What is the coefficient of the term containing (a^2b) in ( (4a-b)^3 )?
Answer and explanation
Correct answer: - (48)
To find the coefficient of the term containing \\(a^2b\\), use the cube identity \\( (u-v)^3=u^3-3u^2v+3uv^2-v^3\\). Here, \\(u=4a\\) and \\(v=b\\). The required term is the second term, because it contains two factors of \\(u\\) and one factor of \\(v\\). Its value is \\(-3(4a)^2b\\). Since \\((4a)^2=16a^2\\), the term becomes \\(-48a^2b\\).
The coefficient is therefore \\(-48\\), so option A is correct. The negative sign is important because this is a cube of a difference, not a cube of a sum. Option B has the correct magnitude but the wrong sign. Option C results from an incorrect calculation, and option D is the coefficient of the first term \\( (4a)^3\\), not the requested term. Hence both the identity and direct expansion confirm A.
Frequently asked questions
What is the correct answer to this question?
- (48)
Why is this the correct answer?
To find the coefficient of the term containing \\(a^2b\\), use the cube identity \\( (u-v)^3=u^3-3u^2v+3uv^2-v^3\\). Here, \\(u=4a\\) and \\(v=b\\). The required term is the second term, because it contains two factors of \\(u\\) and one factor of \\(v\\). Its value is \\(-3(4a)^2b\\). Since \\((4a)^2=16a^2\\), the term becomes \\(-48a^2b\\).
The coefficient is therefore \\(-48\\), so option A is correct. The negative sign is important because this is a cube of a difference, not a cube of a sum. Option B has the correct magnitude but the wrong sign. Option C results from an incorrect calculation, and option D is the coefficient of the first term \\( (4a)^3\\), not the requested term. Hence both the identity and direct expansion confirm A.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Algebraic identities.
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