Question

The real values of 'a' for which the quadratic equation ${\text{2}}{{\text{x}}^{\text{2}}}{\text{ - (}}{{\text{a}}^{\text{3}}}{\text{ + 8a - 1)x + }}{{\text{a}}^{\text{2}}}{\text{ - 4a = 0}}$ possesses roots of opposite sign is given by:
A. a>6
B. a>9
C. 0D. a<0

Answer 1

Hint – In order to solve this problem we need to use the sum of roots and product of roots and follow the condition provided in the question to get the right answer.

Complete step-by-step answer:
The given equation is ${\text{2}}{{\text{x}}^{\text{2}}}{\text{ - (}}{{\text{a}}^{\text{3}}}{\text{ + 8a - 1)x + }}{{\text{a}}^{\text{2}}}{\text{ - 4a = 0}}$
Let us assume that the roots of the equation are p and q as if the quadratic equation is x it will have only two roots.
We know that the sum of roots p + q = $\dfrac{{{\text{ - ( - (}}{{\text{a}}^{\text{3}}}{\text{ + 8a - 1))}}}}{2}$……………(1)
And the product of roots p x q = $\dfrac{{{{\text{a}}^{\text{2}}}{\text{ - 4a}}}}{{\text{2}}}$………..(2)
It is given in the question that the roots are of the opposite sign.
So, we can consider that if p is negative, q is positive and vice-versa.
So, p x q < 0 (always)
From (2) we can say that $\dfrac{{{{\text{a}}^{\text{2}}}{\text{ - 4a}}}}{{\text{2}}}$ < 0
So, we do ${\text{a(a - 4) < 0}}$
We can clearly see that when the value of a is less than 0 the the above term is positive and also when it is 0 or 4 then the term is zero and when it is greater than 4c the term is positive the term is only negative in the interval (0,4).
Hence, ${\text{a}} \in {\text{(0,4)}}$.
So, the Correct option is C.


Note – In this problem you need to know that if the number p belongs to (a, b) then it is not equal to a and b but it can be any number between the numbers a and b. Also whenever you face the equations of inequality then you should try to put the values and check whether the equation is satisfying with that value or not. Proceeding like this you will get the right answer.