Eric Nicholas K's answer to LockB's Secondary 3 A Maths Singapore question.
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For this question, I didn't do the usual approach of "coefficient of 1/x is zero since the term is missing in the final expression".
This is because the question did not state "the first few terms in the expansion of...", but only stated "the expansion of...".
The problem is that if we were to rearrange our expression in descending powers of x from the start, the first term would be -ak^5 x. And because the starting expression at the end is given to be -3/16 (no term in x provided), then -ak^5 would be taken to be zero, but we are given that a and k are non-zero numbers, so the two data pieces are in conflict with each other (one says that -ak^5 must be zero while the other says that neither a nor k is 0).
So, we cannot say that the term -ak^5 x and the term in 1/x is zero. They could be well hidden within the expansion because of this, so we cannot make the assumption that the coefficient of 1/x would be zero.
It turns out to be the case though, but in any case I find this question very badly set. The setter did not put this clearly enough.
This is because the question did not state "the first few terms in the expansion of...", but only stated "the expansion of...".
The problem is that if we were to rearrange our expression in descending powers of x from the start, the first term would be -ak^5 x. And because the starting expression at the end is given to be -3/16 (no term in x provided), then -ak^5 would be taken to be zero, but we are given that a and k are non-zero numbers, so the two data pieces are in conflict with each other (one says that -ak^5 must be zero while the other says that neither a nor k is 0).
So, we cannot say that the term -ak^5 x and the term in 1/x is zero. They could be well hidden within the expansion because of this, so we cannot make the assumption that the coefficient of 1/x would be zero.
It turns out to be the case though, but in any case I find this question very badly set. The setter did not put this clearly enough.
Date Posted:
3 years ago
i still dont really understand the part starting from -ak^5x .....
i am very bad at binomial theorem as i find it extremely confusing and most of the time i dont really know what is going on in this topic...
i am very bad at binomial theorem as i find it extremely confusing and most of the time i dont really know what is going on in this topic...
The main objective of this topic is to expand large expressions without the need to do expansions manually.
In my expansions in the first few lines (before the part "since the constant term is -3/16..."), I expanded the expression for (4 - ax) (k + 1/x)^5 to the desired power of 1/x^2 and grouped the expressions according to the relevant power of x (so I will not have two separate terms in a given power of x later on).
The part "since the constant term is -3/16..." requires us to make comparisons with the relevant powers.
In reality, the question should have mentioned "given that the first two terms in the expansion of". With this, we can deduce that the reason why they skipped the term in 1/x at the end is because the term has zero coefficient.
This part onwards leading to the value of a and of k is basically equating and solving for a and k.
In my expansions in the first few lines (before the part "since the constant term is -3/16..."), I expanded the expression for (4 - ax) (k + 1/x)^5 to the desired power of 1/x^2 and grouped the expressions according to the relevant power of x (so I will not have two separate terms in a given power of x later on).
The part "since the constant term is -3/16..." requires us to make comparisons with the relevant powers.
In reality, the question should have mentioned "given that the first two terms in the expansion of". With this, we can deduce that the reason why they skipped the term in 1/x at the end is because the term has zero coefficient.
This part onwards leading to the value of a and of k is basically equating and solving for a and k.
i still dont really understand the by inspection... part tho
That portion is basically your cubic equation solving. You guess a factor out and then do long division thereafter.
root 1/2 and factor 2k-1 is by guessing or... as i dont really understand where it came from
sorry if i ask too much questions out of a sudden because i have add maths exam on monday...
sorry if i ask too much questions out of a sudden because i have add maths exam on monday...
That one is guessing. You can use your calculator to do the guessing.
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