Question

If $$x$$ is very small in magnitude compared with $$a,$$ then $${\left( {\frac{a}{{a + x}}} \right)^{\frac{1}{2}}} + {\left( {\frac{a}{{a - x}}} \right)^{\frac{1}{2}}}\,$$     can be approximately equal to

A. $$1 + \frac{1}{2}\frac{x}{a}$$
B. $$\frac{x}{a}$$
C. $$1 + \frac{3}{4}\frac{{{x^2}}}{{{a^2}}}$$
D. $$2 + \frac{3}{4}\frac{{{x^2}}}{{{a^2}}}$$  
Answer :   $$2 + \frac{3}{4}\frac{{{x^2}}}{{{a^2}}}$$
Solution :
$$\eqalign{ & {\left( {\frac{a}{{a + x}}} \right)^{\frac{1}{2}}} + {\left( {\frac{a}{{a - x}}} \right)^{\frac{1}{2}}} \cr & = {\left( {\frac{{a + x}}{a}} \right)^{ - \frac{1}{2}}} + {\left( {\frac{{a - x}}{a}} \right)^{ - \frac{1}{2}}} \cr & = {\left( {1 + \frac{x}{a}} \right)^{ - \frac{1}{2}}} + {\left( {1 - \frac{x}{a}} \right)^{ - \frac{1}{2}}} \cr & = \left[ {1 - \frac{1}{2}\frac{x}{a} + \frac{3}{8}\frac{{{x^2}}}{{{a^2}}}} \right] + \left[ {1 + \frac{1}{2}\frac{x}{a} + \frac{3}{8}\frac{{{x^2}}}{{{a^2}}}} \right]\left[ {\because x \ll a,\therefore \frac{x}{a} \ll 1} \right] = 2 + \frac{3}{4} \cdot \frac{{{x^2}}}{{{a^2}}} \cr} $$

Releted MCQ Question on
Algebra >> Binomial Theorem

Releted Question 1

Given positive integers $$r > 1, n > 2$$   and that the co - efficient of $${\left( {3r} \right)^{th}}\,{\text{and }}{\left( {r + 2} \right)^{th}}$$    terms in the binomial expansion of $${\left( {1 + x} \right)^{2n}}$$  are equal. Then

A. $$n = 2r$$
B. $$n = 2r + 1$$
C. $$n = 3r$$
D. none of these
Releted Question 2

The co-efficient of $${x^4}$$ in $${\left( {\frac{x}{2} - \frac{3}{{{x^2}}}} \right)^{10}}$$   is

A. $$\frac{{405}}{{256}}$$
B. $$\frac{{504}}{{259}}$$
C. $$\frac{{450}}{{263}}$$
D. none of these
Releted Question 3

The expression $${\left( {x + {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5} + {\left( {x - {{\left( {{x^3} - 1} \right)}^{\frac{1}{2}}}} \right)^5}$$       is a polynomial of degree

A. 5
B. 6
C. 7
D. 8
Releted Question 4

If in the expansion of $${\left( {1 + x} \right)^m}{\left( {1 - x} \right)^n},$$    the co-efficients of $$x$$ and $${x^2}$$ are $$3$$ and $$- 6\,$$ respectively, then $$m$$ is

A. 6
B. 9
C. 12
D. 24

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