Chapter 2 Quadratic Equations Paper 1 With Answer

7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer

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Chapter 2 Quadratic Equations Paper 11. Rewrite the quadratic equations below in the general form.

(a) !)( 2 =+ x

(b) 1)2"( =− x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

2.&etermine if "'

2

! or 1 is the root of the quadratic equations .1!2

2 =−+ x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

3. Rewrite the quadratic equation below in the general form.

(a) )")(( =−+− x x

(b) !)1)((" =++− x x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

4. &etermine if 2 P or *" p is the root of the quadratic equation p x p x 1!)(2 2 =−− ' gi+en

p is a whole number.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



5. &etermine the roots of the quadratic equations below.

(a) 2 =− x

(b) )!(2 =− x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

6. &etermine the roots of the quadratic equation below

(a) 1!2 =−− x x

(b) , 2=− x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

7. -tate the quadratic equation if the roots are

(a) *2 and !

(b) , onl

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

8. /orm the quadratic equation if the roots are

(a) " and 0 k

(b) p and q' where k ' p and q are constants.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



9. i+en the quadratic equation 2 =−+ x x ' state the sum of the roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

10. -tate the product of the roots of the quadratic equation 1,2 2=−+ x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

11. i+en the quadratic equation 2 =++ cbxax ' state the condition of the discriminant '

ac,b 2 − if there are no real roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

12. i+en the quadratic equation 2 =++ cbxax ' state the condition of the discriminant'

acb 2 − ' if there are two real and different roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

13. i+en the quadratic equation 2 =++ cbxax ' state the condition of the discriminant'

acb 2 − ' if there are two equal roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



14. &etermine the +alue of acb 2 − of the quadratic equation ,2 =+− x x .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

15. &etermine the +alue of the discriminant of the quadratic equation !2 2 =−+− x x .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

16. &etermine the +alue of the discriminant of the quadratic equation 22 2−= x x .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

17.-tate with reasons if the equation "

12

=++ x x

is a quadratic equation.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

18. 3f the quadratic equation 1!112 2 =+− x x is written in the form .))("( =−− q px x

-tate the +alue of p and of q.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



19. -ol+e the equation .,)12)(2( =−+− x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

20. -tate the sum and the product of the quadratic equation 22!" x x =− .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

21. -ol+e the quadratic equation 111 2 =−− x x b factori4ation.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

22. -ol+e the quadratic equation "2 2 =++ x x b using the rele+ant formula. i+e our

answer correct to decimal places.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

23. &etermine the roots of the quadratic equation !2 2 =−+ x x b completing the square.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



24. 3f 5 6 2 is a root of the quadratic equation 2=++ kx x ' find the +alue of k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

25. i+en " and are the roots of the equation 2 =++ q px x ' find the +alue of p and q.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

26. &etermine the tpe of root of the quadratic equation 12"2 =+− x x .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

27. &etermine the tpe of root of the quadratic equation "72 2=−− x x .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

28. &etermine the tpe of root of the quadratic equation .2!12=++ x x

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



29. /ind the +alue of p if the quadratic equation 22 =+− x px has onl one root.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

30. 3f the quadratic equation k x x +=− "2 2 has two real and equal roots' determine the

+alue of k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

31. i+en that p and 2 p are the roots of the quadratic equation 2 =+− k x x where p and k

are constants' find the +alues of p and of k .

#nswer $ p 6 %%%%%%%%%%%%%%%%%%%%%%

k 6 %%%%%%%%%%%%%%%%%%%%%%

32. 3f x 6 a and x 6 " are the roots of the quadratic equation b, x x ",22 −= find the +alues of

a and of b.

#nswer $ a 6 %%%%%%%%%%%%%%%%%%%%%%%

b 6 %%%%%%%%%%%%%%%%%%%%%%%



33. i+en that the two roots of the quadratic equation x( x 8 m) 6 2m 8 " are equal' determine

the possible +alues of m.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

34. 3f α and 9 are the roots of the quadratic equation 1! 2=−+ x x ' find the +alue of

9−α gi+en .β α >

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

35. i+en "+α and 2− β are the roots of the quadratic equation 2 7 1 x x− + = ' find the

+alue of . β +α

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

36. i+en α and α " are the roots of the quadratic equation 7 2 =+− p x x ' find the +alue

of . p+α

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



37. &etermine the +alues of p if the quadratic equation 22 =++− p px x has onl one real

root.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

38. &etermine the ranges of the +alues of p if the quadratic equation 2"2 −= x px has no real

roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

39. &etermine the ranges of the +alue of p if the quadratic equation p x x −= 7 2 has two

real and different roots.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

40. -how that if the quadratic equation 222 =++− p x x has real roots' then . p 1−≤

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



41. 3f the roots of the quadratic equation 22 =+++ nn)x(m x are and *2' find the +alue of

m 0 n.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

42. 3f α and β are the roots of the quadratic equation 1"22 =+− x x ' determine the

quadratic equation if the roots are α 2 and .2 β

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

43. i+e α and β are the roots of the quadratic equation 1! =+ ) x x( ' find the quadratic

equation if the roots are2

α and .

2

β :rite the quadratic equation in the form of

.2 =++ cbxax

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

44. ;he roots of the quadratic equation 22 =−+− k x x is in the ratio " $ 2. &etermine

(a) the roots'

(b) hence the +alue of k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



45. i+en k 8 1 and m 0 2 are the roots of the quadratic equation !12=+− x x '

determine the ratio k $ m where m.k >

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

46. ;he roots of the quadratic equation 1 2 =+− P x x differs b 1. /ind

(a) the roots of the equation

(b) the +alue of P

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

47. i+en that the quadratic equation 22 =++ pqx px has two equal roots' show that

(a) q = 2p'

(b) hence' determine the roots of the equation.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

48. 3f one of the roots of quadratic equation " 2 =++− pkx x is " times the other root'

show that .22 ) )(k (k p −+=

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



49. ;he quadratic equation 222 2 =−−+− )x(p )x(p has two equal roots' find the +alue of

p. <ence' sol+e the equation.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

50. i+en ∝ and9 are the roots of the quadratic equation " 2 =+− p x x . /orm the

quadratic equation if the roots are α and p and =− β α .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

51. ;he quadratic equation !2 =−−+ m )x(m x has two different roots. ;he difference

between the roots is ' find

(a) the roots of the equation

(b) the +alue of m

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

52. ;he quadratic equation 2 ! 2! px mx p+ + = has onl one root' find

(a) m in terms of p'

(b) the roots of the equation

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



53. Pro+e that the quadratic equation "2 2 =−++ p px x has two real and different roots for

all +alues of p.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

54. Pro+e that the quadratic equation "22=−+ m xmx has two real and different roots for

all +alues of m.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

55. ;he quadratic equation p x x −= "22 has real roots' determine the ma5imum +alue of p if

p is an integer.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

56. i+en α is the common root of the quadratic equations !2 =+− k x x and

.k ,k x x 2,2 ≠=+− /ind the +alues of α and k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%



57. 3f one of the roots of the quadratic equation "" 2 =++ pkx x is one0third of the other

root' state p in terms of k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

58. 3f ∝ and9 are the roots of the equation , x x "22 =−+ find the +alue of

(a) β +α and β α

(b) 22 β +α

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

59. α and β are the roots of the quadratic equation 2 =++ px x . α 2 and β 2 are the

roots of the quadratic equation q55 2=+− . /ind the +alues of p and q.

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

60. ;he roots of the quadratic equation 2!2=−+− k x x are in the ratio " $ 2. /ind

(a) the roots'

(b) hence' the +alue of k .

#nswer $ %%%%%%%%%%%%%%%%%%%%%%%%%

#nswer $1)

(a)



2!1

2!!!

)!)(!(

)!(

2

2

2

=++

=+++

=++=+

x x

x x x

x x

x

(b)

12"

12)("

2 =−−

=−

x x

x x

2)

equationquadratictheof rootnot theis "

1!"2()

1!"2(")

1!2 "':hen

2

2

=∴

≠=−+=

−+=

−+=

x

x x x

equationquadratictheof roottheis

2

!

1!2

!

2

!2

1!2 '2

!:hen

2

2

=∴

=

−+

=

−+=

x

x x x

equation.quadratictheof rootnot theis1

12

1!12(1)

1!2 1':hen

2

2

=∴≠−=

−+=

−+=

x

x x x

3)

(a)

17

12"

)")((

2

2

=−+=−−−+

=−+−

x x

x x x

x x

(b)

22"

!17"

!)1)(("

2

2

=−+

=+−−+

=++−

x x

x x x x

x x x

4)



equation.quadratictheof roottheis2

1177

117)2(

1)!)(2()2(2

1!)(2

'2:hen

1!)(2

22

22

2

2

2

p x

p p p p

p p p p

p p p p

p x p x

p x

p x p x

=∴=

−+−=

−+−=

−−−=

−−−

==−−−

equation.quadratictheof rootnot theis"

2!"

11!1217

1"!"2

1!2

":hen

2

22

2

2

p x

p p

p p p p

p p) )( p( p)(

p )x p( x

p- x

−=∴≠−=

−−+=

−−−−−=

−−−

=

5)

(a)

22

2

2

−+=

=

=

=−

or

x

x

x

(b)

or x

or x- x

) x(x

!

! 2

!2

===∴

=−

6)

(a)

or x

x or x

) )(x(x

x x

2,

2,

2,

1!2

−=∴=+=−

=+−

=−−

(b)

,

,

,

, 2

or x

x or x

) x x(

x x

=

=−=∴=−

=−

7)

(a)



1"

!2

!2

2 =−−

=−+=−=

x x

) )(x(x

and xWhen x

(b)

1

,,

,

2 =+−=−−

=

x x

) )(x(x

onlyWhen x

8)

(a)

""

""

"

"

2

2

=−−+

=−−+

=+−−==

k )x(k x

k xkx x

k) )(x(x

k and xWhen x

(b)

2

2

=++−

=+−−

=−−==

pqq)x(p x

pq pxqx x

q) p)(x(x

q p and xWhen x

9)

Compare 2 =−+ x x with −25 (sum of the roots) x + product of the roots =

-um of the roots 6 −

10)

!2

,

1,2

2

2

=−+

=−+

x x

x x

Compare x x !2

,2 =−+ with −2 x (sum of the roots)5 8 product of the roots 6 ' product of the roots

6 *!.

11)

2 <− acb .

12)

2 >− acb .

13)

2 =− acb .

14)



7

27"

,1

,

22

2

=−=

−−=−

=+−

) )( ( )( acb

x x

15)

2"

72!

))(2()!(

!2

2

2

2

−=−=

−−−=

−

=−+−

acb

x x

16)

11

(2)(2))(

2222

22

2

2

=−=

−−=−

=+−−=

acb

x x x x

17)

"1

"1

2"

2

=++

=++

x x

x x

3t is not a quadratic equation because the highest power of 5 is ".

18)

!2"

1!112 2

=−−=+−

) x )( (x

x x

Compare ) x )( (x !2" −− with ,q) )(px(x " =−−. , q p !2 −==

19)

"2

"

""2

")")(5(25

"2

,22

,1)2)(2(

2

2

or x

or x x

x x

x x x

x x

−=

=−=+=−+

=−−

=−−−+

=−+−

20)



2

!

rootstheof product

2

" rootstheof -um

2

!

2

"

!"2

2

2

=

=

=+−

=+−

x x

x x

21)

"

2

2

!

2"!2

2)!)("(2

111 2

−=

=+=−=+−

=−−

or x

x or x

x x

x x

22)

d.p)( . or.

) )( (

a

acbb x

x x

1",!"2!

271

22

"2

2

"2

2

2

2

−−=

−±−=

×−±−

=

−±−=

=++

23)

2

,1

2

,

2

!1

2!11

2

!

2

2

2

22

2

!2

!2

2

222

2

2

±=+∴

=

+=

=−−+

=−−++

=−+

=−+

x

)(x

)( )( x x

x x

x x

2.7,7or.7,72

,1

2

,1

=

−−+−= or x



24) 3f 5 6 2'

then

!=

12=

2=

=(2)22

−=

−=

=++

=++

25) :hen 5 6 " and 5 6 '

then

17

17

17

"

22

2

=−=∴=++=+−

=+−

=−−

, q p

q px with x x xcomparin

x x

) )(x(x

26)

,acb

) )( ( )(

acb

7

12

1"2

2

2

2

<−∴

<−=−=

−−=

−

i.e the roots are imaginar (or no real roots).

27)

'

77

2

"27

2

2

2

>−∴

>+=

−−−=

−

acb

) )( ( )(

acb

i.e the roots are real and different.

28)

,acb

) )( (

acb

11

2!11

2

2

2

=−∴

=−=−=

−

i.e the roots are real and equal

29)

;here is onl one root'



2

71

2

2

2

==−

=−−

=−∴

p

p

)(p)( )(

acb

30)

"2

"2

2

2

=−−−

+=−

k x x

k x x

when the equation has two real and equal roots' then

7

117

7"2

7

2"

2

2

−=

−=

=++=−−−

=−−−−

=−

k

k

k

k)(

k) )( ( )(

acb

31)

7

22

2

2

rootsof product

2

2

rootof -um

2

2

2

==

=

=×∴=

==+∴

==+−

)(

pk

k p p

k

p

p p

k x x

32)

2

1

"2

,

2

,"

2

,rootsof sum

2

"

2

,

",2

",2

2

2

2

=

−=

=+∴

=

=+−

=+−

−=

a

a

b x x

b x x

b x x



1 b

b2

")

2

1"(

b2

""a

b2

" rootsof product

=

=

=∴

=

33)

"2

"2

2 =−−+

+=+

mmx x

mm) x(x

when the roots are equal' then

2

2

2

127

"21

2

2

2

−−=∴=+=+

=++=++

=−−−

=−

orm

or mm

) )(m(m

mm

)m )( ( m

acb

34)

2

!

2

"

2

!

2

"

2

!

2

"

!2"2

!2"2

1! 2

=

−−=−∴∝

−=∝=

−=

=+=−=+−

=−+

)( β

and β i.e

or x

xor x

) x )( x(

x x

35)

,

71

72"

7

172

=+∝=++∝

=−++∴∝=

=+−

β

β

β

t! !"m o# roo

x x

36)

2

7

2

2

=+−

=+−

p x x

p x x

/rom sum of roots'



2

1

2"

=

=+

α

α d

/rom product of roots'

"

2

112

12

"

2

2

2

=

==

=

)(

p

p

α

α

2

1"

"2

1

=

+=+∴∝

p

37)

22 =++− p px x

when the quadratic equation has onl one real root'

2"

2"

2

12

2

2

2

2

−=

=+−=−−

=−−

=+−−

=−

or p

) )(p(p

p p

p p

) )(p( p)(

acthen b

38)

2"

2"

2

2

=+−

−=

x px

x px

when the quadratic equation has no real roots'

7

7

7

2"

then

2

2

>

><−

<−

<−

p

p

p

)(p)( )(

acb

39)

7

7

2

2

=+−

−=

p x x

p x x

the quadratic equation has real and different roots'



1

1

7

then

2

2

<<

>−>−−

>−

p

p

p

)(p)( )(

acb

40)

222 =++− p x x

when the quadratic equation has real roots'

1

7

212

then

2

2

−≤ ≥−

≥−−≥−−

≥+−−

≥−

p p

p

p

) )(p( )(

acb

41)

22 =+++ nn)x(m x

from the sum of roots'

(1) nm

n)(m

2)(n)(m

→−=+

=+−

−+=+−

from the product of roots'2n 12

n (2)

= −= − →

substitute (2) into (1)'

7

(*)*2n*m

2m

m

==∴

=−=−

42)

2

1

2

"

2

1

2

"

1"2

2

2

=

=+

=+−

=+−

β ! $ct o# root #rom prod"

β # root! $ #rom !"m o

x x

x x

α

α

;he quadratic equation with α 2 and β 2 as the roots is



2

2

2

2 2

2

" 12

2 2

" 2

(x )(xβ)

x (β)x β

x x

x x

α

α α

− − =

− + + =

∴ − + = ÷ ÷

− + =

43)

!

!

1

!

!

!

1!

2

2

−=−=+∴

=−+

=−+

=+

β and β

x x

x x

) x x(

α α

when the roots are

2

α and

2

β ' the quadratic equation is

"1

1"

11

!

1

!

1

2

1

1

2

1

22

22

2

2

2

2

2

=−+

=−+

=−+−−

=++−

=+

+−

=−−

x x

x x

)( )( x

β β)( x

β x

β x

) β

)(x(x

α α

α α

α

44)

(a) let the first root be α " and the second root is .2α .

from sum of roots>

" 2 2

the roots are 12 and 7

α α

α

+ ==

∴

(b)

2"=

()=

= ()

=

= )(2"

$rootsof productthefrom

2

2

2

=−=

−=

−=

−=

α

α α

45)



,7

,7

,7!21

11

2!1

!

!

!12

% mk%

and mk m&i'en k

d m an k , then mor i# k

and m k

, then m # k

or x

) )(x(x

x x

=

==∴>

==

=−=+

==

=−=+

=

=−−

=+−

46)

(a) 3f the first root isα ' then the other root is 1+α . 3f 1 2 =+− P x x ' then

2=+− P x x

2

! and

2

" i.e

12

" and

2

" arerootsthe

2

"

"2

1

>rootsof sumthefrom

+∴

=

==++

α

α

α α

(b)

from the product of roots >

" ! P

2 2

1! P

P 1!

= ÷ ÷

=

∴ =

47)

(a) 22 =++ pqx px

the quadratic equation has two equal roots'

pq

pq

pq

p)(p)( q)(

acthen b

2

1

1

2

22

22

2

2

=∴=

=−

=−

=−

(b)



only x

) )(x(x

x x p,

p px px

p p)x( px

p,when q

2

22

22

2

2

2

2

−=∴

=++=++÷

=++

=++

=

48)

"

""

"

2

2

=++−

=++− p

xk

x

pkx x

let the first root beα ' then the other root is α " '

2)2)(= (=

2=

= p

"

p)

"

= "(

"

p

"

"

"

p)("

>rootsof productthefrom

"

=

"

=

"

=

"

>rootsof sumthefrom

22

2

2

2

−+=

−=

−=

+=

+=

+=

=

=

=+

α

α α

α

α

α α

49)

:hen the quadratic equation has two equal roots'

[ ] !!iblei! not po ponly, p

) )(p(p

p p

p p p

) )( (p )(p

acb

2

2

12

17

222

2

2

2

2

=−=

=−+

=−+

=−++−

=−−−−

=−



21

1212

1

277

222

2

2

2

−=

=++=++

=−−−

=−−−+−−

∴

x

) x )( x(

x x

x x

)x( )x(

on become!the eq"ati

50)

2

2

"

2 "

2 1

2

1 2 2

"

1

"

x x p

p x x

#rom the !"m o# root! $

β ( )

i'en β ( )

( ) ( ),

β

#rom the prod"ct o# root!$

p

β

− + =

− + =

+ = →− = →

+ ==

∴ = −

=

" 1"

p ( )

p

− =

= −;he quadratic equation where the roots are α and p is

2,

""

2

2

2

=−+

=−+−−

=∝++∝−

x x

) )( ( )x( x

p p)x( x

51)

3f the first root is α ' then the second root is .+α

(a)

(2)m

m)(

>rootsof productthefrom

(1)12

!2

!)(

roots>of sumthefrom

2 →−=+

−=+

→+−=−+−=

−−=++

α α

α α

α

α

α α

m

m

m



" and 1 i.e1and1 arerootsthe

1

1)1)((

12

12

(2)'and(1)from

2

2

−+−−∴

−==++

=++

−=+

α

α α

α α

α α α

(b)

"

121

21

(1)from

=−−=

−= )(

m α

52)

(a)

pm

pm

p)(p)( m)(

acb

2

12!

2!!

root'oneonlfor

22

2

2

==−

=−

=−

(b)

onl!5

!!

2!1

2!1

2!2!

22!!

2

2

2

2

−==++=++∴

=++=++

==++

) )(x x (

x p, x

p px px

p p)x( px

pand m pmx #rom px

53)

7

7

21

27

"2

"2

2

2

2

2

2

2

22

2

>−

>+−∴

≥−

+−=

+−−=

+−=−−=−

=−++

aci.e b

)(p

)(pl"e! o# p, #or all 'a

)(p

)(p

p p

) )(p( pacb

p px x

since ac b2 >− ' the quadratic equation "2 2 =−++ p px x has two real and different roots.

54)



12

12

12

"2

"2

2

2

2

2

2

2

2

>−∴

>+

>

+=

−−=

−

=−+

acb

mand

ml"e! o# m, #or all 'a

m

m)(m)( )(

acb

m xmx

since 'ac b2 >− the quadratic equation "22=−+ m xmx has two real and different roots.

55)

7

7

2"

"2

"2

2

2

2

2

≤

≥−≥−−

≥−

=+−

−=

p

p

)(p)( )(

root!, #or realacb

p x x

p x x

∴ma5imum +alue of p 6 1 if p is an integer.

56)

?etα and p be the roots of the equation !2 =+− k x x

from sum of roots > p !..........(1)

from product of roots > p =..........(2)

α

α + =

=

let α and m be the root of the equation 2,2 =+− k x x

from sum of roots > m ,..........(")

from product of roots > m 2=..........()

(2) p = '

() m 2=

m 2p (!)

(") (1)' m p 2 ()from (!) and ( )' 2p p 2

p 2

subsitute p 2 in

α

α

α

α

+ ==

=

= →

− − = →− ==

= to (1)'

2 !

"

subsitute " and p 2 into (2)'

" 2 =

=

α

α

α

+ ==

= =× ==

57)



?et α be the first root' then the second not is ."

α

7

"

"

"

""

""

"

""

2

2

2

2

2

k p

p )k

(

p

p ) )( (

root!$rod"ct o# #rom the p

k

k

k

!$"m o# root #rom the !

p xk

x

pkx x

=

=−

=

=

−=

−=

−=+

=++

=++

α

α α

α

α

α α

58)

(a)

2

"

roots>of productthefrom

2

roots>of sumthefrom

2

"2

"2

2

2

−=∝

−=+∝

=−+=−+

β

β

x x

x x

(b)

,

"

2

"22

2

2

2

222

222

=+=

−−−=

−+=+∴

++=+

)( )(

β β)( β

β β β)(

α α α

α α α

59)



2

2

.......... 1

.......... 2

2 2

".......... "

2 2

..........

1

x px

#rom !"m o# root! $

β p ( )

#rom prod"ct o# root! $

β ( )

x x q #rom !"m o# root! $

β

β ( )

#rom prod"ct o# root!$

( )(β) q

β q ( )

#rom ( ) a

α

α

α

α

α

+ + =

+ = −

=

− + =

+ =+ =

==

"

""

2

q 2

nd ( )

p p

#rom ( ) and ( ),

( ) q

− == −

==

60)

(a) 3f the first root is α " ' then the second root is α 2 .

1.and"arerootsthe

1

!2"

>rootstheof sumthefrom

−−−=

−=+α

α α

(b)

1

2

22"

2= ))(2("

>rootstheof productthefrom

−=

−=−=−−

−=

k

k

k ) )( (

α α

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Chapter 2 Quadratic Equations Paper 1 With Answer