Question

8.
(i) Find the first three terms, in descending powers of x, in the expansion (4x- k/x^2)^6.
(ii) Given that the value of the term in the expansion which is independent of x is 240, find possible values of k. I got the answer to the first part fine but I dont understand the second part. One of the terms in the first part doesnt have x but isnt 240 so I dont know what to do with this question. I have the answers if needed and I also can post a picture of my working.

Answer 1

in the expansion the independent term = 3rd term
\[= 15\times \left( 4x \right) ^{4}\times \left( \frac{ k }{ x^{2}} \right) ^{2}\]

Answer 2

Ok, can you explain the how and why of that? I not really following (sorry!)

Answer 3

independent term means it doesn't has x ,

Answer 4

Yes

Answer 5

in third term ,
\[15\times4 ^{4}timesx ^{4}timesk ^{2}divx ^{4}\]

Answer 6

OK and thats where you lose me

Answer 7

\[x^{4}\] cancelled,that means independent term

Answer 8

\[15\times4 ^{4}timesk ^{2} = 240 (given value the problem)\]

Answer 9

\[k^{2} =\frac{ 240 }{ 15\times256 }\]

Answer 10

\[k^{2}=\frac{ 1 }{ 16 }\]

Answer 11

\[k^{2} =(1/4)^{2}\]

Answer 12

k= + or - 1/4

Answer 13

clear

Answer 14

Ok, starting to get it.. but still not really understand why you are using those values or where they came fom, like the 15 and 4^4

Answer 15

15 is coefficient of third term, by pascal triangle

Answer 16

are u clear

Answer 17

Hmmm getting there. They say that the coefficient is 240 but the only term independent of x that I got had a coefficient of 3840. Whats that about?

Answer 18

can u send your solution ,i will try to explain where u gone wrong

Answer 19

The thing is I didnt even get a solution, I had no clue where to start! I figure out all the terms trying to find one with 240 and didnt get there, I can send that?

Answer 20

the expansion is like this
\[\left( 4x \right)^{6}-6\times \left( 4x \right) ^{5}\times \frac{ k }{x ^{2} } + 15\times \left( 4x \right) ^{4}\times \left( \frac{ k }{x ^{2} } \right)^{2}-\]

Answer 21

i wrote first three terms in the expansion

Answer 22

\[3840timesk ^{2}= 15\times4 ^{4}timesk ^{2}\]

Answer 23

which u did is correct

Answer 24

\[3840timesk ^{2}= 240\]
independent term is equal to 240 given in the problem

Answer 25

\[k^{2}= \frac{ 240 }{ 3840 }\]

Answer 26

\[k^{2}= \frac{ 1 }{ 16 }\]
after cancellation

Answer 27

are u following

Answer 28

Oh! I get it now! Thank you so much for your help and patience!

Answer 29

Have a great day!