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334 MATHS SERIES DSE MATHS PREVIEW
VERSION A SAMPLE UNIT & FULL SOLUTION
2
UNIT SAMPLE
1 Quadratic Equations in One Unknown
LLevel 11 1 Solve the following equations:
(a) 0)3()14( xx (b) 0)35()33( xx
2 Solve the following equations:
(a) (b) 0122 xx 035122 xx
3 Solve the following equations, leaving the answers in surd form where appropriate:
(a) (b) 81)15( 2 x 0484)169( 2 x
1
4 What constant should be added to each of following quadratic expressions to obtain a
perfect square? (a) (b) (c) xx 42 xx 62 xx 52
5 Solve the following equations by using the quadratic formula:
(a) (b) 0973 2 xx 04116 2 xx
6 Given the graph of for 210 xxy 100 x .
By adding suitable straight lines, solve the following equations:
(a) (b) 1210 2 xx 010102 xx
2
7 A rectangular flower garden with dimensions 3 m by 7 m is surrounded by a walk of
uniform width. If the area of the walk is , find its width. 2m11
8 The length of a rectangle is more than the width. If the area of the rectangle is
, what are the dimensions of the rectangle?
cm52mc36
9 By finding the discriminant of each of the following quadratic equations, determine the
nature of the roots:
(a) (b) 01202 2 xx xx 18638 2
10 Find the real value(s) of a in the quadratic equation in x if it has equal
roots. Find also the roots.
056 2 axx
3
11 Find the real values of k if the equation has 088 2 kxx
(a) real roots, (b) equal roots, (c) unequal roots.
12 Find the sum of roots and product of roots of the following quadratic equations:
(a) 0 (b) 232 xx 02
22 xx(c) 0 (d) 965 2 xx 34 2 xx
13 If and are the roots of the quadratic equation , find the value of 0142 2 xx
(a) (b)
(c)
11 (d) )1()1(
4
14 If and are the roots of the quadratic equation , find the value of 0532 xx
(a) )1()1( (b) 22
(c)
(d) )1
()1
(
15 Let and be the roots of the quadratic equation , find the value of 0472 xx
(a) (b) 22 2)(
5
16 Form the quadratic equations, with integral coefficients, whose roots are:
(a) 0, 2
3 (b) 2, 5
(c) 3
1 ,
2
3
(d) 5
4 ,
5
4
17 Let and be the roots of the quadratic equation . Form the
quadratic equation whose roots are:
0132 xx
(a) , (b) 2 2 1 , 1
18 Form the quadratic equation, with integral coefficients, whose roots are 32 and
32 .
6
LLevel 21 19 Solve the following equations:
(a) (b) 082816 2 xx 01262519 2 xx
20 Solve the following equations:
(a) 4)31()12( xx (b) 26)3()1( 22 xx
21 Solve 0324)322(2 xx , leaving the answer in surd form.
22 Solve for x: )4()2()4()2( aaxx .
7
23 (a) Draw the graph of for352 2 xxy 24 x on the graph paper
provided.
(b) From the graph in (a), find
(i) the x-intercept(s) and y-intercept(s) of the graph,
(ii) the value of y when 5.2x ,
(iii) the value(s) of x when 4y .
(c) Use the graph drawn in (a) to solve the following equations for : 24 x
(i) (ii) . 0352 2 xx 06104 2 xx
8
24 A square plot of land is covered with cement costing and a fence surrounding it
costs $10 /m. If the total cost is $8 400, find the area of the plot.
2/m8$
25 Find the real value of a in the quadratic equation in x if the equation
has equal roots. Hence find the roots.
0442 axax
26 In order to make each of the following expressions become a perfect square,
what constant should be added?
(a) ab
a2
32 (b) yy 255 2
9
27 Find the value of k if the expression is a perfect square. kxx 43 2
28 Prove that the quadratic equation has real roots for all real values
of k.
0)1(2 kkxx
29 Given the quadratic equation , where k is real. 0242 kkxx
(a) Express its discriminant in terms of k.
(b) The equation has real and equal roots. Write down an equation involving k.
Hence find
(i) the values of k,
(ii) the roots of the equation for the larger value of k.
10
30 Let and be the roots of the equation . Find the value of 0542 2 xx
(a) (b)
(c) )32()32( (d)
11
31 Let and be the roots of the quadratic equation ) . Find
the value of
0643 2 xx ( .
32 Given that 2 and 1 , find the value of . 33
11
33 If 32 is one root of the quadratic equation (k being real), find the
other root and the value of k.
042 kxx
34 Find, by inspection, one root of the quadratic equation , where l and
m are constants. Hence find the other root.
)3(32 mlxlmx
35 If one root of the quadratic equation (k being real) is less than
the other root by 5, find the values of k.
0)13(22 2 kkxx
12
36 Form the quadratic equation, with integral coefficients, whose roots are
(a) a, (b) b2
a,
2
b
37 Form an equation whose roots are the squares of the roots of the quadratic equation
. 0142 xx
13
LMC1 1 The roots of the equation )23()12()2()1( xxxx are
A 1 , 2 .
B 2
1 ,
3
2 .
C 0, 4
5.
D 0, 5
4 .
2 One root of the equation is 3. The other root is 0243 2 kxx
A 3
4.
B 3
8.
C 6. D 8.
3 The reciprocal of a number is 1 less than the original number. The number is A 1. B 1 5 . C 1 5 or 1 5 .
D 1
2
5 or
1 5
2
.
4 If the equation x x k2 6 0 has equal roots, k A . 9 B 9. C 3. D 9 or 9.
14
5 If the quadratic equation ax bx c2 0 has distinct real roots, the roots of the
quadratic equation cx bx2 a2 0 are
A real and distinct.
B real.
C equal.
D unreal.
6 If the graph of mxxy 242 2 touches the x-axis, find the value of m.
A 2 B 4
C 2
D 4
7 The quadratic equation has equal roots, the value of k is kxx 22 )3(
A 1.5.
B 3.
C 4.5.
D 9.
8 The area of a rectangle is . If its length exceeds its width by 2 c , find its
perimeter.
2cm 48 m
A 8 cm
B 14 cm
C 28 cm
D 30 cm
15
9 The product of two consecutive positive odd numbers is 63. Find the larger number.
A 5
B 7
C 8
D 9
10 Which of the following shows the graph of , where and ? cbxaxy 2 a 0 acb 42 A B
C D
11 Given the graph of y x x 2 3 2. Find the equation of the straight line to be added in
order to solve the quadratic equation . xxx 10542 2
A 2
9y
B 2
1y
C 5y
D 7y
16
12
The figure shows the graph of . cbxaxy 2 c
A . 1 B 0.
C 2.
D 3.
13 Which of the following shows the graph of ? 232 xxy
A B
C D
17
14 If and are the roots of the equation , and are 0)(2 abxbax A , a b .
B a , b .
C , b. a D a, b.
15 Form the equation whose roots are 3 and 6.
A 992 xx
B 01892 xx
C xx 3182 D 1832 xx
16 If and are the roots of the equation , 0322 xx 2222
A 2
1.
B 1.
C 2.
D 2
12 .
17 The sum of the roots of the equation equals the product of its
roots.
02)1( 22 kxkx
k
A 1.
B 2
1 or 1.
C 2
1 or 1.
D or 12
1.
18
18 If and are the roots of the equation , 0)1(322 xkxx
22
11
A 3
1
3
22
kk
k.
B 2
2
2
32
kk
k.
C 2
2
2
32
kk
k.
D 3
2
3
22
kk
k.
19 If and are the roots of the equation , form an equation whose
roots are and
02 baxx
3 3 .
A 32 baxx B 0)93()6(2 baxax
C 0)93()6(2 baxax
D 0)93()6(2 baxax
20 Form a quadratic equation whose roots are the reciprocal of the roots of the equation
. 0374 2 xx
A 0 473 2 xx
B 0473 2 xx
C 0473 2 xx
D 0437 2 xx
19
1
FULL SOLUTION
SAMPLE
1 Quadratic Equations in One Unknown Level 1
1 (a) 0)3()14( xx (b) 0)35()33( xx
or 014 x 03 x 033 x or 035 x
4
1x or 3x x 1 or
5
3x
2 (a) 0122 xx (b) 035122 xx 0)3()4( xx 0)5()7( xx
or 04 x 03 x 07 x or 05 x
or 4x 3x 7x or 5x
3 (a) 81)15( 2 x (b) 0484)169( 2 x
9)15(
9)15( 22
x
x 484)169( 2 x
22 22)169( x
22)169( x or 915 x 915 x
22169 x or 22169 x or 2x
5
8x
9
38x or
3
2x
4 (a) 4)2(22)2)((2)2)((24 222222 xxxxxxx
22 )2(4)4( xxx
Constant to be added = 4
(b) 9)3(33))(3(2)()3(26 222222 xxxxxxx
22 )3(9)6( xxx
Constant to be added = 9
(c) 4
25)
2
5()
2
5()
2
5())(
2
5(2))(
2
5(25 222222 xxxxxxx
22 )2
5(
4
25)5( xxx
Constant to be added =4
25
2
5 (a) 6
1577
)3(2
)9()3(4497
x
(b) 2
1
12
511
)6(2
)4()6(412111
x or
3
4
6
(a) 1210 2 xx (b) 010102 xx12y , which is the line to be
added 21010 xx
10y , which is the line to be added
or 8.6 4.1x 2.1x or 8.8
3
7 Let the width be x m 8 Let the width of the rectangle be x cm
1173)23()27( xx Then the length is cm)5( x
36)5( xx 011204 2 xx 0)112()12( xx 03652 xx
0)4()9( xx
2
1x or
2
11x (rejected)
x 4 or 9x (rejected)
the dimensions of the rectangle the width is m
2
1
are cm9cm4 9 (a) 0961)120()2(412 the equation has two real and distinct roots
(b) 063188 2 xx
01692)63()8(4)18( 2 the equation has no real roots
10 aa 2425)()6(452 the equation has equal roots
0
24
25
02425
a
a
When 24
25a , the original equation becomes 0
24
2556 2 xx
025120144 2 xx
0)512( 2 x
the roots are 12
5 ,
12
5
11 kk 3264)()8(482 (a) the equation has real roots (c) the equation has unequal roots
0 0
2
03264
k
k
2
03264
k
k
(b) the equation has equal roots 0
2
03264
k
k
4
12 (a) Sum of roots 31
3
; Product of roots 2
1
2
(b) Sum of roots 1)1
1(
; Product of roots 21
2
(c) Sum of roots 5
6)
5
6(
; Product of roots 5
9
5
9
(d) can be rewritten as 234 2 xx 0234 2 xx
Sum of roots 4
3)
4
3(
; Product of roots 2
1
4
2
13 (a) 22
4
(b) 2
1
2
1
(c) 4
2
1211
(d) 2
31)2(
2
11)()1)(1(
14 3)1
3(
, 51
5
(a) 71351)()1()1(
(b) 19)5(232)( 2222
(c) 5
1922
(d) 95
1)
5
19(5
1)()
1()
1(
15 7 , 4
(a) 57)4(2)7(2)( 2222 (b) 65)4(2572)()( 222
5
16 (a) Method I Method II
The required equation is: The required equation is:
032
0)32(
0)2
3)(0(
2
xx
xx
xx
032
02
3
0)2
3)(0()
2
30(
2
2
2
xx
xx
xx
(b) Method I Method II
The required equation is: The required equation is:
0103
0)5)(2(2
xx
xx
0103
0)5)(2()]5(2[2
2
xx
xx
(c) Method I Method II
The required equation is: The required equation is:
0376
0)32)(13(
02
32
3
13
0)2
3)(
3
1(
2
xx
xx
xx
xx
0376
02
1
6
7
0)2
3)(
3
1()
2
3
3
1(
2
2
2
xx
xx
xx
(d) Method I Method II
The required equation is: The required equation is:
0164025
025
16
5
8
0)5
4)(
5
4()
5
4
5
4(
2
2
2
xx
xx
xx
0164025
025
16
5
8
0)5
4(
2
2
2
xx
xx
x
6
17 3 , 1
(a) The required equation is:
046
0)1(4)3(2
04)(2
0)2)(2()22(
2
2
2
2
xx
xx
xx
xx
(b) The required equation is:
055
0131)23(
01)()2(
0)1)(1()11(
2
2
2
2
xx
xx
xx
xx
18 Sum of roots 4)32()32(
Product of roots 134)3(2)32()32( 22
The required equation is 0142 xx
7
LEVEL 2
19 (a) 082816 2 xx (b) 01262519 2 xx
0)14()2(
0274 2
xx
xx
0)6319()2(
01262519 2
xx
xx
or 2x4
1x 2x or
19
63x
20 (a) 4)31()12( xx (b) 26)3()1( 22 xx
0)1()56(
056 2
xx
xx 269612 22 xxxx
0)2()4(
0822
xx
xx
or 056 x 01 x
04 x or 02 x
6
5x or 1x
4x or 2x
21 Method I
8
0324)322(2 xx
13)1(2
)324)(1(4)322()322( 2
x
Method II
13
0])31([
0)31()31(2
0324)322(
2
22
2
x
x
xx
xx
Note: 324)3(321)31( 22
22 )4()2()4()2( aaxx
0
0
0
)82(82 22 aaxx
)2(2 22 aaxx
)(2)()( axaxax
0)2()( axax
ax or ax 2
23 (a)
(b) (i) x-intercepts: , 0.5 , y-intercept: 3 3
(ii) 3
(iii) , 1 5.3
(c) (i) The roots of the equation are the x-intercepts obtained in 0352 2 xx
(b)(i)
the roots are 3 , 0.5
(ii) 06104 2 xx
0352
0)352(22
2
xx
xx
From (c)(i), the roots are 3 , 0.5
9
25 )4()(4)4( 2 aa 24 Let one side of the plot be x m
8400)4(108 2 xx
0
1616 2
aa
0105052 xx 0)35()30( xx a 1 or a 0 (rejected)
or 30x 35x (rejected) x2 4x 4 0
10
the area of the plot is 2m900 0)2( 2 x
the roots are 2, 2
26 (a) 222222
16
9)
4
3()
4
3(])
4
3()
4
3(2[
2
3
bba
bba
baa
ba
Constant to be added 216
9
b
(b) 4
125)
2
5(5)
2
5(5])
2
5()
2
5(2[5)5(5255 222222 yyyyyyy
Constant to be added 4
125
27 is a perfect square kxx 43 2
The equation has equal roots 043 2 kxx
01216)3(4)4( 2 kk
3
4k
28 0)2(44])1([)1(4 222 kkkkk
the equation has real roots for all real values of k 0)1(2 kkxx
29 (a) 964)24()1(4)( 22 kkkk
(b) the equation has real and equal roots
0
k 2 4k 96 0
(i) 09642 kk 0)8()12( kk
k 12 or 8k
(ii) When k 12, the original equation becomes x2 12x 36 0
0)6( 2 x
the roots are 6, 6
30 (a) 22
4
(b)
2
5
2
5
(c) 9664)32()32( 22
2
126
2
5)2(6
)(6
13]2)[(6
13)(6
2
2
2
22
(d)
)1()1(11
5
62
5
)2
5(22
2
31 9
88)2(4)
3
4(4)()( 222
3
222
3
88 )(
32 )()( 2233
14
68
)2)(1(3)2(
)(3)(
]3)([)(
3
3
2
11
33 Method I
Let be the other root Method II
13483344
0)32(4)32( 2
k
k
41
432
The original equation becomes
0142 xx
32 (1)
kk
1)32( (2)
322
124
x
Sub. (1) into (2):
the other root is 32
134
)32()32(
k
k
34 By inspection, one root of the equation is m
The equation can be rewritten as 03)3(2 lmxmlx
Let be the other root Method I
12
Method II
lmm 3 mlm 3 l3 l3
35 Let and be the roots and
; 5 kk
)2
2(
0)3()9(
0276
2
13)
2
5()5
2
5(
2
13)5(
2
132
5
5
2
kk
kk
kkk
k
k
k
k
or 9k 3k
36 (a) Method I (b) Method I
The required equation is: The required equation is:
0)(24
042
0)2
)(2
()22
(
2
2
2
abxbax
abx
bax
bax
bax
0)(
0))(()(2
2
abxbax
baxbax
Method II
The required equation is:
Method II
0)(
0
0))((
2
2
abxbax
abbxaxx
bxax
The required equation is:
0)(24
0224
0)2)(2(
0)2
2)(
2
2(
0)2
)(2
(
2
2
abxbax
abaxbxx
bxax
bxax
bx
ax
37 Let , be the roots of 0142 xx
, 4 1
142162)( 222 122
The required equation is 01142 xx
13
MC
1 D 2 B
Method I Let be the other root
3
24)3(
)23()12()2()1( xxxx
27623 22 xxxx
045 2 xx 0)45( xx
3
8 or 0x
5
4x
Method II
14
024)3()3(3 2 k 17k 024173 2 xx 0)83()3( xx
3
8x or x = 3
the other root is 3
8
4 B 3 D
Let x be the number
9
0462
k
k
2
51
2
)1(411
01
11
2
x
xx
xx
5 A
For 02 cbxax
6 B
If the graph touches the x-axis,
the discriminant of 0
0
42 acb
0242 2 mxx is 0 For 022 cbxax
0)()2(4)24( 2 m )4(34)2( 222 acbbacb 2 b 0832 m
4m 0
8 C 7 C kxx 22 )3(
2
Let the width be x cm 0962 kxx Then the length is cm )2( x
48)2( xx
5.4
0368
0)9)(2(4)6( 2
k
k
k
4822 xx 04822 xx 0)8()6( xx
6x or 8x (rejected)
Perimeter = 282])26(6[ cm
9 D 10 C Let x be the larger number 0a The smaller number is 2x It opens downward
0)9()7(
0632
63)2(2
xx
xx
xx
042 acb It touches the x-axis (rejected) 7or 9 x
12 C 11 A The curve passes through )2,0( xxx 10542 2
2
)0()0(2 2
c
cb 562 2 xx
2
532 xx
2
9232 xx
the line to be added is 2
9y
13 D 14 C
2or 1
0)2()1(
023
02
x
xx
xx
y
Method I: By inspection
ba
15
it cuts the x-axis at )0,2(,)0,1(
coefficient of 012 x it opens downward
)( , ) )(( ba
a and b are the roots
Method II: Factorization
0))(( bxax
ax or bx
15 D Method I The required equation is:
183
0183
0)6)(3()63(
2
2
2
xx
xx
xx
Method II The required equation is:
183
0183
0)6()3(
2
2
xx
xx
xx
16 A 17 B
2 Sum of roots = Product of roots
2
1
)2(2
)2(222222
0)12)(1(
012
212
2
kk
kk
kk
1k or
2
1k
18 D
0)3()2(
0)1(322
2
kxkx
xkxx
, )2( k )3( k
3
2
3
2
)3(
)3(2)2(
)(
2)(112
2
2
2
2
22
22
22
kk
k
k
kk
19 C
, a b
66)3()3( a
939)(39)(3)3()3( baab
the required equation is 093)6(2 baxax
16
20 B
Let , be the roots of 0374 2 xx
3
7
4
34
711
3
4
4
3111
.1
The required equation is 03
4
3
72 xx
i.e. 0473 2 xx
17
