Additional Maths 360 Solutions

556 Pages·2014·11.798 MB·English

by  Samuel Lim, Daniel Lim

Additional Maths 360 solutions

(Unofficial) [29 Sept 2014]

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Authors:

Daniel and Samuel from Sleight of Math

Disclaimer:

Sleight of Math has no affiliations with the publisher of Additional Maths 360 i.e. Marshall Cavendish

Table of Contents
Ex 1.1 ......................................................................................................... 3 Ex 6.1 ................................................................................................... 143
Ex 1.2 ......................................................................................................... 9 Ex 6.2 ................................................................................................... 149
Ex 1.3 ....................................................................................................... 20 Ex 6.3 ................................................................................................... 153
Ex 1.4 ....................................................................................................... 29 Ex 6.4 ................................................................................................... 162
Rev Ex 1 .................................................................................................. 35 Rev Ex 6 .............................................................................................. 167

Ex 2.1 ....................................................................................................... 42 Ex 7.1 ................................................................................................... 174
Ex 2.2 ....................................................................................................... 51 Ex 7.2 ................................................................................................... 180
Ex 2.3 ....................................................................................................... 55 Ex 7.3 ................................................................................................... 184
Ex 2.4 ....................................................................................................... 60 Ex 7.4 ................................................................................................... 189
Rev Ex 2 .................................................................................................. 63 Ex 7.5 ................................................................................................... 193
Rev Ex 7 .............................................................................................. 196
Ex 3.1 ....................................................................................................... 67
Ex 3.2 ....................................................................................................... 72 Ex 8.1 ................................................................................................... 202
Ex 3.3 ....................................................................................................... 78 Ex 8.2 ................................................................................................... 209
Ex 3.4 ....................................................................................................... 81 Rev Ex 8 .............................................................................................. 224
Ex 3.5 ....................................................................................................... 86
Ex 3.6 ....................................................................................................... 91 Ex 9.1 ................................................................................................... 233
Rev Ex 3 .................................................................................................. 98 Ex 9.2 ................................................................................................... 242
Rev Ex 9 .............................................................................................. 252
Ex 4.1 .................................................................................................... 103
Ex 4.2 .................................................................................................... 108 Ex 10.1 ................................................................................................. 259
Rev Ex 4 ............................................................................................... 114 Ex 10.2 ................................................................................................. 266
Ex 10.3 ................................................................................................. 272
Ex 5.1 .................................................................................................... 118 Rev Ex 10............................................................................................ 280
Ex 5.2 .................................................................................................... 127
Rev Ex 5 ............................................................................................... 138 Ex 11.1 ................................................................................................. 288
Ex 11.2 ................................................................................................. 291
Ex 11.3 ................................................................................................. 298
Rev Ex 11............................................................................................ 319
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Additional Maths 360 solutions

(Unofficial) [29 Sept 2014]

Visit sleightofmath.com for up-to-date pdf and video solutions.

Authors:

Daniel and Samuel from Sleight of Math

Disclaimer:

Sleight of Math has no affiliations with the publisher of Additional Maths 360 i.e. Marshall Cavendish

Ex 12.1 ................................................................................................. 326 Ex 18.1 ................................................................................................. 495
Ex 12.2 ................................................................................................. 330 Ex 18.2 ................................................................................................. 503
Rev Ex 12 ............................................................................................ 339 Ex 18.3 ................................................................................................. 509
Ex 18.4 ................................................................................................. 512
Ex 13.1 ................................................................................................. 347 Rev Ex 18............................................................................................ 517
Ex 13.2 ................................................................................................. 356
Ex 13.3 ................................................................................................. 366 Ex 19.1 ................................................................................................. 523
Rev Ex 13 ............................................................................................ 376 Ex 19.2 ................................................................................................. 530
Rev Ex 19............................................................................................ 535
Ex 14.1 ................................................................................................. 383
Ex 14.2 ................................................................................................. 390 Ex 20.1 ................................................................................................. 539
Ex 14.3 ................................................................................................. 396 Rev Ex 20............................................................................................ 548
Ex 14.4 ................................................................................................. 400
Rev Ex 14 ............................................................................................ 407

Ex 15.1 ................................................................................................. 410
Ex 15.2 ................................................................................................. 418
Ex 15.3 ................................................................................................. 421
Ex 15.4 ................................................................................................. 423
Rev Ex 15 ............................................................................................ 430

Ex 16.1 ................................................................................................. 436
Ex 16.2 ................................................................................................. 445
Rev Ex 16 ............................................................................................ 454

Ex 17.1 ................................................................................................. 460
Ex 17.2 ................................................................................................. 470
Ex 17.3 ................................................................................................. 479
Rev Ex 17 ............................................................................................ 488
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A math 360 sol (unofficial) Ex 1.1
2(a) y = 2−x −(1)
Ex 1.1
2x2+xy+1 = 0 −(2)
1(a) y = 2x+1 −(1)
y = x2+2x−3 −(2) sub (1) into (2):
2x2+x(2−x)+1 = 0
sub (1) into (2): 2x2+2x−x2+1 = 0
2x+1 = x2+2x−3 x2+2x+1 = 0
x2−4 = 0 (x+1)2 = 0
(x+2)(x−2) = 0 x = −1
x = −2 ✓ or x = 2 ✓ y|x=−1 = 2−(−1) = 3
y| = 2(−2)+1 y| = 2(2)+1 ⇒ (−1,3) ✓
x=−2 x=2
= −3 ✓ = 5 ✓
2(b) y = 1−3x −(1)
x2+y2 = 5 −(2)
1(b) y = 2+x −(1)
y = 2x2−5x−6 −(2)
sub (1) into (2):

sub (1) into (2): x2+(1−3x)2 = 5
2+x = 2x2−5x−6 x2+(1−6x+9x2) = 5
2x2−6x−8 = 0 10x2−6x−4 = 0
x2−3x−4 = 0 5x2−3x−2 = 0
(5x+2)(x−1) = 0
(x+1)(x−4) = 0
2
x = −1 ✓ or x = 4 ✓ x = − or x = 1
5
y| = 2+(−1) y| = 2+(4) 2
x=−1 x=4 y| = 1−3(− ) y| = 1−3(1)
= 1 ✓ = 6 ✓ x=−25 5 x=1
11
= = −2
5
1(c) 2x+y = 4 2 11
⇒ (− , ) ✓ ⇒ (1,−2) ✓
y = 4−2x −(1) 5 5

y2−4x = 0 −(2)
2(c) 3x+2y = 1
sub (1) into (2):
2y = 1−3x
(4−2x)2−4x = 0
1−3x
y = −(1)
(16−16x+4x2)−4x = 0 2
4x2−20x+16 = 0
x2−5x+4 = 0 3x2+2y2 = 11 −(2)
(x−1)(x−4) = 0 sub (1) into (2):
2
x = 1 or x = 4 ✓ 3x2+2(1−3x) = 11
2
y| = 4−2(1) y| = 4−2(4)
x=1 x=4 3x2+2(1−6x+9x2) = 11
= 2 ✓ = −4 ✓ 4
6x2+(1−6x+9x2) = 22
15x2−6x−21 = 0
5x2−2x−7 = 0
(5x−7)(x+1) = 0
7
x = or x = −1
5
7
1−3( ) 1−3(−1)
y| = 5 y| =
x=7 2 x=−1 2
5
8
= − = 2
5
7 8
⇒ ( ,− ) ✓ ⇒ (−1,2) ✓
5 5
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A math 360 sol (unofficial) Ex 1.1
3 x2 y 5(ii) Area = 216 −(1)
+ = 3 −(1)
4 3 xy = 216

x+y = 8
Perimeter = 60
y = 8−x −(2)
2x+2y = 60

x+y = 30
sub (2) into (1):
y = 30−x −(2)
x2 (8−x)
+ = 3
4 3
(3x2)+(32−4x) = 36 sub (2) into (1):
3x2−4x−4 = 0 x(30−x) = 216
(3x+2)(x−2) = 0 (30x−x2) = 216
x = −2 or x = 2 ✓ x2−30x+216 = 0
3
(x−12)(x−18) = 0
4(i) x = 12 or x = 18
y| = 30−(12) y| = 30−(18)
x=12 x=18
= 18 = 12
𝑥 𝑦

∴ 12 m by 18 m ✓
4x+4y = 32
x+y = 8
6(a) 3x−2y = 1
y = 8−x ✓ −(1)
2y = 3x−1
3x−1
y = −(1)
4(ii) x2+y2 = 34 ✓ −(2) 2

4(iii) sub (1) into (2): (x−2)2+(2y+3)2 = 26 −(2)
x2+(8−x)2 = 34 sub (1) into (2):
x2+(64−16x+x2) = 34 (x−2)2 +[2(3x−1)+3]2 = 26
2x2−16x+30 = 0 2
x2−4x+4+(3x−1+3)2 = 26
x2−8x+15 = 0
x2−4x+4+(3x+2)2 = 26
(x−3)(x−5) = 0
x2−4x+4+(9x2+12x+4) = 26
x = 3 or x = 5
10x2+8x+8 = 26
y| = 5✓ y| = 3✓
x=3 x=5 10x2+8x−18 = 0
∴ the length of the sides are 3cm & 5cm ✓
5x2+4x−9 = 0
(5x+9)(x−1) = 0
5(i)
𝑦
9
x = − or x = 1 ✓
𝑥 5
9
3(− )−1 3(1)−1
Area = xy ✓ y|x=−9 = 25 y|x=1 = 2
5
Perimeter = 2x+2y ✓ 16
= − ✓ = 1 ✓
5
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A math 360 sol (unofficial) Ex 1.1
6(b) x2−2xy+y2 = 1 −(1) 7(a) xy+20 = 5x −(1)

x−2y = 2 x−2y−3 = 0
−2y = −x+2 −2y = −x+3
1 1 3
y = x−1 −(2) y = x− −(2)
2 2 2
sub (1) into (2):
x2−2x(1x−1) +(1x−1)2 = 1 sub (2) into (1):
2 2 1 3
x( x− )+20 = 5x
x2−x2+2x +(1x2−x+1) = 1 2 2
4 (1x2−3x) +20 = 5x
2x +(1x2−x+1) = 1 2 2
4 1x2−13x+20 = 0
1x2+x+1 = 1 2 2
4 x2−13x+40 = 0
1x2+x = 0
(x−5)(x−8) = 0
4
x2+4x = 0 x = 5 or x = 8
x(x+4) = 0 y| = 1(5)−3 y| = 1(8)−3
x=5 x=8
2 2 2 2
x = 0 ✓ or x = −4 ✓
5
= 1 =
y| = 1(0)−1 y| = 1(−4)−1 2
x=0 2 x=−4 2 ⇒ (5,1) ✓ ⇒ (8,5) ✓
= −1 ✓ = −3 ✓ 2
7(b) 2x−y = 4
6(c) 3y−x = 3
−y = −2x+4
3y = x+3
1 y = 2x−4 −(1)
y = x+1 −(1)
3
2 1
− = 2 −(2) 2x2+4xy−3y = 0 −(2)
3y x
sub (1) into (2):
2 1 sub (1) into (2):
− = 2
3(1x+1) x 2x2+4x(2x−4)−3(2x−4) = 0
3
2 −1 = 2 2x2+(8x2−16x) −6x+12 = 0
x+3 x
(10x2−16x) −6x+12 = 0
2x −(x+3) = 2x(x+3)
2x −x−3 = 2x2+6x 10x2−22x+12 = 0
x−3 = 2x2+6x 5x2−11x+6 = 0
0 = 2x2+5x+3 (5x−6)(x−1) = 0
6
2x2+5x+3 = 0 x = or x = 1
5
(2x+3)(x+1) = 0 6
y| = 2( )−4 y| = 2(1)−4
3 x=6 5 x=1
x = − ✓ or x = −1 ✓ 5
2 = −8 = −2
y| = 1(−3)+1 y| = 1(−1)+1 5
x=−32 3 2 x=−1 3 ⇒ (6,−8) ✓ ⇒ (1,−2) ✓
1 2 5 5
= ✓ = ✓
2 3
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A math 360 sol (unofficial) Ex 1.1
7(c) 3x+y = 1 9
3𝑥
y = 1−3x −(1)
4𝑥

(x+y)(x+2y) = 3 −(2) By Pythagoras Theorem,
sub (1) into (2): (4x)2+(3x)2 = 152
[x+(1−3x)][x+2(1−3x)] = 3 16x2+9x2 = 225
(1−2x)(x+2−6x) = 3 25x2 = 225
(1−2x)(2−5x) = 3 x2 = 9
(5x−2)(2x−1) = 3 x = 3 or x = −3 (rej)
10x2−9x+2 = 3
10x2−9x−1 = 0 Width = 4(3) = 12 inch ✓
(10x+1)(x−1) = 0 Height = 3(3) = 9 inch ✓
1
x = − or x = 1
10
10(i)
1
y| = 1−3(− ) y| = 1−3(1)
x=−110 10 x=1 𝑦 5
13
= = −2
10 𝑥
1 13
⇒ (− , ) ✓ ⇒ (1,−2) ✓
10 10 By Pythagoras’ Theorem,
x2+y2 = 52
8 2x+y = 3 −(1)
y x x2+y2 = 25 ✓ −(1)

3x−y = 2 10(ii) y−x = 1
3x−2 = y y = x+1 −(2)
y = 3x−2 −(2) sub (1) into (2):
x2+(x+1)2 = 25
sub (2) into (1): x2+(x2+2x+1) = 25
2x 3x−2 2x2+2x+1 = 25
+ = 3
3x−2 x 2x2+2x−24 = 0
2x2+(3x−2)2 = 3(3x2−2x)
x2+x−12 = 0
2x2+(9x2−12x+4) = 9x2−6x
(x+4)(x−3) = 0
11x2−12x+4 = 9x2−6x
x = −4 ✓ or x = 3 ✓
2x2−6x+4 = 0
y| = (−4)+1 y| = (3)+1
x2−3x+2 = 0 x=−4 x=3
= −3 ✓ = 4 ✓
(x−1)(x−2) = 0
x = 1 or x = 2
10(iii) ∵ length cannot be negative,
y| = 3(1)−2 y| = 3(2)−2
x=1 x=2 x = 3,y = 4
= 1 = 4
⇒ A(1,1) ✓ ⇒ B(2,4) ✓
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A math 360 sol (unofficial) Ex 1.1
11(i) x2+xy+ay = b 12(i) 1st eqn
at {x = 2,y = 1} 12x2−5y2 = 7
(2)2+(2)(1)+a(1) = b At (1,p),
6+a = b 12(1)2−5(p)2 = 7
b = a+6 −(1) 12−5p2 = 7
−5p2 = −5
2ax+3y = b p2 = 1
at {x = 2,y = 1} p = ±1
2a(2)+3(1) = b
4a+3 = b −(2) 2nd eqn
sub (1) into (2): 2p2x−5y = 7
4a+3 = 6+a At (1,p),
3a = 3 2p2(1)−5(p) = 7
a = 1 ✓ 2p2−5p−7 = 0
b| = 7 ✓ (2p−7)(p+1) = 0
a=1
7
p = or p = −1
11(ii) Put a−1,b = 7 into both equations, 2
∴ p = −1 (common sol) ✓
x2+xy+y = 7 −(1)

12(ii) At p = −1,
2x+3y = 7
12x2−5y2 = 7 −(1)
3y = −2x+7

2 7
y = − x+ −(2) 2x−5y = 7
3 3
−5y = −2x+7
sub (2) into (1): y = 2x−7 −(2)
5 5
x2+x(−2x+7) +(−2x+7) = 7
3 3 3 3
x2+(−2x2+7x) +(−2x+7) = 7 sub (2) into (1):
3 3 3 3
2
1x2+7x +(−2x+7) = 7 12x2−5(2x−7) = 7
3 3 3 3 5 5
1x2+5x+7 = 7 12x2−5(4 x2−28x+49) = 7
3 3 3
25 25 25
1x2+5x−14 = 0
3 3 3 12x2 −4x2+28x−49 = 7
x2+5x−14 = 0 5 5 5
(x+7)(x−2) = 0 56x2+28x−49 = 7
5 5 5
x = −7 or x = 2 (taken)
2 7 56x2+28x−84 = 0
y|x=−7 = − (−7)+ 5 5 5
3 3
= 7 2x2+x−3 = 0
{x = −7,y = 7} ✓ (2x+3)(x−1) = 0
3
x = − or x = 1 (taken)
2
2 3 7
y| = (− )−
3
x=− 5 2 5
2
= −2
3
⇒ (− ,−2) ✓
2
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A math 360 sol (unofficial) Ex 1.1
13(i) 14(ii) If k = 1,
𝑦
h
(7𝑥−20)2+(6𝑦+10)2 = 200
r
Total surface area = 32π
𝑥
2πr2+2πrh = 32π 𝑂
20 5
r2+hr = 16 [shown] ✓ −(1) ( ,− )
7 3
7 1
13(ii) h = 4+r −(2) 𝑦 = − 𝑥+
2 6
sub (2) into (1):
r2+(4+r)r = 16 Line
2r2+4r = 16 21x+6y = 1
2r2+4r−16 = 0 6y = −21x+1
7 1
r2+2r−8 = 0 y = − x+
2 6
(r+4)(r−2) = 0
r = −4 or r = 2 ✓ Ellipse
(rej ∵ r > 0) h| = 4+(2) (7x−20)2 +(6y+10)2 = 200
r=2
= 6 ✓ 72(x−20)2+62(y+5)2 = 200
7 3
14(i) If k = 10, 72(x−20)2 62(y+5)2
7 + 3 = 1
Line 200 200
21x+6y = 10 (x−20)2 (y+5)2
7 + 3 = 1
6y = 10−21x 200 200
72 62
y = 10−621x −(1) (x−270)2 +(y−(−53))2 = 1
2 2
200 200
(√ ) (√ )
Ellipse 72 62
(7x−20)2+(6y+10)2 = 200 −(2) (x−20)2 (y−(−5))2
7 + 3 = 1
2 2
10√2 10√2
( ) ( )
sub (1) into (2): 7 6

2
(7x−20)2+[6(10−621x)+10] = 200 (x−2.86)2+(y−(−1.67))2 = 1
(2.02)2 (2.36)2
(7x−20)2+[(10−21x)+10]2 = 200
20 5
⇒ centre( ,− )
(7x−20)2+(20−21x)2 = 200 7 3
200
(49x2−280x+400) ⇒ horizontal radius = √
= 200 72
+(400−840x+441x2)
200
⇒ vertical radius = √
490x2−1120x+800 = 200 62
Using the graphing calculator to plot curves, the
490x2−1120x+600 = 0
line and ellipse don’t intersect.
49x2−112x+60 = 0
Hence, there are no solutions ✓
−(−112)±√(−112)2−4(49)(60) 112±√784 112±28
x = = =
2(49) 98 98 14(iii) Geometrically, a line and ellipse can only intersect
x = 10 or x = 6 ✓ twice at most. ✓
7 7
10 6
10−21( ) 10−21( )
y| = 7 y| = 7
10 6
x= 6 x= 6
7 7
10 4
= − ✓ = − ✓
3 3
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A math 360 sol (unofficial) Ex 1.2
4(i) 2x2−x−4 = 0
Ex 1.2
i.e. a = 2,b = −1,c = −4
1(a) 3x2+9x = 1 Roots: α & β
3x2+9x−1 = 0
b (−1) 1
i.e. a = 3,b = 9,c = −1 Sum of roots = α+β = − = − =
a (2) 2
Roots: α & β
c (−4)
Product of roots = αβ = = = −2
a (2)
b (9)
Sum of roots = α+β = − = − = −3 ✓
a (3) (α+β)2 = α2+2αβ+β2
c (−1) 1
Product of roots = αβ = = = − ✓ (α+β)2−2αβ = α2+β2
a (3) 3
α2+β2 = (α+β)2 −2αβ
1(b) 4x+2x2 = 3x2+2 1 2
= ( ) −2(−2)
x2−4x+2 = 0 2
1
i.e. a = 1,b = −4,c = 2 = 4 ✓
4
Roots: α & β
4(ii) (α−β)2 = α2−2αβ+β2
b (−4) = (α2+β2) −2αβ
Sum of roots = α+β = − = − = 4 ✓
a (1) 1
= (4 ) −2(−2)
Product of roots = αβ = c = (2) = 2 ✓ 4
a (1) = 33
4
2(i) x2+3x+1 = 0 α−β = ±√33 ✓
2
i.e. a = 1,b = 3,c = 1
Roots: α & β 5(i) x2−4x+c = 0
i.e. a = 1,b = −4,c = c
Sum of roots = α+β = −b = −(3) = −3 Roots: α & (α+2)
a (1)

c (1)
Product of roots = αβ = = = 1
a (1) Sum of roots
b
= α+(α+2) = −
2 2 2β+2α 2(α+β) 2(−3) a
+ = = = = −6 ✓
(−4)
α β αβ αβ (1) ⇒ 2α+2 = −
(1)
2α+2 = 4
2(ii) (2α−1)(2β−1)
2α = 2
= 4αβ −2α−2β +1
= 4αβ −2(α+β) +1 α = 1 ✓
= 4(1) −2(−3) +1 α+2 = 3 ✓
= 11 ✓
5(ii) Product of roots
c
3 40x2−138x+119 = 0 = α(α+2) =
a
i.e. a = 40,b = −138,c = 119 ⇒ α(α+2) = c
Roots: α & β are heights of two men 1(1+2) = c ∵ α = 1
c = 3 ✓
b (−138) 69
Sum of roots = α+β = − = − =
a (40) 20
6(a) Roots: α = 2 & β = 5

Sum of roots = α+β = (2)+(5) = 7
69
Average height = α+β = 20 = 69 ✓ Product of roots = αβ = (2)(5) = (10)
2 2 40

x2−(SOR)x+(POR) = 0
x2−7x+10 = 0 ✓
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A math 360 sol (unofficial) Ex 1.2
6(b) Roots: α = −1 & β = 3 8(i) x2+px+q = 0
Sum of roots = α+β = (−1)+(3) = 2 i.e. a = 1,b = p,c = q
Product of roots = αβ = (−1)(3) = −3 Roots: α & 4α

x2−(SOR)x+(POR) = 0 b
Sum of roots = α+4α = −
x2−2x−3 = 0 ✓ a
= 5α = −p ✓ −(1)

7(i) 1st equation
c
2x2−4x+5 = 0 Product of roots = α(4α) =
a
i.e. a = 2,b = −4,c = 5 = 4α2 = q ✓ −(2)
Roots: α & β
8(ii) From (1):
b (−4) 5α = −p
Sum of roots = α+β = − = − = 2
a (2) α = −p −(3)
c (5) 5 5
Product of roots = αβ = = =
a (2) 2
sub (3) into (2):
2nd equation 1 2
4(− p) = q
Roots: (α−1) & (β−1) 5
4p2
Sum of roots Product of roots = q
25
= (α−1)+(β−1) = (α−1)(β−1) 4p2 = 25q [shown] ✓
= (α+β) −2 = αβ −α−β +1
= (2) −2 = αβ −(α+β) +1 9(i) 2x2−x−2 = 0
= 0 = (5) −(2) +1 i.e. a = 2,b = −1,c = −2
2
Roots: α & β
3
=
2 Sum of roots = α+β = −b = −(−1) = 1
x2−(SOR)x+(POR) = 0 a (2) 2
x2−0x+3 = 0 Product of roots = αβ = c = (−2) = −1
2 a (2)
2x2+3 = 0 ✓
α2+β2
7(ii) 3rd equation = (α+β)2 −2(αβ)
Recall α+β = 2, αβ = 5 1 2
= ( ) −2(−1)
2
2
Roots: 2α & 2β
1
= 2 ✓
4
Sum of roots
= 2α+2β = 2(α+β) = 2(2) = 4 9(ii) β+α = β2+α2 = (214) = −21 ✓
α β αβ (−1) 4
Product of roots
5 9(iii) α4+β4 = (α2+β2)2 −2α2β2
= (2α)(2β) = 4(αβ) = 4( ) = 10
2
= (α2+β2)2 −2(αβ)2

x2−(SOR)x+(POR) = 0 = (21)2 −2(−1)2
x2−4x+10 = 0 ✓ 4
49
= ✓
16
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