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  4. If ((1-3 x)1 / 2+(1-x)5 / 3/√4-x) is approximately equal to a+b x for small values of x, then (a, b)=

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

Binomial Theorem

Solution:

$\frac{(1-3 x)^{1 / 2}+(1-x)^{5 / 3}}{2\left[1-\frac{x}{4}\right]^{1 / 2}}$
$=\frac{\left[1+\frac{1}{2}(-3 x)\right]+\left[1+\frac{5}{3}(-x)\right]}{2\left[1+\frac{1}{2}\left(-\frac{x}{4}\right)\right]}$
(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 $a+b x=1-\frac{35}{24} x$
$ \Rightarrow a=1, b=-\frac{35}{24}$