Mathematics 10 (Science) - KPK - Gain - Solution - PART - 20 - 30 Chapter 2

30 Chapter 2 

Exercise 2.4 

Exp 18i) Form a quadratic equation whose roots are 2a + 1 and 2b +1 

Exercise 2.4 

Sol: Roots of quadratic eq are 2a + 1& 2b +1 

Q1i). If α,β be the roots of ax 2 + bx + c = 

0 , Sum of roots S = 2 a + 1 + 2 b + 

1 S = 2 a + 2 b + 2 Product of roots P = ( 2 a + 1 )( 2 b + 

1 ) find the value of 

α 3 β + β 3 α Sol: Sum of the roots 

α + β = - a bP = 4 ab + 2 a + 2 b + 1 Since x 2 - Sx + P = 0 putting values x 2 - ( 2 a + 2 b + 2 ) x + ( 4 ab + 2 a + 2 b + 1 ) = 0 Exp 18ii) Form a quadratic equation whose 

Product of the roots 

α . β = c a According α 3 β + β 3 α to = αβ the ( α given 2 + 

β 

condition 

2 

) 

roots are a 2and b 2Sol: Roots of quadratic eq are a 2and b 2Sum of roots S = a 2 + 

b 2 = = αβ αβ ( ( α { α 2 + + β β } 22 

+ - 

αβ - 

αβ 

) 

αβ 

) 

Product of roots P = 

a 2 b 2 Putting the values α + β and α,β Since x 2 - Sx + P = 0 putting values x 2 - ( a 2 + b 2 ) x + ab 2 2 = 0 Exp 18iii) Form a quadratic equation whose roots are 1a and 1b Sol: Roots of quadratic eq are 1a and 1b ( )Sum of roots S = 1 a + b 1 = 

a ab+ b Q1ii). If α,β be the roots of , Product of roots P = 1 a × b 1 = 

ab 1 find the value of Since x 2 - Sx + P = 0 putting values 

Sol: As we know & x 2 a ab b x 1 ab 

0 According to the given condition 

Using formula 

Putting the values α + β and α,β 

Q2i). Find quadratic eq whose roots are 

Sol: Sum of the roots 

Product of the roots 

The required equation is given by 

Putting the values α + β and α,β 

Q2ii).Find quadratic eq whose roots are Sol: Sum of the roots Product of the roots 

2 3 3 

222 

2 

2 

3 

2 2 

2 

α β + β α = ⎛ │ │ ⎝ ⎧ ⎨ ⎩ - ⎫ ⎬ ⎭ - ⎛ │ ⎝ ⎞ │ ⎠ ⎞ │ │ ⎠ = ⎛ │ ⎝ - ⎞ │ ⎠ = ⎛ │ ⎝ - ⎞ │ ⎠ 

= 

- ax 2 + bx + c = 

0 ( α - β )2 α + β = - a bα . β = c a ( α - β ) 2 = ( α + β ) 2 - 4 αβ ( ) 

( ) 

( ) 

c b a a 2 c a 

c b a a 2c a . a a c b 2ac a a 

c b 2ac 

a 

- ⎛ │ ⎝ + ⎞ │ ⎠ + = multiply by ab abx 2 - ( a + b ) x + 1 = 0 Exp 18iv) Form a quadratic equation whose roots are 23 and 32 Sol: Roots of quadratic eq are 23 and 32 Sum of roots S = 2 3 + 3 2 = 4 + 6 9 = 

13 6 Product of roots P = 2 3 × 3 2 = 1 Since x 2 - Sx + P = 0 putting values x 2 - 13 6 x + 1 = 0 multiply each term by 6 6 x 2 - 13 x + 6 = 0 Exp 18v) Form a quadratic equation whose roots are 5 2and 52 Sol: Roots of quadratic eq are 5 2and 52 Sum of roots S = 5 2 + 5 2 = 4 10 + 25 = 

10 29 Product of roots P = 2 5 × 5 2 = 1 Since x 2 - Sx + P = 0 putting values x 2 - 10 29 x + 1 = 0 multiply each term by 10 10 x 2 - 29 x + 10 = 

0 - 2 

= ⎛ │ ⎝ 2 - ⎞ │ ⎠ 2 2 22 

2 2 - ⎛ │ ⎝ α β 

b a 4 c a 

⎞ │ ⎠ 

α - β 

= b a-- = 

- 1,2 1 α + β = 1 + 2 1 = 2 2 + 1 = 3 2 α . β = ( 1 ) ⎛ │ ⎝ 2 1 ⎞ │ ⎠ = 2 1 x 2- ( Sumoftheroots ) x + ( Productoftheroots ) = 0 2 

22 4c a . a a α β 

b a 

4ac 

x - 3 2 x + 2 

1 = 0 × by 2 2x - 2. 3 2 x + 2. 2 

1 = 2.0 2x - 3x + 1 = 

0 

- 3, 4 α + β = - 3 + 4 = 1 α . β = ( - 3 )( 4 ) = - 12