Question

Match the following.

Column I		Column II
(i)	${\left(1-2x\right)}^5=$	(p)	$\frac{x^5}{243}-\frac{5x^3}{81}+\frac{10x}{27}-\frac{10}{9x}$ $+\frac{5}{3x^3}-\frac{1}{x^5}$
(ii)	${\left(2x-3\right)}^6=$	(q)	$x^6+6x^4+15x^2+20$ $+\frac{15}{x^2}+\frac{6}{x^4}+\frac{1}{x^6}$
(iii)	${\left(\frac{x}{3}-\frac{1}{x}\right)}^5=$	(r)	$1-10x+40x^2-80x^3$ $+80x^4-32x^5$
(iv)	${\left(x-\frac{1}{x}\right)}^6=$	(s)	$64x^6-576x^5+2160x^4-4320x^3$ $+4860x^2-2916x+729$

• A. $(i)\to (p)$, $(ii)\to (q)$, $(iii)\to (r)$, $(iv)\to (s)$
• B. $(i)\to (r)$, $(ii)\to (s)$, $(iii)\to (p)$, $(iv)\to (q)$
• C. $(i)\to (q)$, $(ii)\to (s)$, $(iii)\to (p)$, $(iv)\to (r)$
• D. $(i)\to (r)$, $(ii)\to (s)$, $(iii)\to (q)$, $(iv)\to (p)$

Answer 1

Answer: (B)

(i) ${\left(1-2x\right)}^5={}^5{C}_0{\left(1\right)}^5-{}^5{C}_1{\left(1\right)}^{4}2x+{}^5{C}_2{\left(1\right)}^3{\left(2x\right)}^2$
$-{}^5{C}_3{\left(1\right)}^2{\left(2x\right)}^3+{}^5{C}_4\left(1\right){\left(2x\right)}^4-{}^5{C}_5{\left(2x\right)}^5$
$=1-5\times 2x+\frac{5\times 4}{2}\times 4x^2-\frac{5\times 4}{2}\times 8x^3$
$+5\times 16x^4-32x^5$
$=1-10x+40x^2-80x^3+80x^4-32x^5$
(ii) ${\left(2x-3\right)}^6={}^6{C}_0{\left(2x\right)}^6-{}^6{C}_1{\left(2x\right)}^5\times 3+{}^6{C}_2{\left(2x\right)}^4{\left(3\right)}^2$
$-{}^6{C}_3{\left(2x\right)}^3{\left(3\right)}^3+{}^6{C}_4{\left(2x\right)}^2{\left(3\right)}^4-{}^6{C}_5\left(2x\right)3^5+{}^6{C}_6{3}^6$
$=64x^6-6\times 32\times 3\times x^5+\frac{6\times 5}{2}\times 16x^4\times 9$
$-\frac{6\times 5\times 4\times 8x^3\times 27}{6}+\frac{6\times 5}{2}\times 4x^2\times 81$
$-6\times 2x\times 243+729$
$=64x^6-576x^5+2160x^4-4320x^3+4860x^2-2916x+729$
(iii) ${\left(\frac{x}{3}-\frac{1}{x}\right)}^5={}^5{C}_0{\left(\frac{x}{3}\right)}^5-{}^5{C}_1{\left(\frac{x}{3}\right)}^4+{}^5{C}_2{\left(\frac{x}{3}\right)}^3{\left(\frac{1}{x}\right)}^2$
$-{}^5{C}_3{\left(\frac{x}{3}\right)}^2{\left(\frac{1}{x}\right)}^3+{}^5{C}_4\left(\frac{x}{3}\right){\left(\frac{1}{x}\right)}^4-{}^5{C}_5{\left(\frac{x}{3}\right)}^5$
$=\frac{x^5}{243}-\frac{5x^3}{81}+\frac{10x}{27}-\frac{10}{9x}+\frac{5}{3x^3}-\frac{1}{x^5}$
(iv) ${\left(x+\frac{1}{x}\right)}^6={}^6{C}_0{x}^6+{}^6{C}_1{x}^5\left(\frac{1}{x}\right)+{}^6{C}_2{x}^4{\left(\frac{1}{x}\right)}^2$
$+{}^6{C}_3{x}^3{\left(\frac{1}{x}\right)}^3+{}^6{C}_4{x}^2{\left(\frac{1}{x}\right)}^4+{}^6{C}_5\left(x\right){\left(\frac{1}{x}\right)}^5+{}^6{C}_6{\left(\frac{1}{x}\right)}^6$
$=x^6+6x^4+15x^2+20+\frac{15}{x^2}+\frac{6}{x^4}+\frac{1}{x^6}$