Question

Let $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ be polynomials of degree $4$ and satisfy the equation $\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=l{x}^4+m{x}^3+n{x}^2+4x+1$ where $a,b,c,p,q,r,l,m,n$ are real constants. Then the value of $\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(0\right)-f^{\prime \prime }\left(0\right)& g^{\prime \prime \prime }\left(0\right)-g^{\prime \prime }\left(0\right)& h^{\prime \prime \prime }\left(0\right)-h^{\prime \prime }\left(0\right)\\ a& b& c\\ p& q& r\end{array}\right|$ is

• A. $3m-n$
• B. $2(3m-n)$
• C. $3(3m+n)$
• D. $3m+n$

Answer 1

The correct option is B $2(3m-n)$

$\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=l{x}^4+m{x}^3+n{x}^2+4x+1$

Differentiating w.r.t. $x,$ we get
$\left|\begin{array}{ccc}{f}^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=4l{x}^3+3m{x}^2+2nx+4$

Again differentiating w.r.t. $x,$ we get
$\left|\begin{array}{ccc}{f}^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=12l{x}^2+6mx+2n\cdots \left(1\right)$

Again differentiating w.r.t. $x,$ we get
$\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)& h^{\prime \prime \prime }\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|=24lx+6m\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$ we have
$\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(x\right)-f^{\prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)-g^{\prime \prime }\left(x\right)& h^{\prime \prime \prime }\left(x\right)-h^{\prime \prime }\left(x\right)\\ a& b& c\\ p& q& r\end{array}\right|$

$=(24lx+6m)-(12l{x}^2+6mx+2n)$

At $x=0,$
$\left|\begin{array}{ccc}{f}^{\prime \prime \prime }\left(0\right)-f^{\prime \prime }\left(0\right)& g^{\prime \prime \prime }\left(0\right)-g^{\prime \prime }\left(0\right)& h^{\prime \prime \prime }\left(0\right)-h^{\prime \prime }\left(0\right)\\ a& b& c\\ p& q& r\end{array}\right|$

$=6m-2n=2(3m-n)$