( a a + x ) 1 2 + ( a a − x ) 1 2 = Options: A . &nbsp 2 + 3 x 2 4 a 2 + ....... B . &nbsp 1 + 3 x 2 8 a 2 + ..... C . &nbsp 2 + x a + 3 x 2 4 a 2 + ...... D . &nbsp 2 - x a + 3 x 2 4 a 2 + .......

Answer: Option A : A ( a + x a ) − 1 2 + ( a − x a ) − 1 2 = ( 1 + x a ) − 1 2 + ( 1 − x a ) − 1 2 =[1 + (- 1 2 )( x a ) + ( − 1 2 ) ( − 3 2 ) 2.1 ( x a ) 2 + ........] + [ 1 + ( − 1 2 ) ( − x a ) + ( − 1 2 ) ( 3 2 ) 2.1 ( − x a ) 2 + . . . . . . . . ] = 2 + 3 x 2 4 a 2 + ...... Here odd terms cancel each other.

(aa+x)12 + (aa−x)12 =

Discussion Forum : Binomial Theorem

Question -

$(\frac{a}{a + x})^{\frac{1}{2}}$ + $(\frac{a}{a - x})^{\frac{1}{2}}$ =

Options:

A . 2 +

\frac{3 x^{2}}{4 a^{2}}

+ .......

B . 1 +

\frac{3 x^{2}}{8 a^{2}}

+ .....

C . 2 +

\frac{x}{a}

\frac{3 x^{2}}{4 a^{2}}

+ ......

D . 2 -

\frac{x}{a}

\frac{3 x^{2}}{4 a^{2}}

+ .......

Answer: Option A
:
A

$(\frac{a + x}{a})^{\frac{- 1}{2}}$ + $(\frac{a - x}{a})^{\frac{- 1}{2}}$ = $(1 + \frac{x}{a})^{\frac{- 1}{2}} + (1 - \frac{x}{a})^{\frac{- 1}{2}}$

=[1 + (- $\frac{1}{2}$ )( $\frac{x}{a}$ ) + $\frac{(- \frac{1}{2}) (- \frac{3}{2})}{2.1}$ $(\frac{x}{a})^{2}$ + ........]

+ $[1 + (- \frac{1}{2}) (- \frac{x}{a}) + \frac{(- \frac{1}{2}) (\frac{3}{2})}{2.1} (- \frac{x}{a})^{2} + . . . . . . . .]$

= 2 + $\frac{3 x^{2}}{4 a^{2}}$ + ......

Here odd terms cancel each other.

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Discussion Forum : Binomial Theorem

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