22 Chapter 2
Exercise 2.1
Q5i). Determine nature of roots of
= ( 0 ) 2 - 4 ( 1 )( -
3
) 3x 2 - 10x + 3 = 0 & verify result by solving them. Sol: To check the nature of the roots
= 0 +
12 =
12 Comparing 3x 2 - 10x + 3 =
0 As b 2- 4ac > 0 , but not a perfect square, with the quadratic equation ax 2 + bx + c =
0 therefor, roots are real “Unequal and irrational” we have a =3 ,b =- 10,c =3
Verification Since we have x 2- 3 = 0 ∴ Discriminant = b 2-
4ac x 2=
3 = ( - 10 ) 2
-
4 ( 3 )( 3
) = 100 -
36 =64= 8 2As b 2- 4ac is a perfect square, therefor, roots are real ”Unequal & rational” Verification
S Set = { 3, - 3 ⇒ } roots x = ±
are 3
Unequal & irrational
Q6i). For what value of k the roots of 2x 2 + 3x + k = 0 are (a) Real (b) Imaginary Sol: Comparing 2x 2 + 3x + k = 0 3x 2- 10x + 3 =
0
with the quadratic equation ax 2 + bx + c = 0 3x 2
- 9x - x + 3 =
0 3x ( x - 3 ) - 1 ( x - 3 )
=
0 ( 3x - 1 )( x - 3 )
=
0 Either Or
- =
we have a =2 ,b =3,c =k (a) if roots are real then b 2- 4ac ≥ 0 ⇒ ( 3 ) 2 - 4 ( 2 )( k ) ≥
0
⇒ 9 - 8k ≥
0
3x 1 0
⇒ 9 ≥
8k
3x =1x = 3 1 - =
=
⇒ 9 8 ≥ k Or k ≤
9 8 (b) if roots are imaginary then b 2- 4ac < 0 Solution Set = ⎧ ⎨ ⎩ 3,3 1 ⎫ ⎬ ⎭x 3 0
x 3
⇒ ( 3 ) 2 - 4 ( 2 )( k )
<
0 Unequal and rational
⇒ 9 - 8k <
0
Q5ii). Determine nature of roots of x 2- 6x + 4 = 0 & verify result by solving them. Sol: To check the nature of the roots
⇒9 < 8k ⇒ 9 8 < k Or k >
9 8 Comparing x 2- 6x + 4 = 0 with the quadratic equation ax 2 + bx + c = 0 we have a =1 ,b =- 6,c =4 Solution Set = ⎧ │ │ ⎨ │ │ ⎩ Re Im al aginary k k ∴ Discriminant = b 2-
4ac ( 6 ) 2 41 ( )( 4 ) 36 16 20 0
≤ > 9 89 8 ⎫ │ │ ⎬ │ │ ⎭ Q6ii). For what value of k the roots of kx 2 + 2x + 1 = 0 are (a) Real (b) Imaginary = - -
Sol: Comparing kx 2 + 2x + 1 = 0 = -
with the quadratic equation ax 2 + bx + c = 0 = >
we have a =k ,b =2,c =1 As b 2- 4ac > 0 , but not a perfect square,
(a) if the roots are real then b 2- 4ac ≥ 0 therefor, the roots are real “Unequal and irrational” Verification Using quadratic formula x b b 2 4ac 2a
⇒ ( 2 ) 2 - 4 ( k )( 1 ) ≥
0
= - ± - Putting values of a,b & c
⇒ 4 - 4k ≥
0 ⇒ 4 ≥
4k ⇒ 1 ≥ k Or k ≤
1
x =
- ( - 6 ) ± ( - 6 ) 2 - 4 ( 1 )( 4 ) 2 ( 1
)(b) if roots are imaginary then b 2- 4ac <
0 ⇒ ( ) - ( )( ) <
x =
6 ± 36 - 20 2 x = 6 ± 2 ⇒ - <
20 =
6 ± 2
2 5 ⇒< ⇒ < > x = SS = 3 { 3 ±
+ 5
5,3 - 5 } roots are Unequal & irrational
Q5iii). Determine nature of roots of x 2- 3 = 0 & verify result by solving them. Sol: To check the nature of the roots Comparing x 2- 3 = 0 with the quadratic equation ax 2 + bx + c = 0 soa =1 ,b =0,c =-3 ∴ Discriminant = b 2-
4ac 2 2 4 k 1 0
4 4k 0 4 4k 1 k Or k 1
Solution Set = ⎧ │ ⎨ │ ⎩ Re Im al aginary k k ≤ > 1 1
⎫ │ ⎬ │ ⎭ Q6iii). For what value of k the roots of x 2+ 5x + k = 0 are (a) Real (b) Imaginary Sol: Comparing x 2+ 5x + k = 0 with the quadratic equation ax 2 + bx + c = 0 we have a =1 ,b =5,c =k (a) if the roots are real then b 2- 4ac ≥
0
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