Moderate -0.3 This is a straightforward binomial expansion problem requiring students to expand (1+x/2)^6 using the binomial theorem, multiply by (4+ax), collect x^2 terms, and solve a linear equation for a. It's slightly easier than average because it's a standard textbook exercise with clear steps and no conceptual surprises, though it does require careful algebraic manipulation.
1 The coefficient of \(x ^ { 2 }\) in the expansion of \(( 4 + a x ) \left( 1 + \frac { x } { 2 } \right) ^ { 6 }\) is 3 . Find the value of the constant \(a\). [4]
OE. In or from a correct expansion. Can be implied by correct equation.
\(\times(4+ax) \rightarrow 3a + 15 = 3\)
M1
2 terms in \(x^2\) equated to 3 or \(3x^2\). Condone \(x^2\) on one side only.
\(a = -4\)
A1
CAO
## Question 1:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{6x}{2}$, $15 \times \frac{x^2}{4}$ | B1 B1 | OE. In or from a correct expansion. Can be implied by correct equation. |
| $\times(4+ax) \rightarrow 3a + 15 = 3$ | M1 | 2 terms in $x^2$ equated to 3 or $3x^2$. Condone $x^2$ on one side only. |
| $a = -4$ | A1 | CAO |
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1 The coefficient of $x ^ { 2 }$ in the expansion of $( 4 + a x ) \left( 1 + \frac { x } { 2 } \right) ^ { 6 }$ is 3 . Find the value of the constant $a$. [4]\\
\hfill \mbox{\textit{CAIE P1 2019 Q1 [4]}}