book costs
I(40) 5540
40 40 = 136 AUD
d 10 or 150
Exercise 4F
a 50 kg = 110 pounds b y
y= 2.2x
Kilograms (kg)
Pounds(lb)
x0 20 40 60 80 100
50
100
150
200
250
c gradient = 2.2 p (x) = 2.2x
d p (75) = 165 p (125) = 275 e y= 2.2x x= y
2 2. k(x) =
x
2 2.
f k(75) = 34.1 k(100) = 45.5
1 = S$2.05
a 50 = S$102.5
b yy= 2.05x
GBP ()
S
GD
($)
x0 20 40 60 80 100
50
100
150
200
250
c gradient = 2.05 s(x) = 2.05x
d s (80) = 164 s (140) = 287
e p (x) =x
2 05. p (180) = 87.8
1 = $1.55
a 60 GBP = 93 USD
b y y= 1.55x
GBP ()
USD($)
x0 20 40 60 80
20
40
60
80
100
120
140
c gradient = 1.55 u (x) = 1.55x
d u(300) = 465 u (184) = 285.2 e p(x) =
x
1 55.
f p (250) = 161 p (7750) = 5000
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
4
Exercise 4G
a
0 21 3 5 7 94 6 8
10050
150200250300
400350
Temperature(C)
Time (minutes)
b 10C c T (x) = 40x+ 10
a
0 20 40 60 80 100
20
40
60
80
Lengthofspring(mm)
Weight (g)
b 18 mm c 20g
d 0.5 mm e L (x) = 0.5x+ 18
b T (x) =2
3x+ 10
c 66.7C
b 20 cm
c 20 cm
d 350 g
e L (x) = 0.08x+ 20
Exercise 4H
a Flour = 80s+ 60f
b Fat = 50s+ 90f
c 80s+ 60f= 820
50s+ 90f= 880
s= 5 f= 7 5 sponge cakes, 7 fruit cakes
8t+ 3c= 51
100t+ 30c= 570
t= 3 c= 9 3 tables, 9 chairs
3v+ 5c= 59
7v+ 3c= 70
v= 6.65 c= 7.81 7 vans, 8 cars
80p+ 50t= 620
10p+ 25t= 190
p= 4 t= 6 4 passenger planes, 6 transport planes
70x+ 40y= 1440
x= 2y
140y+ 40y= 1440
180y= 1440
y= 8 x= 16
16 volume 1, 8 volume 2
Exercise 4I
0 1 2 3123
24
6
8
10
12
14
16
18
(0, 1)x
y= 2x2+1
y
0 2 424
2
4
2
4(0, 3)
y= x2+ 3
x
y
(3, 0)(3, 0)
0 2 424
20
10
(0, 2)
y= 3x2 2
x
y
( ) 23 , 0( )23
, 0
0 21 32 13
10
8
6
4
2
8
6
4
2
(0, 7)
y= 2x2+ 7
x
y
( )72 , 0( )
72 , 0
Exercise 4J
(3, 2) x= 2
(5, 4) x= 5
(4, 1) x= 4
(5, 7) x= 5
(3, 4) x= 3
Exercise 4K
y=x(x 4)
a x= 2 b (0, 0) (4, 0) c (2, 4)
y=x(x+ 6)
a x= 3 b (0, 0) (6, 0) c (3, 9)
y= 8xx 2=x (8 x)
a x= 4 b (0, 0) (8, 0) c (4, 16)
y= 3xx 2=x (3 x)
a x= 32 b (0, 0) (3, 0) c 3 9,
2 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
5
y=x22x=x (x 2)
a x= 1 b (0, 0) (2, 0) c (1, 1)
y=x 2 x=x (x1)
a x= 12 b (0, 0) (1, 0) c 1 1,
2 4
y=x 2 + 4x=x(x+ 4)
a x= 2 b (0, 0) (4, 0) c (2, 4)
y=x 2 +x=x(x+ 1)
a x= 1
2 b (0, 0) (1, 0) c 1 1,
2 4
y= (x+ 1) (x 3)
a x= 1 b (1, 0) (3, 0) c (1, 4)
y= (x 5) (x+ 3)
a x= 1 b (5, 0) (3, 0) c (1, 16)
y= (x 2) (x 6)
a x= 4 b (2, 0) (6, 0) c (4, 4)
y= (x+ 2) (x 4) a x= 1 b (2, 0) (4, 0) c (1, 9)
Exercise 4L
y=x 2 2x+ 3
a x= 1 b no points c (1, 2)
y=x 2+ 4x 5 = (x+ 5) (x 1)
a x= 2 b (5, 0), (1, 0) c (2, 9)
y=x 2+ 6x+ 4
a x= 3 b (0.764, 0), (5.24, 0)
c (3, 5)
y= 3x 2 6x+ 2
a x= 1 b (0.423, 0), (1.58, 0)
c (1, 1)
y= 2x 2 8x 1
a x= 2 b ( 0.121, 0), (4.12, 0) c (2, 9)
y= 2x 2+ 6x 7
a x= 3
2
b (0.898, 0), (3.90, 0) c3 23
,2 2
y= 0.5x 2x+ 2
a x= 1 b no points c 32
1,
y= 0.5x 2+ 3x 4
a x= 3 b (1.12, 0), (7.12, 0) c17
23,
Exercise 4M
f (x) =x 2+ 2x 3 = (x+ 3) (x 1)
a i (0, 3)
ii x= 1
iii (1, 4)
iv (3, 0), (1, 0)
v y 4
b
0 21 32 135 4 2
4
6
10
8
4
2
6
(1, 4)
y=x2+ 2x 3
x
y
f (x) =x 2+ 8x+ 7 = (x+ 1) (x+ 7)
a i (0, 7) ii x= 4
iii (4, 9) iv (7, 0), (1, 0)
v y 9
b
0 12139 8 7 6 5 42
4
10
8
6
10
8
4
2
6
(4, 9)
y=x2+ 8x + 7
x
y
f (x) =x 2 6x 7 = (x 7) (x+ 1)
a i (0, 7) ii x= 3
iii (3, 16) iv (1, 0), (7, 0)
v y 16
b
04 4 8 124
8
16
12
20
16
8
4
12
(3, 16)
y=x2 6x 7
x
y
f (x) =x 2 3x 4 = (x 4) (x+ 1)
a i (0, 4) ii x=3
2
iii3
2
25
4,
iv (1, 0), (4, 0)
v y 254
b
02 2 4 644
8
8
12
4
y=x2 3x 4
( )32254
,
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
6
f (x) =x 2 3x 10 = (x 5) (x+ 2)
a i (0, 10) ii x=3
2
iii 3 49,2 4
iv (2, 0), (5, 0)
v y 494
b
02 2 4 644
8
12
4
16
12
8
y=x2 3x 10
( )32494
,
x
y
f (x) = 2x 2+x 3 = (2x+ 3) (x 1)
a i (0, 3) ii x= 14
iii 1 25,4 8
iv3
40,
, (1, 0)
v y 258
b
02 2 4 644
8
12
4
16
y= 2x2+x 3
( )14 258, x
y
f (x) = 2x 2+ 5x 3 = (2x 1) (x+ 3)
a i (0, 3) ii x= 54
iii 5 49,4 8
iv 12
0,
, (3, 0)
v y 498
b
02 2 448
8
4
8
12
4
16y= 2x2+ 5x 3
( )54498
,
x
y
f (x) = 3x 2x 4 = (3x 4) (x+ 1)
a i (0, 4) ii x= 16
iii1
6,
49
12
iv 3
40,
, (1, 0)
v y 4912
b
02 2 44
10
20
10
y= 3x2x 4
( )1
6
49
12,
x
y
Exercise 4N
02 2
(1,3)
(3,0)(1,0)
44
2
4
4
2
x
x= 1
y
02 2
(1,2)
44
4
8
12
4
x
y
x= 1
02 2 44
4
8
4
x
y
(0,1)
x= 0
02 2 4 x
y
2
4
2
(1.5,1)(3,0)(0,0)
x= 1.5
0
(2,2)
(4,2)(0,2)2 2 4 x
y
4
8
12
4x= 2
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
7
0 x
y
2 4 6 8
4
6
8
2
(5,0)(2,0)
(0,4)
x= 3.5
Exercise 4O
f (x) =x 2+ 3x 5 =g (x)x 2
a (3, 5), (1, 1)
b x 2+ 3x 5 =x 2
x 2+ 2x 3 = 0
(x+ 3) (x 1) = 0
x= 3 or 1 (Same as part a)
c h(x) = 2x 3 x 2+ 3x 5 = 2x 3
x 2+x 2 = 0
(x+ 2) (x 1) = 0
x= 2 or 1
(2, 7), (1, 1)
f (x) =x 2+ 3x 5 5 x 2
a x+y+ 5 = 0
y= 5 x
x 2+ 3x 5 = 5 x
x2
+ 4x= 0 x(x+ 4) = 0
x= 0 or 4 (0, 5) (4, 1)
a f (x) = 5 + 3xx 2 g(x) = 1
5 + 3xx 2= 1
x 2 3x 4 = 0
(x 4)(x+ 1) = 0
x= 4 or 1 (4, 1), (1, 1)
b f (x) = 5 + 3xx 2 h(x) = 2x+ 3
5 + 3xx 2= 2x+ 3
x 2x 2 = 0
(x 2)(x+ 1) = 0
x= 2 or 1 (2, 7), (1, 1)
b f: range = {y: 3.125 y 18}
g: range = {y: 2 y 4}
c x= 1 or 2
e f (x) = h(x)
2x 2x 3 = 2x+ 2
2x 2 3x 5 = 0
(2x 5)(x+ 1) = 0
x= 52or 1
f (2, 7), (2, 3)
a (2.12, 1.5), (2.12, 1.5)
f (x)
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
8
f (x) = ax 2+ bx+ 2
b
a2= 1 b= 2a
(1, 5) 5 = a b+ 2
a b= 3
a= 3
a= 3 b= 6 f (x) = 3x 2 6x+ 2
g(x) = ax 2+ bx 3
ba2
= 2 b= 4a
(2, 5) 5 = 4a 2b 3
2a b= 4
2a= 4
a= 2 b= 8 g(x) = 2x 2 8x 3
f (x) = ax2+ bx
b
a2=
12
b= a
1 1 1 1 1,2 2 2 4 2
a b
2 = a 2b
2 = a
a= 2 b= 2 f (x) = 2x 2+ 2x
g(x) = ax 2+ bx+ 3
b
a2= 0 b= 0
(1, 2) 2 = a+ 3
a= 1 g(x) = x 2+ 3
Exercise 4Q
a 2l+ 2w= 170
l+ w= 85
l= 85 w
A= lw= w(85 w)
A= w(85 w)
For maximum area, w= 42.5, l= 42.5
length = 42.5 m, width = 42.5 m
b 2l+ w+ (w 15) = 110 2l+ 2w= 125
l= 62.5 w
A= w(62.5 w)
For maximum area, w= 31.25, l= 31.25
length = 31.25 m, width = 31.25 m
p (u) = 0.032u 2+ 46u 3000
a 13531.25 riyals
b 3000 riyals
c 69 units
H (t) = 37t t2
a 270 m
b 342.25 m
c 37 s
Exercise 4R
For all questions:yintercept is (0, 1), horizontal
asymptote isy= 0
Exercise 4S
f (x) = 2x a (0, 1) b y= 0
0 x
y
y =2x
1 1 322
8
2
4
6
f (x) = 6x a (0, 1) b y= 0
0 x
y
y =6x
1 1 22
10
20
30
f (x) =1
3
x
a (0, 1) b y= 0
0 x
y
1 1 22
8
2
4
6
y = ( )x13
f (x) = 15
x
a (0, 1) b y= 0
0 x
y
1 1 22
20
5
10
15
y = ( )x15
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
9
f (x) = 3(2)x+ 4 a (0, 7) b y= 4
0 x
y
1 1 2 323
20
5
15
25
10
y =3(2)x+ 4
f (x) = 2(4)x 1 a (0, 3) b y= 1
0 x
y
2 211
10
30
20
y =2(4)x 1
f (x) = 1(2)x
+ 3 a (0, 2) b y= 3
0 x
y
2 2
2
4
2y =1(2)x+ 3
f (x) = 4(3)x 2 a (0, 2) b y= 2
0 x
y
23 21 31
30
10
20
y =4(3)x 2
f (x) = 0.5(2)x+ 3 a (0, 3.5) b y= 3
0 x
y
1 1 2 3 423
8
2
6
10
4y =0.5(2)x+ 3
f (x) = 2(0.5)x+ 1 a (0, 3) b y= 1
0 x
y
1 1 2 323
15
3
12
18
69
y = 2(0.5)x+ 1
f (x) = 0.4x+ 1 a (0, 2) b y= 1
y =0.4x+ 1
0 x
y
1 1 223
15
3
12
18
6
9
f (x) = 2(0.1)x 1 a (0, 1) b y= 1
0 x
y
1 1 2 323
40
10
30
60
50
20
y =2(0.1)x 1
Exercise 4T
f (x) = 4(2)x+ 2 a (0, 6) b y= 2
0 x
y
1 1 223
15
3
12
18
6
9y =4(2)x+ 2
f (x) = 4x+ 1 a (0, 0) b y= 1
0 x
y
2123 1
15
12
3
3
9
6y =4x+ 1
f (x) = 2(2)x+ 3 a (0, 1) b y= 3
0 x
y
2123 1
12
3
3
9
6
y =2(2)x
+ 3
f (x) = 3(2)x 2 a (0, 1) b y= 2
0 x
y
2 1 21 3 4
2
1
3
4
2
1
3
y =3(2)x 2
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
10
f (x) = 0.5(3)x+ 2 a (0, 2.5) b y= 2
0 x
y
23 1 21 3 44
8
10
12
14
4
2
6
y =0.5(3)x+ 2
f (x) = 0.5x+ 1 a (0, 2) b y= 1
0 x
y
2 2 44
4
6
8
2
y =0.5x+ 1
f (x) = 2(0.1)x 1 a (0, 1) b y= 1
0 x
y
2 2 44
8
12
16
4
4
y =2(0.1)x 1
f (x) = 0.4x+ 2 a (0, 3) b y= 2
0 x
y
2 2 44
8
12
16
4y =0.4x+ 2
f (x) = 3(0.2)x+ 4 a (0, 7) b y= 4
0 x
y
23 1 214
812
16
20
24
28
32
4
y = 3(0.2)x+ 4
f (x) = 5(3)x 2 a (0, 3) b y= 2
0 x
y
23 1 21 3 44
8
12
16
20
4
4
8
y = 5(3)x 2
Exercise 4U
0 x
y
2 2 44
2
4
6
8
g(x)= 2x+ 0.5
f(x)= 2x+ 0.5
a (0, 1.5)
b y= 0.5
v(t) = 26000x t
a 26000 euros
b 26000x= 22100 x= 0.85
c v (t) = 26000 (0.85)t
v (9) = 6022.04 v(10) = 5118.73 10 years
M(t) = 150(0.9)
t
a
0
M(t)
t4020 60 80 100
40
80
120
160
M(t)= 150(0.9)t
b M = 0
c M(20)= 18.2g
d M(6) = 79.7 M(7) = 71.7 7 years
A (t) = 50(1.06)t
a
A(t)= 50(1.06)t
0
A(t)
t44 8 16 2012
40
80
120
160
b days before 1st June.
c A (14)= 113 m2
d A (8) = 79.7 A (9) = 84.5 t= 8
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8/10/2019 Maths Studies topic 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
11
k+ c = 5 and 0 = 2k+ c, so c= 10 k= 5
T (t) = 18 + 60(2)t
a
T(t)= 18 + 60(2)t
0
T(t)
t4 8 10
20
40
60
80
b 78C
c T (5)= 19.875C
d T (1.4) = 40.7 T (1.5) = 39.2 1.5 minutes
e 18C As tincreases Tgets closer to 18C
(T= 18 is an asymptote). D (t) = 18000 (0.9)t
a 18000 USD
b D (5) = 10628.82 USD
c D (6) = 9565.94 D (7) = 8609.34 7 years
f (x) =2x
a (0, 6) (2, 0.8)
b=1a 0.8 =
4a a= 5 b= 0.2
y= 2x+ 3 A (0, a) B (1, b)
a a= 20+ 3 a= 4
b= 21+ 3 b= 5
b y= 3
a a= 1.667 b= 19
b
0 x
f(x)
2 2
4
8
12
16
f(x)= 2(3)x
+ 1
c range = {y:y> 1} (orf (x) > 1)
Exercise 4V
a f (x) = 0.0015x4+ 0.056x3 0.60x 2+ 1.65x+ 4
b 8.77 hours
c 1.80 hours, 17.4 hours
a 6
b (x 2)4+ 6 = 6
(x 2)4= 0
x= 2
c f (x) 6
Exercise 4W
b 28.9C
c 50 = 21 +79
x
29 =79
x x= 2.72 minutes
d x= 0
e y= 21
f 21C
f (x) = 100 100
x
b 90C
c 100 100
x = 30
100
x = 70 x= 1.43 minutesd 100C
b 8 = 25
x x 2=
5
8 x= 0.791
c x= 0,y= 0
d f(x) > 0
b 3.75
c 5 = 3 +6
x
6
x= 2 x= 3
d x= 0,y= 3
e range is all real numbers except 3
{y:x,y 3}
Exercise 4X
b minimum value = 17.5 (whenx= 1.71)
c 75.3 ms1
d 50 =20
x+2x2
2x 3 50x+ 20 = 0
x= 0.403s, 4.79s
a v=x (2x)y= 2x 2y
b y= 2 2300 50
2x x
A= 2x 2+xy+xy+2xy+ 2xy
A= 2x 2+ 6xy
A= 2x 2+ 6x1
= +
50 9002
22x x
x
d For minimum area,x= 6.0822,y=2
50
6.0822
= 4.0548length = 6.08 cm, breadth = 12.2 cm,
height = 4.05 cm
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8/10/2019 Maths Studies topic 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
12
a v x h= 13
2
b
hl
x
2
l 2= h 2+2 2
2
2 2
x xhl
c A=x 2+4
2
xl=x 2+ 2xl= x x h x2
22 2
2
+ +
d 1500 =2
3
x h h= 2
4500
x
l=2 2
4
4500
4
x
x A=x 2+ 2x
2 2
4
4500
4
x
x
f For minimum area,x= 14.7084, h=2
4500
14.7084
= 20.8009
side length = 14.7 m, height = 20.8 m
Let width =x, length = 2x, height = h
320 = 12x+ 4h
h= 80 3x
viewing area = 2xh+ 2xh+xh
= 5xh
A= 5x(80 3x)
maximum viewing area = 2666.67
= 2670 cm 2
Exercise 4Y
f(x) = 1 +2
x,x 0
a {x:x,x 0}
b x 10 5 4 2 1 0.5 0.2 0
f (x) 0.8 0.6 0.5 0 1 3 9
x 0.2 0.5 1 2 4 5 10
f (x) 11 5 3 2 1.5 1.4 1.2
c
0 x
y
4 26 42 6 8 10810
2
4
6
810
8
10
6
4
2
y = 1 +2x
d x= 0
e y= 1
f(x) = 8x 1+ 3,x 0
a {x:x,x 0}
x 10 8 5 4 2 1 0
f (x) 2.2 2 1.4 1 1 5 0
b x 1 2 4 5 8 10
f (x) 11 7 5 4.6 4 3.8
c
0 x
y
4 26 42 6 8 10810
2
4
6
8
10
8
10
6
4
2
y = + 38x
d x= 0
e y= 3
Exercise 4Z
Sketch graphs
Range :y 1.81
y
y
y 1.25
y< 0 or y 2.98
Exercise 4AA
a
0 x
y
4 26 42 6 88
2
2
4
6
8 y =x2
y = 4 1x)(
b (0.254, 0.0646), (1.86, 3.46), (2.11, 4.47)
a, c
0 x
y
4 26 42 6 88
2
2
4
6
8
6
4
g(x)= 3x
f(x)= 1 +4x
b y= 1,x= 0
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8/10/2019 Maths Studies topic 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
13
d 1 +4
x= 3x x= 1 or 1.33
e {y:y,y 1}
a (0.366, 0.669), (0.633, 2.01)
b y= 0
a
0 x
y
2 13 21 3 44
1
1
2
3
4
3
4
2
g(x)=x3
f(x)=3x
b3
xx 3= 0
3
x=x 3 2 solutions
c 1.32 or 1.32
0 x
y
2 1 21 3
2
2
4
6
8
6
4
y =x3 3x2+ 2x
y = 3x 4
(1.11, 7.34), (1.25, 0.238), (2.86, 4.58)
0 x
y
2 13 21 3 44
2
2
4
6
8
6
8
4
y =x3+x2 6x
y = 2x
(2.99, 0.126), (0.147, 0.903), (2.41, 5.31)
0 x
y
2 13 21 3 44
1
1
2
3
4
3
4
2
y =x+ 2
y =5x
a x= 1.45 or 3.45b y= 0
c x= 0
Exercise 4AB
a time in hours, water consumption in litres
b 0700 2000
c 0700 1200, 1400 1600
d 1200 1400, 1600 2000
e 1200 (local maximum at 1600)
f 0700, 2000 (local minimum at 1400)
a time in minutes, temperature in C
b 100C
c 35C
d1
2minute
e no
f approximately 22C
at 0 5 10 15 20
N 1 2 4 8 16
b 13s
c 2(60 5)= 212= 4096
a 45 m
b 1.5 s and 5.5 s
c 0 3.5 s
d 3.5 s 7 s
e 90 m, 3.5 s
f ball returns to ground level
a i 3.8 m ii 2.2 m iii 0200 and 0600b 2 < t< 6
a twice
b 0400 0900
c 1600
d 5C
e 1100 1600
f 1300 and 1930
g no, the temperature at the start of the
following day is 1C whereas it was 3C at the
start of this day.
a x 2y= 16 y x=16
2
b x 0.5 1 2 4 8 10
y = f (x) 64 16 4 1 0.25 0.16
c
0 x
y
1 2 3 4 5 6 7 8 9
40
10
30
60
50
20y =
16
x2
d tends to zero
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8/10/2019 Maths Studies topic 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
14
a 3000 cm3
b x 2y= 3000 yx
= 30002
c A=x 2+ 4xy=x 2+ 4x2
3000
x
212000
xA x
d
x(m) 5 10 15 20 25 30 35
A(x)(cm2)
(2sf )2400 1300 1000 1000 1100 1300 1600
e
0 x
A
5 10 15 20 25 30 35
2000
500
1500
1000
A=x2+12000
x
f x= 18.2
Review exercisePaper 1 style questions
a 0000 0600
b 1130 1700
c 13 C
c= nr+s
a 35000 = 6r+s
116000 = 24r+s
18r= 81000
r= 4500 SGDb 35000 = 6 4500 +s
s= 8000 SGD
a x 2+ 5x=x(x+ 5)
b
0
(0, 0)(5, 0)
(2.5, 6.25)
x= 2.5
y=x2+ 5x
x
y
2 13 216 5 4
2
8
10
6
4
2
h(t) = 30t 5t2 0 t 6
a h(4) = 40 m
b 45 m
c from t= 1 to t= 5, 4 s
a f (x) =2x
m
(3, 1.6) 1.6 = 23
m m=
8
1.6= 5
b f (x) =2
5
x
(0, n) n= 15
f(2) =2
5
2
= 45
a x 2 2x 15 = (x 5)(x+ 3)
b i At A,x= 3 A = (3, 0)
ii At B,x= 1 B = (1, 16)
a ii
b i
c iii
d iv a i A (1.68, 1.19)
ii B (2.41, 1.81)
b f (x)
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8/10/2019 Maths Studies topic 4
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Worked solutions: Chapter 4 Oxford University Press 2012: this may be reproduced for class use solely for the purchasers institute
WORKED SOLUTIONS
15
b y= 0,x= 0
c
0 x
y
2 13 21 3 44
2
2
4
6
8
6
8
4
g(x)= 2x
d x= 1.41
e {y:y,y 0}
f (x) = 2(1.5)x+ 3
a a= 4.33 b= 7.5
b
0 x
y
2 13 21 3
1
1
2
3
4
5
7
6
2
f(x)= 2(1.5)x+ 3
c f (x) > 3
d x= 3.09
e y= 3
f (t) = 21 + 77(0.8)t
a f (0) = 21 + 77 = 98C
b y= 21
c 21C
d f (8) = 33.9C
a
0 x
y
4 26 42 6 8 10810
2
2
4
6
8
6
8
4
f(x)=x2x 6
g(x)= 1 2x
b (0.5, 6.25)
c 2
d (0, 1)
e (2.19, 3.39),
(3.19, 7.39)
f x= 2.19, 3.19
a
0 x
y
2 13 21 3 44
2
2
4
6
8
10
6
4
f(x)=x2
3x
b x= 0
c
0 x
y
2 13 21 3 44
2
2
4
6
8
10
6
4 y = 3(2)x+ 9
d y= 9
e (2.73, 8.55), (0.454, 6.81), (1.53, 0.362)
P =2
10
x+ 10x 60
a x 0 10 20 30 40 50 60 70 80 90
P 60 30 100 150 180 190 180 150 100 30
b
0x
y
20 40 60 80
50
50
100
150
200
P = + 10x 60x2
10
c i 190 euros ii 50 iii 33 or 67
iv 60 euros
a
0 x
y
2 134 21
(2.65, 0) (2.65, 0)
(0, 7)
3 4
2
2
4
68
6
8
4
y =x2 7
(0, 7), (2.65, 0), (2.65, 0)
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8/10/2019 Maths Studies topic 4
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WORKED SOLUTIONS
b
0 x
y
2 134 21
(2.65, 0) (2.65, 0)
3 4
2
2
4
6
8
6
8
4
(0, 7)
y = 7 x2
c x= 2.65
d c= 1, 2, 3, 4, 5, 6
a2 2
2 2
x x +2x
x2 2x= 0
x(x 2) = 0
x= 0 or 2 (0, 0), (2, 2)
b x=
=b
a2
2
1= 2 x= 2
c k= 2
d
0(0, 0)(4, 0)
x
y
2 13 21 3 4 5
1
1
2
3
4
3
4
2
f(x)=x2
2
g(x)= + 2xx2
2
e f(x)