Question 4 Exercise 7.3

Solutions of Question 4 of Exercise 7.3 of Unit 07: Permutation, Combination and Probablity. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Q4 If $x$ is such that $x^2$ and higher of $x$ may be neglected, then show that $$ \sqrt{\frac{1-3 x}{1+4 x}}=1-\frac{7 x}{2} $$

Solution: Given that $$ \sqrt{\frac{1-3 x}{1-4 x}}=(1-3 x)^{\frac{1}{2}}(1+4 x)^{-\frac{1}{2}} $$

Applying binomial expansion and neglecting $x^2$ and higher powers of $x$. $$ \begin{aligned} & =\left(1-\frac{3 x}{2}\right) \times\left(1-\frac{4 x}{2}\right) \\ & =\left(1-\frac{3 x}{2}\right)(1-2 x) \end{aligned} $$

Multiplying and neglecting $x^2$ and higher powers of $x$ $$ \begin{aligned} & =1-2 x-\frac{3 x}{2} \\ & =1 \cdot \frac{4 x+3 x}{2} \\ & =1 \cdot \frac{7 x}{2} . \end{aligned} $$