Question

If $\frac{(1-3x{)}^{1/2}+(1-x{)}^{5/3}}{\sqrt{4-x}}$ is approximately equal to $a+bx$ for small values of $x$, then $(a,b)=$

• A. $\left(1,\frac{35}{24}\right)$
• B. $\left(1,-\frac{35}{24}\right)$
• C. $\left(2,\frac{35}{12}\right)$
• D. $\left(2,-\frac{35}{12}\right)$

Answer 1

Answer: (B)

$\frac{(1-3x{)}^{1/2}+(1-x{)}^{5/3}}{2{\left[1-\frac{x}{4}\right]}^{1/2}}$
$=\frac{\left[1+\frac{1}{2}(-3x)\right]+\left[1+\frac{5}{3}(-x)\right]}{2\left[1+\frac{1}{2}\left(-\frac{x}{4}\right)\right]}$
(Neglecting higher powers of $x$
$=\frac{\left[1-\frac{19}{12}x\right]}{\left[1-\frac{x}{8}\right]}$
$=\left[1-\frac{19}{12}x\right]{\left[1-\frac{x}{8}\right]}^{-1}$
$=\left[1-\frac{19}{12}x\right]\left[1+\frac{x}{8}\right]=1-\frac{35}{24}x$
then $a+bx=1-\frac{35}{24}x$
$\Rightarrow a=1,b=-\frac{35}{24}$