View
219
Download
3
Category
Preview:
Citation preview
1
SHW 2-01 Total: 20 marksReview Exercise 2 (p. 2.5)
1.
6 or 21
06or 0120)6)(12(06112 2
xx
xxxxxx
1M+1A2.
34 or
34
043or 0430)43)(43(016249 2
xx
xxxxxx
∴34
x (repeated)
1M+1A3.
32 or
23
023or 0320)23)(32(06136
061362
2
xx
xxxxxx
xx
1M+1A4. Using the quadratic formula,
12or 14
2
2622
6762
)1(2)168)(1(422 2
x
1M+1A5. Using the quadratic formula,
1435
)7(2)1)(7(455 2
x
∵ 3 is not a real number.∴ The equation has no real roots.
1M+1A6. Using the quadratic formula,
4331
)2(2)4)(2(411 2
x
1M+1A10.
520
55
4164451645
kk
k
∴ 4k1M+1A
11.
26
22
17121721
kkk
∴ 3k1M+1A
12.
88
88
80888032248032)3(8
kk
kk
∴ 1k1M+1A
13.
64
66
4044603610)12(31
kk
kk
∴32
k
1M+1A
2
SHW 2-A1 Total: 23 marks1. ∵ The equation kx2 – 4x + 1 0 has a double real root.
∴ Δ 0i.e.
404160)1)((4)4( 2
kk
k
1M+1A2. ∵ The equation 5x2 + 2x – k 0 has two distinct real roots.
∴ Δ 0i.e.
51
02040))(5(42 2
k
kk
∴ The range of possible values of k is51
k .
1M+1A3. ∵ The graph of y 2x2 4x + (k 1) does not intersect the
x-axis.∴ Δ 0i.e.
10880)1(8160)1)(2(4)4( 2
kk
kk
∴ The range of possible values of k is k 1.1M+1A
4. ∵ The graph of y x2 – px + 25 touches the x-axis at onepoint.
∴ Δ 0i.e.
10or 10100
0100
0)25)(1(4)(
2
2
2
ppp
p
p
1M+1A5. ∵ The equation 2x2 3x + k 0 has real roots.
∴ Δ 0i.e.
890890))(2(4)3( 2
k
kk
∴ The range of possible values of k is89
k .
1M+1A6. (a) ∵The equation x2 – 12x + 4k 0 has a repeated
real root.∴ Δ 0i.e.
90161440)4)(1(4)12( 2
kkk
1M+1A
(b) Root of the equation
6)1(2
12
1A
7. (a) ∵ The graph of y 2x2 + 5x + k cuts the x-axis at twopoints.
∴ Δ 0i.e.
825
08250))(2(452
k
kk
∴The range of possible values of k is825
k .
1M+1A(b) The smallest integral value of k is 3.
For k 3, the corresponding quadratic equation is
23or 1
0)32)(1(0352 2
xx
xxxx
∴The x-intercepts of the corresponding graph are
1 and23 .
1M+1A8. ∵ The graph of y x2 – 6x – m does not intersect the x-axis.
∴ Δ 0i.e.
904360))(1(4)6( 2
mmm
∴ The range of possible values of m is m 9.1M+1A
9. (a) ∵The graph of kxxy 231 2 touches the x-axis at
one point.∴ Δ 0
i.e.
3
0344
0)(314)2( 2
k
k
k
1M+1A(b) y-intercept
3 k
∴The length of OP 3 – 0 3
1M+1A
3
SHW 2-A2 Total: 24 marks1. ∵ The graph of y 3x2 + 6x + k has only one x-intercept.
∴ Δ 0i.e.
3012360))(3(46 2
kkk
1M+1A2. ∵ The equation –4x2 + 8x + k 0 has no real roots.
∴ Δ 0i.e.
4016640))(4(482
kkk
∴ The range of possible values of k is k 4.1M+1A
3. ∵ The graph of y 6x2 3x – (k + 1) has two x-intercepts.∴ Δ 0i.e.
811
024330)1(2490)]1()[6(4)3( 2
k
kkk
∴ The range of possible values of k is8
11k .
1M+1A4. ∵ The equation px2 + 8x – (p + 8) 0 has two equal real
roots.∴ Δ 0i.e.
40)4(
0168
0)8(4640)]8()[(48
2
2
2
pp
pp
pppp
1M+1A5. ∵ The graph of y 5x2 + 10x – (2k – 1) has x-intercepts.
∴ Δ 0i.e.
2040800)12(201000)]12()[5(410 2
kk
kk
∴ The range of possible values of k is k 2.1M+1A
6. (a) ∵ The graph of y 3x2 + (p 3) x + 12 has only onex-intercept.
∴ Δ 0i.e.
15or 9123
144)3(
0)12)(3(4)3(2
2
ppp
p
p
1M+1A(b) For p 9,
the x-intercept of the graph
2)3(239
For p 15,
the x-intercept of the graph
2)3(2315
1A+1A
7. (a) ∵ The equation 3x2 10x + k + 1 0 has two distinctreal roots.
∴ Δ 0i.e.
322012880)1(121000)1)(3(4)10( 2
k
kkk
∴The range of possible values of k is3
22k .
1M+1A(b) The largest integral value of k is 7.
For k 7, the corresponding quadratic equation is
2or 340)2)(43(08103
0171032
2
xx
xxxx
xx
1M+1A8. ∵ The graph of y –4x2 – 4x + k cuts the x-axis at two
points.∴ Δ 0i.e.
1016160)4(4)4( 2
kkk
∴ The range of possible values of k is k 1.1M+1A
9. (a)kxx
kxxy
22
)1(22
∵The graph of y –2x2 – 2x + k touches the x-axis atone point.∴ Δ 0i.e.
21
0840))(2(4)2( 2
k
kk
1M+1A
(b) x-intercept
21
)2(22
∴The length of21
210
OQ
1M+1A
4
SHW 2-B1 Total: 25 marks
1. The required quadratic equation is
023
022
0)2)(1(
2
2
xx
xxx
xx
1M+1A+1A2. The required quadratic equation is
020
02054
0)5)(4(0)]5()[4(
2
2
xx
xxx
xxxx
1M+1A+1A3. The required quadratic equation is
0189
01863
0)6)(3(0)]6()][3([
2
2
xx
xxx
xxxx
1M+1A+1A4. The required quadratic equation is
0132
0122
0)12)(1(
)0(221)1(2
021)1(
2
2
xx
xxx
xx
xx
xx
1M+1A+1A5. The required quadratic equation is
02320
028520
0)25)(14(
)0(20525
414
052
41
052
41
2
2
xx
xxx
xx
xx
xx
xx
1M+1A+1A6. (a)
8or208or02
0)8)(2(
xxxx
xx
1M+1A(b) The roots of the required equation are 2 3 and
8 3, i.e. 5 and 5.The required quadratic equation is
025
0)5)(5(0)5)](5([
2
x
xxxx
1M+1A+1A
7. (a)
6or23
06or0320)6)(32(018152 2
xxxx
xxxx
1M+1A
(b) The roots of the required equation are
23
1
and
61
, i.e.
32
and61
.
The required quadratic equation is
021518
0231218
0)16)(23(
)0(18616
323
061
32
061
32
2
2
xx
xxx
xx
xx
xx
xx
1M+1A+1A
5
SHW 2-B2 Total: 8 marks1. The required quadratic equation is
06
0632
0)3)(2(0)3)](2([
2
2
xx
xxx
xxxx
1A2. The required quadratic equation is
0158
01553
0)5)(3(
2
2
xx
xxx
xx
1A3. The required quadratic equation is
01610
01682
0)8)(2(0)]8()][2([
2
2
xx
xxx
xxxx
1A
4. The required quadratic equation is
06173
06183
0)6)(13(
)0(3)6(313
0)6(31
0)6(31
2
2
xx
xxx
xx
xx
xx
xx
1A5. The required quadratic equation is
031616
0312416
0)34)(14(
)0(16434
414
043
41
043
41
2
2
xx
xxx
xx
xx
xx
xx
1A6. (a)
4or704or07
0)4)(7(02832
xxxx
xxxx
1M+1A(b) The roots of the required equation are (7) and 4,
i.e. 7 and 4.The required quadratic equation is
0283
02847
0)4)(7(0)]4()[7(
2
2
xx
xxx
xxxx
1A
6
SHW 2-C1 Total: 50 marks
1. Sum of roots 414
Product of roots 212
1A+1A
2. Sum of roots43
4)3(
Product of roots 248
1A+1A
3. Sum of rootskk2
36
Product of roots31
3
kk
1A+1A
4. (a) Sum of roots
12
26
26
)(
k
k
k
(b) Product of roots
16126
66
k
1A+1A
5. (a) Sum of roots
284183
1M+1A
(b)1
3roots ofProduct k
122)3(2
kk
1A
6.
3712
2)14()1(roots of Sum
83242
3)13(3
23)1(roots ofProduct
kk
k
k
1M+1M+1A+1A
7. (a) 212
1A
(b) 515
1A(c)
61)2(5
1)1)(1(
1M+1A(d)
381)2(3)5(9
1)(391339)13)(13(
1M+1A
8.
34
43
12
1A+1A
(a)
3344
11
1M+1A
(b)
49
)3(43
)3(
341
1111122
1M+1A9. (a) Let and – 1 be the roots of the equation.
23212
5)10()1(roots of Sum
∴The larger root of the equation is23 .
1M+1A
(b)
43
34
155
3123
23
53)1(roots ofProduct
k
k
k
k
1M+1A
7
10. ∵ 4 and 2 are the x-intercepts of the graph ofy x2 + mx + n.
∴ 4 and 2 are the roots of x2 + mx + n 0.
6
)2()4()1(
roots of Sum
m
m
8
)2)(4(1
roots ofProduct
n
n
1M+1A+1A11. (a) Since and are the roots of 2x2 – 6x + 1 0,
we have
21
32
)6(
1A+1A(i)
583
8)()4()4(
1A(ii)
29
16)3(421
16)(41644)4)(4(
1M+1A(b) For the required quadratic equation,
sum of roots 5)4()4(
product of roots29)4)(4(
∴The required quadratic equation is
09102
029)5(
2
2
xx
xx
1A
12. Since and are the roots of 2x2 + 4x – 3 0, we have
23
224
1AFor the required quadratic equation,sum of roots
7232)2(
2)(
2)2(
2
2
22
22
1M+1Aproduct of roots
49
23
)(2
2
22
1A∴ The required quadratic equation is
09284
0497
2
2
xx
xx
1A13. Let and be the roots of 5x2 4x 2 0, then
52
54
)5()4(
1A+1A
The roots of the required quadratic equation are1 and
1 .
Sum of roots
25254
11
Product of roots
2552
1
1
11
1A+1A∴ The required quadratic equation is
0542
0252
2
2
xx
xx
1A
8
SHW 2-C2 Total: 25 marks
1. Sum of roots35
3)5(
Product of roots31
1A+1A
2. Sum of roots 326
Product of roots27
1A+1A
3. Sum of roots2
2)2(2
442)4(
kkk
Product of roots21
)2(22
422
kk
kk
1A+1A
4. (a) Product of roots
525
2)5(
kkk
kk
1M+1A
(b) Sum of roots
25
10
10
k
1A
5.36
2roots ofProduct
(rejected) 2or 24
22
2
2
1M+1A
3)(
2roots of Sum k
93
3
3222
k
k
k
1M+1A
6. (a) 91
)9(
(b) 616
1A+1A(c)
1569
)()1(
1A(d)
396
)()1(
1A
7.
21
22
)4(
1A(a)
42
)(22
222
1A(b)
29
214
)2( 2222
1M+1A8. (a) Let and 2 be the roots of the equation.
393192roots of Sum
∴ The roots of the equation are 3 and 6.1M+1A
(b)
1818
1)6)(3(roots ofProduct
kk
k
1M+1A
9
SHW 2-C3 Total: 22 marks
9. (a) Let and + 3 be the roots of the equation.
(rejected) 1or 40)1)(4(043
4328)3(roots ofProduct
2
2
∴The smaller root of the equation is 4.
1M+1A
(b)
25102
5)34(4
2)5()3(roots of Sum
kk
k
k
1M+1A10. (a) ∵ M(6, 0) is the mid-point of AB.
∴
12
62
1A+1A(b) ∵ and are the x-intercepts of the graph of
.931 2 kxxy
∴ and are the roots of .0931 2 kxx
4312
31
)(roots of Sum
kk
k
1M+1A11. (a) Since and are the roots of x2 – 7x + 3 0, we have
313
71
)7(
1A(i)
14)7(2
)(222
(ii)
12)3(4
422
1A+1A(b) For the required quadratic equation,
sum of roots 2 + 2 14product of roots 1222 ∴The required quadratic equation isx2 – 14x + 12 0
1M+1A
12. Since and are the roots of 3x2 2x 6 0, we have
236
32
3)2(
1A
For the required quadratic equation,
sum of roots
54
3534
1322
232
1)(2)1)(1(
111
11
1
1M+1A
product of roots
5335
1)1)(1(
11
11
1
1A∴ The required quadratic equation is
0345
053
54
2
2
xx
xx
1A13. Let and be the roots of 6x2 8x 3 0, then
21
63
34
68
1AThe roots of the required quadratic equation are 3and 3.Sum of roots
4343
)(333
Product of roots
29
219
933
1A+1A∴ The required quadratic equation is
0982
029)4(
2
2
xx
xx
1A
10
SHW 2-D1 Total: 20 marks1.
ii
iiii
31)36()98(
3968)39()68(
1A2.
(a)
ii
iiii
73)1(37
37)37( 2
1A(b)
17)1(16
4)4)(4( 22
iii
1A3.
(a)
iiiii
ii
7
7
77
2
1A
(b)
i
i
iii
ii
ii
51
52
52
)1(42
22
22
21
21
22
1M+1A4. (a)
i
i
x
12
222
42
)1(2)2)(1(422 2
1M+1A(b)
i
i
x
222
442
164
)1(2)8)(1(4)4()4( 2
1M+1A
5.
ii
iii
ii
79)1(3763296
)3)(32(2
1A
6. (a)8)510()23(8)52()103(
iyx
yiix
∴
2823
xx and
20510
yy
1M+1A+1A(b)
yiixxyiixxyiixiix
yiiix
1)32()32(1)1(3)32(213322
1)32)((2
∴
1132
x
x and
5)1(32
32
yyyx
1M+1A+1A
7. ∵ 3i is a root of the quadratic equation02 qpxx .
∴
03)9(039
0)3()3(2
2
piqqpii
qipi
∴
003
pp and
909
1M+1A+1A
11
SHW 2-D2 Total: 22 marks1.
ii
iiii
175)89()61(
8691)86()91(
1A2.
(a)
ii
iiii
104)1(410
410)25(2 2
1A(b)
ii
iii
43)1(44))(2(22)2( 222
1A
3. (a)
i
ii
ii
ii
25
2525
25
2
1A
(b)
i
i
iiiiii
ii
ii
59
5310
186)1(9
)1(6183
61833
36
36
22
2
1M+1A4.
ii
iiiii
5548)1(245524
2464924)83)(38(
2
1A5.
(a)
iiyxixiyiixiyi
56)3()22(5623256)23()2(
∴
2622
xx and
25)3(
yy
1M+1A+1A
(b)
iyxixiyxxiiyixi
iiyxi
63263)1(2
)2(3)2(2
32
∴
362
xx and
13)3(3
yyyx
1M+1A+1A
6. (a) Putx = 3 + i into 02 qpxx , we have
0)6()38(0369
0)3()3(2
2
ipqpqpipii
qipi
1A
∴
06038
pqp
6p and 10q .
(b) ∵ Sum of roots of the equation =16
= 6
∴ The other root of the equation = )3(6 ii 3
1M+1A+1A+1M+1A
7. (a) ∵ kx2 + 8x = 1 = 0 has no real roots.
160)1)((4)8(
0Δ2
kk
(b)
1717434
2834
48
)17(2)1)(17(488
018172
2
i
i
x
xx
1M+1A+1M+1A
12
SHW 2-D3 Total: 20 marks1. (a)
i
i
iiii
31
)118()32(
11382)113()82(
(b)
i
i
iiii
108
)64()53(
6543)65()43(
1A+1A
2. (a)
i
i
i
i
ii
i
i
i
i
i
i
2
1
2
1
2
1
)1(1
)1(
221
21
1
11
(b)
i
i
i
i
ii
i
i
i
i
i
i
5
3
5
4
25
2015
)1(169
)1(2015
2)4(23
22015
43
43
43
5
43
5
2A+3A3. (a)
i
i
ii
iii
iii
1339
)3)](1(49[
)3](2)2(23[
)3)(23)(23(
)23)(3)(23(
(b)
i
i
iii
ii
ii
iiiiiii
2515
)1(52520
2552020
)4)(55(
)4)](1(352[
)4)(2362()4)(2)(31(
2A+3A
4. (a)
i
i
i
i
iii
i
i
i
i
i
i
i
i
iii
i
ii
i
13
12
13
5
26
2410
)1(251
)1(5245
2)5(2)1(
25255
51
51
51
5
51
5
)1(352
5
23322
5
)32)(1(
5
(b)
i
i
i
i
iii
i
i
i
i
i
i
i
i
i
iii
i
ii
1
25
2525
)1(916
)1(32528
2)3(24
2321428
34
34
34
7
34
7
34
)1(6
34
2326
34
)2)(3(
2A+3A
5. (a)
4
)1(3
22)3()3)(3(
iii
(b)
i
i
iiiii
16316
2)4)(3(
2)]3)(3)[(3(2)3(3)3(
1A+1M+1A
13
SHW 2-Revision9. (a) ∵ The graph of 18242 2 xkxy has only one
x-intercept.∴ 0
i.e.
401445760)18)(2(4)24( 2
kk
k
1M+1A(b) For k = 4, the corresponding quadratic equation is
230)32(
09124
018248
01824)4(2
2
2
2
2
x
x
xx
xx
xx
∴ The x-intercept of the graph is23 .
1M+1A
19. 23
)6(
33kk
1A∵ 1)21)(21( ∴
3
0344
03
4)2(2
14)(2114221
k
k
k
1M+1A21. ∵ and are the roots of 01352 xx .
∴ 515
13113
1AFor the required quadratic equation,
135135
11
131131
111
1A+1A∴ The required quadratic equation is
01513
0131
135
2
2
xx
xx
1A
28. (a)
ii
iiii
26)13()17(
137)1()37(
1A(b)
ii
iiii
105)55()32(
5352)53()52(
1A(c)
ii
iiiii
iiiii
8)43()26()1(2436
2436
)2)(2()3)(2()23)(2(2
1A(d)
ii
ii
iii
247)1(16249
16249
)4()4)(3(23)43(2
222
1A30. (a)
ixix
x
x
x
5353
253
25)3(
025)3(2
2
1M+1A(b) Using the quadratic formula,
i
i
x
232
862
86
)1(2)11)(1(4)6()6( 2
1M+1A(c) Using the quadratic formula,
i
i
x
32
32
684
684
)3(2)2)(3(4)4()4( 2
1M+1A31. (a)
0232
3222
2
kxx
kxx
788169
)2)(2(4)3( 2
kk
k
1M+1A
(b) If k is a negative integer, then 0 , i.e. the equationhas no real roots.∴ Peter’s claim is agreed.
1A+1A
14
Exam-type Questions (p. 2.50)1. (a) ∵ The graph of nmxxy 2 cuts the y-axis at
C(0, –5).∴ By substituting x = 0 and y = –5 into
nmxxy 2 , we have
5)0(05 2
nnm
1M+1A
(b) ∵ and are the x-intercepts of the graph of
52 mxxy .∴ and are the roots of 052 mxx .
mm
1
515
1A
20
)5(4)(
4)(
4)2(
2
)(
2
2
2
22
22
2
m
m
1M+1A∵ units sq. 15 of Area ABC
∴
)rejected( 4or 416
3620
620
6
155)(21
units sq. 1521
2
2
2
mm
m
m
OCAB
1M+1A
2. (a) ∵ The graph of )52(202 kxkxy cuts they-axis at C(0, –5).
∴ By substituting x = 0 and y = –5 into)52(202 kxkxy , we have
5525
)52()0(20)0(5 2
kk
kk
1M+1A
(b) ∵ and are the x-intercepts of the graph of
5205 2 xxy .∴ and are the roots of 05205 2 xx .
4)5(
20
1A
(c) Coordinates of M
)0 ,2(
0 ,24
0 ,2
1ADistance between M and C
629
)]5(0[)02( 22
∴ The distance between M and C is not greater than6.
1M+1A
3. (a)
i
iii
ii
ii
53
51
)1(9162
9162
3131
312
312
2
1M+1A
(b) (i) ∵i31
2
is a root of the equation
0442 xpx .
∴
05
12256
258
516
05
125
16256
258
045
1254
259
256
251
0453
514
53
51
2
2
ipp
iip
iiip
iip
1M+1ABy comparing the real parts, we have
10258
516
0258
516
p
p
p
1M+1A(ii) When 10p , the quadratic equation becomes
044)10(
44102
22
xxr
rxxx
1A∵ The equation 044)10( 2 xxr has two
distinct real roots.∴ 0
914416016160160)4)(10(4)4( 2
rrr
r
∴ The range of values of r is 9r .1M+1A
15
Recommended