The term independent of x in the expansion of 1 60 - x 8 81 . 2 x 2 - 3 x 2 6 is equal to

Question

Aicharya · Accepted Answer

To find the coefficient of term independent of

x

in given expression, first let's find constant term and coefficient of

x^{- 8}

in the expansion of

{(2 x^{2} - \frac{3}{x^{2}})}^{6}

T_{r + 1} =^{6} C_{r} {(2 x^{2})}^{6 - r} \times {(- \frac{3}{x^{2}})}^{r}

T_{r + 1} =^{6} C_{r} 2^{6 - r} \times {(- 3)}^{r} \times x^{12 - 4 r}

For constant term:

r = 3

, hence constant term =

^{6} C_{3} 2^{3} \times {(- 3)}^{3} \times x^{0}

For coefficient of

x^{- 8} : r = 5

, hence coefficient of

x^{- 8} =^{6} C_{5} \times 2^{1} {(- 3)}^{5} \times x^{- 8}

Given expression can be written as

(\frac{1}{60} - \frac{x^{8}}{81}) (^{6} C_{3} \times 2^{3} \times {(- 3)}^{3} \times x^{0} + \dots +^{6} C_{5} \times 2^{1} \times {(- 3)}^{5} \times x^{- 8} + \dots)

Hence, coefficient of term independent of

x

= \frac{1}{60} \times^{6} C_{3} \times 2^{3} {(- 3)}^{3} - \frac{1}{81} \times^{6} C_{5} \times 2^{1} \times {(- 3)}^{5}

= - 72 + 36 = - 36

Solution