15. Find the expansion of (3x²– 2ax + 3a²)³ using binomial theorem.

Kindly consider the following figure

  $T_{r+1}{=}^{60}{C}_r{\left(x^{\frac{1}{2}}\right)}^{60-r}{\left(x^{-\frac{1}{3}}\right)}^r{\left(5^{\frac{-1}{4}}\right)}^{60-r}{\left(5^{\frac{1}{2}}\right)}^r$

for

$x^{10}\frac{60-r}{2}-\frac{r}{3}=10$    

$\Rightarrow 180-3r-2r=60$             

->r = 24

k = 3 + exponent of 5 in 

= $3+\left(\left[\frac{60}{5}\right]+\left[\frac{60}{5^2}\right]-\left[\frac{24}{5}\right]-\left[\frac{24}{5^2}\right]-\left[\frac{36}{5}\right]-\left[\frac{36}{5^2}\right]\right)$  

= 3 + (12 + 2 – 4 – 0 – 7 – 1)

= 3 + 2 = 5

14. For (a – b) to be a factor of aⁿ – b ⁿwe need to show (aⁿ – bⁿ) = (a – b)k as k is a natural number.

We have, for positive n

aⁿ = ${(a-b+b)}^n$ = ${\left[\left(a-b\right)+b\right]}^n$

=>aⁿ = ⁿC₀(a – b)ⁿ + ⁿC₁(a – b)^{n -1}b + ⁿC₂(a – b)^{n – 2}b² + ………… +ⁿC_n-1 $\left(a-b\right) b^{n-1}$ + ⁿC_nbⁿ

=>aⁿ= ${\left(a-b\right)}^n$ + ⁿC₁ ${\left(a-b\right)}^{n-1} b$ + ⁿC₂ ${\left(a-b\right)}^{n-2} b^2$ + …………….…+ ⁿC_n-1 $\left(a-b\right) b^{n-1}$ + $b^n$ [Since, ⁿC₀ = 1 and

14. For (a – b) to be a factor of aⁿ – b ⁿwe need to show (aⁿ – bⁿ) = (a – b)k as k is a natural number.

We have, for positive n

aⁿ = ${(a-b+b)}^n$ = ${\left[\left(a-b\right)+b\right]}^n$

=>aⁿ = ⁿC₀(a – b)ⁿ + ⁿC₁(a – b)^{n -1}b + ⁿC₂(a – b)^{n – 2}b² + ………… +ⁿC_n-1 $\left(a-b\right) b^{n-1}$ + ⁿC_nbⁿ

=>aⁿ= ${\left(a-b\right)}^n$ + ⁿC₁ ${\left(a-b\right)}^{n-1} b$ + ⁿC₂ ${\left(a-b\right)}^{n-2} b^2$ + …………….…+ ⁿC_n-1 $\left(a-b\right) b^{n-1}$ + $b^n$ [Since, ⁿC₀ = 1 and

13. The general term of the expansion ${\left(3+ax\right)}^9$ is

T_r+1 = ⁹C_r $3^{\left(9-r\right)}$ ${\left(ax\right)}^r$

= ⁹C_r $3^{\left(9-r\right)}{a}^r{x}^r$

At r = 2,

T₂₊₁ = ⁹C₂ $3^{\left(9-2\right)}{a}^2{x}^2$

= $\frac{9!}{2!\left(9-2\right)!}$ 3⁷a²x²

= $\mathrm{9\prime }$$\mathrm{8\prime }$$7!$/$\mathrm{2\prime }$$\mathrm{1\prime }$$7!\mathrm{}$ 3⁷a²x²

= 36 ×3⁷a²x²

At r = 3,

T₃₊₁ = ⁹C₃ $3^{\left(9-3\right)}{a}^3{x}^3$

= $\frac{9!}{3!\left(9-3\right)!}$ 3⁶a³x³

= 9'8'7'6!/3'2'1'6! 3⁶a³x³

= 84 ×3⁶a³x³

Given that,

Co-efficient of $x^2$ = co-efficient of $x^3$

=> 36 × $3^7{a}^2$ = 84 ×