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The number of all possible positive inte...

The number of all possible positive integral values of `alpha` for which the roots of the quadratic equation `6x^2-11x+alpha=0` are rational numbers is : (a) 3 (b) 2 (c) 4 (d) 5

A

5

B

2

C

4

D

3

Text Solution

Verified by Experts

The correct Answer is:
D
For the roots of quadratic equation `ax^(2)+bx+c=0` to be rational `D=(b^(2)-4 ac)` should be perfect square. In the equation `6x^(2)-11x+alpha=0`
`therefore` For root to be rational
`D=(-11)^(2)-4(6)(alpha)` should be a perfect square.
`implies D(alpha)=121-24alpha` should be a perfect square Now,
`D(1) =121-24=97` is not a perfect equare.
`D(2)=121 -24xx2=73` is not a perfect square.
`D(3)=121-24xx3=49` is a perfect square.
`D(4) =121-24xx4=25` is a perfect square.
`D(5)=121-24xx5=1` is a perfect square.
and for `alpha ge 6, D(alpha) lt 0,` hence imaginary roots.
`therefore` For 3 values of `alpha (alpha =3, 4, 5),` the roots are rational.
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