2 Quadratic Equations in One Unknown (II) 2 Quadratic Equations in One Unknown (II) Review Exercise 2 (p. 2.5) 1. 2. ∴ (repeated) 3. 4. Using the quadratic formula, 5. Using the quadratic formula, ∵ is not a real number. ∴ The equation has no real roots. 6. Using the quadratic formula, 7. The x-intercepts of the graph of are –2.0 and 3.0. Therefore, the roots of are –2.0 and 3.0. 35
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2 Quadratic Equations in One Unknown (II)
2 Quadratic Equations in One Unknown (II)
Review Exercise 2 (p. 2.5)
1.
2.
∴ (repeated)
3.
4. Using the quadratic formula,
5. Using the quadratic formula,
∵ is not a real number.∴ The equation has no real roots.
6. Using the quadratic formula,
7. The x-intercepts of the graph of are –2.0 and 3.0.Therefore, the roots of are –2.0 and 3.0.
35
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
8. The x-intercept of the graph of is 0.5.
Therefore, the root of is 0.5.
9. (a) The graph of intersects the x-axis at two points.
Therefore, the equation has two unequal real roots.
(b) The graph of touches the x-axis at one point.Therefore, the equation has one double real root.
(c) The graph of does not intersect the x-axis.
Therefore, the equation has no real roots.
10.
∴
11.
∴
12.
∴
13.
∴
To Learn More
To Learn More (p. 2.23)
(a) Mid-point of PQ
(b) Length of PQ
To Learn More (p. 2.30)(a)
(b)
(c)
(d)
Classwork
Classwork (p. 2.7)
Value of ( = b2
– 4ac)
Nature of roots
2 unequal real roots
1 doublereal root
No realroots
(a) 0 (b) 24 (c) – 31 (d) – 400 (e) 25
Classwork (p. 2.10)1. (a) 0 (b) < 0
2. (a) 2 (b) > 0
3. (a) 0 (b) < 0
4. (a) 1 (b) = 0
5. (a) 1 (b) = 0
6. (a) 2 (b) > 0
Classwork (p. 2.18)
Sum of roots Product of roots
(a)
(b)
(c)
(d)
(e)
(f)
36
2 Quadratic Equations in One Unknown (II)
Classwork (p. 2.29)1. (a)
(b)
(c)
(d)
2. (a)
(b)
Classwork (p. 2.31)
Real part Imaginary part
(a) 3 – 4
(b) – 5 7
(c) 8 8 0
(d) 0
(e)
Classwork (p. 2.32)(a) (b) (c) (d)
Quick Practice
Quick Practice 2.1 (p. 2.7)∵ The equation has one double real root.∴ = 0
i.e.
Quick Practice 2.2 (p. 2.8)∵ The equation has no real roots.∴
i.e.
∴ The range of values of k is .
Quick Practice 2.3 (p. 2.9)(a) For the equation ,
(b) ∵ is a quadratic equation.
∴ The coefficient of cannot be zero.
i.e.
(i) ∵ The equation has two distinct real roots.
∴
∴ The range of values of k is except .
(ii) ∵ The equation has real roots.
∴
∴ The range of values of k is except .
Quick Practice 2.4 (p. 2.11)(a) ∵ The graph of touches the x-axis at one point P.
∴
i.e.
(b) For m = 9, the corresponding quadratic equation is
∴ The coordinates of P are .
Quick Practice 2.5 (p. 2.11)(a) ∵ The graph of has two
x-intercepts.∴
37
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
i.e.
∴ The range of values of m is .
(b) The smallest integral value of m is 0.For m = 0, the corresponding quadratic equation is
∴ The x-intercepts of the graph are and 1.
Quick Practice 2.6 (p. 2.15)(a) The required quadratic equation is:
(b) The required quadratic equation is
Quick Practice 2.7 (p. 2.16)
(a) Sum of roots
Product of roots
∴ The required quadratic equation is
(b) Sum of roots
Product of roots
∴ The required quadratic equation is
Quick Practice 2.8 (p. 2.19)(a) For the equation ,
∴
(b)
Quick Practice 2.9 (p. 2.20)
Quick Practice 2.10 (p. 2.21)
(a)
(b)
38
2 Quadratic Equations in One Unknown (II)
(c)
Quick Practice 2.11 (p. 2.22)
(a)
(b)
Quick Practice 2.12 (p. 2.23)∵ and are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
Quick Practice 2.13 (p. 2.29)
(a)
(b)
Quick Practice 2.14 (p. 2.32)
By comparing the real parts, we have
By comparing the imaginary parts, we have
Quick Practice 2.15 (p. 2.33)
(a)
(b)
Quick Practice 2.16 (p. 2.34)
(a)
(b)
39
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(c)
Quick Practice 2.17 (p. 2.35)
(a)
(b)
(c)
Quick Practice 2.18 (p. 2.36)Using the quadratic formula,
Quick Practice 2.19 (p. 2.37)∵ is a root of the equation .
∴
∴From (2),
By substituting into (1), we have
Further Practice
Further Practice (p. 2.12)
1.
40
2 Quadratic Equations in One Unknown (II)
∵ The equation has two unequal real roots.∴
i.e.
∴ The range of values of m is m < 8.
2. (a) ∵ The graph of touches the x-axis at one point P.
∴
i.e.
(b) For k = 5, the corresponding quadratic equation is
∴ The coordinates of P are .
For , the corresponding quadratic equation is
∴ The coordinates of P are .
Further Practice (p. 2.23)
1.
2.
(a)
(b)
Further Practice (p. 2.35)
1.
2. (a)
By comparing the real parts, we have
By comparing the imaginary parts, we have
(b)
By comparing the real parts, we have
41
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
By comparing the imaginary parts, we have……(1)
By substituting into (1), we have
Exercise
Exercise 2A (p. 2.12)Level 11. For the equation ,
∵∴ The equation has two unequal real roots.
2. For the equation ,
∵∴ The equation has no real roots.
3. For the equation ,
∵∴ The equation has one double real root.
4. Consider .
∴ The graph of has one x-intercept.
5. Consider .
∴ The graph of has no x-intercepts.
6. Consider .
∴ The graph of has two x-intercepts.
7. ∵ The equation has one double real root.
∴
i.e.
8. ∵ The equation has two equal real
roots.∴
i.e.
42
2 Quadratic Equations in One Unknown (II)
9. ∵ The equation has two unequal real roots.
∴
i.e.
∴ The range of values of m is .
10. ∵ The equation has two unequal real roots.
∴
i.e.
∴ The range of values of m is .
11. ∵ The equation has no real roots.∴
i.e.
∴ The range of values of k is .
12. ∵ The equation has no real roots.
∴
i.e.
∴ The range of values of k is k > 5.
13. ∵ The graph of has only one x-intercept.
∴
i.e.
14. ∵ The graph of has no x-intercepts.∴
i.e.
∴ The range of values of m is .
43
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
15. ∵ The equation has real roots.
∴
i.e.
∴ The range of values of k is .
16. ∵ The equation has real roots.
∴
i.e.
∴ The range of values of k is .
17. ∵ The graph of touches thex-axis at only one point.
∴
i.e.
18. ∵ The graph of cuts the x-axis at two distinct points.
∴
i.e.
∴ The range of values of m is .
Level 2
19.
∵ The equation has one double real root.
∴
i.e.
20.
∵ The equation has two distinct real roots.
∴
i.e.
∵ is a quadratic equation.∴ The coefficient of x2 cannot be zero.i.e.
∴ The range of values of m is except .
21. (a) ∵ The graph of touches the x-
axis at one point P.∴
i.e.
(b) For k = 9, the corresponding quadratic equation is
∴ The coordinates of P are (–6, 0).
22. (a)
∵ The graph of touches the x-axis at one point Q.
∴
i.e.
(b) For , the corresponding quadratic equation is
∴ The coordinates of Q are .∴ Length of OQ
44
2 Quadratic Equations in One Unknown (II)
23.
∵ The graph of has no x-intercepts.
∴
i.e.
∴ The range of values of p is .
24.
∵ The equation has real roots.∴
i.e.
∴ The largest value of k is .
25.
∵ The graph of intersects the x-axis.
∴
i.e.
∴ The smallest value of m is .
26. (a)
∵ The graph of cuts the x-axis at two points.
∴
i.e.
∴ The range of values of p is .
(b) The largest integral value of p is 0.For p = 0, the corresponding quadratic equation is
∴ The x-intercepts of the graph are and 1.
27. (a)
∵ The equation has two unequal real roots.
∴
i.e.
∴ The range of values of k is .
(b) The smallest integral value of k is .For , the corresponding quadratic equation is
28. For the equation ,
∴ The equation has two unequal real roots for any positive values of p.
29. ∵ The equation has two distinct real roots.
∴
i.e.
∴ Any pair of integral values of a and c such that
is acceptable.
45
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
∴
(or any other reasonable answers)
30. ∵ The graph of touches the x-axis at one point.
∴
i.e.
∴ Any pair of integral values of m and n such that
m2 = 4n is acceptable.
∴
(or any other reasonable answers)
31. (a)
(i) ∵ The equation has
two distinct real roots.∴
i.e.
∴ The range of possible values of k is
.
(ii) The only possible negative integral value of k is .For , the corresponding quadratic
equation is
(b)
Exercise 2B (p. 2.17)Level 11. The required quadratic equation is
2. The required quadratic equation is
3. The required quadratic equation is
4. The required quadratic equation is
5. The required quadratic equation is
6.
∴ The required quadratic equation is
7.
∴ The required quadratic equation is
8.
46
2 Quadratic Equations in One Unknown (II)
∴ The required quadratic equation is
9.
∴ The required quadratic equation is
10.
∴ The required quadratic equation is
11.
∴ The required quadratic equation is
Level 2
12. (a)
(b) The roots of the required quadratic equation areand , i.e. 0 and 6 respectively.
∴ The required quadratic equation is
13. (a)
(b) The roots of the required quadratic equation are
and , i.e. and respectively.
∴ The required quadratic equation is
47
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
14. (a)
(b) The roots of the required quadratic equation are
and , i.e. and respectively.
∴ The required quadratic equation is
15. (a) Using the quadratic formula,
(b) The roots of the required quadratic equation are
, i.e. .
∴ The required quadratic equation is
16. (a)
Using the quadratic formula,
(b) The roots of the required quadratic equation are
, i.e. .
48
2 Quadratic Equations in One Unknown (II)
∴ The required quadratic equation is
17. (a) The required quadratic equation is
(b) When m = 1, n = 6 or m = 6, n = 1.The required quadratic equation is
When m = 2, n = 3 or m = 3, n = 2.The required quadratic equation is
Exercise 2C (p. 2.24)Level 1
1.
2.
3.
4.
5. For the equation mx2 + 9x + n = 0,
6. Let be the other root.
∴ The other root is 5.
7. (a)
(b)
8. (a)
(b)
9.
49
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
10.
When ,
When ,
∴ ,
11.
(a)
(b)
12.
(a)
(b)
13.
(a)
(b)
14. ∵ and are the roots of x2 – 5x – 3 = 0.
∴
(a) For the required quadratic equation,
∴ The required quadratic equation is
(b) For the required quadratic equation,
∴ The required quadratic equation is
15. ∵ and are the roots of .
50
2 Quadratic Equations in One Unknown (II)
∴
(a) For the required quadratic equation,
∴ The required quadratic equation is
(b) For the required quadratic equation,
∴ The required quadratic equation is
16. (a) Sum of roots =
Product of roots =
(b) ∵ –2 and 3 are the x-intercepts of the graph of y = px2 + qx – 12.
∴ –2 and 3 are the roots of px2 + qx – 12 = 0.
17. (a) (i) By substituting (0, 15) into y = –x2 + mx + n,we have
(ii) ∵ 5 is one of the x-intercepts of the graph ofy = –x2 + mx + n.
∴ 5 is one of the roots of –x2 + mx + n = 0.Let be the other root.
∴ The coordinates of P are (–3, 0).
51
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(b)
Level 2
18.
19.
20. (a)
∵ Sum of roots = product of roots + 6
∴
(b) By substituting k = 5 into the equation, we have
21. Let and be the roots of the equation.
52
2 Quadratic Equations in One Unknown (II)
22. Let and be the roots of , where.
∵ The difference between the roots is 3.
∴
23.
(a)
∴
(b)
24.
(a)
(b)
25.
(a)
(b)
53
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
26. Sum of roots =
Product of roots =
(a)
(b)
(c) From (b), (rejected ∵ )
(d)
27.
(a)
(b)
(c)
(d) From (c), (rejected∵ )
28. ∵ and are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
29. ∵ and are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
30.
∵
54
2 Quadratic Equations in One Unknown (II)
∴
31.
∵
∴
32. (a) (i) ∵ is a root of .
∴
(ii)
(b) ∵ is also a root of .
∴
33. Let and be the roots of rx2 – (r + 2)x + 2 = 0, where < .∴ The coordinates of A and B are (, 0) and (, 0)
respectively.
(a) ∵
∴
(b) By substituting into the equation, we have
∴ The coordinates of A and B are (1, 0) and (4, 0) respectively.
34. Let and be the roots of , where < .∴ The coordinates of A and B are (, 0) and (, 0)
respectively.
55
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(a) ∵ P(5, 0) is the mid-point of AB.
∴
(b) By substituting into the equation, we have
∴ The coordinates of A and B are (2, 0) and (8, 0) respectively.
∴ The length of AB
Exercise 2D (p. 2.37)Level 1
1. (a)
(b)
2. (a)By comparing the real parts, we have
By comparing the imaginary parts, we have
(b)
By comparing the real parts, we have
By comparing the imaginary parts, we have
3.
4.
5.
6.
7.
By comparing the real parts, we have
By comparing the imaginary parts, we have
8.
By comparing the real parts, we have
By comparing the imaginary parts, we have
9.
10.
11.
12.
56
2 Quadratic Equations in One Unknown (II)
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
57
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
24.
25. Using the quadratic formula,
26. Using the quadratic formula,
Level 227.
28.
29.
30.
31.
By comparing the real parts, we have
By comparing the imaginary parts, we have
32.
By comparing the real parts, we have
By comparing the imaginary parts, we have
58
2 Quadratic Equations in One Unknown (II)
33.
∴From (1),
By substituting into (2), we have
34.
∴From (2),
By substituting into (1), we have
35. (a)
(b)
(c)
(d)
36. ∵ is a root of the equation .
∴
∴From (2),
By substituting into (1), we have
37. ∵ is a root of the equation .
∴
∴From (2),
By substituting into (1), we have
38. (a)
(b)
∵ is a root of the equation
.
∴
∴From (2),
By substituting into (1), we have
39. (a) ∵ is a root of the equation .
∴
∴If , then
59
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
∴ is also a root of .∴ Sharon’s claim is agreed.
(b)
Check Yourself (p. 2.41)1. (a) (b) (c)
(d) (e)
2.
3. ∵ The graph of has only one x-intercept.
∴
i.e.
4. ∵ The equation has no real roots.∴
i.e.
∴ The range of values of k is .
5. (a)
(b)
(c)
(d)
6. (a) The required quadratic equation is
(b) The required quadratic equation is
7. (a)By comparing the real parts, we have
60
2 Quadratic Equations in One Unknown (II)
By comparing the imaginary parts, we have
(b)
By comparing the real parts, we have
By comparing the imaginary parts, we have
8. (a)
(b)
(c)
(d)
Revision Exercise 2 (p. 2.42)Level 11. For the equation ,
∴ The equation has two unequal real roots and the corresponding graph has two x-intercepts.
2. For the equation ,
∴ The equation has one double real root and the corresponding graph has one x-intercept.
3. For the equation ,
∴ The equation has no real roots and the corresponding graph has no x-intercepts.
4. For the equation ,
∴ The equation has two unequal real roots and the
corresponding graph has two x-intercepts.
61
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
5. ∵ The graph of has two x-intercepts.
∴
i.e.
∴ The range of values of p is p < 1.
6. ∵ The graph of does not intersect the x-axis.
∴
i.e.
∴ The range of values of p is .
7. ∵ The equation has real roots.
∴
i.e.
∴ The range of values of p is .
8.
(a) ∵ The equation has two unequal real roots.
∴
i.e.
∴ The range of values of k is k < 6.
(b) ∵ The equation has no real roots.
∴
i.e.
∴ The range of values of k is k > 6.
9. (a) ∵ The graph of has only one
x-intercept.∴
i.e.
62
2 Quadratic Equations in One Unknown (II)
(b) For k = 4, the corresponding quadratic equation is
∴ The x-intercept of the graph is .
10. (a)
∵ The equation has two equal real roots.
∴
i.e.
(b) By substituting into the equation, we have
11.
12.
13.
14. (a)
(b)
15.
16. Let and be the roots of .
When ,When ,∴
17.
63
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(a)
(b)
(c)
18.
(a)
(b)
(c)
19.
∵
∴
20. (a) The required quadratic equation is
64
2 Quadratic Equations in One Unknown (II)
(b) The required quadratic equation is
(c) The required quadratic equation is
21. ∵ and are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
22. ∵ and are the roots of .
∴
For the required quadratic equation,
65
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
∴ The required quadratic equation is
23. For the equation ,
∴ The required quadratic equation is
24. Let and be the roots of .
For the required quadratic equation,
∴ The required quadratic equation is
25. ∵ The graph of cuts the x-axis at
and .
∴ The roots of are and .
26. (a) (i) ∵ The graph oftouches the x-axis at one point.
∴
i.e.
66
2 Quadratic Equations in One Unknown (II)
(ii) By substituting into, we have
∴ The coordinates of P are .
∵ The graph of cuts the y-axis at Q.
∴ By substituting into , we have
∴ The coordinates of Q are .
(b)
27. (a)
(b)
28. (a)
(b)
(c)
(d)
(e)
(f)
29. (a)
By comparing the real parts, we have
By comparing the imaginary parts, we have
(b)
By comparing the real parts, we have
By comparing the imaginary parts, we have
(c)
∴From (2),
By substituting into (1), we have
67
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
30. (a)
(b) Using the quadratic formula,
(c) Using the quadratic formula,
Level 2
31. (a)
(b) If k is a negative integer, then , i.e. the equationhas no real roots.∴ Peter’s claim is agreed.
32. (a) ∵ The equation has one double real root.
∴
i.e.
(b) By substituting into , we have
∴ The x-intercept of the graph is –2.
33. (a) ∵ The graph of
cuts the x-axis at two points.∴ is a
quadratic equation.∴ The coefficient of cannot be zero.
∴
(b) ∵ The graph of
cuts the x-axis at two points.∴
i.e.
∴ The range of values of m is except
.
34. (a) ∵ The graph of y = ax2 + 8x + c touches the x-axis at only one point.
∴i.e.
Since a and c are positive integers,when a = 1 and c = 16, a + c = 1 + 16 = 17;when a = 2 and c = 8, a + c = 2 + 8 = 10;when a = 4 and c = 4, a + c = 4 + 4 = 8;when a = 8 and c = 2, a + c = 8 + 2 = 10;when a = 16 and c = 1, a + c = 16 + 1 = 17.∴ The possible values of a + c are 8, 10 and 17.
(b) For the equation 2x2 + (a + c)x + 10 = 0,Δ = (a + c)2 – 4(2)(10) = (a + c)2 – 80When a + c = 8,Δ = 82 – 80 = –16 < 0In this case, the equation 2x2 + (a + c)x + 10 = 0 has no real roots.∴ Ken’s claim is not correct.
35.
68
2 Quadratic Equations in One Unknown (II)
For the equation ,
∴ The quadratic equationhas two distinct real roots for any real values of k.
36. (a) ∵ is a quadratic equation.
∴
∵ Sum of roots is equal to product of roots.
∴
(b) If the sum of its roots is equal to the product of its roots, then .By substituting into
, we have
∴ The roots of the quadratic equation must be distinct.
∴ Kelvin’s claim is agreed.
37.
(a)
(b)
(∵ )
69
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(c)
38.
∵
∴
39.
∴
40.
For the quadratic equation ,
41. ∵ and are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
42. (a)
(b) The required quadratic equation is
70
2 Quadratic Equations in One Unknown (II)
43. ∵ 2 and 2 are the roots of .
∴
For the required quadratic equation,
∴ The required quadratic equation is
44. ∵ The graph of intersects the x-axis at two points.
∴ The equation has two distinct real roots.
∴
i.e.
Also, the y-intercept of the graph is positive.
∴
∴
∴ The possible integral values of k are 1, 2 and 3.
45. (a) ∵ The graph of does not intersect the x-axis.∴
71
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
i.e.
∴ The range of values of k is .
(b) When x = 1,
For , .
∴ It is possible that lies on the graph.
46. Consider the quadratic equation.
∴ The quadratic equation has two distinct real roots.∴ The graph of has
two x-intercepts.∴ The graph is not correctly sketched.
47. (a) Let and be the roots of ,where .
∵∴
(b) x-coordinate of M
∴ The coordinates of M are .
48. (a)
(b)
(c)
(d)
49. (a)
(b)
(c)
(d)
72
2 Quadratic Equations in One Unknown (II)
(e)
50. (a)
∴From (1),
By substituting into (2), we have
(b)
∴(1) + (2):
By substituting into (1), we have
51. (a)
∵ The imaginary part of z is 2.
∴
(b) By substituting into (1), we have
52. Let , where a and b are real numbers.
∵ is an imaginary number.
∴
73
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
∵ is a real number.
∴
∴
53.
(a) ∵ The equation has two non-real roots.∴
i.e.
∴ The range of values of m is m > 6.
(b) The smallest integral value of m is 7.By substituting m = 7 into , we have
Using the quadratic formula,
54. (a) ∵ is a root of the equation.
∴
∴From (2),
By substituting into (1), we have
(b) By substituting and into
, we have
∵ The equation has real roots.∴
74
2 Quadratic Equations in One Unknown (II)
i.e.
∴ The range of values of k is .
55. (a)
(b)
56. (a)
(b)
Multiple Choice Questions (p. 2.47)1. Answer: C
I.
II.
III.
∴ Only II and III have real roots.∴ The answer is C.
2. Answer: D∵ The equation has no real roots.
∴i.e.
∴ The range of values of p is p > 16.3. Answer: D
∵ The equation has one double real root.
∴
4. Answer: A∵ The graph of cuts the x-axis at two
points.∴i.e.
∴ The possible values of c are 15 and 20.
5. Answer: D
∵
∴
6. Answer: D∵∴ a and b are the roots of .
75
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
7. Answer: D∵ is a root of .
∴
∵ and are the roots of .
∴
8. Answer: C
The required quadratic equation is
9. Answer: B
For the required quadratic equation,
∴ The required quadratic equation is
10. Answer: A
∴ The real part is .
11. Answer: D
By comparing the real parts, we have
By comparing the imaginary parts, we have
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2 Quadratic Equations in One Unknown (II)
12. Answer: BLet , where a and b are real numbers.
∴
By substituting a = 3 into (1), we have
∴
HKMO (p. 2.48)1. ∵ 1 is a root of .
∴
∵ a and b are roots of .
∴
∴
2. Consider .
Consider .
∴ The roots of are 2006 and 2007.
3.
∴
Exam Focus
Exam-type Questions (p. 2.50)1. (a) ∵ The graph of cuts the y-axis at
C(0, –5).∴ By substituting x = 0 and y = –5 into
, we have
(b) ∵ and are the x-intercepts of the graph of .
∴ and are the roots of .
∵
∴
2. (a) ∵ The graph of cuts the
y-axis at C(0, –5).∴ By substituting x = 0 and y = –5 into
, we have
77
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
(b) ∵ and are the x-intercepts of the graph of .
∴ and are the roots of .
(c) Coordinates of M
Distance between M and C
∴ The distance between M and C is not greater than 6.
3. (a)
(b) (i) ∵ is a root of the equation
.∴
By comparing the real parts, we have
(ii) When , the quadratic equation becomes
∵ The equationhas two
distinct real roots.∴
∴ The range of values of r is .
78
2 Quadratic Equations in One Unknown (II)
4. Answer: A
∵ The equation has a repeated root.
∴
5. Answer: A
6. Answer: A
I.
II.
III.
∴ Only I and II are real.∴ The answer is A.
Investigation Corner (p. 2.53)
1.
79
NSS Mathematics in Action (2nd Edition) 4A Full Solutions
2. Let x and y be the length and the width of the rectangle respectively.From the question, we have and .Solution Steps done by
BabyloniansAlgebraic Notations added to facilitate understanding