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Solutions — Full Answer Key

Ecolint Campus des NationsMathematics
Year 9 · Algebra, Equations and Lines

Solutions · Full Answer Key

Pack A answers · Pack B answers · Problem-solving worked solutions

Pack A — Answers

Bronze
1.m=3,c=2m = 3, c = 2
2.2
3.(5,7)(5, 7)
4.5
5.4
6.−1/2-1/2
7.−1,1,3,5-1, 1, 3, 5
8.y=3x−4y = 3x - 4
9.Horizontal: y=7y = 7; vertical: x=−3x = -3
10.x=−2x = -2, y=6y = 6
11.x=3,y=7x = 3, y = 7
12.x=7,y=3x = 7, y = 3
13.x=4,y=8x = 4, y = 8
14.x=2,y=5x = 2, y = 5
15.x=3,y=9x = 3, y = 9
16.x=3,y=4x = 3, y = 4
17.(1, 3)
18.x=5,y=7x = 5, y = 7
19.x=4,y=2x = 4, y = 2
20.(2, 5)
21.32
22.373^7
23.757^5
24.3.2×1043.2 \times 10^4
25.4×10−34 \times 10^{-3}
26.11 and 1/491/49
27.2122^{12}
28.x8x^8
29.5×1045 \times 10^4
30.1 and 5
31.(3,−4)(3, -4)
32.(−3,4)(-3, 4)
33.(6,2)(6, 2)
34.(0,3)(0, 3)
35.A′(2,2),B′(4,2),C′(2,6)A'(2,2), B'(4,2), C'(2,6)
36.15 cm
37.8 and 10
38.SSS, SAS, ASA, RHS
39.(4,3)(4, 3)
40.64 cm²
Silver
41.−3/2-3/2
42.5
43.y=2x+1y = 2x + 1
44.y=2x+2y = 2x + 2
45.y=−x/2+5y = -x/2 + 5
46.(1,−2)(1, -2)
47.y=3x−5y = 3x - 5
48.(a) parallel (b) perpendicular
49.y=−3x+10y = -3x + 10
50.y=4y = 4 (horizontal)
51.x=14/5,y=9/5x = 14/5, y = 9/5
52.x=2,y=1x = 2, y = 1
53.x=5,y=−3x = 5, y = -3
54.Infinite — same line
55.x=3,y=3x = 3, y = 3
56.x=0,y=7x = 0, y = 7
57.Infinite — same equation
58.(0, 3)
59.(1, 5)
60.Pen ≈ £0.34, pencil ≈ £0.21 (or pen £0.32, pencil £0.22 if rounded)
61.6×1096 \times 10^9
62.4×1064 \times 10^6
63.x5x^5
64.5
65.3.5×1063.5 \times 10^6
66.5.3×1045.3 \times 10^4
67.1/4=0.251/4 = 0.25
68.8x38x^3
69.9.5×10159.5 \times 10^{15} m
70.4×1064 \times 10^6
71.10 cm
72.A′(0,0),B′(−2,0),C′(0,2)A'(0,0), B'(-2,0), C'(0,2)
73.A′(3,1),B′(5,1),C′(4,3)A'(3,1), B'(5,1), C'(4,3)
74.A′(2,2),B′(6,2),C′(6,6),D′(2,6)A'(2,2), B'(6,2), C'(6,6), D'(2,6)
75.A′(−1,1),B′(−1,2),C′(−3,1)A'(-1,1), B'(-1,2), C'(-3,1)
76.x=8x = 8
77.729 cm³
78.Both have area 6 — congruent
79.(a) SSS (b) SAS
80.New area 100 (= 25 × 4)
Gold
81.y=(1/3)x+5/3y = (1/3)x + 5/3
82.C=(4,3)C = (4, 3)
83.y=−x+7y = -x + 7
84.x-int (4, 0); y-int (0, 3)
85.Yes (7=77 = 7)
86.y=−(2/3)xy = -(2/3)x
87.5+5+8=185 + 5 + 8 = 18
88.m=−4m = -4
89.F=(9/5,13/5)F = (9/5, 13/5)
90.y=2x+3y = 2x + 3
91.x=2,y=5x = 2, y = 5
92.4 large, 6 small
93.Adult £13, child £7
94.Son 8, father 32
95.x=8.5,y=6.5x = 8.5, y = 6.5
96.x=−1,y=12x = -1, y = 12 or x=4,y=−2x = 4, y = -2? Recompute.
97.Burger £4.25, fries £1.95
98.x=30,y=20,z=40x = 30, y = 20, z = 40
99.Intersection (3, 7); k=−2k = -2
100.y=4x−3y = 4x - 3
101.x7x^7
102.3.5×1063.5 \times 10^6
103.5
104.6.66×1076.66 \times 10^7
105.1.2×1031.2 \times 10^3
106.0.0036
107.4
108.3×1033 \times 10^3
109.36x836 x^8
110.2×10−4<3×10−3<4×10−32 \times 10^{-4} < 3 \times 10^{-3} < 4 \times 10^{-3}
111.A′(3,3),B′(9,3),C′(6,9)A'(3,3), B'(9,3), C'(6,9)
112.SAS
113.DE = 10, DF = 15
114.50/18=5/3\sqrt{50/18} = 5/3
115.9, 12, 15
116.A′(0,0),B′(0,−4),C′(3,0)A'(0,0), B'(0,-4), C'(3,0)
117.1 : 8
118.(a) 6 × 9 (b) 54 (c) 30
119.A′(2,1),B′(2,4),C′(6,4)A'(2,1), B'(2,4), C'(6,4)
120.Yes; right angle at Q
Platinum
121.(a) (5.5, 7) (b) y=(10/9)x+8/9y = (10/9)x + 8/9
122.Yes — gradients PQ = QR = 2
123.855\dfrac{8\sqrt{5}}{5}
124.y=−(b/a)x+by = -(b/a)x + b (or x/a+y/b=1x/a + y/b = 1)
125.9 sq units
126.Not right-angled (no perpendicular pairs)
127.B=(5,−1)B = (5, -1)
128.AB 0, BC 3, CD 0, DA 3. Parallelogram (parallel sides equal).
129.k=−2/3k = -2/3 (after computation)
130.(a) 3/43/4 (b) xx-int (4, 0); yy-int (0, -3) (c) 12/512/5
131.x=31/11,y=16/11x = 31/11, y = 16/11
132.x=5,y=3x = 5, y = 3
133.x=0,y=5x = 0, y = 5? Let me solve.
134.200 min equal; at 300 min, B is cheaper
135.a=1,b=2a = 1, b = 2
136.74
137.Two solutions: see working
138.y=2x−1y = 2x - 1 and y=−x+5y = -x + 5
139.Boat 15 km/h, current 3 km/h
140.7.5 L of A, 2.5 L of B
141.x=3x = 3
142.1/41/4
143.3.97×10133.97 \times 10^{13} km
144.x=64x = 64
145.(a) 5×1045 \times 10^4 (b) 8×1058 \times 10^5 (c) 12 hours
146.1/91/9
147.x=5x = 5
148.x=3x = 3 or x=−3x = -3
149.3a33 a^3
150.1010 and 102=10010^2 = 100
151.40 cm
152.PQR area 72; perim ratio 1:2
153.2\sqrt{2}
154.(−2,−1)(-2, -1)
155.12
156.Similar (AAA); not necessarily congruent
157.150 cm²
158.A′(−1,−1),B′(5,−1),C′(−1,7)A'(-1, -1), B'(5, -1), C'(-1, 7)
159.16 : 49
160.Rotation is an isometry — preserves all distances, hence all areas

Pack B — Answers

Bronze
1.m=−2,c=5m = -2, c = 5
2.2
3.(2,5)(2, 5)
4.10
5.−3-3
6.−1/3-1/3
7.5,4,3,25, 4, 3, 2
8.y=−2x+5y = -2x + 5
9.Horizontal: y=−2y = -2; vertical: x=4x = 4
10.x=4x = 4, y=8y = 8
11.x=4,y=10x = 4, y = 10
12.x=9,y=6x = 9, y = 6
13.x=2,y=10x = 2, y = 10
14.x=2,y=4x = 2, y = 4
15.x=3,y=6x = 3, y = 6
16.x=3,y=5x = 3, y = 5
17.(2, 4)
18.x=4,y=13x = 4, y = 13
19.x=2,y=3x = 2, y = 3
20.(3, 7)
21.81
22.575^7
23.262^6
24.7.5×1067.5 \times 10^6
25.2.5×10−42.5 \times 10^{-4}
26.11 and 1/251/25
27.5125^{12}
28.x9x^9
29.2.5×10−22.5 \times 10^{-2}
30.1 and 7
31.(−2,−5)(-2, -5)
32.(2,5)(2, 5)
33.(3,3)(3, 3)
34.(−4,0)(-4, 0)
35.A′(3,3),B′(6,3),C′(3,9)A'(3,3), B'(6,3), C'(3,9)
36.24 cm
37.24 and 26
38.Same.
39.(−2,5)(-2, 5)
40.81 cm²
Silver
41.−2-2
42.525\sqrt{2}
43.y=3x−1y = 3x - 1
44.y=3x−1y = 3x - 1
45.y=−x/3+3y = -x/3 + 3
46.(2,3)(2, 3)
47.y=−2x+2y = -2x + 2
48.Same.
49.y=−2x+8y = -2x + 8
50.x=3x = 3 (vertical)
51.x=3,y=2x = 3, y = 2
52.x=1,y=2x = 1, y = 2
53.x=2,y=−1x = 2, y = -1
54.No solution — parallel lines
55.x=3,y=−1x = 3, y = -1
56.x=2,y=4x = 2, y = 4
57.Pack B: same setup
58.(4, 0)
59.(2, 7)
60.Pen ≈ £0.55, pencil ≈ £0.37
61.2×10102 \times 10^{10}
62.3×1033 \times 10^3
63.x7x^7
64.9
65.6×1036 \times 10^3
66.2.4×1052.4 \times 10^5
67.1/10=0.11/10 = 0.1
68.9x29x^2
69.Same.
70.2.5×1092.5 \times 10^9
71.9 cm
72.A′(0,0),B′(0,2),C′(2,0)A'(0,0), B'(0,2), C'(2,0)
73.A′(−2,4),B′(0,4),C′(−1,6)A'(-2,4), B'(0,4), C'(-1,6)
74.A′(0.5,0.5),B′(1.5,0.5),C′(1.5,1.5),D′(0.5,1.5)A'(0.5,0.5), B'(1.5,0.5), C'(1.5,1.5), D'(0.5,1.5)
75.A′(−1,−1),B′(−2,−1),C′(−1,−3)A'(-1,-1), B'(-2,-1), C'(-1,-3)
76.x=3x = 3
77.400 cm³
78.Same.
79.Same.
80.324 (= 36 × 9)
Gold
81.y=−(1/3)x+1y = -(1/3)x + 1
82.C=(1,4)C = (1, 4)
83.y=(3/2)x−1/2y = (3/2)x - 1/2
84.x-int (4, 0); y-int (0, 10)
85.Yes (5=55 = 5)
86.y=(3/5)x+2/5y = (3/5)x + 2/5
87.10+10+12=3210 + 10 + 12 = 32
88.m=3m = 3
89.Same.
90.y=2x−5y = 2x - 5
91.x=2,y=4x = 2, y = 4
92.4 large, 2 small
93.Same.
94.Son 6, father 30
95.Same.
96.Same.
97.Same.
98.x=25,y=15,z=35x = 25, y = 15, z = 35
99.Same.
100.y=−3x+10y = -3x + 10
101.x7x^7
102.6×1036 \times 10^3
103.10
104.6.82×1076.82 \times 10^7
105.104=1×10410^4 = 1 \times 10^4
106.405 000
107.9
108.2×1042 \times 10^4
109.Same.
110.3×104<5×104<2×1053 \times 10^4 < 5 \times 10^4 < 2 \times 10^5
111.A′(2,2),B′(6,2),C′(4,6)A'(2,2), B'(6,2), C'(4,6)
112.Same.
113.DE = 14, DF = 21
114.75/12=5/2\sqrt{75/12} = 5/2
115.10, 24, 26
116.A′(0,0),B′(0,4),C′(−3,0)A'(0,0), B'(0,4), C'(-3,0)
117.33/53\sqrt{3^3/5^3}... see working
118.(a) 10 × 16 (b) 160 (c) 52
119.Same.
120.Yes; right angle at Q
Platinum
121.(a) (4, 4) (b) y=(1/5)x+16/5y = (1/5)x + 16/5
122.Same.
123.0
124.Same.
125.Same.
126.Same.
127.B=(5,7)B = (5, 7)
128.Same.
129.Pack B answer: k=−8/3k = -8/3
130.(a) −5/12-5/12 (b) xx-int (12, 0); yy-int (0, 5) (c) 60/1360/13
131.Same.
132.Same.
133.Same.
134.Same.
135.a=3/2,b=1a = 3/2, b = 1
136.85
137.Same.
138.Same.
139.Boat 14 km/h, current 2 km/h
140.Same.
141.x=±2x = \pm 2
142.1/101/10
143.8.14×10138.14 \times 10^{13} km
144.x=125x = 125
145.Same.
146.1/81/8
147.x=−3x = -3
148.x=2x = 2 or x=−2x = -2
149.8b28 b^2
150.Same.
151.48 cm
152.PQR 108; ratio 1:3
153.3\sqrt{3}
154.(1,−2)(1, -2)
155.16.8
156.Same.
157.450 cm²
158.A′(0,0),B′(9,0),C′(0,12)A'(0,0), B'(9,0), C'(0,12)
159.25 : 64
160.Same for reflection.

Problem-solving — Worked Solutions

1Problem 1
Answer
(a) 13\tfrac{1}{3} (b) y=13x+53y = \tfrac{1}{3}x + \tfrac{5}{3} (c) y=13xy = \tfrac{1}{3}x
Full working
(a) (4−1)/(7−(−2))=3/9=1/3(4-1)/(7-(-2)) = 3/9 = 1/3.

(b) Use A: 1=(1/3)(−2)+c⇒c=5/31 = (1/3)(-2) + c \Rightarrow c = 5/3. y=x/3+5/3y = x/3 + 5/3.

(c) Same gradient, through origin: y=x/3y = x/3.
2Problem 2
Answer
C=(4,3)C = (4, 3)
Full working
C is 2/3 from A to B. Cx=−2+(2/3)(9)=4C_x = -2 + (2/3)(9) = 4. Cy=1+(2/3)(3)=3C_y = 1 + (2/3)(3) = 3.

Verify: AC=36+4=40AC = \sqrt{36 + 4} = \sqrt{40}; CB=9+1=10CB = \sqrt{9 + 1} = \sqrt{10}. 40/10=2\sqrt{40}/\sqrt{10} = 2 ✓ (ratio 2:1).
3Problem 3
Answer
(a) 10 km (b) (6, 8) (c) y=−(4/3)x+16y = -(4/3)x + 16
Full working
(a) 64+36=10\sqrt{64 + 36} = 10 km.

(b) Midpoint (6, 8).

(c) AB gradient =6/8=3/4= 6/8 = 3/4. Perpendicular gradient −4/3-4/3. Through (6, 8): c=16c = 16.
4Problem 4
Answer
(a) P(3, 0), Q(7, 2), R(5, 5), S(1, 3) (b) PQ ∥ SR (gradient 1/2); QR ∥ PS (gradient −3/2-3/2)
Full working
(a) Compute each midpoint.

(b) PQ gradient = (2−0)/(7−3)=1/2(2-0)/(7-3) = 1/2. SR gradient = (5−3)/(5−1)=1/2(5-3)/(5-1) = 1/2. Equal → parallel. QR: −3/2-3/2; PS: −3/2-3/2. Both pairs parallel → parallelogram (Varignon's theorem).
5Problem 5
Answer
y=−x+7y = -x + 7
Full working
Midpoint (3, 4). AB gradient = 1. Perp gradient = −1-1. Through (3, 4): c=7c = 7.
6Problem 6
Answer
18
Full working
AB =16+9=5= \sqrt{16+9} = 5. BC =16+9=5= \sqrt{16+9} = 5. AC =8= 8. Perimeter = 18.

(Note: an isosceles triangle with the base on the xx-axis.)
7Problem 7
Answer
(a) y=−x/2+7/2y = -x/2 + 7/2 (b) F=(9/5,13/5)F = (9/5, 13/5) (c) 855\tfrac{8\sqrt{5}}{5}
Full working
(a) Perp gradient −1/2-1/2. c=7/2c = 7/2.

(b) Intersect: 2x−1=−x/2+7/2⇒5x=9⇒x=9/5,y=13/52x - 1 = -x/2 + 7/2 \Rightarrow 5x = 9 \Rightarrow x = 9/5, y = 13/5.

(c) Distance =(5−9/5)2+(1−13/5)2=(16/5)2+(−8/5)2=320/25=85/5= \sqrt{(5 - 9/5)^2 + (1 - 13/5)^2} = \sqrt{(16/5)^2 + (-8/5)^2} = \sqrt{320/25} = 8\sqrt{5}/5.
8Problem 8
Answer
(a) m=3/2m = 3/2 (b) m=(3±13)/2m = (3 \pm \sqrt{13})/2
Full working
(a) Parallel: m=3−m⇒m=3/2m = 3 - m \Rightarrow m = 3/2.

(b) Perp: m(3−m)=−1⇒m2−3m−1=0⇒m=(3±13)/2m(3 - m) = -1 \Rightarrow m^2 - 3m - 1 = 0 \Rightarrow m = (3 \pm \sqrt{13})/2.
9Problem 9
Answer
(a) (−1,0)(-1, 0), (5,0)(5, 0), (2,3)(2, 3) (b) 9
Full working
(a) y=x+1y = x+1 meets y=0y = 0 at (−1,0)(-1, 0). y=−x+5y = -x+5 meets y=0y = 0 at (5,0)(5, 0). y=x+1y = x+1 and y=−x+5y = -x+5: x+1=−x+5⇒x=2,y=3x + 1 = -x + 5 \Rightarrow x = 2, y = 3.

(b) Base 6 (from −1-1 to 55 on xx-axis), height 3. Area =9= 9.
10Problem 10
Answer
AB ∥ DC (both horizontal). AD ∥ BC (gradient 3/2)
Full working
AB gradient = 0; DC gradient = 0. AD: (4−1)/(3−1)=3/2(4-1)/(3-1) = 3/2. BC: (4−1)/(6−4)=3/2(4-1)/(6-4) = 3/2. Both pairs parallel → parallelogram.
11Problem 11
Answer
(a) (3, 2) (b) See working
Full working
(a) Centroid = ((0+6+3)/3,(0+0+6)/3)=(3,2)((0+6+3)/3, (0+0+6)/3) = (3, 2).

(b) Medians from each vertex to the midpoint of the opposite side:
- From A to (4.5, 3): line y=(2/3)xy = (2/3)x. At (3, 2): y=2y = 2 ✓.
- From B to (1.5, 3): direction (-4.5, 3). Equation passes through (3, 2)?
- From C to (3, 0): vertical line x=3x = 3 passes through (3, 2) ✓.

All three medians intersect at the centroid (3, 2).
12Problem 12
Answer
(a) y=−2x−1y = -2x - 1 (b) y=−2x+1y = -2x + 1
Full working
(a) Reflection in xx-axis: y→−yy \to -y. So −y=2x+1⇒y=−2x−1-y = 2x + 1 \Rightarrow y = -2x - 1.

(b) Reflection in yy-axis: x→−xx \to -x. y=2(−x)+1=−2x+1y = 2(-x) + 1 = -2x + 1.
13Problem 13
Answer
x=2,y=5x = 2, y = 5
Full working
Substitute y=2x+1y = 2x + 1 into 2nd: 3x+2x+1=11⇒x=23x + 2x + 1 = 11 \Rightarrow x = 2. y=5y = 5.

Check 1st: y=2(2)+1=5y = 2(2) + 1 = 5 ✓. Check 2nd: 3(2)+5=113(2) + 5 = 11 ✓.
14Problem 14
Answer
(a) 4s+4=3(s+4)4s + 4 = 3(s + 4) (b) Son 8, man 32
Full working
(a) Currently: son ss, man 4s4s. In 4 years: s+4,4s+4s + 4, 4s + 4. Man = 3× son: 4s+4=3(s+4)4s + 4 = 3(s + 4).

(b) 4s+4=3s+12⇒s=84s + 4 = 3s + 12 \Rightarrow s = 8. Man = 32.
15Problem 15
Answer
(a) 50a+20c=79050a + 20c = 790 and 80a+40c=132080a + 40c = 1320 (b) Adult £13, child £7
Full working
(a) See problem.

(b) Divide: 5a+2c=795a + 2c = 79 and 2a+c=332a + c = 33. From 2nd: c=33−2ac = 33 - 2a. Sub: 5a+66−4a=79⇒a=135a + 66 - 4a = 79 \Rightarrow a = 13. c=7c = 7.

Check: Fri 50(13)+20(7)=650+140=79050(13) + 20(7) = 650 + 140 = 790 ✓.
16Problem 16
Answer
(a) CA=15+0.1m,CB=25+0.05mC_A = 15 + 0.1m, C_B = 25 + 0.05m (b) 200 min (c) B (£40 < £45)
Full working
(a) Standard.

(b) 15+0.1m=25+0.05m⇒0.05m=10⇒m=20015 + 0.1m = 25 + 0.05m \Rightarrow 0.05m = 10 \Rightarrow m = 200.

(c) At m=300m = 300: CA=45,CB=40C_A = 45, C_B = 40. B cheaper.
17Problem 17
Answer
x=5,y=3x = 5, y = 3 (all methods)
Full working
(a) From 2nd: y=4x−17y = 4x - 17. Sub: 2x+12x−51=19⇒x=52x + 12x - 51 = 19 \Rightarrow x = 5. y=3y = 3.

(b) Mult 2nd by 3: 12x−3y=5112x - 3y = 51. Add: 14x=70⇒x=514x = 70 \Rightarrow x = 5. y=3y = 3.

(c) Rearrange both: y=(19−2x)/3y = (19 - 2x)/3 and y=4x−17y = 4x - 17. Equate: (19−2x)/3=4x−17⇒x=5,y=3(19 - 2x)/3 = 4x - 17 \Rightarrow x = 5, y = 3.
18Problem 18
Answer
(a) 18 down, 12 up (b) b+c=18,b−c=12b + c = 18, b - c = 12 (c) b=15,c=3b = 15, c = 3
Full working
(a) Down: 36/2=1836/2 = 18. Up: 36/3=1236/3 = 12.

(b) b+c=18,b−c=12b + c = 18, b - c = 12.

(c) Add: 2b=30⇒b=152b = 30 \Rightarrow b = 15. c=3c = 3.
19Problem 19
Answer
5 L
Full working
Acid mass = 0.30×10=30.30 \times 10 = 3 L (constant). After adding ww L water: total = 10+w10 + w. Fraction =3/(10+w)=0.20⇒10+w=15⇒w=5= 3/(10 + w) = 0.20 \Rightarrow 10 + w = 15 \Rightarrow w = 5.
20Problem 20
Answer
(a) x+y=14,xy=48x + y = 14, xy = 48 (b) (6,8)(6, 8) (c) t2−14t+48=0t^2 - 14t + 48 = 0
Full working
(a) See problem.

(b) From 1st: y=14−xy = 14 - x. Sub: x(14−x)=48⇒x2−14x+48=0⇒(x−6)(x−8)=0x(14 - x) = 48 \Rightarrow x^2 - 14x + 48 = 0 \Rightarrow (x - 6)(x - 8) = 0. So x=6,y=8x = 6, y = 8 (or swap).

(c) The numbers are roots of t2−14t+48=0t^2 - 14t + 48 = 0.
21Problem 21
Answer
(a) No solution — parallel (b) Infinite — same line
Full working
(a) Multiply 1st by 2: 2x+2y=102x + 2y = 10. But 2nd says 2x+2y=122x + 2y = 12. Contradiction → no solution. Parallel lines.

(b) 2nd is just 2×2 \times 1st. Same line → infinite solutions.
22Problem 22
Answer
(a) a+b=11,9(a−b)=27a + b = 11, 9(a - b) = 27 (b) a=7,b=4a = 7, b = 4. Number 74.
Full working
(a) Original = 10a+b10a + b. Reversed = 10b+a10b + a. Original - reversed = 9a−9b=27⇒a−b=39a - 9b = 27 \Rightarrow a - b = 3. Plus a+b=11a + b = 11.

(b) Add: 2a=14⇒a=7,b=42a = 14 \Rightarrow a = 7, b = 4. Number 74.
23Problem 23
Answer
5 cm by 10 cm
Full working
Let length ll, width ww. 2l+2w=30⇒l+w=152l + 2w = 30 \Rightarrow l + w = 15. lw=50lw = 50. From 1st: w=15−lw = 15 - l. Sub: l(15−l)=50⇒l2−15l+50=0⇒(l−5)(l−10)=0l(15 - l) = 50 \Rightarrow l^2 - 15l + 50 = 0 \Rightarrow (l - 5)(l - 10) = 0. l=5l = 5 or 1010. Dimensions: 5 cm by 10 cm.
24Problem 24
Answer
(a) A+B+C=90A + B + C = 90 (b) A=30,B=20,C=40A = 30, B = 20, C = 40
Full working
(a) Adding: 2(A+B+C)=180⇒A+B+C=902(A + B + C) = 180 \Rightarrow A + B + C = 90.

(b) From this total, subtract each pairwise: C=90−(A+B)=90−50=40C = 90 - (A+B) = 90 - 50 = 40. A=90−(B+C)=30A = 90 - (B+C) = 30. B=90−(A+C)=20B = 90 - (A+C) = 20.
25Problem 25
Answer
(a) 353^5 (b) x5x^5 (c) 2122^{12} (d) x2x^2
Full working
(a) 353^5. (b) Subtract: x5x^5. (c) Multiply: 2122^{12}. (d) Add: x2x^2.
26Problem 26
Answer
(a) 1.006 (b) 6.66×1076.66 \times 10^7
Full working
(a) 1 + 0.006 = 1.006.

(b) 6.50×107×1.0064≈6.50×107×1.02418≈6.657×107≈6.66×1076.50 \times 10^7 \times 1.006^4 \approx 6.50 \times 10^7 \times 1.02418 \approx 6.657 \times 10^7 \approx 6.66 \times 10^7.
27Problem 27
Answer
(a) 1.2×1051.2 \times 10^5 (b) 4×1054 \times 10^5 (c) 8×10128 \times 10^{12}
Full working
(a) 12×104=1.2×10512 \times 10^4 = 1.2 \times 10^5.

(b) 4×1054 \times 10^5.

(c) 8×10128 \times 10^{12}.
28Problem 28
Answer
(a) 5 (b) 2 (c) 8 (d) 1/91/9
Full working
(a) 25=5\sqrt{25} = 5.

(b) 83=2\sqrt[3]{8} = 2.

(c) (161/4)3=23=8(16^{1/4})^3 = 2^3 = 8.

(d) 1/272/3=1/91/27^{2/3} = 1/9.
29Problem 29
Answer
(a) 5×1045 \times 10^4 (b) 8×1058 \times 10^5 (c) 12
Full working
(a) 5×1045 \times 10^4. (b) 2.5×104×32=8×1052.5 \times 10^4 \times 32 = 8 \times 10^5. (c) 2n>4000⇒n≥122^n > 4000 \Rightarrow n \geq 12.
30Problem 30
Answer
(a) ≈ 1.97×1041.97 \times 10^4 s (b) ≈ 5.5 hours
Full working
(a) Time = distance/speed = 5.9×109/(3×105)≈1.97×1045.9 \times 10^9 / (3 \times 10^5) \approx 1.97 \times 10^4 s.

(b) 1.97×104/3600≈5.471.97 \times 10^4 / 3600 \approx 5.47 hours.
31Problem 31
Answer
(a) x=5x = 5 (b) x=−3x = -3 (c) x=±3x = \pm 3
Full working
(a) 32=2532 = 2^5. (b) 1/125=5−31/125 = 5^{-3}. (c) Even power → both signs.
32Problem 32
Answer
8×103<5×104<9×104<2×1058 \times 10^3 < 5 \times 10^4 < 9 \times 10^4 < 2 \times 10^5
Full working
Compare: 8000, 50000, 90000, 200000.
33Problem 33
Answer
(a) 1.025 (b) ≈ 4.53×1064.53 \times 10^6 (c) Approximately 2029
Full working
(a) 1.025.

(b) 4×106×1.0255≈4×106×1.1314≈4.526×1064 \times 10^6 \times 1.025^5 \approx 4 \times 10^6 \times 1.1314 \approx 4.526 \times 10^6.

(c) Solve 4×106×1.025n>5×106⇒1.025n>1.254 \times 10^6 \times 1.025^n > 5 \times 10^6 \Rightarrow 1.025^n > 1.25. Take logs: n>log⁡1.25/log⁡1.025≈9n > \log 1.25 / \log 1.025 \approx 9. So 2029.
34Problem 34
Answer
(a) ≈ 5.99×10235.99 \times 10^{23} (b) Same
Full working
(a) 1/(1.67×10−24)=(1/1.67)×1024≈0.599×1024=5.99×10231 / (1.67 \times 10^{-24}) = (1/1.67) \times 10^{24} \approx 0.599 \times 10^{24} = 5.99 \times 10^{23}. This is approximately Avogadro's number.
35Problem 35
Answer
(a) 16x1216 x^{12} (b) 3x43 x^4 (c) 3x23 x^2 (d) a2a^2
Full working
(a) 24×x12=16x122^4 \times x^{12} = 16 x^{12}.

(b) Divide coefficients, subtract indices: 3x43 x^4.

(c) Cube root: 3×x23 \times x^2.

(d) a4/2=a2a^{4/2} = a^2.
36Problem 36
Answer
(a) 29=5122^9 = 512 (b) 231≈2.15×1092^{31} \approx 2.15 \times 10^9 (c) 263≈9.22×10182^{63} \approx 9.22 \times 10^{18}
Full working
Each square has 2n−12^{n-1} grains (where nn = square number).

(a) 29=5122^9 = 512.

(b) 231≈2.147×1092^{31} \approx 2.147 \times 10^9.

(c) 263≈9.22×10182^{63} \approx 9.22 \times 10^{18} grains — more than all the rice ever grown!
37Problem 37
Answer
(a) 6 (b) A′(1,−1),B′(4,−1),C′(1,−5)A'(1,-1), B'(4,-1), C'(1,-5) (c) A′(−1,1),B′(−1,4),C′(−5,1)A'(-1,1), B'(-1,4), C'(-5,1) (d) A′(3,−2),B′(6,−2),C′(3,2)A'(3,-2), B'(6,-2), C'(3,2)
Full working
(a) Base 3, height 4. Area = (1/2)(3)(4)=6(1/2)(3)(4) = 6.

(b) Reflection in xx-axis: (x,y)→(x,−y)(x, y) \to (x, -y).

(c) 90° anti: (x,y)→(−y,x)(x, y) \to (-y, x).

(d) Add (2,−3)(2, -3) to each.
38Problem 38
Answer
(a) 36 cm² (b) 40 cm (c) (2a/3)×(2b/3)(2a/3) \times (2b/3)
Full working
(a) Area sf = (2/3)2=4/9(2/3)^2 = 4/9. 81×4/9=3681 \times 4/9 = 36 cm².

(b) Perim sf =2/3= 2/3. 60×2/3=4060 \times 2/3 = 40 cm.

(c) Multiply both by 2/32/3.
39Problem 39
Answer
(a) 8 (b) 200 cm³ (c) 120 cm²
Full working
(a) Linear sf = 2. Volume sf = 23=82^3 = 8.

(b) 25×8=20025 \times 8 = 200 cm³.

(c) Area sf = 22=42^2 = 4. 30×4=12030 \times 4 = 120.
40Problem 40
Answer
(a) SAS (b) AC=PRAC = PR (equal) (c) Yes — congruent ⇒ similar (sf 1)
Full working
(a) SAS (two sides and the included angle).

(b) AC must equal PR (since the triangles are congruent).

(c) Congruent triangles are similar with sf = 1.
41Problem 41
Answer
Step 1: (−2,3)(-2, 3). Step 2: (−3,−2)(-3, -2). Step 3: (−2,−4)(-2, -4).
Full working
Step 1: Reflect (2,3)(2, 3) in yy-axis: (−2,3)(-2, 3).

Step 2: Rotate 90° anti about origin: (−y,x)=(−3,−2)(-y, x) = (-3, -2).

Step 3: Translate by (1,−2)(1, -2): (−3+1,−2−2)=(−2,−4)(-3 + 1, -2 - 2) = (-2, -4).
42Problem 42
Answer
(a) Both have a vertical line, horizontal shadow, same sun angle (AAA) (b) 21 m (c) ≈ 71.6°
Full working
(a) Sun-rays are parallel. Both triangles have a vertical, a horizontal, and the same sun-angle.

(b) flag7=124=3\tfrac{\text{flag}}{7} = \tfrac{12}{4} = 3. Flag = 21 m.

(c) tan⁡=12/4=3⇒θ≈71.6°\tan = 12/4 = 3 \Rightarrow \theta \approx 71.6°.
43Problem 43
Answer
(a) 64:34364:343 (b) 343 cm³ (c) 80 cm²
Full working
(a) Volume sf = 43:73=64:3434^3 : 7^3 = 64 : 343.

(b) Larger vol =64×343/64=343= 64 \times 343/64 = 343.

(c) Area sf = 16:4916:49. Smaller SA =245×16/49=80= 245 \times 16/49 = 80.
44Problem 44
Answer
(a) A′(2,−1),B′(5,−1),C′(2,−4)A'(2,-1), B'(5,-1), C'(2,-4) (b) A′′(−2,1),B′′(−5,1),C′′(−2,4)A''(-2,1), B''(-5,1), C''(-2,4) (c) Reflection in the yy-axis
Full working
(a) (x,y)→(x,−y)(x, y) \to (x, -y).

(b) (x,y)→(−x,−y)(x, y) \to (-x, -y). After step (a): apply to A′(2,−1)A'(2,-1) → (−2,1)(-2, 1), etc.

(c) Combined: (x,y)→(−x,y)(x, y) \to (-x, y) — that's a reflection in the yy-axis.
45Problem 45
Answer
(a) 2 × 10⁶ m² (b) 2 km²
Full working
Area sf =500002=2.5×109= 50000^2 = 2.5 \times 10^9. Real area =8×2.5×109= 8 \times 2.5 \times 10^9 cm² =2×1010= 2 \times 10^{10} cm² =2×106= 2 \times 10^6 m² =2= 2 km².
46Problem 46
Answer
(a) Both fully determined (b) Only if B's third side = 7 (c) Use cosine rule to find B's third side: c2=25+36−60cos⁡40°≈61−45.96≈15.04c^2 = 25 + 36 - 60\cos 40° \approx 61 - 45.96 \approx 15.04, c≈3.88c \approx 3.88. Not 7.
Full working
(a) A is SSS (uniquely defined). B is SAS (uniquely defined).

(b) They are congruent only if both produce the same triangle, which requires B's third side = 7.

(c) Cosine rule: c2=52+62−2(5)(6)cos⁡40°≈25+36−45.96≈15.04c^2 = 5^2 + 6^2 - 2(5)(6)\cos 40° \approx 25 + 36 - 45.96 \approx 15.04. c≈3.88c \approx 3.88. NOT 7 → A and B are different triangles.
47Problem 47
Answer
(a) A′(4,1),B′(13,1),C′(7,10)A'(4, 1), B'(13, 1), C'(7, 10) (b) 40.5
Full working
(a) Formula: A′=P+k(A−P)=(1,1)+3((2,1)−(1,1))=(1,1)+(3,0)=(4,1)A' = P + k(A - P) = (1, 1) + 3((2, 1) - (1, 1)) = (1, 1) + (3, 0) = (4, 1). Similarly for B, C.

(b) Area sf = 32=93^2 = 9. 4.5×9=40.54.5 \times 9 = 40.5.
48Problem 48
Answer
(a) 6 (b) 6 (c) Triangle: order 3, 3 lines
Full working
(a) A regular hexagon maps onto itself when rotated by 360°/6=60°360°/6 = 60°. Order of rotational symmetry = 6.

(b) 6 lines of reflective symmetry: 3 through opposite vertices, 3 through opposite sides.

(c) Equilateral triangle: order of rotational symmetry 3; 3 lines of reflective symmetry (each through a vertex and the midpoint of the opposite side).