7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 1/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 2/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 3/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 4/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 5/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 6/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 7/32
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 %%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 8/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 9/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 10/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 11/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 12/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 13/32
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 $ %%%%%%%%%%%%%%%%%%%%%%%%%
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 14/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 15/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 16/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 17/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 18/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 19/32
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
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 20/32
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'
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 21/32
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
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 22/32
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'
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 23/32
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'
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 24/32
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
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 25/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 26/32
,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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 27/32
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
=−=
=−+
=−+
=−++−
=−−−−
=−
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 28/32
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
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 29/32
" 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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 30/32
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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 31/32
?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)
7/23/2019 Chapter 2 Quadratic Equations Paper 1 With Answer
http://slidepdf.com/reader/full/chapter-2-quadratic-equations-paper-1-with-answer 32/32
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