Question

Let $a{x}^2+a^{2}x+2=0$ be a quadratic equation, $a\in R$ and $S$ be the set of values of $a$ for which roots of this equation are imaginary or equal, then answer the following questions:

If $\alpha$ and $\beta$ are respectively sum and product of roots of the given equation, then $-\alpha \beta$ is

• A. a prime number
• B. an odd integer
• C. an irrational number
• D. dependent on value of a

Answer 1

The correct option is A a prime number

$a{x}^2+a^{2}x+2=0$

Hence, from theory of equations, we get

$\alpha =\frac{-a^2}{a}=-a$

$\beta =\frac{2}{a}$

Hence

$-\alpha .\beta =-a.\left(\frac{-2}{a}\right)$

$=2$

$\to$ Prime number.