Question

The value of K for which one of the roots of x² - x + 3 k = 0, is
double of one of the roots of x² - x + k = 0 is
(a) 1 (b) - 2 (c) 2 (d) none of these

Answer 1

Answer:

b). - 2

Note:

• The possible values of unknown (variable) for which the equation is satisfied are called its solutions or roots .

• If x = a is a solution of any equation in x , then it must satisfy the given equation otherwise it's not a solution (root) of the equation.

Solution:

Here,

The given quadratic equations are :

x² - x + 3k = 0 ---------(1)

x² - x + k = 0 --------(2)

It is given that ,

One of the roots of eq-(1) is double of one of the roots of eq-(2).

Let , x = a be a root of eq-(2) then the x = 2a will be the root of eq-(1).

Since,

x = a is a root of eq-(2) , thus x = a must satisfy eq-(2) ;

Thus,

=> a² - a + k = 0

=> k = a - a² --------(3)

Also,

x = 2a is a root of eq-(1) , thus x = 2a must satisfy eq-(1) ;

=> (2a)² - (2a) + 3k = 0

=> 4a² - 2a + 3k = 0

=> 4a² - 2a + 3(a - a²) = 0

=> 4a² - 2a + 3a - 3a² = 0

=> a² + a = 0

=> a(a + 1) = 0

=> a = 0 , -1

Now,

If a = 0 , then using eq-(3) , we have ;

=> k = a - a²

=> k = 0 - 0²

=> k = 0

Also,

If a = -1 , then using eq-(3) , we have ;

=> k = (-1) - (-1)²

=> k = -1 -1

=> k = -2

Thus,

Required value of k are 0 , -2

Hence,

The required answer is : b). -2

Answer 2

$\huge\mathfrak\blue{Answer:-}$

Let one root of x2 - x + 3k = 0 is 2m.

Subtracting 2m in the equation,

4m2 - 2m + 3k = 0 ______ eq (1)

Now, let one root of equation x2 - x + k = 0 is m

Sub m in the equation

m2 - m + k = 0 _________ eq (2)

Multiply the eq (2) with 4

4m2 - 4m + 4k = 0 _______eq (3)

Now, (1) = (3)

4m2 - 2m + 3k = 4m2 - 4m +4k

2m - k = 0

m = k/2

Substitute the value of m in eq (2)

(k/2)2 - k/2 + k = 0

(k2 - 2k + 4k) /2 = 0

k2 + 2k = 0

k(k+2) = 0

k = - 2

Hence, option b is correct.