Question

If $f,g,h$ are differentiable functions of $x$ and $\Delta =\left|\begin{array}{ccc}f& g& h\\ (xf{)}^{\prime }& (xg{)}^{\prime }& (xh{)}^{\prime }\\ (x^{2}f{)}^{\prime \prime }& (x^{2}g{)}^{\prime \prime }& (x^{2}h{)}^{\prime \prime }\end{array}\right|$ then

• A. $\Delta =\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^3{f}^{\prime }{)}^{\prime }& (x^3{g}^{\prime }{)}^{\prime }& (x^3{h}^{\prime }{)}^{\prime }\end{array}\right|$
• B. $\Delta =\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x{f}^{\prime \prime }{)}^{\prime }& (x{g}^{\prime \prime }{)}^{\prime }& (x{h}^{\prime \prime }{)}^{\prime }\end{array}\right|$
• C. $\Delta =\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^2{f}^{\prime \prime }{)}^{\prime }& (x^2{g}^{\prime \prime }{)}^{\prime }& (x^2{h}^{\prime \prime }{)}^{\prime }\end{array}\right|$
• D. none of these

Answer 1

Answer: (A)

$\Delta =\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^3{f}^{\prime }{)}^{\prime }& (x^3{g}^{\prime }{)}^{\prime }& (x^3{h}^{\prime }{)}^{\prime }\end{array}\right|$

$\left|\begin{array}{ccc}f& g& h\\ (xf{)}^{\prime }& (xg{)}^{\prime }& (xh{)}^{\prime }\\ (x^{2}f{)}^{\prime \prime }& (x^{2}g{)}^{\prime \prime }& (x^{2}h{)}^{\prime \prime }\end{array}\right|$
$=\left|\begin{array}{ccc}f& g& h\\ x{f}^{\prime }+f& x{g}^{\prime }+g& x{h}^{\prime }+h\\ \left(\right(x^2{f}^{\prime })+2xf{)}^{\prime }& \left(\right(x^2{g}^{\prime })+2xg{)}^{\prime }& \left(\right(x^2{h}^{\prime })+2xh{)}^{\prime }\end{array}\right|$
$=\left|\begin{array}{ccc}f& g& h\\ x{f}^{\prime }+f& x{g}^{\prime }+g& x{h}^{\prime }+h\\ (x^2{f}^{\prime \prime }+2x{f}^{\prime }+2x{f}^{\prime }+2f)& (x^2{g}^{\prime \prime }+2x{g}^{\prime }+2x{g}^{\prime }+2g)& (x^2{h}^{\prime \prime }+2x{h}^{\prime }+2x{h}^{\prime }+2h)\end{array}\right|$
At $R_3=R_3-R_2-R_1,$
$=\left|\begin{array}{ccc}f& g& h\\ x{f}^{\prime }+f& x{g}^{\prime }+g& x{h}^{\prime }+h\\ (x^2{f}^{\prime \prime }+3x{f}^{\prime })& (x^2{g}^{\prime \prime }+3x{g}^{\prime })& (x^2{h}^{\prime \prime }+3x{h}^{\prime })\end{array}\right|$
At$R_2=R_2-R_1$,
$=x\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^2{f}^{\prime \prime }+3x{f}^{\prime })& (x^2{g}^{\prime \prime }+3x{g}^{\prime })& (x^2{h}^{\prime \prime }+3x{h}^{\prime })\end{array}\right|$
$=\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^3{f}^{\prime \prime }+3x^2{f}^{\prime })& (x^2{g}^{\prime \prime }+3x^2{g}^{\prime })& (x^3{h}^{\prime \prime }+3x^2{h}^{\prime })\end{array}\right|$
$=\left|\begin{array}{ccc}f& g& h\\ f^{\prime }& g^{\prime }& h^{\prime }\\ (x^3{f}^{\prime }{)}^{\prime }& (x^3{g}^{\prime }{)}^{\prime }& (x^3{h}^{\prime }{)}^{\prime }\end{array}\right|$