pre-IB Mathematics
ANSWERS
Krzysztof Sikora
April 11, 2016
CC
BY
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c 2013 - 2016 by Krzysztof Sikora.
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2
C
Contents
1 Numbers
1.1 Primes, factors and divisibility . . . . . . . . . . . . . .
1.2 Fractions and decimals . . . . . . . . . . . . . . . . . . .
1.3 Subsets of real numbes set . . . . . . . . . . . . . . . . .
1.4 Absolute value . . . . . . . . . . . . . . . . . . . . . . .
1.5 Percentages . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Approximations. Decimal places and significant figures.
1.7 Exponents and roots . . . . . . . . . . . . . . . . . . . .
1.8 Expantions. Pascal’s triangle and binomial coefficients. .
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2 Logic
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3 Sets
3.1 Sets and subsets . . . . . . .
3.2 Venn diagrams . . . . . . . .
3.3 Operations on sets . . . . . .
3.4 Chapter review (sets & logic)
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12
12
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14
15
4 Statistics
4.1 Types of data . . . . . . .
4.2 Averages, range, quartiles
4.3 Groued data, frequencies .
4.4 Miscelaneous problems . .
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16
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5 Linear function
5.1 Basic concepts . . . . . . . . . . . . . . . .
5.2 Slope-intercept equation of a line . . . . . .
5.3 General equation of a line . . . . . . . . . .
5.4 Vectors . . . . . . . . . . . . . . . . . . . .
5.5 Simultaneous equations . . . . . . . . . . .
5.6 Applications of linear equations and vectors
5.7 Chapter review . . . . . . . . . . . . . . . .
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20
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6 Functions
6.1 Basic properties . . . . . . . . . . . . .
6.2 Transformations of graphs of functions
6.3 Equations and inequalities . . . . . . .
6.4 chapter review . . . . . . . . . . . . .
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25
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27
35
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7 Quadratic function
7.1 Solving quadratic equations . . . . . . . . . . . .
7.1.1 Factorisation . . . . . . . . . . . . . . . .
7.1.2 Completing the square . . . . . . . . . . .
7.1.3 Quadratic formula . . . . . . . . . . . . .
7.2 Parabola . . . . . . . . . . . . . . . . . . . . . . .
7.3 Applications of quadratics . . . . . . . . . . . . .
7.3.1 Quadratic inequalities . . . . . . . . . . .
7.3.2 Problems involving quadratics . . . . . .
7.3.3 Investigating graphs of rational functions
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38
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8 Trigonometry
8.1 Degrees and radians . . .
8.2 Trigonometric ratios . . .
8.3 Trigonometric functions .
8.4 Trigonometric equations .
8.5 Trigonometry in geometry
8.6 Arcs, sectors, segments . .
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42
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45
46
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3
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Contents
9 Geometry
9.1 Polygons . . . . . . . . .
9.2 Circles . . . . . . . . . .
9.3 Similarity . . . . . . . .
9.4 Solid geometry . . . . .
9.5 Miscellaneous problems
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46
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47
10 Numbers II
10.1 Factorials and binomial theorem . . . . .
10.2 Logarithms . . . . . . . . . . . . . . . . .
10.2.1 Algebra of logarithms . . . . . . .
10.2.2 Logarithmic equations . . . . . . .
10.2.3 Aplications . . . . . . . . . . . . .
10.3 Absolute value equations and inequalities
10.4 Complex numbers . . . . . . . . . . . . .
10.5 Mathematical induction . . . . . . . . . .
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49
49
11 Quadratics and polynomials
11.1 Vieta’s formulae for quadratics
11.2 Algebraic fractions . . . . . . .
11.3 Equation of a circle . . . . . . .
11.4 Polynomials . . . . . . . . . . .
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49
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4
Chapter 1
Numbers
1.1
Primes, factors and divisibility
Q1. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
(6) 3
(11) 3
(2) co-prime
(7) co-prime
(12) 13
(3) co-prime
(8) co-prime
(4) 7
(9) 26
Q2. (1) 3
(13) 12
(10) 2
(5) co-prime
(14) co-prime
Q3. (1) HCF: 6, LCM: 90
(12) HCF: 1, LCM: 4008003
(2) HCF: 16, LCM: 96
(13) HCF: 2, LCM: 2002000
(3) HCF: 12, LCM: 72
(14) HCF: 17, LCM: 595
(4) HCF: 18, LCM: 216
(15) HCF: 14, LCM: 210
(5) HCF: 18, LCM: 630
(16) HCF: 7, LCM: 245
(6) HCF: 8, LCM: 168
(17) HCF: 22, LCM: 1452
(7) HCF: 7, LCM: 504
(18) HCF: 125, LCM: 5000
(8) HCF: 9, LCM: 648
(19) HCF: 25, LCM: 10000
(9) HCF: 8, LCM: 448
(20) HCF: 16, LCM: 640
(10) HCF: 99, LCM: 198
(21) HCF: 24, LCM: 1728
(11) HCF: 1, LCM: 4006002
(22) HCF: 13, LCM: 936
Q4. (1) 2 × 32
(2) 23 × 3
(13) 24 × 5
(25) 54
(14) 34
(26) 23 × 53
2
(37) 232
(38) 26 × 32
(3) 2 × 3 × 5
(15) 2 × 3 × 5
(27) 3 × 11 × 61
(4) 25
(16) 32 × 11
(28) 22 × 23
(39) 33 × 52
(5) 22 × 32
(17) 32 × 13
(29) 25 × 3
(40) 22 × 132
(6) 2 × 3 × 7
(18) 27
(30) 22 × 52
(7) 24 × 3
(19) 26 × 3
(31) 24 × 32
(8) 2 × 33
(20) 22 × 72
(32) 2 × 34
(42) 32 × 112
(9) 23 × 7
(21) 23 × 33
(33) 25 × 7
(43) 32 × 53
2
(10) 3 × 7
6
2
4
(22) 2 × 11
(11) 2
(23) 2 × 3 × 7
(12) 23 × 32
(24) 24 × 52
(41) 7 × 11 × 13
(34) 2 × 3 × 5
2
(35) 172
(36) 22 × 112
(44) 23 × 172
(45) 22 × 32 × 52 × 7
Q5. (1) 1
(5) 3
(9) 6
(13) 3
(2) 2
(6) 5
(10) 4
(14) 7
(3) 3
(7) 3
(11) 1
(15) 0
(4) 2
(8) 0
(12) 6
(16) 10
5
Chapter 1. Numbers
Q6. (1) 3,5
(4) 3,9
(7) 2,4
(10) 2,4
(2) 2,3,4,5,6,10
(5) none
(8) 2,5,10
(11) 2,3,4,6,9
(3) 3,5
(6) 2
(9) 5
(12) 2,3,6
1.2
Fractions and decimals
Q7. (1) recurring
(6) terminating
(11) terminating
(16) recurring
(2) terminating
(7) recurring
(12) recurring
(17) recurring
(3) terminating
(8) terminating
(13) recurring
(18) terminating
(4) terminating
(9) recurring
(14) terminating
(19) terminating
(15) recurring
(20) terminating
(19) 3 19
30
(5) terminating
(10) terminating
6
Q8. (1) 1 25
(7) 12 18
(13) 14 14
99
(2) 2 17
20
(8) 75 78
(14)
(9) 2 94
(15)
4
(10) 3 33
(16)
(11) 5 59
(17)
(12) 7 19
33
(18)
(3)
(4)
(5)
(6)
1
160
3 38
5 58
7 11
25
Q9. (1) 0.625
(5) 0.024
107
333
202
4 333
68
5 333
23
45
19
1 90
(9) 0.2̇7̇
(20) 5 173
330
1
(21) 6 66
523
(22) 1 1665
817
(23) 7 3330
1
(24) 9 1665
(13) 0.416̇
(2) 0.1875
(6) 0.7̇
(10) 0.2̇85714̇
(14) 0.13̇6̇
(3) 0.875
(7) 0.2̇
(11) 0.83̇
(15) 0.0̇6̇
(4) 0.275
(8) 0.1̇8̇
(12) 0.27̇
(16) 0.1̇35̇
1.3
Subsets of real numbes set
Q10. (1) R \ Q
(4) Q
(7) Q
(10) R \ Q
(2) R \ Q
(5) Q
(8) Q
(11) R \ Q
(3) Q, Z, N
(6) Q
(9) Q, Z, N
(12) R \ Q
1.4
Absolute value
Q11. (1) 7.2
(2) 3.4
(3) 3.4 − π
√
(4) 5 − 2
(5)
3√
3−2 2
Q12. (1) 4.5; 0.5
(2)
(3)
1
3; 1
1
1
4 ; −1 4
Q13. (1) −1; −5
(2) −1; −3 32
(3)
1 34 ; 1 14
√
(11) 3 2 − 4
√
(12) 2 7 − 5
√
(13) 5 2 − 7
√
(14) 3 9 − 2
√
(15) 5 − 2 6
(6) 10 − π 2
√
(7) 2 3 − 3
(8) 16
(9) 8
(10) 0
√
(16) 2 10 − 6
(17) π − 3
(18) 13
√
(19) 7 − 4 3
√
(20) 10 − 3 11
(7) − 53 ; 1 45
(10) −0.5; 5.5
(8) no solution
(11)
(9) −1; −1 32
(12) no solution
(4) 3 45 ; 2 15
(7) no solution
(10)
(5) 1 31 ; − 13
(8) 0; − 67
(11) no solution
(4) −7; −3
(5)
(6)
(6)
2 31
1 34 ; −1 14
− 25 ; −3 35
(9)
Q14.
6
7 5
9; 9
1
5
11 ; 11
11 12
23 ; 23
(12) no solution
Chapter 1. Numbers
(1) − 21 , 12
1
2, 1
(5)
(3) 0, 2
(6)
(2)
Q15. (1) −1.5, 3
(2) − 34 , 8
1.5
(4) − 23 , 2
(7) −4, 0
1 3
6, 2
2
5, 4
(8) 2, 4 23
(9) −7, −2.5
(3) −1, 3
(5) 0, 2.4
(4) 1, 7
(6) − 32 ≤ x ≤
(7) − 14
1
2
(8) −1, 97
Percentages
Q16. (1) 48
(7) 22.4
(2) 67
Q21. (1) 28.9
(3) 42
(2) 0.65
(4) 68
(3) 65
(5) 65
(4) 14%
Q28. 1188 zl
(5) 12%
Q29. 20%
Q22. (1) 18%
Q30. 10%
Q17. (1) 6%
(2) 25%
Q25. 550
Q26. 528 zl
Q27. 182.50 zl
(3) 18%
(2) 12.5%
(4) 14.2%(14 16 %)
(3) 82%
(5) 225%
(4) 84%
Q32. 60 zl
(6) 240%
(5) 76%
Q33. 3120 zl
Q31. 10%
Q18. (1) 65
(6) 34%
(2) 85
(7) 42%
(3) 32
(8) 22.4%
Q35. 5.60 zl
(4) 480
(9) 456
Q36. 25%
(5) 25
(10) 255
Q37. increased by 12.5%
(11) 70
Q19. (1) 16%
(2) 16.7%
(12) 85
Q38. 301 zl
(3) 16%
(13) 65
Q39. 18%
(4) 16.7%
(14) 165
(5) 25%
(15) 65
(6) 20%
(16) 11.9
Q41. on average, 2.04%
(17) 44.1
Q42. 8.25%
Q20. (1) 33.3%
Q40. 378 zl
(18) 143
(2) 7.2
Q43. 7.96%
(19) 25%
(3) 237.5%
Q44. 800, final smaller by 14.5%
(20) 20%
(4) 15.2
1.6
Q34. 850 zl
(5) 77.8%
Q23. 45
Q45. 150%
(6) 22.4
Q24. 14%
Q46. 300, final smaller by 4%
Approximations. Decimal places and significant figures.
Q47. (1) 102.44
(2) 2.01
Q48. (1) 10
(2) 20
(3) 3.61
(5) 14.14
(7) 0.01
(4) 3.90
(6) 30.00
(8) 0
(3) 6710
(5) 30
(7) 650
(4) 340
(6) 430
(8) 110
7
Chapter 1. Numbers
Q49. (1) 2000
(3) 12000
(5) 0
(7) 43000
(4) 4000
(6) 130000
(8) 62000
(4) 0.0004
(7) 0.0021
(10) 0.00208
(2) 20000
(5) 4000000
(8) 0.00025
(11) 0.000255
(3) 0.002
(6) 25
(9) 25.4
(12) 45600000
(2) 6521000
Q50. (1) 20000
Q51. (1) 2335000 ≤ a < 2345000
(7) 18.95 ≤ a < 19.05
(2) 932.5 ≤ a < 933.5
(8) 32450 ≤ a < 32550
(3) 4045000 ≤ a < 4055000
(9) 0.09985 ≤ a < 0.09995
(4) 0.01225 ≤ a < 0.01235
(10) 0.002455 ≤ a < 0.002465
(5) 0.004495 ≤ a < 0.004505
(11) 0.4045 ≤ a < 0.4055
(6) 1995 ≤ a < 2005
(12) 0.06995 ≤ a < 0.07005
1.7
Exponents and roots
Q52. (1) 5
(6) 3
(11)
(2) 3
(7) 7
(12)
(3) 6
(8) 2
(13)
(4) 3
(9)
(14)
(5) 2
(10)
2
3
5
2
(15)
4
3
7
2
3
2
5
3
7
3
Q53. (1) 25
>
52
(7) (− 21 )5 <
−( 21 )6
(2) 25
>
(−2)5
(8) (−2)5 <
24
(3) (−2)5 <
(−2)4
(9) (−2)5 <
(−2)6
(4) 40
04
(10) (−2)5 >
−26
(5) (−2)5 >
(−2)7
(11) (−2)5 =
−25
(6) ( 12 )5
( 12 )6
(12) (−2)4 >
−24
>
>
(16)
3
2
(17)
5
2
(18)
4
3
(19)
3
2
(8) 57
(15) 219
(22) 5−2
(2) 36
(9) 32
(16) 38
(23) 74
(3) 777
(10) 33
(17) 9−1
(24) 3−7
(4) −211
(11) 220
(18) 90 = 1
(25) 3−13
(5) 230
(12) 416
(19) 21 = 2
(26) 2−19
(6) −235
(13) 228
(20) 26
(27) 5−17
(7) 36
(14) 311
(21) 2−35
(28) 2−4
(13) n3
(19) b−3
3
2
(20) x−2
Q54. (1) 314
Q55. h < f = g < b = d < a = e < c < i = j
Q56. f = j < g = i < a = b < c < d = e < h
Q57. (1) x3.5
(2) a
2
3
(7) b5
(8) c
14
3
(14) n
(3) a6
(9) y 3
(15) a5
(21) y −5.5
8
9
4
(16) a3
(22) t 3
8
(23) w 3
1
(4) a 3
(10) d
(5) a5
(11) s 5
(17) p−3
(6) x5
(12) t2
(18) s− 3
14
Q58.
8
1
(24) a−7
Chapter 1. Numbers
(1) x12
3 2
2a
2
(5)
4
(2) 3p
(6) 3s
(3) 3x
(7)
(4)
2 2
3x
(6)
(4) 32
(8)
(2)
(3)
(11)
3
(12)
64
27
2
5
16
81
27
512
(5)
16
9
1
32
(10)
3
2n
(8) 6w
Q59. (1) 0.00001
(9) 2a
(7)
(9) 1000000
(12)
16
3
3
16
16
49
(11)
2
5
(12)
81
16
(13)
1
3
(10)
(11)
√
Q60. (1) 2
(2) 4
(3)
1
8
(4) 27
(5)
81
16
(2)
(3)
(4)
2
2
√
3
3
√
6
3
√
6
2
Q62. (1) 6
√
(2) 4 2
(3) 15
Q63. (1)
(2)
1.8
√
√
(14) 1024
√
(5) 2 3
√
(6) 3 2
√
(7) 2 7
√
(8) 7 2
√
Q61. (1)
3
3
9
4
1
4
1
5
2
3
(6)
(7)
(8)
(9)
(10)
(9)
3−1
√
14 3
3
√
(13)
√
(10) 5 3
(11)
(12)
√
(7) 6 2
(5) 28
√
(6) 12 2
(8) 0
10
5
√
√
5 6
2
√
15 2
2
√
(4) −4 3
(14)
10
2
√
(15)
30
2
√
(10) −7 3
√
(11) −4 2
√
(12) 10 3
√
(9) 6 5
√
(3) 4 3 − 6
√
(4) −7 − 4 3
2+1
7 3
2p
3 2
2a
1 5
40 s
(5) −1
√
√
(6) 92 2 − 3 3
√
√
(7) −3 2 − 14
(8)
√
√
2 10− 15
5
Expantions. Pascal’s triangle and binomial coefficients.
Q64. (1) x2 − 2x + 1
(6) x2 − 10x + 25
(11) x2 − 43 x +
(2) x2 + 4x + 4
(7) x2 − 3x + 2.25
(12) 4x2 − 4x + 1
2
2
(3) x − 6x + 9
(8) x + 5x + 6.25
2
2
(9) x + x +
(4) x + 8x + 16
2
(5) x − 8x + 16
2
(10) x +
4
3x
(6)
(3) 4x2 s2 − 12xs4 + 9s6
(7)
(4) 9a4 b6 + 12a3 b7 + 4a2 b8
(8)
Q67. (1)
(2)
(3)
(4)
(5)
x+1
(19) 25x − 20x + 4
2
(20) 36x2 + 12x + 1
6x + 4
(5) 4p2 q 4 − 12p4 q 3 + 9p6 q 2
(2) x6 + 4x4 y 2 + 4x2 y 4
√
Q66. (1) 6 + 4 2
√
(2) 11 − 6 2
√
(3) 43 − 24 3
1 2
4x −
9 2
4x +
2
(18)
(15) 9x + 12x + 4
Q65. (1) a4 − 2a2 b + a2 b2
(17)
2
(14) 4x − 12x + 9
4
9
(16) 4x2 + 20x + 25
2
(13) 9x + 6x + 1
1
4
+
4
9
√
(4) 30 − 12 6
√
(5) 122 − 56 3
√
(6) 182 + 96 3
s2 t4
3 3
4 2
4 − 2s t + 4s t
9 2 2
4 2 10
2 6
4 a c + 2a c + 9 a c
4
9 6 6
4 3
9 a − a c + 16 a c
√
(7) 99 + 60 2
√
(8) 201 − 126 2
√
(9) 304 − 60 15
a2 + 2ab + 2ac + b2 + 2bc + c2
a4 − 2a2 b + 4a2 c + b2 − 4bc + 4c2
a4 + 2a3 b + 3a2 b2 + 2ab3 + b4
a4 + 2a3 − a2 − 2a + 1
25x2 y 2 + 20x2 y + 4x2 − 30xy 2 − 12xy + 9y 2
(6)
(7)
(8)
(9)
(10)
9
√
(10) 114 + 36 10
√
(11) 55 − 22 6
(12)
49
2
9a2 b2 + 12a2 bc + 4a2 c2 − 6ab2 c − 4abc2 + b2 c2
4a4 b2 + 12a3 b3 + 9a2 b4 + 4a2 b + 6ab2 + 1
9s2 t2 − 12s2 t + 4s2 − 12st3 + 8st2 + 4t4
√
√
√
9+4 2+4 3+2 6
√
√
√
9+4 2−4 3−2 6
√
√
√
(11) 10 + 2 6 + 2 10 + 2 15
√
√
√
(12) 11 + 6 2 − 4 3 − 2 6
√
(7) 54 + 30 3
√
√
(8) 11 2 + 9 3
√
√
(9) 9 3 − 11 2
√
√
(10) 132 3 − 162 2
√
√
(11) 21 3 + 15 6
√
√
(12) 12 6 − 20 2
Q68. (1) a3 − 3a2 c + 3ac2 − c3
(2) a6 + 6a4 b + 12a2 b2 + 8b3
(3) a6 − 3a5 b + 3a4 b2 − a3 b3
(4) a6 + 6a5 + 12a4 + 8a3
(5) 8x6 y 3 + 36x5 y 4 + 54x4 y 5 + 27x3 y 6
(6) 27x6 y 3 − 54x5 y 4 + 36x4 y 5 − 8x3 y 6
Q69. (1) a4 − 4a3 c + 6a2 c2 − 4ac3 + c4
(2) x4 + 8x3 y + 24x2 y 2 + 32xy 3 + 16y 4
√
(3) 193 − 132 2
(4) a5 − 5a4 c + 10a3 c2 − 10a2 c3 + 5ac4 − c5
(5) x5 + 10x4 y + 40x3 y 2 + 80x2 y 3 + 80xy 4 + 32y 5
√
(6) 843 − 589 2
√
(7) 485 + 198 6
(8) 64a6 b6 − 576a6 b5 + 2160a6 b4 − 4320a6 b3 + 4860a6 b2 − 2916a6 b + 729a6
Q70. (1) 63
(2) 0
Q71. (1)
(2)
(3)
(4)
(5)
(6)
(7)
(3) 30
(5)
15
16
(7) 11
(4) 58
(6) 28
(8) 24
P20
r=1 r
P22
2r
Pr=1
n
r=1 r
P26
r=1 (2r − 1)
P10 r
r=0 2
P9
3−r
r=1 3
P40
r=1 (5 − 2r)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
P20
r=1 (4r − 1)
P16
r=1 (13 − 3r)
P17 1+r
r=1 ( 3 )
P11
r
r=0 (−2)
P8
r
r+1
)
r=1 (3 (−1)
P12
−r
r=0 (−2)
P20 (−1)r+1
)
r=1 (
r
Q72. −4320
Q81. −1 + 3x − 10x3
Q73. −20
Q82. −1 − 5x + 40x3
Q74.
560
3
Q75. 160
(15)
P25
(16)
P11
(17)
P99
(18)
P26
(19)
P21
(20)
P18
r=1
r2
r=1 (r
3
(−1)r+1 )
r
r=1 r+1
r=1
(−1)r+1
2r
r=1 (r(r
+ 1))
2r
r=1 (2r−1)(2r+1)
Q83. 2 + 12x − 21x2
Q84. 360
Q76. 840
Q77. −489888
Q85. − 1567
9
Q78. 2048x11 − 16896x10 + 63360x9
Q86. 2
Q79. −1152x2 + 1152x − 512
Q87. ±3
Q80. 8 − 20x + 18x2
Q88. −1
Chapter 2
Logic
Q1. (1) For every real number there exists an integer smaller than the real number.
(2) All natural odd powers of −1 are equal −1.
(3) All natural even powers of −1 are equal 1.
(4) There exists such natural number n that each real number s is greater or equal to the sum of n and s.
10
Chapter 2. Logic
(5) For every positive real number there is exactly one real number whose square is equal to the number
considered.
(6) If 2 divides a natural number then 4 divides it, too.
(7) There is a natural number such that if it is divisible by 2 then it is divisible by 4, too.
(8) There exists a natural number that is not divisible by 2 but it is divisible by 4.
(9) For every integer its power of 2 is an integer, too.
(10) A number is rational whenever its power of 2 is rational.
(10) ∀x ∈ R (x < x2 ) ⇒ (x < 0), FALSE
Q2. (1) ∀n ∈ N n ∈ Z or n ∈ N ⇒ n ∈ Z, TRUE
(2) ∀n ∈ Z n ∈ Q or n ∈ Z ⇒ n ∈ Q, TRUE
(11) ∀x, n ∈ Z x2n > 0, FALSE
(3) ∃x ∈ R ¬(n > 0) ∧ ¬(n < 0), TRUE
(4) ∃n ∈ N ¬(n > 0), TRUE
(12) ∃x, n ∈ Z x2n+1 ≤ 0, TRUE
(13) ∀x, y ∈ R (x < y) ⇒ (x2 < y 2 ), FALSE
(5) ∀n ∈ Z (4 | n) ⇒ (2 | n), TRUE
(14) ∃x ∈ R+ x < x2 ,TRUE
(6) ∀n ∈ Z ((2 | n) ∧ (3 | n)) ⇒ (6 | n), TRUE
(7) ∀x ∈ R+ ∃y ∈ R x = y 2 , TRUE
(15) ∀x ∈ R+ x > x2 ,FALSE
(8) ∀x ∈ Z+ ∃y ∈ Z x = y 2 , FALSE
(16) ∀n ∈ Z (2 | n) ⇒ (4 | n2 ), TRUE
(9) ∀x, y ∈ R ∃z ∈ Z+ z < |x − y|, FALSE
(17) ∀n ∈ Z (4 | n) ⇒ (16 | n2 ), FALSE
Q3. (1) tautology
(6) tautology
(11) tautology
(16) tautology
(2) contradiction
(7) tautology
(12) tautology
(17) tautology
(3) tautology
(8) tautology
(13) tautology
(18) tautology
(4) tautology
(9) tautology
(14) tautology
(10) tautology
(15) tautology
Q4. (1) tautology
(2) tautology
(3) tautology
(4) tautology
(5) tautology
(6) tautology
Q5. (1) ¬p ∧ ¬q
(5) ¬p ∨ q
(9) ¬p ∧ ¬q
(5) contradiction
(7) contradiction
(13) ∃x (¬p ∨ q)
(2) p ∧ ¬q
(6) p ∨ q
(10) ∃x ¬p
(14) ∀x (p ∧ q)
(3) p ∧ q
(7) p ∧ q
(11) ∃x (p ∧ ¬q)
(15) ∀x (p ∧ ¬q)
(4) ¬p ∨ ¬q
(8) ¬p ∧ ¬q
(12) ∃x (¬p ∧ ¬q)
(16) ∀x (¬p ∨ ¬q)
Q6. (1)
(2)
(3)
(4)
(5)
(6)
(7)
∃x ∈ R (x2 ≤ 0)
∃x ∈ N ((x ≤ 0) ∧ (x 6= 0))
∃x ∈ Z ((2 | x) ∧ (4 - x2 ))
∃x ∈ Z ((2 - x) ∧ (2 | x))
∃x ∈ N ((x < 0) ∨ (x2 ≤ x))
∃x ∈ Z ((x > 0) ∧ (x 6∈ N))
∃x ∈ R ((x ≥ 0) ∧ (x ≤ 0))
(8) ∃x ∈ R ((x2 ≤ x − 1) ∨ (x2 < −x))
(9) ∀x ∈ N (x > 0)
(10) ∀x ∈ N ((x ≤ 0) ∧ (x 6= 0))
(11) ∀x ∈ Z ((x2 + 4x = 0) ∧ (x ≥ 0))
(12) ∀x ∈ R ((x2 6∈ Z) ∨ (x ∈ Z))
(13) ∀x ∈ R ((x2 ≥ 0) ∧ (|x| ≥ 1))
Q7. (1) Quadrilateral ABCD is neither a rhombus nor a rectangle.
(2) Quadrilateral ABCD is not a parallelogram or it does not have an axis of symmetry.
(3) A triangle has all sides of equal length and not all of its angles are equal to 60◦ .
(4) There is an integer that has exactly three prime factors and that is not a square of an integer.
(5) There is an integer that is divisible by 2 while its square is not divisible by 4.
(6) There is an integer divisible by 6 whose square is not divisible by 36.
(7) There is an integer divisible by 2 whose square is not divisible by 4.
(8) There is an integer divisible by both 2 and 3 that is not divisible by 6.
(9) There exists a real number that is neither positive nor negative.
(10) There exists a real number whose square is negtive.
Q8.
11
(1) ¬(p ∧ q)
(4) p ∨ ¬q
(7) ¬(p ∧ ¬q)
(10) ¬p ∧ ¬q ∧ p
(2) ¬p ∧ q
(5) p ∨ q
(8) ¬(¬p ∨ q)
(11) ¬p ∨ ¬q ∧ p
(3) p ∧ q
(6) ¬(p ∨ q)
(9) ¬(¬p ∧ ¬q)
(12) ¬p ∧ ¬q ∨ p
Chapter 3
Sets
3.1
Sets and subsets
Q1. (1) {−5, −4, 4, 5}
(7) {−1, 0, 1, 3}
(2) {−1, 0, 1}
(8) {−1, 0, 1, 2, 3, 4}
(3) {−4, −3, −2, 2, 3, 4}
(9) {−3, −1, 1}
(4) {−4, −3, 3, 4}
(10) {−2, − 13 , 1}
(5) {0, 1, 2, 3, 4}
(11) ∅
(6) {2, 3, 4, 5}
(12) {−8, −7, −6, −5, −4, −3, −2, −1, 0, 1, 2}
Q2. (1) {0, 1, 2, 3, 4}
(7) ∅
(2) {−8, −7, −6, −5, −4, −3, −2, −1}
(8) {−5, −4, −3, −2, −1}
(3) {−2, −1, 0, 1, 2}
(9) {0, 1, 2, 3}
(4) ∅
(10) {−5, −4, −3, 3, 4, 5}
(5) Z
(11) {−6, 0, 6}
(6) N
(12) {−8, −4, 0, 4, 8}
Q3. ∅, {1}, {3}, {1, 3}
Q4. ∅, {2}, {5}, {8}, {2, 5}, {2, 8}, {5, 8}, {2, 5, 8}
Q5. {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}
Q6. {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 3}, {3, 4}
Q7. {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5}
Q8. {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5}
Q9. {0, 2}, {0, 4}, {0, 8}, {0, 2, 4}, {0, 2, 8}, {0, 4, 8}, {0, 2, 4, 8}
Q10. 15
Q11. 16
Q12. 8
3.2
Venn diagrams
A
Q13. (1)
B
A
A ∩ B0
(2)
12
B
A ∪ B0
Chapter 3. Sets
A
B
A
B
(A ∩ B 0 )0
(3)
A
C
(11)
B
A
A0 ∪ (B ∩ C)
B
(A \ B)0
(4)
A
B
C
(12)
A
(A ∪ B) \ C
B
(5)
(A ∪ B) \ (A ∩ B)
A
B
C
(13)
A
A ∪ (B \ C)
B
(A ∪ B) \ A
(6)
A
B
C
(14)
(7)
A
(A ∩ B) \ (A ∪ B)
A
(A ∩ B) \ C
B
B
C
(15)
A ∩ (B \ C)
B \ (A ∩ B)
(8)
A
A
B
C
(9)
A
C
C
(16)
(A ∪ B) ∩ C 0
B
(10)
B
A
(A ∩ B) ∪ C
A \ (B ∩ C)
B
C
(17)
A \ (B ∪ C)
Q14. (1) true
(3) false
(5) false
(7) true
(9) true
(2) false
(4) false
(6) true
(8) true
(10) true
Q15. (1)
ski
25
Q16. (1)
biology
9 38 − x
(2) 25
snowboard
6
(3) 17
17
(2) 11
physics
(3) 27
x 14 − x
9 + 38 − x + x + 14 − x = 50
13
Chapter 3. Sets
Q17. (1) 6
(2) 2
Q18. 1
Q20. (1) 28
(2) 3
Q21. (1) 12
(2) 14
Q19. (1) 3
(2) 8
Q22. (1) 160
(2) 12
...............................................................................................................
Q23. (1) 20
(2) 4
(3) 11
Q24. (1) 30
(2) 0
(3) 10
Q25. (1) 12
(4) 2
(2) 1
Q26. Biology - 5, Chemistry - 1, English - 2
Q27. 40
Q28. 4
Q29. 74
3.3
Operations on sets
Q30. (1) A = {1, 3, 5, 7, 9, 11}
(8) {0, 2, 4, 6, 8, 10, 12}
(15) {12}
(2) B = {0, 2, 4, 6, 8, 10}
(9) {1, 3, 5, 7, 9, 11, 12}
(16) 6
(3) {1, 5, 7, 11}
(10) {0, 1, 2, . . . , 10, 11}
(17) 6
(4) {3, 9}
(11) {3, 9}
(18) 4
(5) ∅
(12) {12}
(19) 0
(6) {3, 9}
(13) {12}
(20) 8
(7) {0, 6}
(14) {12}
(21) 9
Q31. (1) 11
(2) {2, 4, 12}
(3) {3, 7, 9}
(4) {0, 6, 8, 10, 14}
Q32. (1) 11
(2) {3, 6, 18}
(3) {4, 10, 13}
(4) {0, 9, 12, 15, 21}
Q33. (1)
(2)
(3)
(4)
(5)
(6)
(7)
false
false
true
false
true
false
11
(14) {±14, ±13, ±11, ±10,
±7, ±5, ±2, ±1}
(8) 7
(9) 3
(10) {±15, ±9, ±6, ±3}
(15) B
(11) {−8, −4, 4, 8}
(16) C
(12) ∅
(17) A
(13) ∅
(18) C
(6) {2, 4, 8, a, b, c, d}, a, b, c, d ∈ U \ C
Q34. (1) 2
(2) 10
(7) {1, 12}
(3) 8
(8) {12}
(4) {6, 8, a, b}, where a, b ∈ U \ A
(5) {1, 6, 10, a, b, c}, a, b, c ∈ B
(9) {1, 12}
Q35. (1) [−5, 5]
−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 9 10 11
(2) ] − ∞, −5[∪]5, 6]
−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 9 10 11
(3) [6, 11[
−7−6−5−4−3−2−1 0 1 2 3 4 5 6 7 8 9 10 11
14
Chapter 3. Sets
Q36. (1) {−3}
−8−7−6−5−4−3−2−1 0 1 2 3 4
(2) [−8, −3[
−8−7−6−5−4−3−2−1 0 1 2 3 4
(3) ] − 3, 2[
−8−7−6−5−4−3−2−1 0 1 2 3 4
Q37. (1) ] − ∞, 0[∪]0, 3[∪]3, 9[
−2−1 0 1 2 3 4 5 6 7 8 9
(2) {−2, 5}
−2−1 0 1 2 3 4 5 6 7 8 9
(3) ] − 2, 0[∪]3, 5[∪[7, 9]
−2−1 0 1 2 3 4 5 6 7 8 9
(4) ] − ∞, −2[∪]0, 3[∪]5, 7[
−2−1 0 1 2 3 4 5 6 7 8 9
Q38. (1) ∅
(2) ] − ∞, −3]
−3−2−1 0 1 2 3 4 5 6 7
(3) [−1, 3]
−3−2−1 0 1 2 3 4 5 6 7
(4) ] − 3, 1]∪]3, 5]∪]7, +∞]
−3−2−1 0 1 2 3 4 5 6 7
(5) ] − 3, −1]
−3−2−1 0 1 2 3 4 5 6 7
(6) B =] − 1, 3]∪]5, 7]
−3−2−1 0 1 2 3 4 5 6 7
(7) ]3, 5]∪]7, +∞[
−3−2−1 0 1 2 3 4 5 6 7
(8) ] − ∞, −3] ∪ [2, +∞[
−3−2−1 0 1 2 3 4 5 6 7
(9) ] − 3, −1]
−3−2−1 0 1 2 3 4 5 6 7
(10) ] − ∞, −3]
−3−2−1 0 1 2 3 4 5 6 7
Q39. (1) (A ∪ B)0 = A0 ∩ B 0
(5) A0 ∪ B 0 = (A ∩ B)0
(2) (A ∩ B)0 = A0 ∪ B 0
(6) A0 ∩ B 0 = (A ∪ B)0
(3) (A0 ∩ B)0 = A ∪ B 0
(7) A0 ∪ B = (A ∩ B 0 )0
(4) (A ∪ B 0 )0 = A0 ∩ B
(8) A ∩ B 0 = (A0 ∪ B)0
3.4
Chapter review (sets & logic)
Q1. (1) tautology
(2) tautology
Q2. (1) True: all values but p = q = r = 0.
(2) False: p = q = r = 0.
Q3. (1)
(i)
(ii)
(iii)
(iv)
∀x ∈ Z (2 | x) ⇒ (4 | x)
∃x ∈ Z (2 | x) ∧ (4 - x)
There exists an even number that is not divisible by 4.
negation
(2)
(i)
(ii)
(iii)
(iv)
∀x ∈ Z (2 | x) ⇒ (4 | x2 )
∃x ∈ Z (2 | x) ∧ (4 - x2 )
There exists an even number whose square is not a multiple of 4.
the statement
15
(3)
(i)
(ii)
(iii)
(iv)
∀x ∈ R ∃n ∈ Z n2 ≥ x3
∃x ∈ R ∀n ∈ Z n2 < x3
There is a real number such that its cube is larger than a square of any integer.
the statement
(4)
(i)
(ii)
(iii)
(iv)
∀n ∈ Z ((12 | n) ∨ (18 | n)) ⇒ (9 | n)
∃n ∈ Z ((12 | n) ∨ (18 | n)) ∧ (9 - n)
There is an integer that is a multiple of 12 or of 18 but not of 9.
negation
Q8. e.g. (A\(B∪C))∪(C \(A∪B)) or (A∪C)\(A∩C)\B
Q4. (1) (¬p ∧ ¬q) ∨ r
(2) ¬(p ∧ (¬q ∨ r))
Q5. (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
] − ∞, −4] ∪ {−3}
{−1, 0, 1, 2}
{3}
{0, 1, 2, 3}
D =] − ∞, −5[∪[3, +∞[
[−5, 3[
] − ∞, −4[∪[−3, 3[∪]3, +∞[
[−5, 3[
W
R
25 − 16 = 9
16
16 − 9 = 7
(30 − 7 = 23)
Q9.
Q6. {a}, {◦}, {4}, {a, ◦}, {a, 4}, {◦, 4},
{a, ◦, 4}.
E has 32 subsets.
M
U
6−x
Q7. A0 ∩ (B ∪ C)0
A
18 − (10 − x)
B
0
x
4−x
C
22 − (12 − x)
(A \ B)0 ∩ C
S
A
8−x
B
Q10.
15 − (22 − (12 − x))
From total no of students equal 38 we obtain x = 3
and hence:
(a) 13
(b) 3
C
(c) 2
Chapter 4
Statistics
4.1
Q1. (1)
(2)
(3)
(4)
Types of data
qualitative
quantitative
quantitative
quantitative
(5)
(6)
(7)
(8)
quantitative
qualitative
quantitative
quantitative
(9)
(10)
(11)
(12)
16
qualitative
quantitative
quantitative
quantitative
(13) quantitative
Chapter 4. Statistics
Q2. (1)
(2)
(3)
(4)
(5)
4.2
Q3.
(6)
(7)
(8)
(9)
(10)
discrete
continuous(?)
discrete
continuous(?)
continuous(?)
(11)
(12)
(13)
(14)
(15)
continuous
continuous
continuous
continuous
continuous
(16) discrete
discrete
continuous
discrete
discrete
continuous
(17) continuous
Averages, range, quartiles
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
mean
142
1(0.9997)
8.5
1000(999.75)
3.25
83.4(83.375)
1.82
3.1
Q4. (1) (7)
(2) (3)
Q5. 169cm
mode
141
0.995
7
998
4
90
3
3
median
141
1
8.5
999.5
3.5
84.5
2
3
(3) (1)
Q6. 61.6kg
Q1
141
0.995
7
998
2
79.5
0
2
Q2
143
1.003
9.5
1001
4
88.5
3
4
(4) (5)
(5) (8)
Q7. 187.2cm
Q10. (1) 55.5
range
5
0.01
3.5
5
5
18
3
6
(7) (4)
(8) (6)
Q9. 64.1kg
(3) 46.7
Q12. 161.4cm
Q14. (1) 172.5
(6) (2)
Q8. 169cm
(2) 56.5
Q11. 2 min 22 sec
I.Q.R.
2
0.008
2.5
3
2
9
3
2
Q13. 71.3kg
(2) 170.5
(3) 22
Q15. (1) mean: 73.8, mode: none, median: 84; mean takes all data into consideration, median is not affected
by the ”outliers”
(2) mean: 46.6, mode: 25, median: 22; median best, mode reasonable, too
(3) mode: trainers (the only)
Q16. (1) 70
(2) 65
(3) 79
Q17. 164, 178
Q18. (1)
(2)
(3)
(4)
(5)
a = 3, b = 5
a = 7, b = 8
a = 3, b = 9
a = 3, b = 6 or a = 4, b = 5
a = 13, b = 14, c = 16, d = 17, e = 19
Q19. mode = 1.99zl, median = 2.05zl, mean = 2.14zl
Q20.
test
1
2
4.3
Q21.
h
152
153
157
160
163
165
168
range
71
62
mode
21
46
median
44
67
Q1
29.5
48.5
Q3
76.5
81.5
I.Q.R.
47
33
Groued data, frequencies
f
4
2
3
3
6
3
3
h×f
608
306
471
480
978
495
504
∴ mean =
3842
24
≈ 160
17
mean
53.2
66.7
Chapter 4. Statistics
Q22. a = 12, b = 4.
Q23. 6
Q24. 25
Q25. a = b = 5, median = 11
Q26. a = 5, b = 8
Q27. (1) 3
(2) 6
Q28. (1) 6
(2) 15
(3) 5
(5) 1
(4) Q1 = 5, Q3 = 6
(6) 5.44
(3) 15
(5) 4
(4) Q1 = 13, Q3 = 17
(6) 13.4
Q29. a = b = 9
Q30. (1)
class
130 < h ≤ 140
140 < h ≤ 150
150 < h ≤ 160
160 < h ≤ 170
170 < h ≤ 180
180 < h ≤ 190
f
2
14
31
33
16
4
100
h×f
270
2030
4805
5445
2800
740
16090
h (approx.)
135
145
155
165
175
185
h
h ≤ 140
h ≤ 150
h ≤ 160
h ≤ 170
h ≤ 180
h ≤ 190
c.f.
2
16
47
80
96
100
(2) 160 < h ≤ 170
(3) 150 < Q1 ≤ 160, 160 < Q3 ≤ 170
(4) 161 cm
Q31. (1) (6),(7)
(4) (5)
(6) (a)
(b)
(c)
(d)
(5) (6)
(7) no
(2) (7), 6
(3) (7), 4
7
0
4
1
(8) yes
(9) (a)
(b)
(c)
(d)
Q32. (1) A: 2.2, B: 2.45
(2) A&B: 2
(3) A: 68, 85, 94, 100, B: 8, 25, 59, 75, 88, 100
(4)
median
Q1
Q3
A
2
1
3
B
2
1.5
3.5
A
B
(5)
0
(6) T - true, N
A
(a) T
(b) N
(c) T
(d) N
(e) T
(f) T
1
2
3
4
5
- does not have to be true, F - false
B
T
N
T
T
T
T
Q33.
18
5
0
3
1
Chapter 4. Statistics
(a) FALSE
Q34. (1) 4
(b) FALSE
(c) FALSE
(2) 68
(3) 43-16=27
Q35. (1) 50
(2) 17
(3) 35
(4) 68
(10)
(5) 30th : 81 − 83, 70th : 119 − 120
(6) 12 (11-13)
(7) 125
(8) 75
(9) 50%
time
0 < t ≤ 20
20 < t ≤ 40
40 < t ≤ 60
60 < t ≤ 80
80 < t ≤ 100
100 < t ≤ 120
120 < t ≤ 140
140 < t ≤ 160
160 < t ≤ 180
180 < t ≤ 200
(d) FALSE
no of students
2
4
11
18
25
25
18
11
4
2
(11) 100 minutes
time
0<t≤4
4<t≤8
8 < t ≤ 12
12 < t ≤ 16
16 < t ≤ 20
20 < t ≤ 24
24 < t ≤ 28
28 < t ≤ 32
32 < t ≤ 36
36 < t ≤ 40
Q36. (1) 20-21 minutes
(2) 11
(3) 24
(4) 16
(5) mean ≈ 20
no of students
1
2
3
4
5
5
4
3
2
1
Q37. (1) 22hrs
(2) 8 or 7
(3) 15th : 17, 65th : 24
(4) 27
(5) 16
(6) c = 16, d = 27
Q38. (1) 164cm
(2) 173-156=17
(3) 173
(4) 154
(5) c = 154, d = 173
4.4
Miscelaneous problems
Q39. (1) mean = 171, range = 12
(2) mean = 172, range = 10 or 12 or 14
Q43. a = 6, b = 7, c = 7, d = 10
Q44. a = 4, b = 6, c = 8, d = 8
(3) mean = 171.5, range = 10 or 12 or 14
Q45. a = 6, b = c = d = 8, e = f = g = 9
Q40. 960 ml
Q46. x ∈ {8, 9, 10, 11, 12, 13}, y = 14 − x
Q41. (1) 3.25l
(2) 4.5l
Q42. (1) 26.40 zl
(2) 24 zl
Q47. x = 3
Q48. median =9.4, Q1 = 9.2, Q3 = 9.85,
range = 1.3, I.Q.R. = 0.65,
mode = 9.2, mean ≈ 9.468
19
Q49. 4,6,6,6,7
(4) 86
Q50. 3,4,5,6,6,7
(5) 67
(6) c = 67, d = 86
Q51. 2,4,5,5,6,6,7
Q52.
grade
4
5
6
7
min no
1
0
3
2
(7) 76kg
max no
2
3
5
2
Q54. (1) 4000AM
(2) 1900AM
(3) 30th : 3400AM, 70th : 4900AM
(4) 6500AM
Q53. (1) 76kg
(5) 2800AM
(2) 10-11kg
th
(3) 30
: 72kg, 70
(6) c = 2800, d = 6500
th
: 79kg
(7) 4400AM
Chapter 5
Linear function
5.1
Basic concepts
Q1. (1) (6, 7)
(2) (8, −1)
Q2. (1) 5
(2) 13
(3) 10
5.2
(3) (3, −5)
(5) (7, −1.5)
(7) (0.5, 6.5)
(4) (−1, 2)
(6) (−8, −3)
(8) (34, −7)
(4)
(5)
(6)
√
4 2
√
3 10
√
5 2
√
(7) 5 5
√
(8) 4 5
√
(9) 10 2
(10) 17
Slope-intercept equation of a line
Q3. (1)
(2)
4
3
− 43
(3) 1
(4)
1
3
(5) − 13
(7) −2
(6) 2
(8) − 32
(6) e.g.(−1, −3), (1, −4), (3, −5)
Q4. (1) e.g.(3, 2), (6, 4), (9, 6)
(2) e.g.(2, −3), (4, −6), (6, −9)
(7) e.g.(2, 0), (6, 3), (10, 6)
(3) e.g.(2, 0), (3, 2), (4, 4)
(8) e.g.(2, 0), (5, −5), (8, −10)
(4) e.g.(−3, −2), (−2, −5), (−1, −8)
(9) e.g.(−5, −5), (−2, 2), (1, 9)
(10) e.g.(−3, 3), (2, 1), (7, −1)
(5) e.g.(6, 0), (10, 1), (14, 2)
Q5. Use a GDC to check your answers.
Q6. (1) yes
(2) yes
(3) no
(4) yes
(5) yes
(6) yes
(7) yes
(8) no
(9) yes
(10) yes
Q7. (1) 7
(2) 6.5
(3) 3
(4) 0
(5) 5.5
(6) −3
(7) 14
(8) −6
(9) 6
(10) −4.5
Q8. (1) y = 2x − 3
(2) y = −3x − 1
(3) y = 41 x + 2
(4) y = − 12 x − 4
(5) y = 43 x − 6
(6) y = − 35 x + 5
(7) y = 37 x + 3
(8) y = − 52 x + 2
(9) y = 23 x + 1
(10) y = − 23 x + 3
20
Chapter 5. Linear function
Q9. (1) y = 2x − 2
(2) y = −3x + 3
(3) y = 14 x − 32
(4) y = − 12 x +
(5) y = 34 x − 21
(6) y = − 53 x +
5
2
(7) y = 73 x + 1
(8) y = − 52 x −
(9) y = 23 x + 3
1
3
(10) y = − 23 x − 4
8
5
Q10. (1) y = − 12 x + 3
(2) y = 13 x − 7
(3) y = −4x − 14
(4) y = 2x + 6
(5) y = − 43 x + 12
(6) y = 53 x + 5 52
(7) y = − 73 x − 1 67
(8) y = 25 x − 20 12
(9) y = − 23 x + 12
Q11. (1) m > 0
(2) m > −1
(3) m > 2
(4) m > 2.5
Q12. (1) m > 4
(2) m < − 43
(3) no such m
(4) −2 < m < 2
Q13.
(10) y = 23 x + 3 23
(i) 91.4 km
h
(ii) between 1.5 and 2hrs: 96 km
h
(iii) between 1.5 and 2.5hrs: 95 km
h
5.3
General equation of a line
Q14. (1) y = − 32 x − 2 12
(2) y = −2x + 1 13
(3) y = 12 x + 1 12
(4) y = − 43 x − 31
(5) y = 3x − 2
(6) y = 53 x − 1 31
(7) y = 25 x − 3 12
(8) y = − 37 x − 12
(9) y = − 14 x + 12
(10) y = 32 x + 1 13
Q15. (1) 6x − 3y − 4 = 0
(2) 3x + y − 2 = 0
(3) x − 4y + 2 = 0
(4) x + 2y + 3 = 0
(5) 9x − 12y + 4 = 0
(6) 5x + 3y − 4 = 0
(7) 14x − 6y − 3 = 0
(8) 4x + 10y − 35 = 0
(9) 4x − 6y − 15 = 0
(10) 9x + 6y + 8 = 0
Q16. (1) 4x + 6y + 15 = 0
(2) 6x + 2y − 3 = 0
(3) x − 2y + 3 = 0
(4) 5x + 6y + 4 = 0
(5) 3x − y − 2 = 0
(6) 5x − 3y − 4 = 0
(7) 4x − 5y − 35 = 0
(8) 14x + 6y + 3 = 0
(9) x + 4y − 2 = 0
(10) 9x − 8y + 6 = 0
(6) 5x − 3y + 29 = 0
Q17. (1) 4x + 6y − 20 = 0 or 2x + 3y − 10 = 0
(2) 6x + 3y = 0 or 2x + y = 0
(7) 4x − 10y − 2 = 0 or 2x − 5y − 1 = 0
(3) x − 2y − 1 = 0
(8) 14x + 6y − 138 = 0 or 7x + 3y − 69 = 0
(4) 9x + 12y + 27 = 0 or 3x + 4y + 9 = 0
(9) x + 4y + 2 = 0
(5) 3x − y − 14 = 0
(10) 9x − 6y + 42 = 0 or 3x − 2y + 14 = 0
Q18. (1) 6x − 4y − 4 = 0 or 3x − 2y − 2 = 0
(6) 3x + 5y − 3 = 0
(2) 3x − 6y − 45 = 0 or x − 2y − 15 = 0
(7) 10x + 4y + 24 = 0 or 5x + 2y + 12 = 0
(3) 2x + y + 8 = 0
(8) 6x − 14y − 26 = 0 or 3x − 7y − 13 = 0
(4) 12x − 9y + 36 = 0 or 4x − 3y + 12 = 0
(9) 4x − y − 43 = 0
(5) x + 3y − 18 = 0
Q19. (1) 10
√
(2) 3 13
√
(3) 2 5
(4) 19.5
(10) 6x + 9y + 15 = 0 or 2x + 3y + 5 = 0
(5)
√
√
(9) 2 17
√
(10) 3 29
√
(11) 3 17
√
(12) 12 401
10
√
9
(6) 2 45
√
(7) 2 116
√
(8) 2 13
21
√
(13) 2 101
√
(14) 2 37
(15)
√
13
Chapter 5. Linear function
5.4
Vectors
−4
4
2
(2)
4
3
6
1
(4)
−3
Q20. (1)
2
6
−3
(6)
7
−5
−10
8
(8)
−10
(3)
(5)
(7)
(3) (1, 0)
(5) (1, 3)
(7) (−6, 2)
8
3
−4
(10)
8
(9)
Q21. Find point B.
(1) (−18, −1)
(2) (3, −7)
−5
Q22. (1)
4
−2
(2)
−6
−8
Q23. (1)
11
5
(2)
−7
(6) (−1, −9)
−1
(3)
−1
1
(4)
−13
−33
(5)
−2
−7
(6)
−13
(4) (6, 10)
−3
3
−1
(4)
1
(3)
(9) (−5, 10)
(8) (0, −1)
(10) (15, 4)
1
−10
−12
−9
(5)
(6)
13
5
15
(8)
−20
2
2
−23
(10)
−37
(7)
(9)
~ v
u+
Q24. (1)
~ v
u−
~u
~u
~ v
u−
~ v
u+
~v
~v
~ v
u+
(7)
(2)
~v
~v
~u
~ v
u−
~ v
u−
~u
~ v
u+
~v
(8)
~ v
u+
~ v
(3) u −
~ v
u+
~u
~v
~u
~ v
u+
(4)
~ v
u−
~u
~ v
u−
~v
(9)
~v
~ v
u−
~u
~v
(5)
~ v
u+
~ v
u+
~u
~ v
u+
(10)
~u
~ v
u−
~v
(6)
22
~ v
u−
Chapter 5. Linear function
Q25. (1) 10 left, 4 up
(6) 18 right, 11 down
(2) 19 right, 18 up
(7) 6 left, 6 down
(3) 4 left, 4 up
(8) 8 left, 7 up
(4) 7 right, 22 down
(9) 7 left, 9 up
(5) 5 left, 6 up
6
Q26. (1) ±
−8
−48
20
√ 6 √5
(4) ±
12 5
(3) ±
12
(2) ±
9
Q27. (1) 3
(3) 5
(4) −1
(2) 2
(2)
Q29. (1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
5.5
12
9
15
36
√ 8 √5
6
(4) ±
±
−8
−4 5
0
3
6
e.g.
,
,
−1
1
3
0
2
4
e.g.
,
,
−3
−6
−9
0
1
2
e.g.
,
,
−3
−1
1
0
1
2
e.g.
,
,
6
3
0
0
4
8
e.g.
,
,
−9
−8
−7
0
2
4
e.g.
,
,
4
3
2
0
4
8
e.g.
,
,
8
11
14
0
3
6
e.g.
,
,
1
−4
−9
0
3
6
e.g.
,
,
3
10
17
0
5
10
e.g.
,
,
−4
−6
−8
Q28. (1) ±
(10) 1 right, 9 down
!
√ 18
√
−4√ 13
5
(7)
±
(5) ± √
24
6 13
5
√ √ 2 √5
6 10
(8) ±
(6) ± √
2 10
−6 5
(3) ±
√ −3√ 2
3 2
3√
(10) ±
−3 3
(9) ±
(5) 3
(6) − 10
3
!
√
9√ 12
√
9√13
2 √2
5
(7)
±
(9)
±
(5) ±
9
6 13
− √95
2 2
√ √ √ 3 √10
12√ 5
4 3
(6) ±
(8) ±
(10) ±
−9 10
4 5
4
0
3
6
(11) e.g.
,
,
3
−1
−5
0
2
4
(12) e.g.
,
,
2
−3
−8
0
−2
−4
(13) e.g.
,
,
−5
−6
−7
0
12
24
(14) e.g.
,
,
1
−6
−13
0
−1
−2
(15) e.g.
,
,
−2
−5
−8
0
−3
−6
(16) e.g.
,
,
−4
−9
−14
0
−10
−20
(17) e.g.
,
,
2
−2
−6
0
5
10
(18) e.g.
,
,
6
3
0
0
4
8
(19) e.g.
,
,
−3
−4
−5
0
−6
−12
(20) e.g.
,
,
−1
−10
−19
Simultaneous equations
(5) x = − 23 , y = 2
(9) x = −1, y = 3
(2) x = 1, y = 2
(6) x = −6, y = 9
(10) x = −1, y = 3
12
(14) x = − 13
,y =
(3) x = 2, y = −3
(7) x = 3, y = 2
(11) x = 0, y = 2
(15) x = − 92 , y = 1
(4) x = −2, y = 1
(8) x = −3, y = 2
(12) x = −1, y = 0
(16) x = 8, y = −7
Q30. (1) x =
10
7 ,y
=
12
7
Q31. (1) x = −1, y = 1
(2) x = −0.5, y = 1.5
(3) x = 2, y = 1
(5) x = − 11
5 ,y =
(4) x = 1, y = 1
(6) x = −2, y = 4
23
(13) x = −4, y = −3
18
5
34
13
(7) no solutions
(8) x = − 27 , y = − 43
Chapter 5. Linear function
5.6
Applications of linear equations and vectors
Q32.
(i) A(−4, −3), B(4, 1)
√
(ii) 4 5
(iii) 30
Q33.
(i) A(−4, 2), B(8, −2)
√
(ii) 4 10
(iii) 24
Q34.
(i) −6, 4.5
(ii) (3, 3)
(iii) 15.75
Q35.
√
(i) 12 5
(ii) 30
Q36.
√
(i) 12 10
(ii) 60
Q37. 22.5
Q45. (1.59, −2.55)
Q38. 19.5
Q46.
(i) y = 2.8x + 8
Q39. 12 girls and 8 boys
(ii) 41.60 pln
Q40. 38 cars, 14 motorcycles
(iii) 7.85 km
Q41. 18
Q47.
(i) 42.6 mln
Q42. 34
(ii) 28.4
Q43. 6 buses, 24 cars
(iii) 48.3
Q44. y = 0.358x − 1.59
(iv) 2203
Q48.
(i) c - 7.21mph
(ii) 12 miles East, 10 miles North of O, YES - at 2pm
(iii) 15 miles East, 12 miles North of O, NO - c will arrive first
Q49.
(i) 11 km 200 m
(ii) 17 minutes
Q50.
(i) 347 km
(ii) 3h 48mins
Q51. 20
5.7
Q52. (6, 5) and (−2, 9)
(iii) 91.2 km
h
Q53. (2, 4) or (8, −8)
Q54. (−3, 3)
Chapter review
non-calculator questions
Q1.
Q2.
(i)
(ii)
(iii)
(iv)
(2.5, −4.5)
(0, −12)
75
37.5
(i)
(ii)
(iii)
(iv)
(v)
2x + y + 3 = 0
(−2, 1)
A : −12, B : 0.5
C : 4, D : −3.5
ABD;
they have the same base (BD),
but the heights:√AP >√CP
(either AP =5 5> 3 5 = CP or
~ | = | 10 | > 10
e.g. |AP
5
6
|<9
3
so AP > 10 > 9 > CP )
and |P~S| = |
(vi) (−8.5, 4)
√
√
Q3. (i) 12 5 + 3 10
(ii) (10.5, −1)
5
(iii) ( 20
3 , −3)
(iv) y = 2x − 15
√
√
(v) 8 5 + 2 10
√
Q4. (i) 3 13
(ii) (−5, 1)
(iii) (−8, −3.5) and (−2, 5.5)
24
calculator questions
Q5.
1.75
(ii) ±
8.77
0
(iii)
0
−4
(iv)
−8
(i) A(−6, −2.5), B(4, 2.5)
(ii) (1, 5)
(iii) x − 2y + 9 = 0 or y = 12 x +
9
2
(iv) (−3.54, 2.73)
(v) 25.9
(vi) 3.58
Q7.
(vii) 29.1
6.84
Q6. (i) ±
13.7
(i) A(12.2, 0), B(0.935)
(ii) (6.12, 4.68)
(iii) 114
Chapter 6
Functions
6.1
Basic properties
Q1. (1) yes
Q2. (1)
(2)
(3)
(2) yes
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−4 < x ≤ 4
−3 ≤ y ≤ 3
−2, 2.5
] − 5, 1]
[1, 4]
—
0, 1.5
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−5 ≤ x < 5
−3 ≤ y < 2
3
[−5, −1], [2, 5[
—
[−1, 2]
[−1, 2]
−5 < x ≤ 5
1≤y<2
—
—
] − 5, −3], ] − 3, −1], ] −
1, 1], ]1, 3], ]3, 5]
(vi) —
(vii) —
(3) yes
(6)
(7)
(i)
(ii)
(iii)
(iv)
(v)
(8)
(iii)
(iv)
(v)
(vi)
(vii)
−4
[−5, −3]
[−3, −1]
x ≥ −1
−3.5 < x < −1
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−4 < x ≤ 2
−3 < y ≤ 3
−1
−4 < x ≤ 2
—
—
−4 < x ≤ −2
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−5 ≤ x ≤ 2
−2 ≤ y ≤ 3
−3.5, −1
[−5, −2[, [−2, 2]
—
—
1
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
x ≥ −5
y ≥ −1
−3, −1.5
[−2, −1], [4, +∞[
[−5, −2]
[−1, 4]
[−3, −1.5]
−5 ≤ x < 5
{−2, −1, 0, 1, 2, }
[−1, 1[
—
—
[−5, −3[,
[−3, −1[, (9) (i)
[−1, 1[, [1, 3[, [3, 5[
(ii)
(vii) —
(iii)
(iv)
(5) (i) x ≥ −5
(v)
(ii) −2 ≤ y ≤ 2
(4)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(4) no
[−4, 5[
[−1, 2[
1
− 10
3 , −2, 4
[−4, −2[, [−2, 1[
[3, 5[
25
(5) no
(6) yes
(vi) [1, 3[
(vii) − 38 , 3
(10)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
x≤5
y≤2
−3.5, −1.5
x ≤ −3
[−2, −1[, [−1, 0]
[−3, −2] ∪ [0, 5]
x ≤ −4
(11)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
x ≥ −4
y≤4
3
[−2, −1]
[−4, −2], [−1, +∞[
–
−4, −1.5, 0
(12)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−4 < x ≤ 5
−2 ≤ y < 3
−2.5, −1, 2
[−2, 1], [4, 5]
] − 4, −2], [1, 4]
—
−2, 3, 5
(13)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
x≤4
y≤3
−4, −2
] − ∞, −3], [−2, 0], [3, 4]
[−3, −2], [0, 3]
—
−3, −1, 1, 3.5
(14)
(i) −4 ≤ x < 5
Chapter 6. Functions
−2 < y ≤ 2
(vii) ] − 5, −3[∪] − 3, 0[∪]2, 3]
(iv) [−4, −1], [0, 1[
−2, −1, 4
(v) ] − 1, 0], [1, 4]
(20) (i) −5 ≤ x ≤ 3
[2, 3]
(vi)
—
(ii) −2 ≤ y ≤ 2
[−4, −3[, [−3, −2], ] −
(vii) [−4, 1[∪[1, 2]
(iii) 2, between −4 and −3,
2, −1], ] − 1, 0[, [0, 2],
between −2 and −1
(26) (i) −4 < x ≤ 3
[3, 5[
(iv) [−3, 0], [2, 3]
(ii) −2 ≤ y ≤ 2
(vi) —
(v) [−5, −3], [0, 2]
(iii) −3.5, 2
(vii) −4, −3, 0, 3
(vi) —
(iv) ] − 4, 1], ] − 1, 1]
(15) (i) −5 < x < 3
(vii) ] − 5, −1[∪]1, 3[
(v) [1, 3]
(ii) −2 ≤ y ≤ 2
(21) (i) −4 < x ≤ 3
(vi) —
(iii) −4, −1, 2
(ii) 0 ≤ y ≤ 2
(vii) −4 < x ≤ 3
(iv) ] − 5, −3], [1, 3[
(iii) 3
(27) (i) −3 ≤ x < 2,
(v) [−3.1]
(iv) ] − 4, −3]
2<x≤3
(vi) —
(v) [−3, −2], [1, 3]
(ii) y = −1, 0 ≤ y ≤ 2
(vii) −3
(vi) [−2, 1[
(iii) −2
(16) (i) −4 < x < 4
(vii) [−2, 1[∪{2}
(iv) [−2, −1]
(ii) −3 < y ≤ 3
(v) [1, 2[
(22) (i) x ≤ 2
(iii) −3, 1
(vi) [−3, −2[, [−1, 1],
(ii) y ≥ −1
(iv) ] − 4, −1], [2, 3]
]2, 3]
(iii) −3, −1
(v) [−1, 2], [3, 4[
(vii)
[−3, −2]∪]2, 3]
(iv) [−2, 0]
(vi) —
(28) (i) −4 < x ≤ −1,
(v) ] − ∞, −2]
(vii) —
0<x≤2
(vi) [0, 2]
(ii) −3 < y ≤ 2
(17) (i) −4 ≤ x ≤ 3
(vii) ] − 3, −1[
(iii)
−2, 2
(ii) −2 ≤ y ≤ 3
(23) (i) −4 < x ≤ 4
(iv) ] − 4, −1], ]0, 1]
(iii) −3.5, 1
(ii) −1 < y ≤ 1
(v) [1, 2]
(iv) [−4, −2]
(iii) −3, −1, 1, 3
(vi) —
(v) [−2, 3]
(iv) ] − 4, −2], ] − 2, 0],
(vii)
] − 2, −1]∪]0, 2[
(vi) —
]0, 2], ]2, 4]
(vii) [−4, 3[∪]0, 3]
(29) (i) x > −4
(v) —
(ii)
(iii)
(iv)
(v)
(18)
(19)
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−4 < x ≤ 3
−2 ≤ y ≤ 2
−2, 0, 3
] − 4, −1], [1, 3]
[−1, 1]
—
] − 4, −3[∪] − 0.5, 2[
(i)
(ii)
(iii)
(iv)
(v)
(vi)
−5 < x ≤ 3
1≤y≤3
—
] − 5, −3], [−1, 1]
[−3, −1], [1, 3]
—
Q3. (1) domain: R,
(vi) —
(vii) {−2, 0, 2, 4}
(24)
(25)
−3 ≤ x < 1
{−1, 0, 1, 2}
[−1, 0[
—
—
[−3, −2[, [−2, −1[,
[−1, 0[, [0, 1[
(vii) [−1, 1[
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(30)
(i) −4 ≤ x ≤ 4
(ii) −3 ≤ y ≤ 2
(iii) −1, 0, 3
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
y ≥ −1
0
] − 3, −1], [1, +∞[
] − 4, −3], ]0, 1]
—
{−1}∪]0, +∞[
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
−2 < x ≤ 3
−2 ≤ y ≤ 1
2
[1, 3]
] − 2, −1]
[−1, 0], ]0, 1]
[−1, 0]
range: [0, +∞[
(4) domain: [0, +∞[,
range: [0, +∞[
(2) domain: R,
range: R
(5) domain: R,
range: R
(3) domain: R \ {0},
range: R \ {0}
(6) domain: R,
range: [0, +∞[
Q4. (1) − 25
(2) 3
(3)
1
3
(4) ±3
(5)
1
2
(6) −3, 1
(7) − 3π
2
(8)
− 31
(9) −5.5
26
(10) ±2
√
(13) −2 3
(16) 1
(11) −3, 1
(14) 4.5
(17)
(12) —
(15) 2
(18) 9
1 7
2, 2
Chapter 6. Functions
Q5. (1) domain: R,
(9) domain: R,
range: R
range: [−9, +∞[
(2) domain: ] − ∞, 3], range: [0, +∞[
(10) domain: R,
range: [3, +∞[
(3) domain: R \ {1},
range: R \ {3}
(11) domain: R,
range: R
(4) domain: R,
range: [−4, +∞[
(12) domain: [0, +∞[,
range: [−3, +∞[
(5) domain: R,
range: [−2, +∞[
(13) domain: R,
range: [0, +∞[
(6) domain: R,
range: R
(14) domain: R,
(7) domain: [− 13 , +∞[, range: [0, +∞[
(15) domain: [0,
range: ] − ∞, 4[∪[7, +∞[
(8) domain: R \ {−4}, range: R \ {−2}
Q6. (1) 1
(4) −40
(10) − 11
9
(7) 5π
13
9
(2) 3
(5) 16
(8) 5
(11)
(3) 3.5
(6) 0
(9) −3
(12) 4.2
(7) f (−x) = − 32 x + π
√
(8) f (−x) = 1 − 3x
Q7. (1) f (−x) = −2x + 5
√
(2) f (−x) = 3 + x
(3) f (−x) = 3 −
2
x+1
range: ] − ∞, 3]
1
1
4 [∪] 4 , +∞[,
(9) f (−x) =
3
x−4
2
−2
(13) 18
√
(14) 2 6 − 3
(16) −27
(15) −16
(18) − 13
(17)
2
3
√
(13) f (−x) = − 3x + 6
√
(14) f (−x) = −2x − 3
(15) f (−x) = −x3 − 8
(4) f (−x) = 9 − x2
(10) f (−x) = x − 4
(16) f (−x) = −(x + 1)3
(5) f (−x) = (2x + 1)2
(11) f (−x) = (1 − x)2 − 4
(17) f (−x) = 3 − |2x + 4|
(6) f (−x) = |1 − x| − 2
(12) f (−x) = 3 + |4 + 2x|
(18) f (−x) =
√
√−x−3
2 −x−1
Q8. (1) O
(4) E
(7) O
(10) E
(13) E
(16) N
(2) N
(5) E
(8) O
(11) N
(14) O
(17) E
(3) O
(6) N
(9) N
(12) E
(15) E
(18) N
6.2
Transformations of graphs of functions
Q9. Graphs of y = f (x) (black, dashed) and y = −f (x) (red, solid)
y
y
x
x
(1)
(4)
y
y
x
x
(2)
(5)
y
y
x
x
(3)
(6)
27
Chapter 6. Functions
Q10. Graphs of y = f (x) (black, dashed) and y = f (−x) (red, solid)
y
y
x
x
(4)
(1)
y
y
x
x
(2)
(5)
y
y
x
x
(3)
(6)
Q11. Graphs of y = f (x) (black, dashed), y = 2f (x) (red, solid) and y = 21 f (x) (blue, solid)
y
y
x
x
(4)
(1)
y
y
x
x
(2)
y
(5)
y
x
x
(3)
(6)
28
Chapter 6. Functions
Q12. Graphs of y = f (x) (black, dashed), y = f (2x) (red, solid) and y = f ( 12 x) (blue, solid)
y
y
x
x
(1)
(4)
y
y
x
x
(2)
(5)
y
y
x
x
(3)
(6)
Q13. Graphs of y = f (x) (black, dashed) and y = g(x) (red, solid).
y
y
x
x
(4)
(1)
g(x) = (x − 3)2 − 1
x ∈ R, y ≥ −1
√
g(x) = x + 1 + 2
x ≥ −1, y ≥ 2
y
y
x
x
(2)
(5)
g(x) = (x + 2)3 − 3
x, y ∈ R
g(x) = − 32 x − 3
x, y ∈ R
y
y
x
x
(6)
(3)
1
g(x) = x−4
+1
x ∈ R \ {4}, y ∈ R \ {1}
g(x) = |x + 5| + 1
x ∈ R, y ≥ 1
29
Chapter 6. Functions
y
y
x
x
(7)
g(x) = (x + 4)2 + 2
x ∈ R, y ≥ 2
(10)
y
√
g(x) = x + 4 − 2
x ≥ −4, y ≥ −2
y
x
x
(8)
(11)
g(x) = 21 x + 3
x, y ∈ R
g(x) = (x + 2)3 + 1
x, y ∈ R
y
y
x
x
(9)
(12)
1
g(x) = x+1
−2
x ∈ R \ {−1}, x ∈ R \ {−2}
g(x) = |x + 3| − 4
x ∈ R, y ≥ −4
Q14. y = f (x) (black, dashed), y = |f (x)| (red, solid)
y
y
x
x
(1)
(4)
y
y
x
x
(5)
(2)
y
y
x
x
(6)
(3)
30
Chapter 6. Functions
Q15. graph of y = f (x) (dashed, black) and of y = f (|x|) (solid, red)
y
y
x
x
(4)
(1)
y
y
x
x
(5)
(2)
y
y
x
x
(3)
Q16. (1)
(6)
(i) y = x2
(8)
(2)
(ii) translation by
2
−4
(iii) y = (x − 2)2 − 4
(3)
(i) y = x2
(11)
(i) y = x3
(ii) translation by
(7)
(i) y = x3
(ii) reflection in x-axis or in y-axis
−5
followed by translation by
1
3
(iii) y = −(x + 5) + 1
2
−3
(iii) y = |x − 2| − 3
(12)
−5
(ii) translation by
1
3
(iii) y = (x + 5) + 1
(i) y = x3
(ii) reflection in x-axis or in y-axis
(iii) y = −x3
0
−3
(i) y = |X|
(i) y = x3
(6)
−5
−2
followed by translation by
√
(iii) y = −x − 3
4
(ii) translation by
1
3
(iii) y = (x − 4) + 1
(5)
x
−1
(ii) translation by
−4
2
(iii) y = (x + 1) − 4
(4)
√
(ii) translation by
√
(iii) y = x + 5 − 2
√
(9) (i) y = x
(ii) reflection in y-axis
√
(iii) y = −x
√
(10) (i) y = x
(ii) reflection in y-axis
(i) y = x2
(i) y =
3
(ii) translation by
1
2
(iii) y = (x − 3) + 1
(i) y = |X|
(ii) reflection in x axis
followed by translation by
2
1
(iii) y = −|x − 2| + 1
√
(13) (i) y = x
−2
(ii) translation by
−2
√
(iii) y = x + 2 − 2
√
(14) (i) y = x
(ii) reflection in y-axis
followed by translation by
31
2
−2
Chapter 6. Functions
−2
−2
followed by reflection in y-axis
√
(iii) y = −x + 2 − 2
√
(15) (i) y = x
(ii) reflection in x-axis
−4
followed by translation by
1
√
(iii) y = − x + 4 + 1
or translation by
(16)
(17)
(i) y =
(iii) y =
(i) y =
(iii) y =
1
x−2
1
x
(20)
(21)
(22)
1
x+2
1
x
−2
−1
−1
(i) y =
(ii) reflection in x-axis or in y-axis
−2
followed by translation by
−1
1
(iii) y = − x+2 − 1
(28)
1
x
(i) y =
(ii) reflection in x-axis or in y-axis
1
followed by translation by
2
1
(iii) y = − x−1
+2
(29)
(i) y = x2
(ii) vertical stretch by −2
3
(iii) translation by
3
(30)
(i) y = x2
(ii) vertical stretch by
(i) y = |x|
(ii) vertical stretch by − 23
3
(iii) translation by
2
(i) y = x3
(ii) vertical stretch by − 14
−3
(iii) translation by
2
√
(25) (i) y = x
(ii) vertical stretch by −2
0
(iii) translation by
−2
√
(26) (i) y = x
(ii) vertical stretch by 2
−1
(iii) translation by
−3
√
(27) (i) y = x
(ii) reflection in x-axis
(iii) horizontal stretch by 2
0
(iv) translation by
3
2
0
−1
−2
(24)
1
x
(ii) translation by
(19)
(23)
(i) y = x1
(ii) reflection in x-axis or in y-axis
(iii) y = − x1
(ii) translation by
(18)
(iii) translation by
(i) y = x1
(ii) vertical / horizontal stretch by
0
(iii) translation by
−1
1
2
(i) y = x1
(ii) vertical / horizontal stretch by −2
2
(iii) translation by
1
(i) y = x2
3
−4
(iii) reflection of the part below x-axis in the
axis
(ii) translation by
1
2
Q17. Graphs of y = f (x) (black, dashed) and y = g(x) (red, dotted).
(1) vertical stretch by 2, translation 3 right
y
(3) reflection in y-axis, translation 2 left
y
x
x
(2) vertical stretch by 3, translation 1 left
y
(4) translation 2 right, 2 down
y
x
x
32
Chapter 6. Functions
(5) reflection in x axis of the part for x > 0
y
(12) vertical stretch by 2 followed by
translation 1 left, 3 down
y
x
x
(6) vertical stretch by −2, translation 3 right
y
(13) vertical stretch by −2 followed by
translation 3 left, 2 up
y
x
x
(7) horizontal dilation by
y
1
2,
translation 1 up
(14) vertical dilation by −2 followed by
translation 1 left, 1 up
y
x
x
(8) horizontal dilation by 13 , translation 2 up
y
(15) vertical dilation by −2 followed by
translation 3 right 1 up
y
x
x
(9) reflection in y-axis, translation 1 down
y
(16) reflection in x-axis followed by
translation 2 right 2 down
y
x
(10) reflection in x/y-axis followed by
translation 3 down
y
x
x
(17) reflection in x-axis of the part y < 0
followed by vertical stretch by 2
y
(11) reflection in y-axis of the part x > 0
followed by reflection in x-axis
of the part y < 0
y
x
x
(18) horizontal dilation by 12 ,
translation 1 down
y
x
33
Chapter 6. Functions
(26) vertical dilation by − 12 followed by
translation 3 left, 2 up
y
(19) reflection in x-axis followed by
translation 3 right, 3 up
y
x
x
(20) vertical stretch by 2 followed by
translation 1 right, 2 up
y
(27) vertical dilation by 3 followed by
translation 1 left, 3 down
y
x
x
(28) vertical dilation by −2 followed by
translation 2 left, 4 up
y
(21) reflection in x-axis followed by
translation 2 right 2 up
y
x
x
(22) vertical dilation by 21 followed by
translation 1 left 2 down
y
(29) reflection of the part left of y-axis in the axis
y
x
x
(30) reflection of the part below x-axis in the axis
y
(23) reflection in x-axis of the part y < 0
followed by reflection in x-axis
y
x
x
2
(31) shift by
followed by reflection of the part
(24) reflection in x-axis and horizontal dilation by 2
−1
followed by translation 3 up
right of y-axis in the axis
y
y
x
x
(25) vertical dilation by 21 followed by
translation 3 right, 1 down
y
−2
followed by reflection of the part
−4
below x-axis in the axis
y
(32) shift by
x
x
34
Chapter 6. Functions
Q18. (1) A0 = (4, 4),
(2) A0 = (−3, 2),
Q19.
(3) A0 = (−1.5, 3),
(5) A0 = (2, 7),
(4) A0 = (−3, −4),
(6) A0 = (−8, −5).
(3) y = f ( x2 )
(i) (1) y = f (−x) − 2
y
y
x
x
(4) y = 12 f (x − 1) + 2
y
(2) y = −f (x + 1) − 1
y
x
x
(ii) (1) y = −f (x) − 1
(2) y = f (2x) − 1
(3) y = 21 f (−2x)
6.3
Equations and inequalities
Q20. (1) −1, 31
(3) − 74 , 34
(2) −1, 2
− 83 , 43
Q21.
(i) −2, 2
(4)
Q22.
(ii) 0, 4
(5) 2
(7) 11
(6) −1
(i) −3
(8)
Q23.
(ii) −5
Q25. (1) 0: —
(ii)
(9) −1
(11) 3
(10) 1
1
2
7
2
(5) 3: −0.0644, 3.17, 4.89
Q24.
(12)
(i) 9
(ii) 4.5
(9) 2: −1.15, 1.15
(2) 1: −1.31
(6) 1: 2.21
(10) 2: −1.22, 0.549
(3) 2: −1, 1.54
(7) 1: −1.52
(11) 2: 0.780, 5.55
(4) 1: 0.0605
(8) 3: −0.481, 1.31, 3.17
(12) 2: −5.24, −0.764
Q26. (1) x ≤ −0.861 or 0.746 ≤ x ≤ 3.11
(8) −2 ≤ x ≤ −1.41 or −1 < x ≤ 1.41
(2) −4.59 < x < −0.887 or x > 1.47
(9) −3 < x ≤ −2 or −1.41 ≤ x < −1
or 1.41 ≤ x < 3
(3) −0.535 ≤ x ≤ 0.444 or x ≥ 3.69
6.4
(i)
4
3
(4) −1.65 < x < 1.27 or 2 < x < 2.38
(10) −0.562 < x < 1 or 3.56 < x ≤ 4
(5) −0.618 < x < 0 or 1.62 < x ≤ 2
(11) 1 ≤ x ≤ 3.56
(6) 0 ≤ x ≤ 1
(12) −2.41 < x < −0.305
(7) −1.88 ≤ x < −1 or 0.347 ≤ x ≤ 1.53
(13) 0.918 ≤ x ≤ 2.66
chapter review
non-calculator questions
35
1
2
Chapter 6. Functions
y
vertical dilation by 13
followed by
shift 3 left and 1 down
Q1. (1)
x
y
horizontal dilation by −2
followed by
shift 1 up
(2)
x
y
(3)
vertical dilation by − 12
followed by
shift 3 right and 1 down
x
y
vertical dilation by −3
followed by
shift 1 left and 2 up
x
(4)
y
vertical (or horizontal) dilation by −3
followed by
shift 2 left and 1 down
x
(5)
y
vertical (or horizontal) dilation by − 12
followed by
shift 2 up
(6)
x
36
Chapter 6. Functions
y
reflection of the part right of y-axis in the axis
x
(7)
y
shift 3 left and 3 down
followed by
reflection of the part below the x-axis in th axis
(8)
x
−1
(4) y = − 32 |x + 3| + 2
√
(5) y = −2x − 1
− 2)3 + 1
(6) y = ||x + 3| − 2|
Q2. (1) y = −2(x + 3)2 + 3
(2) y =
(3) y =
−2
x−1
1
2 (x
Q3. (1) even
(2) odd
(3) neither
y
y
x
x
(5)
y
Q4.
(i) (1)
y
x
(6)
y
x
(2)
y
(7)
(ii) (1)
(2)
(3)
(4)
(5)
(6)
(7)
x
(3)
y
y
y
y
y
y
y
y
= f (x − 2) + 1
= −f (x + 1) + 1
= −2f (x − 1)
= 21 f (x) − 1
= |f (x) + 1|
= f (2x) + 1
= f (|x|)
x
(4)
f4 : x ∈ R
f8 : x ≤ 3, x 6= −3
f3 : y ≥ −1
f1 : x ≤ − 23
f5 : x ∈ R, x 6= −1
RANGE:
f4 : y ≥ −5
f2 : x ∈ R
f6 : x ∈ R
f1 : y ≤ 0
f5 : y ∈ R, y 6= 3
f3 : x ∈ R
f7 : x > −3
f2 : y ≤ 4
f6 : y ≤ 2
Q5. DOMAIN:
37
x
function
domain
range
zeroes
decreasing
increasing
constant
even
odd
one-to-one
Q6.
(1)
[−5, −1[∪]1, 5]
[0, 2]
−4, 4
[−5, −4], [2, 4]
[−4, −2], [4, 5]
[−2, −1[, ]1, 2]
yes
no
no
(2)
] − 5, 4[
] − 4, 2]
−3, 0, 3
[−1, 1]
] − 5, −1], [1, 5[
∅
no
no
no
(3)
[−5, 5]
[−2, 2]
−4, 0, 4
[−5, −4], [−2, 2], [4, 5]
[−4, −2], [2, 4]
∅
no
yes
no
calculator questions
Q7. (1) (−3.59, −0.279), (−0.549, −1.82), (10.1, 0.0986)
(2) (4.24, 9.48), (8.83, 18.7)
(3) (−13.4, −0.590), (0.561, 20.1), (15.3, −0.416)
(4) (−16.4, −6.82), (0.382, 0.159), (16.0, 6.66)
Q8. (1) x ∈ [−20, −3.59[∪] − 0.549, 0[∪]10.1, 20] / −20 ≤ x < −3.59 or −0.549 < x < 0 or 10.1 < x ≤ 20
(2) x ∈ [−20, 4.24[∪]8.83, 20] / −20 ≤ x < 4.24 or 8.83 < x ≤ 20
(3) x ∈ [−20, −13.4]∪]0, 0.561] ∪ [15.3, 20] / −20 ≤ x ≤ −13.4 or 0 < x ≤ 0.561 or 15.3 ≤ x ≤ 20
(4) x ∈ [−20, −16.4] ∪ [0.382, 16.0] / −20 ≤ x ≤ −16.4 or 0.382 ≤ x ≤ 20
(answer can be given in any of the two forms shown above)
Chapter 7
Quadratic function
7.1
Solving quadratic equations
7.1.1
Factorisation
Q1. (1) x2 + 3x + 2
(5) x2 − 6x + 8
(9) x2 + 2x − 8
(2) x2 + 4x + 3
(6) x2 − 13x + 12
(10) x2 − 9
(3) x2 + 7x + 10
(7) x2 + x − 6
(11) x2 − x − 12
(4) x2 − 4x + 3
(8) x2 − x − 6
(12) x2 + 4x − 12
Q2. (1) 2x2 + 3x + 1
(5) 2x2 − x − 1
(9) 6x2 + 13x − 5
(2) 2x2 + 9x + 10
(6) 3x2 + x − 2
(10) 6x2 − 13x + 5
(3) 2x2 − 7x + 3
(7) 3x2 − 5x − 2
(11) 6x2 + 7x − 5
(4) 2x2 − 5x + 3
(8) 3x2 − x − 2
(12) 15x2 − 16x + 4
Q3. (1)
(2)
(3)
(4)
(5)
(6)
(x)(x − 2)
(x + 2)(x + 1)
(x + 3)(x + 2)
(x + 3)(x + 1)
(x + 4)(x)
(x + 4)(x + 1)
(7)
(8)
(9)
(10)
(11)
(12)
(x + 4)(x + 2)
(x + 4)(x + 3)
(x + 5)(x + 1)
(x + 5)(x + 2)
(x + 6)(x + 2)
(x + 1)(x + 12)
(13)
(14)
(15)
(16)
(17)
(18)
(x − 1)(x − 3)
(x − 1)(x − 2)
(x − 2)(x − 3)
(x − 1)(x − 6)
(x − 2)(x − 4)
(x − 1)(x − 8)
Q4. (1) (x + 3)(x − 2)
(5) (x + 1)(x − 8)
(9) (x + 2)(x − 6)
(2) (x + 2)(x − 3)
(6) (x + 3)(x − 3)
(10) (x + 6)(x − 2)
(3) (x + 4)(x − 2)
(7) (x + 6)(x − 1)
(11) (x + 4)(x − 3)
(4) (x + 2)(x − 4)
(8) (x + 3)(x − 4)
(12) (x + 4)(x − 6)
38
(13) x2 − 2x − 24
(14) x2 + 5x − 24
(15) x2 + 4x − 21
(13) 15x2 − 17x − 4
(14) 15x2 + 4x − 4
(15) 12x2 − 25x + 12
(19) (x − 3)(x − 4)
(20) (x − 2)(x − 6)
(21) (x − 1)(x − 12)
(13) (x + 2)(x − 12)
(14) (x + 8)(x − 3)
(15) (x + 24)(x − 1)
Chapter 7. Quadratic function
Q5. (1) (x + 1)(x + 2)
(8) (x + 3)(x + 4)
(15) (x − 2)(x − 2)
(22) (x + 3)(x − 4)
(2) (x + 2)(x + 3)
(9) (x + 2)(x + 6)
(16) (x − 3)(x − 5)
(23) (x − 3)(x + 5)
(3) (x + 1)(x + 3)
(10) (x − 1)(x − 2)
(17) (x − 3)(x − 4)
(4) (x + 2)(x + 4)
(11) (x − 2)(x − 3)
(18) (x − 2)(x − 6)
(5) (x + 1)(x + 4)
(12) (x − 1)(x − 8)
(19) (x + 2)(x − 4)
(6) (x + 2)(x + 3)
(13) (x − 2)(x − 4)
(20) (x + 1)(x − 8)
(26) (x + 4)(x − 6)
(7) (x + 3)(x + 5)
(14) (x − 1)(x − 4)
(21) (x + 6)(x − 2)
(27) (x − 1)(x + 24)
(24) (x + 2)(x − 12)
(25) (x − 3)(x + 8)
Q6. (1)
(2)
(3)
(4)
(5)
(6)
(x + 1)(2x + 1)
(2x + 1)(x + 2)
(2x + 3)(x + 1)
(2x + 3)(x + 2)
(x + 3)(2x + 1)
(2x + 5)(x + 2)
(7)
(8)
(9)
(10)
(11)
(12)
(2x − 1)(x − 1)
(2x − 1)(x − 2)
(2x − 1)(x − 3)
(2x − 3)(x − 1)
(x + 1)(2x − 1)
(2x + 1)(x − 1)
(13)
(14)
(15)
(16)
(17)
(18)
(x + 2)(3x − 1)
(x + 1)(3x − 2)
(3x + 1)(x − 2)
(3x + 2)(x − 1)
(3x − 4)(x − 2)
(3x − 2)(x − 4)
(19) (3x − 1)(x − 8)
Q7. (1)
(2)
(3)
(4)
(5)
(6)
(3x − 1)(2x − 1)
(2x − 1)(3x − 2)
(3x + 2)(2x − 1)
(3x + 1)(2x − 3)
(3x + 2)(2x − 3)
(3x + 2)(3x − 1)
(7)
(8)
(9)
(10)
(11)
(12)
(2x + 5)(3x − 1)
(3x − 5)(2x − 1)
(3x + 5)(2x − 1)
(5x − 1)(3x − 4)
(5x − 2)(3x − 2)
(5x + 1)(3x − 4)
(13)
(14)
(15)
(16)
(17)
(18)
(3x + 2)(5x − 2)
(3x + 5)(2x − 1)
(2x + 5)(3x − 1)
(3x + 1)(2x − 5)
(2x + 1)(3x − 5)
(2x + 3)(5x − 3)
(19) (5x + 9)(2x − 1)
7.1.2
Completing the square
Q8. (1) x2 − 2x + 3
(2) x2 − 4x + 0
(3) 2x2 + 4x − 2
Q9. (1)
(2)
(3)
(4)
(5)
(6)
Q10. (1)
(2)
(3)
(x + 1)2 + 1
(x + 1)2 − 1
(x − 2)2 − 3
(x + 2)2 + 1
(x − 3)2 + 1
(x + 3)2 − 2
(i) (x + 1)2 + 2 = 0
(ii) no solutions
(i) (x + 2) − 5 = 0
√
(ii) −2 ± 5
(6)
(i) 2(x + 2)2 − 5 = 0
(7)
7.1.3
(13)
(14)
(15)
(16)
(17)
(18)
√
10
2
2
(8)
(i) 3(x + 1) − 3 = 0
(ii) −2, 0
(i) −5(x + 1)2 + 9 = 0
√
3 5
5
2
(20) (5x + 1)(2x − 9)
(21) (5x + 2)(3x − 1)
− 23 (x
(ii) −3 ±
(10)
(19) 2(x + 1)2 + 4
(20) −2(x + 12 )2 +
(21) − 13 (x −
1
2
3 2
5
)
+
2
4
(i) − 34 (x − 23 )2 −
(ii) no solutions
(i)
4
3 (x
2
3
+ 6)2 − 10 = 0
√
30
2
− 53 (x − 65 )2 + 1 =
√
15
6
±
5
5
2
2
5 (x + 2) − 2 = 0
2
+ 3) + 5 = 0
√
30
2
(11)
(i)
(ii)
(12)
(i)
(ii) −2 ±
√
5
Quadratic formula
Q11. (1) −2.62, −0.382
(5) −3.41, −0.586
(9) −5.16, 1.16
(2) no solution
(6) −3.62, −1.38
(10) −6.61, 0.606
(14) 0.614, 4.89
(3) −7.16, −0.838
(7) −1.24, 3.24
(11) −1.47, 7.47
(15) −3, 0.667
(4) 0.764, 5.24
(8) −5.19, 0.193
(12) −5.46, 1.46
(16) no solution
39
=0
(ii) −6 ±
(i) 13 (x + 3) + 21 = 0
(ii) no solutions
(i)
(9)
3
2
(ii) −1 ±
√
2(x − 2)2 + 4
2(x − 1.5)2 − 3.5
3
1
2
2 (x − 1) + 2
−2(x + 25 )2 + 29
2
−(x + 2)2 + 6
− 21 (x − 2)2 + 1
(i) −4(x − 1)2 + 3 = 0
(ii) 1 ±
2
(ii) −2 ±
(4)
(x + 4)2 − 8
(x − 1.5)2 − 5.25
(x + 2.5)2 − 0.25
2(x − 1)2
2(x + 1)2 − 2
2(x + 2)2 − 6
(5)
(21) (4x − 3)(3x − 4)
(7) − 12 x2 + 2x − 27
(8) 23 x2 − 38 x + 76
(9) 14 x2 + 1x + 3
(4) 3x2 + 24x + 28
(5) −4x2 − 4x + 3
(6) 5x2 − 10x − 5
(7)
(8)
(9)
(10)
(11)
(12)
(20) (3x + 2)(x − 4)
(13) −0.851, 2.35
0
Chapter 7. Quadratic function
(17) −5.26, 0.76
(19) 2.2
(21) −2.77, 1.44
(23) −2
(18) −2.82, 1.07
(20) 1.16, 4.49
(22) −2.26, 0.591
(24) no solution
√
√
−3− 5 −3+ 5
,
2
2
Q12. (1)
(2)
(3)
(4)
(5)
(8) −2,
3
2
√
(14)
√
no solution
(9) −3−2 23 , −3+2 23
√
√
√
√
−4 − 10, −4 + 10
√
√
(10) 3−2 31 , 3+2 31
3 − 5, 3 + 5
√
√
√
√
−2−2 7 −2+2 7
,
−2 − 2, −2 + 2 (11)
3
3
√
√
(12) no solution
(6) −5−2 5 , −5+2 5
−1
(7) 2 , 4
(13) no solution
7.2
(15)
1
2,
2
3,
−6
(20)
−3
(21)
(16) no solution
(17)
(18)
(19)
1
2,
(22)
−5
√
√
−7− 65 −7+ 65
,
8
8
√
√
−1− 6 −1+ 6
,
5
5
(23)
(24)
√
√
2− 7 2+ 7
3 ,
3
√
√
−9− 89 −9+ 89
,
4
4
√
√
−5− 37 −5+ 37
,
6
6
√
√
−5− 13 −5+ 13
,
4
4
√
√
2− 7 2+ 7
,
2
2
Parabola
Q13. (1) x-int.: (−2, 0), (−1, 0),
vertex: (−1.5, −0.25),
y-intercept: (0, 2)
(5) x-int.: (−3.41, 0), (−0.586, 0),
vertex: (−2, −2),
y-intercept: (0, 2)
(9) x-int.: (−0.225, 0), (2.22, 0),
vertex: (1, 3),
y-intercept: (0, 1)
(2) x-int.: (−1, 0), (3, 0),
vertex: (1, −4),
y-intercept: (0, −3)
(6) x-int.: (−8.87, 0), (−1.13, 0), (10) x-int.: (−1, 0), (4, 0),
vertex: (−5, −7.5),
vertex: (1.5, 6.25),
y-intercept: (0, 5)
y-intercept: (0, 4)
(3) x-int.: (−7.16, 0), (−0.838, 0),
vertex: (−4, −10),
y-intercept: (0, 6)
(7) x-int.: (−1, 0), (2, 0),
vertex: (0.5, −4.5),
y-intercept: (0, −4)
(11) x-int.: (−0.333, 0), (1, 0),
vertex: (0.333, 1.333),
y-intercept: (0, 1)
(4) x-int.: none,
vertex: (1.5, 3.75),
y-intercept: (0, 6)
(8) x-int.: (0.209, 0), (4.79, 0),
vertex: (2.5, 5.25),
y-intercept: (0, −1)
(12) x-int.: (−0.667, 0), (0.5, 0),
vertex: (−0.083, 2.042),
y-intercept: (0, 2)
Q14. y = −x2 − 2x + 8
Q22. y = − 32 x2 + 9x −
Q15. y = −2x2 + 12x − 10
Q23. y = −2x2 − 4x + 6
Q16. y = 21 x2 − 12 x − 3
Q24. y = x2 − 2x − 1
Q17. y = −2x2 + 8x −
Q18. y = − 32 x2 +
21
2 x
7
2
− 15
2
Q19. y = x + 10x + 24
Q20. y = − 32 x2 + 6x − 1
2
Q21. y = −1x + 8x − 18
Q33.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
7.3
7.3.1
25
2
Q29. y = −3x2 − 6x − 49 ,
y = −3(x + 21 )(x + 32 )
Q30. y = 12 x2 − 2x,
y = 21 (x)(x − 4)
2
Q25. y = 2x + 12x + 17
Q26. y = − 21 x2 − 2x + 3
Q31. y = − 21 x2 − 3x − 4,
y = − 21 (x + 3)2 + 21
Q27. y = (x + 1)2 − 4,
y = (x + 3)(x − 1)
Q28. y = 2(x − 34 )2 − 27
8 ,
y = 2(x + 12 )(x − 2)
y=
x2 − 2x − 15
2x2 − 12x + 10
2x2 + 5x − 3
x2 + 4x + 1
−2x2 − 18x − 28
−4x2 + 28x − 49
− 12 x2 − 2x − 3
y=
(x − 1)2 − 16
2(x − 3)2 − 8
2(x + 54 )2 − 49
8
(x + 2)2 − 3
−2(x + 92 )2 + 25
2
−4(x − 72 )2
− 12 (x + 2)2 − 1
−4x2 + 8x − 2
√
−2x2 + 2x 2 + 7
−4(x − 1)2 + 2
√
−2(x −
2 2
)
2
+8
Q32. y = −2x2 − x + 3,
y = −2(x + 41 )2 + 3 18
y=
(x − 5)(x + 3)
2(x − 1)(x − 5)
2(x − 12 )(x + 3)
√
√
(x + 2 − 3)(x + 2 + 3)
−2(x + 2)(x + 7)
−4(x − 27 )2
—
−4(x − 1 +
√
−2(x −
2
2
Applications of quadratics
Quadratic inequalities
Q34.
40
√
2
)(x
2
√
−1−
+ 2)(x −
√
2
2
2
)
2
− 2)
−3, 5
1, 5
−3, 12
√
−2 ± 3
−2, −7
3.5
none
√
1±
√
2
2
(1, −16)
(3, −8)
(− 54 , − 49
)
8
(−2, −3)
(− 92 , 25
)
2
( 72 , 0)
(−2, −1)
64
64
49
12
100
0
−2
(1, 2)
32
2
2
±2
(
√
2
, 8)
2
64
Chapter 7. Quadratic function
(1) ] − ∞, −2] ∪ [−1, +∞[
(5) ] − ∞, −1] ∪ [− 13 , +∞[
(2) [−3, − 13 ]
(6) ] − ∞, − 37 ] ∪ [1, +∞[
(3)
] 21 , 2[
(4) ] −
(7) ] −
∞, −2[∪] 12 , +∞[
(8) ] −
(10) no solutions
3, − 13 [
1 3
2, 2[
Q35. (1) ] − ∞, −3.41[∪] − 0.586, +∞[
(9) [− 12 , 1]
(11) R
(12) ] − ∞, − 12 [∪] − 13 , +∞[
(10) ] − 2.57, 1.07[
(2) ] − ∞, −3.28[∪]0.61, +∞[
(11) [−2.23, 0.897]
(3) ] − ∞, 0.634] ∪ [2.37, +∞[
(12) ] − ∞, −0.693] ∪ [1.44, +∞[
(4) no solutions
(13) ] − 12, −2.87[
(5) R
(14) ] − ∞, −1.43[∪]0.904, +∞[
(6) R
(15) [−0.48, 0.956]
(7) [0.219, 2.28]
(16) ] − ∞, −2.26] ∪ [0.591, +∞[
(8) ] − ∞, −0.897] ∪ [2.23, +∞[
(17) R
(9) ] − ∞, 0.719[∪]2.78, +∞[
(18) no solutions
7.3.2
Problems involving quadratics
Q36.
(i) 12 hours
(ii) 6 hours
(iii) 72 km
Q37.
(i) 27 km
(ii) 9 hours
(iii) 4 hours
(iv) 75 km
Q38.
(i) 12 m
(ii) 2.58 s
(iii) 0.816 s
(iv) 15.3 m
Q39. 50m × 100m
Q44.
Q40. both 11; 121
Q41. 11 and 5.5; 60.5
√
(i) y = 6; 2 10
√
(ii) y = 8; 2 11
√
(iii) p = 2; 2 5
Q42. 4 and 6; 24
√
Q45.
Q43. 0.5
Q46.
(i) no
2
2
(ii) 3m4cm
(iii) 2m26cm
Q47. 3.01m
Q49. 6.5m × 13m
Q48.
Q50. 0.75m2 = 7500cm2
(i) 2400m
(ii) 5100m
Q51. 9mm
(iii) 5420m
(iv) 14 hrs 20 mins
Q52. 20.7cm or 54.3cm
(v) 23 hrs 20 mins
Q53. 85cm
Q54. yes: edge 1m long veritcally, edge 85cm long across the ditch, edge 2m long along the ditch
7.3.3
Investigating graphs of rational functions
Q55. (1) 1.
2.
3.
4.
5.
6.
—
(1, 0)
(0, 12 )
x=2
y=1
—
(2) 1. —
2. (−2, 0)
3. (0, −4)
4. x = 1
5. y = 2
6. —
(3) 1.
2.
3.
4.
5.
6.
(4) 1.
2.
3.
4.
5.
6.
—
(3, 0)
(0, −3)
x = −2
y=2
—
—
(2, 0)
(0, − 12 )
x = −2
y = 21
—
(5) 1. —
2. (−1, 0)
3. (0, − 31 )
41
4. x = 1
5. y = 13
6. —
(6) 1.
2.
3.
4.
5.
6.
y=2
none
(0, 2)
none
none
—
(7) 1.
2.
3.
4.
5.
6.
1
y = x+3
none
(0, 13 )
x = −3
y=0
—
(8) 1.
2.
3.
4.
5.
6.
1
y = x+3
none
(0, 13 )
x = −3
y=0
—
(9) 1.
2.
3.
4.
5.
6.
1
y = x−2
none
(0, − 12 )
x=2
y=0
—
(10) 1. y =
2.
3.
4.
5.
6.
(11) 1.
2.
3.
4.
5.
6.
(−2, 0)
(0, −1)
x=2
y=1
—
y = x+3
x+2
(−3, 0)
(0, 1.5)
x = −2
y=1
—
(12) y =
1.
2.
3.
4.
5.
6.
x+2
x−2
2.
3.
4.
5.
6.
(14) 1. y =
2x+1
x−1
—
none
(0, −1)
x=1
y=2
—
(13) 1. y =
(−3, 0), (2, 0)
(0, 1)
x = −2, x = 3
y=1
—
3. (0, 23 )
4. x = −3, x = 2
5. y = 2
6. —
(x−3)(x−2)
(x+3)(x+2)
(17) 1. y =
2.
3.
4.
5.
6.
(2, 0), (3, 0)
(0, 1)
x = −3, x = −2
y=1
—
(15) 1.
y = (x+3)(2x−1)
(x−3)(x+1)
(−3, 0), ( 12 , 0)
2.
3.
4.
5.
6.
(x+3)(x−2)
(x−3)(x+2)
2. (−2, 0), (1, 0)
2. (−3, 0), (2.5, 0)
3. (0, 1)
4. x = −1.5, x = 5
5. y = 1
6. —
(18) 1. y =
x2 −4
x+1
2. (±2, 0)
(0, 1)
x = −1, x = 3
y=2
—
(16) 1. y =
(x+3)(2x−5)
(x−5)(2x+3)
3. (0, −4)
4. x = −1
5. none
2(x−1)(x+2)
(x−2)(x+3)
6. —
Chapter 8
Trigonometry
8.1
Degrees and radians
Q1. (1)
(2)
(3)
π
2
π
4
π
3
(4)
(5)
(6)
Q2. (1) 3.14
π
6
π
12
3π
4
(7)
(8)
(9)
2π
3
3π
2
π
9
(10)
(11)
(12)
5π
18
5π
12
11π
6
(13)
(14)
(15)
7π
12
7π
6
5π
6
(3) 1.40
(5) 1.75
(7) 1.36
(9) 3.49
(11) 1.26
(2) 1.57
(4) 0.209
(6) 0.995
(8) 1.89
(10) 0.314
(12) 5.10
Q3. (1) 30◦
(2) 120◦
(3) 225◦
(4) 22.5◦
(5) 75◦
(6) 105◦
(7) 300◦
(8) 40◦
(9) 100◦
(10) 150◦
(11) 172◦
(12) 90.0◦
(13) 57.3◦
(14) 50.0◦
(15) 69.9◦
8.2
Trigonometric ratios
θ
Q4.
sin θ
cos θ
tan θ
π
6
π
4
π
3
30◦
45◦
60◦
√
√
1
2
√
3
2
√1
3
2
2
√
2
2
1
3
2
1
2
√
3
42
Chapter 8. Trigonometry
√
sin θ
cos θ
tan θ
(6) sin α =
40◦
0.643
0.766
0.839
(7) sin α =
60◦
0.866
0.5
1.73
(8) sin α =
1◦
0.0175
1
0.0175
(9) sin α =
1
0.841
0.54
1.56
0.233
0.972
0.24
1.5
0.997
0.0707
14.1
0.8
0.717
0.697
1.03
(2) cos α = √213 ,
tan α = 32
8◦
0.139
0.99
0.141
(3) sin α =
1.57
1
0.000796
1260
13.5
Q5.
√
5
5
2
3 , cos α = 3 , tan α = 2
15
8
15
17 , cos α = 17 , tan α = 8
√
√
7
7
3
4 , cos α = 4 , tan α = 3
√
√
11
11
5
6 , cos α = 6 , tan α = 5
θ
◦
Q7. (1) cos α = 54 ,
tan α = 34
(4)
0.3
0.296
0.955
0.309
1.2
0.932
0.362
2.57
1.2◦
0.0209
1
0.0209
Q6. (1) sin α = 35 , cos α = 45 , tan α =
(4) sin α =
(5) sin α =
(6)
3
4
4
3
(2) sin α = 54 , cos α = 35 , tan α =
(3) sin α =
(5)
5
12
5
13 , cos α = 13 , tan α = 12
√5 , cos α = √4 , tan α = 5
4
41
41
3
1
√ , cos α = √ , tan α = 3
10
10
Q9. (1) 0.748 or 42.8◦
(7) cos α = √15 ,
tan α = 2
(8) sin α = 31 ,
1
tan α = 2√
2
(9) sin α = 32 ,
√
cos α = 35
√3 ,
√21
tan α = 23
9
sin α = 41
,
9
tan α = 40
sin α = 12
13 ,
5
cos α = 13
sin α = √12 ,
cos α = √12
Q8. (1) 15
(2) 12
(3) 3.4
(5) 1.55 or 88.6◦
√
(10) cos α = 3 5 2 ,
√
tan α = 614
(11) sin α = 78 ,
tan α = √715
5
,
(12) sin α = 11
√
4 6
cos α = 11
√
(7) 6 10
√
(8) 6 5
(4) 4.5
(5) 7.5
√
(6) 29
(9) 12.5
(9) 1.25 or 71.6◦
(2) 1.44 or 82.2◦
(6) 1.19 or 68◦
(10) 0.527 or 30.2◦
(3) 0.779 or 44.6◦
(7) 0.89 or 51◦
(11) 0.503 or 28.8◦
(4) 0.308 or 17.6◦
(8) no such angle
(12) 0.202 or 11.6◦
Q10. 22.9cm, 23.9cm
Q11. 4.10cm, 9.35cm
Q12. 3.77cm, 4.60cm
Q13. 4.35cm, 4.83cm
(4) 0.0463
(7) 20.5
(10) 25.7
(13) 5.6
(16) 159
(2) 23.3
(5) 2.08
(8) 41.8
(11) 28.7
(14) 13.3
(17) 1.82
(3) 1.98
(6) 95.8
(9) 26.2
(12) 71.8
(15) 15.1
(18) 27.6
Q14. (1) 8.75
Q15. 205
Q21. 12.4
Q16. 36.5
Q18. 28.2 m
√
Q19. 9 3 ≈ 15.6
Q17. 23.0 m
Q20. 8.10
Q23. 4.7 m
8.3
Q24. 4.62 m
Q22. 84.3
Q25. 12.8
Trigonometric functions
Q26.
sin A
√
3
2
√
− 22
√
2
2
1
2
− 12
√
− 22
√
2
2
(1) −
(2)
(3)
(4)
(5)
(6)
(7)
cos A
1
2
√
2
2
√
2
2
√
− 23
√
− 23
√
2
2
√
− 22
−
tan A
√
− 3
−1
(8) −1
0
none
(9) 1
0
none
(10)
−1
√
3
3
(12)
3
3
(13)
−1
(14)
−1
(15)
43
√
2
2
√
− 22
− 12
− 12
√
− 23
(11) −
√
−
1
2
√
− 23
√
− 22
√
− 22
√
− 23
√
3
2
− 12
√
−
3
3
1
1
√
3
3√
−
√
3
3
3
Chapter 8. Trigonometry
√
(16)
3
2
√
3
2
− 21
√
2
2
√
− 23
− 12
√
3
2
1
2 √
(17) −
(18)
(19)
(20)
(21)
(22)
√
(25)
3
2
√
2
2
√
(26) −
√
(27)
(28)
(30)
3
3
(33) −
√
− 3
−1
0
(34)
(35)
(36) −
1
(37)
2
2
2
2
−
1
(38)
3
2
1
2
− 12
√
− 3
(39)
√
−
3
3
(40)
√
3
2
−
Q27. (1) cos α = − 45 , tan α = − 34
(2) cos α = − √213 , tan α =
√
(5) sin α =
1
3
2
(6) sin α =
2
2
√
√1 , tan α
5
√1
2
= −2
(3) sin α = −
(7) cos α =
(4) sin α =
1
(8) sin α = 13 , tan α = − 2√
2
21
3
7 , tan α = − 2
9
9
− 41
, tan α = 40
Q28. (1) π2 or 90◦
(2) − π2 or −90◦
(3) 0
◦
(4) − 3π
4 or −135
(5) − π6 or −30◦
◦
(6) − 5π
6 or −150
◦
(7) 5π
6 or 150
3π
(8) 4 or 135◦
−1
−1
√
−
3
3
−1
1
√
− 3
1
2
√
√
3
2
√
3
2
3
3
√
− 33
−
(10)
(11)
(12)
(13)
2π
3
or 120◦
(14) π or 180◦
(17) 0.307 or 17.6◦
(2) −0.561 or −32.2◦
(10) −0.263 or −15◦
(18) −1.38 or −78.8◦
(3) −2.84 or −163◦
(11) 0.374 or 21.4◦
(19) −2.07 or −119◦
(4) −2.3 or −132◦
(12) 2.44 or 140◦
(20) −0.767 or −43.9◦
(5) −2 or −115◦
(13) −2.67 or −153◦
(21) −2.78 or −159◦
(6) −0.114 or −6.55◦
(14) 0.61 or 34.9◦
(22) −1.88 or −107◦
(7) 0.326 or 18.7◦
(15) −0.69 or −39.5◦
(23) −0.46 or −26.4◦
(8) 0.403 or 23.1◦
(16) 0.543 or 31.1◦
(24) 2.45 or 140◦
8.4
√
5
3
√
√
cos α = 3 5 2 , tan α = − 614
sin α = − 78 , tan α = √715
√
5
sin α = 11
, cos α = − 4116
(9) sin α = − 23 , cos α = −
(9) π3 or 60◦
(10) − π3 or −60◦
(11) − π4 or −45◦
◦
(12) − 2π
3 or −120
(9) 2.65 or 152◦
Q29. (1) 2.42 or 139◦
3
0
2
2
√
− 22
√
− 23
√
2
2
√
− 22
5
− 13
=
3
√
2
2
√
− 22
√
− 23
− 21
− 12
12
13 , cos α =
− √12 , cos α
1
√
− 12
2
2
1
2
√
√
− 3
√
2
2
√
3
2
√
− 12
− 12
2
2
√
3
3
√
1
2
√
(32) 0
√
3
2
3
2
√
2
2
√
(31) −
−1
√
3
− 12
√
−
√
(29)
√
− 23
√
− 22
(23) 0
(24)
√
− 3
√
− 3
− 12
Trigonometric equations
π
6,
π
3,
π
2,
5π
6
5π
3
5π
2
(8)
(4) π, 3π
(10)
Q30. (1)
(2)
(3)
(5)
(6)
Q31. (1)
(2)
(3)
(7)
(9)
π 3π
4, 4
π 11π
6, 6
π
4
π
3
π
6
(11)
(12)
7π 11π
6 , 6
2π 4π
3 , 3
5π 7π
4 , 4
5π
7π
6 , 6
−5π −π 7π 11π
6 , 6 , 6 , 6
−4π −2π 2π 4π 8π
3 , 3 , 3 , 3 , 3
(13)
(14)
(15)
(16)
(17)
(18)
(7) 5π
6
3π π
(8) − 7π
4 , − 4 , 4
−2π π
(9) −5π
3 , 3 , 3
(4) 0
(5) 3π
4
(6) 2π
3
Q32.
44
π 2π
3, 3
π 7π
4, 4
4π 5π
3 , 3
3π
5π
4 , 4
−11π −10π −5π −4π
3 ,
3 , 3 , 3
−15π −9π −7π −π
4 , 4 , 4 , 4
(10)
−11π −5π π
6 , 6 , 6
(11)
−5π −π 3π
4 , 4 , 4
Chapter 8. Trigonometry
(1) 30◦ , 150◦
(6) 120◦ , 300◦
(11) 60◦ , 300◦
(16) 270◦
(2) 120◦ , 240◦
(7) 90◦
(12) 150◦ , 330◦
(17) 150◦ , 210◦
(3) 45◦ , 225◦
(8) 180◦
(13) no solutions
(18) 30◦ , 210◦
(4) 240◦ , 300◦
(9) 0◦ , 180◦ , 360◦
◦
(5) 30 , 330
◦
◦
◦
(10) 0 , 180 , 360
Q33. (1)
(2)
(3)
(4)
(5)
(6)
0.412, 2.73
1.23, 5.05
0.433, 2.71
0.635, 5.65
3.66, 5.76
1.88, 4.41
Q34. (1)
(2)
(3)
(4)
1.11
1.15
0.464
1.89
(7)
(8)
(9)
(10)
(11)
(12)
(5)
(6)
(7)
(8)
(14) 90◦ , 270◦
(15) 120◦ , 300◦
◦
3.99, 5.44
1.85, 4.43
0.789, 2.35, 7.07, 8.64
1.17, 5.11, 7.45, 11.4
−5.6, −3.82, 0.682, 2.46
−5.73, −0.555, 0.555, 5.73
(13) 3.59, 5.84, 9.87, 12.1
(14) 1.79, 4.49, 8.08, 10.8
(15) −2.74, −0.401, 3.54, 5.88
(16) −3.77, −2.51, 2.51, 3.77
(9) −4.95, −1.8, 1.34
2.85
1.91
1.75
−5.03, −1.89, 1.25
(10) −5.7, −2.55, 0.588
(11) −7.28, −4.14, −0.998
Q35. (1) 11.5◦ , 168◦
(4) 204◦ , 336◦
(7) 26.1◦ , 154◦
(10) −169◦ , −11◦
(2) 72.5◦ , 287◦
(5) 139◦ , 221◦
(8) −76.7◦ , 76.7◦
(11) −112◦ , 112◦
(3) 42◦ , 222◦
(6) 108◦ , 288◦
(9) −120◦ , 59.5◦
(12) −66.5◦ , 113◦
π 5π
6, 6
7π 11π
6 , 6
Q36. (1) 0, π, 2π,
(2)
(3)
(4)
(5)
(7) 0.34, 2.8, 5.55, 3.87
(8) π6 , 5π
6 , 0.73, 2.41
7π 11π
(9) 6 , 6
5π
(10) π3 , π2 , 3π
2 , 3
4π 5π
(11) π3 , 2π
3 , 3 , 3
(12) 0, π, 2π
7π 11π
(13) π6 , 5π
6 , 6 , 6
π 5π
6, 6 ,
π 3π
2, 2
π 2π 4π 5π
3, 3 , 3 , 3
7π 11π
6 , 6 , 5.76,
3.67, 0.412, 2.73
(6)
8.5
π 5π 3π
6, 6 , 2
(14)
(15)
π 2π
3 , 3 , 1.98, 4.3
4π
0, 2π
3 , 3 , 2π
(16) 1.91,
5.44
(17)
(18)
(19)
0.841,
2π 4π
3 , 3 , 2.3,
π 5π
3, 3
0, π4 , π, 5π
4
(20)
(21)
4.37, (22)
3.98
(23)
(24)
(25)
π 3π 5π 7π
4, 4 , 4 , 4
π 2π 4π 5π
3, 3 , 3 , 3
π 5π 7π 11π
6, 6 , 6 , 6
3π 7π
4 , 4 , 1.11, 4.25
π 5π
4 , 4 , 0.464, 3.61
3π 7π
4 , 4 , 2.36, 5.5
Trigonometry in geometry
(6) c ≈ 6.96, a ≈ 0.625
(11) b ≈ 2.22, a ≈ 4.42
(16) a ≈ 4.7, c ≈ 3.69
(2) b ≈ 4.89, c ≈ 6.63
(7) b ≈ 4.31, a ≈ 11.6
(12) b ≈ 11.3, c ≈ 1.21
(17) a ≈ 12.1, b ≈ 4.67
(3) a ≈ 8.96, c ≈ 10.5
(8) a ≈ 4.52, c ≈ 2.82
(13) b ≈ 14.1, a ≈ 16.9
(18) b ≈ 13.7, c ≈ 13
(4) c ≈ 8.5, b ≈ 4.14
(9) a ≈ 3.19, b ≈ 6.36
(14) a ≈ 6.67, c ≈ 11.6
(19) b ≈ 2.09, a ≈ 4.94
(5) a ≈ 15.1, b ≈ 5.76
(10) c ≈ 2.68, a ≈ 4.41
(15) c ≈ 9.66, b ≈ 12.6
(20) c ≈ 11.1, a ≈ 10.8
Q37. (1) b ≈ 5.53, c ≈ 7.04
Q38. (1) B ≈ 49.9◦ , C ≈ 85.1◦ or B ≈ 130◦ , C ≈ 4.92◦
◦
◦
◦
◦
(11) no such triangle
(2) B ≈ 61.1 , C ≈ 61.9 or B ≈ 119 , C ≈ 4.11
(12) B ≈ 66.1◦ , C ≈ 41.9◦
(3) A ≈ 6.28◦ , C ≈ 161◦
(13) B ≈ 49.1◦ , A ≈ 86.9◦ or B ≈ 131◦ , A ≈ 5.07◦
(4) C ≈ 13.4◦ , B ≈ 155◦ or C ≈ 167◦ , B ≈ 1.37◦
(14) A ≈ 29.4◦ , C ≈ 109◦
(5) no such triangle
(15) C ≈ 56.9◦ , B ≈ 82.1◦ or C ≈ 123◦ , B ≈ 15.9◦
(6) C ≈ 51.8◦ , A ≈ 74.2◦
(16) A ≈ 79.5◦ , C ≈ 26.5◦ or A ≈ 101◦ , C ≈ 5.45◦
(7) B ≈ 36◦ , A ≈ 83◦
(17) A ≈ 31.9◦ , B ≈ 117◦ or A ≈ 148◦ , B ≈ 0.924◦
(8) A ≈ 57◦ , C ≈ 78◦ or A ≈ 123◦ , C ≈ 12◦
(18) B ≈ 67.4◦ , C ≈ 57.6◦ or B ≈ 113◦ , C ≈ 12.4◦
(9) no such triangle
(19) B ≈ 14.7◦ , A ≈ 142◦
(10) C ≈ 67.1◦ , A ≈ 36.9◦
(20) C ≈ 60.9◦ , A ≈ 62.1◦ or C ≈ 119◦ , A ≈ 3.88◦
Q39.
45
(1) c ≈ 6.88
(5) b ≈ 5.56
(9) b ≈ 5.47
(13) a ≈ 1.88
(17) b ≈ 3.89
(2) a ≈ 3.14
(6) a ≈ 2.29
(10) a ≈ 8.82
(14) c ≈ 4.9
(18) c ≈ 9.33
(3) c ≈ 3.2
(7) a ≈ 6.6
(11) a ≈ 4.7
(15) b ≈ 2.94
(19) a ≈ 1.75
(4) b ≈ 4.19
(8) c ≈ 7.77
(12) c ≈ 2.22
(16) c ≈ 7.34
(20) b ≈ 8.53
Q40. (1) A ≈ 46◦
(5) B ≈ 22.5◦
(9) B ≈ 117◦
(13) C ≈ 24.1◦
(17) A ≈ 25.8◦
(2) C ≈ 61.8◦
(6) C ≈ 18.6◦
(10) C ≈ 17.6◦
(14) B ≈ 54.8◦
(18) C ≈ 2.1◦
(3) A ≈ 87.1◦
(7) A ≈ 47.5◦
(11) A ≈ 14.6◦
(15) A ≈ 59.8◦
(19) A ≈ 65.2◦
(4) B ≈ 88.1◦
(8) C ≈ 57.5◦
(12) B ≈ 12.1◦
(16) B ≈ 51◦
(20) B ≈ 23.2◦
(8) c ≈ 11.4
c ≈ 3.75 or c ≈ 26
(9) b ≈ 5.78 or b ≈ 29.9
a ≈ 17.9
(10) a ≈ 9.16 or a ≈ 23.1
no such triangle
(11) a ≈ 30.6
b ≈ 6.32 or b ≈ 12.4
(12) c ≈ 1.06 or c ≈ 6.56
b ≈ 0.984 or b ≈ 11.1
(13) a ≈ 3.26 or a ≈ 6.94
a ≈ 1.5 or a ≈ 28.3
(14) no such triangle
no such triangle
√
3) ≈ 17.1
Q46. (i) 199◦
(i) 25
4 (1 +
√
√
√
5
(ii) 019◦
(ii) 2 (4 + 2 3 + 6 − 2)
≈ 21.2
(iii) 75.2 km
Q41. (1)
(2)
(3)
(4)
(5)
(6)
(7)
Q42.
Q43.
(i) 9
√
√
(ii) 9 6 − 3 2 ≈ 17.8
Q48.
(i) 56.4 km
(ii) 55.4 km
(iii) 72.8 km
Q45. 6.47
8.6
(16) c ≈ 11.8 or c ≈ 24.3
(17) b ≈ 14.1
(18) c ≈ 2.41 or c ≈ 4.8
(19) a ≈ 14.8 or a ≈ 46.1
(20) b ≈ 1.3 or b ≈ 5.64
(iv) 044◦
(v) 274◦
Q49. 20 km or 52.3 km
Q47. 44.7 km, 84.9 km
Q44. 146◦
(15) b ≈ 11.9
Q50. (1) both 66.5 km
(2) 60.9 km and 30.5 km
Arcs, sectors, segments
Q51. (1) 10.2
(3) 6.08
(5) 20.6
(7) 5.06
(2) 18.3
(4) 16.6
(6) 11.9
(8) 15.3
Q52. (1) 1.11
(2) 2.15
(3) 40.6
(4) 18.4
(5) 0.117
(6) 0.301
Q53. (1) 3.05
(2) 50.4
(3) 1.53
(4) 0.285
(5) 9.88
(6) 0.622
Q54. (1) 100◦
(2) 66.2◦
(3) 101◦
(4) 66.7◦
(5) 43.5◦
Q55.
2
3
rad or 38.2◦
Q56. 26.4
Q57.
(i) 12.5
(ii) 11.6
(6) 36◦
√
Q58. 18.00l
Q60. 100( 2π
3 − 3)cm2 ≈
3 +
2
≈ 82.6cm
Q59. 50(π − 2)cm2 ≈ 57.1cm2
Chapter 9
Geometry
9.1
Polygons
Q61. (1) true
(6) false
(11) true
Q62. 20cm2
(2) true
(7) false
(12) false
(3) true
(8) false
(13) false
(4) false
(9) true (?)
(14) true
Q63. 252cm2
√
Q64. 32 3 ≈ 55.4
(5) true
9.2
Q65. 64
(10) false
Circles
46
Q66. 67◦ or 113◦
Q72. E F̂ G = 80◦ ,
E D̂G = 100◦
or E F̂ G = 100◦ ,
E D̂G = 80◦
Q67. 156
√
Q68. 27 3cm
√
Q69. 24 3 ≈ 41.6
Q85. 2 : 1
Q86. A = 30◦ ,
B = 60◦ , S = 90◦
Q80. 9cm2
√
Q81. 8 2cm2
Q75. A = 31◦ ,
√
Q82. 32 3cm2
Q83. 128cm2
√
Q84. 27 3
Q77. 78.5◦
√
Q78. 16 2
√
Q79. 24 3cm2
Q74. A = 58◦ ,
B = C = 61◦
Q70. 16cm
√
Q71. 36 3 ≈ 62.4
Q87. —
Similarity
Q88. (1) 4.5
Q89.
Q76. 180cm2
Q73. 34◦
2
9.3
B = 59◦ , C = 90◦
2
3r
Q90.
(2) —
Q91.
140
3
Q92. —
9.4
2
3r
√
√
16
3
6
Q93. 43.56
Q96.
3
Q94. 4 : 1
Q97. equal
Q95. 6.4
Q98. 3 : 1
Q99.
(i) 16
(ii) 5 : 3
Q100. —
Solid geometry
√
Q111. cube, 3 2 : 16
√
Q101. V = 9 2 2 ,
√
A=9+9 3
√
Q102. 36 + 36 7
Q106. 25m40cm
√
Q107. 3 3 : 1
√
Q108. 3 3 : 1
Q112. cube, 2 : 9
√
Q113. V = 83 , A = 4 3
Q103. 224cm3
√
Q104. 1.024 6 ≈ 2.51(m3 ) Q109. 1 : 6
√
Q105. 141cm
Q110. 3 : 9
9.5
Q114. regular octahedron,
1:2
Q115. regular tetrahedron,
1 : 27
Q116. 30.2%
Q117. 12.3%
√
(i) 2 2
Q118.
√
(ii)
2
3
Miscellaneous problems
Q119. 2
√
Q120. 81( 2 − 1)
√
Q121. (1) 3 2 − 4
Q124. 1.55
Q127. 5
Q125. (1) 1
Q128.
(2) 1.44
Q122. —
Q123. 1.46
Q126.
√
2
3 (2
√
10
3
Q134.
2
2
Q129. 65
(3) 0.95262
√
2
Rr
√
(4) √R+
r
(2) —
√
Q133. 5 2
3 − 3)
Q135.
Q130. —
25π
4 (2
−
√
Q131. 36 3
Q136. 154.9m2
Q132. —
Q137. 50%
√
Chapter 10
Numbers II
10.1
Factorials and binomial theorem
Q1. (1) 6
(2) 24
(3) 120
(4) 720
(5) 5040
Q2. (1) 7
(2) 8
(3) 12
(4) 110
(5) 380
Q3. (1) 7
(2) 5
(3) 8
Q4. (1) 3
(4) 10
(7) 6
(10) 15
(13) 35
(2) 4
(5) 10
(8) 15
(11) 21
(14) 56
(3) 6
(6) 5
(9) 20
(12) 35
(15) 28
Q5.
47
3) ≈ 5.26
Chapter 10. Numbers II
n(n+1)
2
(1)
(2)
3n(3n−1)
2
(3)
10.2
Logarithms
10.2.1
Algebra of logarithms
Q6. (1) 3
(2)
1
4
n(n−1)
2
(4)
n(n+1)
2
(5)
(4)
25
2
(5) 46 = 212
√
2
2
(3)
Q7. (1) 3
(6) − 12
(11) 2.5
3
2
2
3
3
2
1
4
(7) − 32
(12)
(8) −3
(13) −5
(18)
(9) − 74
(14) 5
(19) − 83
(15) −2
(20)
(2)
(3)
(4)
(5)
(10) 3.5
Q8. (1) 2b
(2) a + b
Q9. (1) a + 2b
(16) − 32
5
4
(17)
(22) − 74
3
5
1
2
(24)
1
5
6
(25) 2 × 3 4
(6) 3 + a + 3b
(8) 3 + 2a
(5) a + 6b
(7)
(6) 6b
(8) a
+b
1
a
(5)
3
a
(4) 1 +
1
2a
(6)
Q11. (1) −a − 3b
(2)
Q12. (1) a − 2b − 12 c
(2) 3a + 12 b − 2c)
−b
1
2a
1
3a
(7)
(8)
1
2a
(3) a − 21 b
(4) 1 − 4a − 32 b
(3) 2.5a − 1.5b − c
(4) −4a − 2.5b − 43 c
(3) a = 4, b = 1
(5) a = 3, b = 1
(2) a = 2, b = 1
(4) a = 3, b = 3
(6) a = 4, b = 2
3
4 (2a
Q15. (1)
√
− 1)
(2)
3
4
(4) 3( 3 )
Q16. (1) 2a
(5)
a
2
(6) a
(3) 32a
(4) 8a2
10.2.2
Q17. (1)
(7) a3
(8)
(3) 8
Q18. (1) no solutions
10.2.3
Q19.
14−8a
3a−3
√
(5) 2 2
(7)
1
4
(6)
(9) c2
(13)
(10) a4
(14)
1
a4
3
(15)
(11)
1
b2
(4)
(12) b
(16)
Logarithmic equations
27
64
(2) 27
(2)
1 3
3b
2
3
2a−1
(3)
(3) 8
(2) 25
(2)
2a+3
4a−2
11
4
+ 4b
2+5a
3
5
2a
Q13. (1) a = 2, b = 2
Q14. (1)
1
12500
√2
3
(23)
(4) 3 + b
(3)
(2) 2a
3
4
(21)
(7) 3a
1
2a
1
9
(8) 8
1 2
16 a
1
c
16
a
729
b2
√
3
(17) c4
(18)
(19)
(20)
(21)
1 6
8a
256
a12
(22)
(23) ac
6
b
243
2
(24)
(10) x =
64
π3
1
8
1
5 or
(5) 55 = 3125
√
(6) 3 2
(8)
(3) 7
(5) − 12
(7) 0.1
(6) 9
(8) no solutions
1
2
(11) x =
(9) no solutions
17
4
(12) x =
Q20.
(i) 15 litres
Q21.
(i) 0.242
≈ 0.167
(ii) 34.3 minutes
(ii) 965 thousands
(iii) 39.9 years
(iii) 228 minutes
(iii) 29
(ii)
a2 b4
18
(7)
Aplications
1
6
1
3
25
c5
(4) 9
(4)
(i) 5 milion
(n+2)(n+1)n
6
(6) 11 × 3n
(5) 2 + a + b
(4) a + 2b
Q10. (1) −a
(6)
(3) 2a + b
(3)
(2) 2a + 4b
2n(2n−1)
2
48
x=5
10.3
Absolute value equations and inequalities
Q22. (1)
(2)
(3)
(4)
(5)
(6)
(7)
x 6= −2
x ∈] − ∞, 1[∪]2, +∞[
x ∈] − ∞, −2] ∪ [ 43 , +∞[
x∈R
x ∈] − 94 , − 14 [
x∈∅
x ∈ [− 25 , − 12 ]
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(2) −2 < x < 0
(3)
(4)
≤x≤0
≤x≤5
Q24. (1) 1.6 ≤ x ≤ 4
10.4
(6)
− 25
< x < 2, x 6=
(7)
− 32
≤ x ≤ − 61 , x 6= − 21
5
2
(15) x =
(16) x ∈ R
(17) x ∈] − ∞, 1] ∪ [4, +∞[
(18) x ∈ [ 34 , 2]
(19) x ∈] − ∞, − 72 [∪] 12 , +∞[
(20) x ∈] − 43 , 83 [
(5) x < −4 or −4 < x < − 52
or x > 2
Q23. (1) x < 0 or x > 2
− 23
− 15
x∈∅
x∈R
x ∈] − 12 , 92 [
x ∈ [− 32 , 12 ]
x ∈] − ∞, − 32 ] ∪ [ 29 , +∞[
x ∈] − ∞, 23 [∪]2, +∞[
x∈∅
(8) 0 < x < 4
(9) x < −1 or −1 < x ≤ − 81
or x ≥ 2.5
1
2
(10)
4
5
≤x≤
(3) − 72 < x < −1
(2) x < 1 or x > 3.5
16
3
(4) −1 ≤ x ≤ 3
Complex numbers
Q25. (1) 2 + 11i
(2) 11 + 13i
Q26. (1)
2
5
− 15 i
(2) 1 − 17i
(3) 20
(5) −9 + 2i
(7) 9 + 12i
(4) 13i
(6) 19 − 9i
(8) 8 + 0.25i
(5) 7 + 6i
(7) −1 − 34 i
(3)
+ 45 i
(4) −12 + 5i
Q27. (1) 2 − 11i
(2) −11 − 2i
10.5
3
5
(6) 1 + 21i
(8)
−8
41
+
31
41 i
(9) −6 − 8i
(10) −10 + 10i
(9) 6 − 8i
(10)
7
10
+
√
(5) −8 − 8i 3
√
(6) −16 + 16i 2
(3) −4
(4) −64
1
10 i
(11) 13
(12) 4 − 7i
(11) −5 + 12i
(12) 8 − i
√
(7) −16 + 16i 3
(8) 1
Mathematical induction
Q28. —
Chapter 11
Quadratics and polynomials
11.1
Vieta’s formulae for quadratics
Q1. (1) 2; different signs
(6) 2; different signs
(11) 2; both positive
(16) 2; different signs
(2) 2; both negative
(7) 1; positive
(12) 2; different signs
(17) 2; both positive
(3) 0
(8) 2; both positive
(13) 2; both negative
(18) 2; both negative
(4) 1; negative
(9) 0
(14) 1; positive
(19) 2; different signs
(15) 2; both positive
(20) 2; both negative
(5) 2; both negative
(10) 2; both positive
Q2. 0 < m < 1
Q3. −2 < m < 0
Q7.
Q6.
2
9
<m<
1
3
<m≤
Q8. 0 ≤ m ≤
Q4. no such m
Q5. −2.5 < m ≤ −0.5 or m ≥ 0.5
1
3
√
2+ 6
4
√
−7+5 2
2
Q9. −2 < m < − 12
√
Q10. m ≤ −10 − 6 3 or m > 2
49
Chapter 11. Quadratics and polynomials
Q11.
√
18−8 3
11
Q12.
4
3
≤m<
≤m<
2
3
(ii) −1 < m ≤
3
2
(i) m < 0 or m ≥
Q13.
11.2
Q15. (1)
(2)
Q14.
√
7+4 2
3
√
7−4 2
,
3
x 6=
1
3
√
(i) m < −2 or m ≥ 10 + 6 3
√
(ii) −1 < m ≤ 10 − 6 3, and m 6= − 21
Algebraic fractions
x+2
2x+1
2x−3
x+2
(3)
(4)
3x−2
2x+5
2x+1
2x+3
(5)
(6)
3x−2
2x+3
3x−2
2x−3
(8)
5x+1
5x−1
2x+5
5x+2
(7)
(9)
(10)
8x+2
x−2
6x−5
2x+1
(11)
(12)
2x−5
5x+2
−3x+4
2x+1
(4)
24x2 −22x+4
5
(6)
2x2 −3x+1
2x2 +3x+1
(8)
2x2 +7x+6
2x2 −7x+6
(5)
6
6x2 +x−1
(7)
−4x+6
6x+3
(9)
6x2 −x−1
6x2 +x−1
5x+12
(x+2)(x+3)
(4)
−12
(2x+1)(2x−3)
(7)
17x+7
(5x+1)(x+2)
(10)
3x−1
(6x−5)(3x−2)
(2)
−x
(2x−3)(x−2)
(5)
4x+3
(3x−4)(2x−1)
(8)
8x+14
(2x+5)(2x−1)
(11)
4x+19
(2x−5)(5x+2)
(3)
24x
(3x−2)(3x+2)
(6)
14x+16
(3x−2)(2x+5)
(9)
−18x+5
(6x+2)(3−2x)
(12)
−6x+7
(4−3x)(2x−3)
7x2 +x−5
(2x+1)(x−2)(x+3)
(4)
−14x−8
(2x+3)(3x−2)(2x−3)
(7)
10x2 +29x−1
(5x−1)(3x+2)(x+2)
(10)
12x2 −6x−2
(2x+1)(3x+1)(3x−2)
(2)
x2 +7x+13
(x+2)(x+3)(x−2)
(5)
14x2 −3x+3
(2x+3)(2x−3)(2x+1)
(8)
11x2 +7x−4
(5x+2)(3x)(2x−1)
(11)
−19x2 −25x
(5x+2)(3x)(5x+2)
(3)
−36x+5
(2x+5)(2x−5)(3x−2)
(6)
21x2 −20x−1
(2x−3)(5x−1)(2x+5)
(9)
20x2 −2x−4
(x−2)(3x−2)(2x−3)
(12)
−15x2 +18x+10
(2x+1)(5x+2)(2x−3)
4x+8
Q16. (1) − 3x−2
(2) − 23
Q17. (1)
Q18. (1)
11.3
(3)
2
3
Equation of a circle
Q19. (1) centre: (2, 1), radius =5
(2) centre: (0, −2), radius =4
(3) centre: (3, 0), radius =3
(4) centre: (4, −4), radius =4
(5) centre: (1.5, −2.5), radius =3
Q20. (1)
(2)
(3)
(4)
(5)
(6)
(5, 5), (−2, 4)
(4, −5), (5, −2)
(4, 7), (8, −5)
(6, 5), (−2, 3)
(−3.5, 2.5), (0.5, −9.5)
(3, 1), (−9, 9)
(10) centre: (−1.5,
(6) centre: (−5, 2), radius =7
√ −0.5),
√
radius
=2
2
(7) centre: (−3, 1), radius =3 2
√
√
(8) centre: (1, 1.5), radius =4 3 (11) centre: (2, 0.5), radius =5 3
(7)
(8)
(9)
(10)
(11)
(12)
(−10, −3), (−2, −7)
(8, 0.5), (6, −3.5)
(2, 6.5), (−10, 0.5)
(5.5, −6.5), (0.5, −9.5)
(9, −3.5), (1, 8.5)
(9.5, −1), (7.5, −9)
(2) (6, 2), (9, −2)
Q28.
(3) (−2, −2), (6, 2)
(14) (1, 1)
(15) (−1, −3)
(16) (−2, 1)
√
√
13 or 5 13
√
Q29. b = 2 or b = −4, r = 4 2
(4) (−2, −2), (0, 4)
Q22. (x − 8)2 + (y − 1)2 = 25
√
Q30. a = 2 or a = −10, r = 4 5
Q23. (x + 2.5)2 + (y + 0.5)2 = 32.5
625
16
Q31. 2 or
1
2
Q25. (x − 4)2 + (y − 10)2 = 180
Q32. −3 or − 31
Q26. (−1.5, 4) or (−0.5, 2)
Q33. −6 or 14
11.4
(13) (2, 1)
Q27. (−5.5, −3.5) or (3.5, −0.5)
Q21. (1) (8, 1), (7, 2)
Q24. (x − 3)2 + (y − 34 )2 =
(12) centre: (0.5,
√ −3),
radius =2 10
(9) centre: (−3,
√ 1.5),
radius =3 2
Polynomials
Q34.
50
Chapter 11. Quadratics and polynomials
(1) x ∈]1, +∞[
(7) x ∈] − ∞, −3] ∪ [−1, 1]
(13) x ∈] 23 , 32 [∪] 32 , +∞[
(2) x ∈] − ∞, 1]
(8) x ∈] 13 , 21 [∪] 12 , +∞[
(14) x ∈ { 35 }[ 34 , +∞[
(3) x ∈] − ∞, 0[
(9) x ∈] − ∞, − 13 [
(15) x ∈] − ∞, −2[∪] 14 , 3[
(4) x ∈ {−1}[0, +∞[
(10) x ∈] − ∞, − 23 [∪] − 23 , − 23 [
(16) x ∈] − ∞, −5] ∪ { 57 }
(5) x ∈] − ∞, −1] ∪ [0, 3]
(11) x ∈ [−1, 12 ] ∪ [2, +∞[
(17) x ∈ { 73 }[ 12
5 , +∞[
(6) x ∈] − 1, 1[∪]1, +∞[
(12) x ∈] − ∞, 32 ] ∪ { 32 }
(18) x ∈] − ∞, − 32 [∪] − 23 , 32 [
Q35. (1) x ∈] 15 , +∞[
(7) x ∈] − ∞, −3] ∪ [−1, 0]
(13) x ∈]0, 32 [∪] 32 , +∞[
(2) x ∈] − ∞, 52 ]
(8) x ∈]0, 12 [∪] 12 , +∞[
(14) x ∈ [0, 53 ] ∪ [ 34 , +∞[
(3) x ∈] − ∞, 0[
(9) x ∈] − 31 , 0[∪] 32 , +∞[
(15) x ∈] − ∞, −2[∪]0, 14 [
(4) x ∈ {−2} ∪ [0, +∞[
(10) x ∈] − ∞, − 23 [∪]0, 13 [
(16) x ∈] − ∞, −5] ∪ [0, 57 ]
(5) x ∈] − ∞, −1] ∪ [0, 2]
(11) x ∈] − ∞, −1] ∪ [0, 12 ]
7
(17) x ∈ [− 10
3 , 0] ∪ [ 3 , +∞[
(6) x ∈] − 1, 0[∪]1, +∞[
(12) x ∈] − ∞, 0] ∪ [ 32 , 32 ]
(18) x ∈] − ∞, − 32 [∪]0, 32 [
(7) x ∈ [−4, −2] ∪ [4, +∞[
(13) x ∈] − 23 , 32 [∪] 32 , +∞[
(2) x ∈] − ∞, −2] ∪ [1, 2]
(8) x ∈] − ∞, − 32 [∪] 12 , 23 [
(14) x ∈] − ∞, 34 ]
(3) x ∈] − 12 , 14 [∪] 12 , +∞[
(9) x ∈] − ∞, − 23 [∪] 12 , 23 [
(15) x ∈] − 14 , 15 [∪] 14 , +∞[
Q36. (1) x ∈] − 3, 2[∪]3, +∞[
(4) x ∈ [− 32 , − 12 ] ∪ [ 23 , +∞[
(10) x ∈] − ∞, − 13 [
(16) x ∈] − ∞, − 57 ] ∪ [ 43 , 75 ]
(5) x ∈] − ∞, − 52 ] ∪ [1, 52 ]
(11) x ∈ [− 12 , 25 ] ∪ [ 12 , +∞[
(17) x ∈ [− 73 , 32 ] ∪ [ 73 , +∞[
(6) x ∈] − 53 , 32 [∪] 53 , +∞[
(12) x ∈ {− 32 }[ 32 , +∞[
(18) x ∈] − ∞, − 32 [
(7) quotient: x3 − x2 − 2x + 3,
remainder: 2x − 1
Q37. (1) quotient: x3 + 3x2 − 2x − 1,
remainder: 1
(2) quotient: x3 − x2 − 2x + 3,
remainder: −1
(8) quotient: x3 + 2x2 − 2x + 3,
remainder: 2x + 3
(3) quotient: 2x4 − 2x3 − 3x2 + 6x + 3,
remainder: 6
(9) quotient: x4 + x3 − x2 − 2x + 3,
remainder: −1
(4) quotient: 3x4 − x3 − 2x + 3,
remainder: −5
(10) quotient: 2x4 + 5x3 − x2 − 2x + 3,
remainder: 2x − 3
(5) quotient: −x4 + 2x2 − 2x + 1,
remainder: −7
(11) quotient: 3x4 + x3 + 2x2 − 2x + 3,
remainder: 3x − 2
(6) quotient: −2x4 + 3x2 − 2x + 1,
remainder: −4
(12) quotient: 2x4 + x3 − 2x2 + 3,
remainder: 4x
2x+1
3x2 −2
2x + 3 + x23x−1
+x−2
x+4
x + 3 + x2 −2x−3
3x2 + x + 3x3x−5
2 +x−2
Q38. (1) x3 + 3x2 − 2x − 1 +
(2) 4x2 −
(3) 2x3 −
3
(4) 2x −
Q39. (1) −3
(4)
(2) 3
(5)
(3) 12
(6)
−x+4
x2 −3x+2
4
2x3 − 4x2 + 3x + 1 + x2 −2x+1
x3 − 6x2 − x + 1 + 2x25x+6
−3x−2
3
2
2x − x − x − 1 + −3x2−8
+2x−1
(5) 3x3 − 5x2 + 4x − 2 +
(6)
(7)
(8)
7
2
5
3
281
8
(11)
(12)
(10)
(8) − 89
(11) − 11
32
(9) − 34
(12)
(i) − 12
Q43.
(ii) (x + 1)(2x + 1)(x − 2)
(i) 2
(ii) (2x − 5)2 (x − 2)
Q41.
(i) − 13
Q44.
(ii) (x + 2)(x − 3)(3x + 1)
(i) −4
(ii) (x + 4)(x − 5)(3x − 4)
Q42.
(i) 32
(ii) (2x − 3)2 (2x − 3)
(i) 2
(ii) (3x + 5)2 (x − 2)
51
(10)
4
(7) − 25
Q40.
Q45.
5x+6
x2 −3x+5
4x3 − 3x2 + 2x − 1 + x2−x+5
−2x+3
3x+3
3
2
2x − x + 3 + 2x2 +2x−1
x3 + 2 + x22x−1
+x−1
(9) x3 + 2x2 + 3x + 4 +
Q46.
9
4
40
9
(i) − 43
(ii) (4x − 1)(4x + 3)(2x − 1)
Q47.
(i) − 34
(ii) (3x − 2)(3x + 1)(3x + 4)