Standard +0.3 This is a straightforward binomial coefficient problem requiring students to set up two expressions using the binomial theorem, form an equation, and solve a quadratic. It's slightly above average difficulty due to the two-part setup and algebraic manipulation, but follows a standard pattern with no conceptual surprises.
The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\).
Find the possible values of the constant \(a\). [4]
Equating their p + their q to 276 leading to an equation in a2 only
M1
No x terms and no extra terms. If p and q are not identified
then it needs to be clear from the expansion that the
appropriate coefficients are being used.
69a2 276 implies the first 3 marks.
Answer
Marks
Guidance
a2
A1
CAO
4
Answer
Marks
Guidance
Question
Answer
Marks
Question 2:
2 | [Coefficient of x4 = p =]1 5a2 | B1 | May be seen in an expansion or with x4.
[Coefficient of x2 = q =] 54a2 | B1 | May be seen in an expansion or with x2.
Equating their p + their q to 276 leading to an equation in a2 only | M1 | No x terms and no extra terms. If p and q are not identified
then it needs to be clear from the expansion that the
appropriate coefficients are being used.
69a2 276 implies the first 3 marks.
a2 | A1 | CAO
4
Question | Answer | Marks | Guidance
The coefficient of $x^4$ in the expansion of $(x + a)^6$ is $p$ and the coefficient of $x^2$ in the expansion of $(ax + 3)^4$ is $q$. It is given that $p + q = 276$.
Find the possible values of the constant $a$. [4]
\hfill \mbox{\textit{CAIE P1 2023 Q2 [4]}}