Question

If $f\left(x\right),g\left(x\right),h\left(x\right)$ are three polynomials of degree $3$, then $\varphi \left(x\right)=\left|\begin{array}{ccc}{f}^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ f^{\prime \prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)& h^{\prime \prime \prime }\left(x\right)\end{array}\right|$ a polynomial of degree:

• A. 3
• B. 4
• C. 5
• D. None of these

Answer 1

The correct option is A 3

Writing degrees in the determinant,

$\left|\begin{array}{ccc}2& 2& 2\\ 1& 1& 1\\ 0& 0& 0\end{array}\right|$

Hence as we expand the determinant, we would have terms with $x^2\times x\times \lambda$ form, $\Rightarrow \varphi \left(x\right)$ would be of $3$ degree $\Rightarrow \left(A\right)$