Question

From the roots of $f\left(x\right)\cdot g\left(x\right)\cdot h\left(x\right)=0,$ the maximum number of different matrices is formed (where $f\left(x\right),g\left(x\right),h\left(x\right)$ are quadratic polynomials). If no roots in common for these quadratic equations, then which of the following statement(s) is(are) always correct

• A. All quadratic equation have two different real roots.
• B. Atleast one quadratic equations have equal roots.
• C. Atleast one quadratic equation have no real roots.
• D. All quadratic equations have non-real roots.

Answer 1

Answer: (A), (D)

All quadratic equations have non-real roots.

For maximum number of matrices, there should be more different elements.
$\Rightarrow$ all roots of the equations $f\left(x\right)=0,g\left(x\right)=0,h\left(x\right)=0$ should be different, and each equation should give two different roots.
So, options $\left(a\right)$ and $\left(d\right)$ are correct.