Binomial Theorem And Its Simple Applications Question 123
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) $ =
Options:
A) $ ( 1,\frac{35}{24} ) $
B) $ ( 1,-\frac{35}{24} ) $
C) $ ( 2,\frac{35}{12} ) $
D) $ ( 2,-\frac{35}{12} ) $
Show Answer
Answer:
Correct Answer: B
Solution:
- $ \frac{{{(1-3x)}^{1/2}}+{{(1-x)}^{5/3}}}{2{{[ 1-\frac{x}{4} ]}^{1/2}}} $
$ =\frac{[ 1+\frac{1}{2}(-3x)+\frac{1}{2}( -\frac{1}{2} )\frac{1}{2}{{(-3x)}^{2}}+…. ]+[ 1+\frac{5}{3}(-x)+\frac{5}{3}\frac{2}{3}\frac{1}{2}{{(-x)}^{2}}+…. ]}{2[ 1+\frac{1}{2}( -\frac{x}{4} )+\frac{1}{2}( -\frac{1}{2} )\frac{1}{2}{{( -\frac{x}{4} )}^{2}}+…. ]} $
$ =\frac{[ 1-\frac{19}{12}x+\frac{53}{144}x^{2}-…. ]}{[ 1-\frac{x}{2}-\frac{1}{8}x^{2}-…. ]}=1-\frac{35}{24}x+…. $
Neglecting higher powers of $ x $ , then $ a+bx=1-\frac{35}{24}x\Rightarrow a=1,b=-\frac{35}{24} $ .
BETA
