When expanded, which expression has a non-zero constant term?
A - HSC - SSCE Mathematics Extension 1 - Question 9 - 2017 - Paper 1
Question 9
When expanded, which expression has a non-zero constant term?
A. $(x + \frac{1}{x^2})^7$
B. $(x^2 + \frac{1}{x^3})^7$
C. $(x^3 + \frac{1}{x^4})^7$
D. $(x^4 + \fr... show full transcript
Worked Solution & Example Answer:When expanded, which expression has a non-zero constant term?
A - HSC - SSCE Mathematics Extension 1 - Question 9 - 2017 - Paper 1
Step 1
A. $(x + \frac{1}{x^2})^7$
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Answer
The constant term can be found by applying the binomial theorem: the constant term occurs when the power of x is zero. We look for terms where the exponent from x cancels with the exponent from x21:
Choose k from x and (7−k) from x21:
xk(x21)(7−k)=xk−2(7−k)=x3k−14
Setting 3k−14=0 gives k=314. No integer solution means no constant term.
Step 2
B. $(x^2 + \frac{1}{x^3})^7$
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Answer
Using the same method:
x2k(x31)(7−k)=x2k−3(7−k)=x5k−21
Setting 5k−21=0 gives k=521. No integer solution means no constant term.
Step 3
C. $(x^3 + \frac{1}{x^4})^7$
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Answer
Applying the binomial theorem again:
x3k(x41)(7−k)=x3k−4(7−k)=x7k−28
Setting 7k−28=0 gives k=4. A valid integer solution indicates there is a constant term.
Step 4
D. $(x^4 + \frac{1}{x^5})^7$
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Answer
Again:
x4k(x51)(7−k)=x4k−5(7−k)=x9k−35
Setting 9k−35=0 gives k=935. No integer solution means no constant term.
