Question

If the roots of the equation 3x^2 + 2x + a^2-a=0 is x are of opposite signs, then a lies in the interval
A) (-1,0)
B) (0,1)
C) (1,3)
D) (-0,-1)​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{The roots of the equation}$

$\mathsf{3x^2+2x+a^2-a=0}\;\textsf{are of opposite sign}$

$\underline{\textsf{To find:}}$

$\textsf{The interval in which a lie}$

$\underline{\textsf{Solution:}}$

$\textsf{Since the roots are of opposite signs, the product of}$

$\textsf{the roots is negative}$

$\mathsf{\dfrac{a^2-a}{3}\,<\,0}$

$\mathsf{a^2-a\,<\,0}$

$\mathsf{a(a-1)\,<\,0}$

$\mathsf{(a-0)(a-1)\,<\,0}$

$\implies\textsf{a lies between 0 and 1}$

$\implies\mathsf{a\,{\in}\,(0,1)}$

$\textsf{Hence, option(B) is correct}$

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