Review Exercise 1 (p - Tak Sun Secondary Schoolpersonal.tsss.edu.hk/roh/F4/Solutions (SHW2).pdf1 SHW2-01Total:20marks ReviewExercise2(p.2.5) 1. or 6 2 1 2 1 0 or 6 0 (2 1)( 6) 0 2

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


1435

)7(2)1)(7(455 2

x

∵ 3 is not a real number.∴ The equation has no real roots.


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


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


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


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


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


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


0189

01863

0)6)(3(0)]6()][3([

2

2

xx

xxx

xxxx


0132

0122

0)12)(1(

)0(221)1(2

021)1(

2

2

xx

xxx

xx

xx

xx


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


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


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


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


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


9. (a) Let and + 3 be the roots of the equation.

(rejected) 1or 40)1)(4(043


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


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

qq

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


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

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Review Exercise 1 (p - Tak Sun Secondary Schoolpersonal.tsss.edu.hk/roh/F4/Solutions (SHW2).pdf1 SHW2-01Total:20marks ReviewExercise2(p.2.5) 1. or 6 2 1 2 1 0 or 6 0 (2 1)( 6) 0 2