Moderate -0.5 This is a straightforward binomial expansion problem requiring students to expand (1-2x)^5 using the binomial theorem, multiply by (1+kx), collect x^3 terms, and solve a linear equation for k. It's slightly easier than average as it involves routine application of a standard technique with no conceptual challenges, though it requires careful algebraic manipulation.
Can be seen in an expansion but must be simplified correctly
Coefficient of \(x^2\) in \((1-2x)^5\) is \(40\)
B1
Coefficient of \(x^3\) in \((1+kx)(1-2x)^5\) is \(40k - 80 = 20\)
M1
Uses the relevant two terms to form an equation \(= 20\) and solves to find \(k\). Condone \(x^3\) appearing in some terms if recovered
\(k = \dfrac{5}{2}\)
A1
Total: 4 marks
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Coefficient of $x^3$ in $(1-2x)^5$ is $-80$ | B1 | Can be seen in an expansion but must be simplified correctly |
| Coefficient of $x^2$ in $(1-2x)^5$ is $40$ | B1 | |
| Coefficient of $x^3$ in $(1+kx)(1-2x)^5$ is $40k - 80 = 20$ | M1 | Uses the relevant two terms to form an equation $= 20$ and solves to find $k$. Condone $x^3$ appearing in some terms if recovered |
| $k = \dfrac{5}{2}$ | A1 | |
| **Total: 4 marks** | | |
1 The coefficient of $x ^ { 3 }$ in the expansion of $( 1 + k x ) ( 1 - 2 x ) ^ { 5 }$ is 20 .\\
Find the value of the constant $k$.\\
\hfill \mbox{\textit{CAIE P1 2020 Q1 [4]}}