Standard +0.3 This is a straightforward application of the binomial expansion requiring students to expand (1+4x)^(1/2), multiply by (a+bx), collect coefficients of x and x², then solve two simultaneous linear equations. While it involves multiple steps, each step is routine and the problem-solving approach is standard for this topic type, making it slightly easier than average.
6 When \(( a + b x ) \sqrt { 1 + 4 x }\), where \(a\) and \(b\) are constants, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x ^ { 2 }\) are 3 and - 6 respectively.
Find the values of \(a\) and \(b\).
State or imply \(1 + 2x\) as first terms of the expansion of \(\sqrt{1+4x}\)
B1
Allow for correct unsimplified expression
State or imply \(-2x^2\) as third term of the expansion of \(\sqrt{1+4x}\)
B1
Allow for correct unsimplified expression
Form an expression for the coefficient of \(x\) or coefficient of \(x^2\) in the expansion of \((a+bx)\sqrt{1+4x}\) and equate to given coefficient
M1
All relevant terms considered
Obtain \(2a + b = 3\), or equivalent
A1
One correct equation
Obtain \(-2a + 2b = -6\) or equivalent
A1
Second correct equation
Obtain answer \(a = 2\) and \(b = -1\)
A1
## Question 6:
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $1 + 2x$ as first terms of the expansion of $\sqrt{1+4x}$ | B1 | Allow for correct unsimplified expression |
| State or imply $-2x^2$ as third term of the expansion of $\sqrt{1+4x}$ | B1 | Allow for correct unsimplified expression |
| Form an expression for the coefficient of $x$ or coefficient of $x^2$ in the expansion of $(a+bx)\sqrt{1+4x}$ and equate to given coefficient | M1 | All relevant terms considered |
| Obtain $2a + b = 3$, or equivalent | A1 | One correct equation |
| Obtain $-2a + 2b = -6$ or equivalent | A1 | Second correct equation |
| Obtain answer $a = 2$ and $b = -1$ | A1 | |
6 When $( a + b x ) \sqrt { 1 + 4 x }$, where $a$ and $b$ are constants, is expanded in ascending powers of $x$, the coefficients of $x$ and $x ^ { 2 }$ are 3 and - 6 respectively.
Find the values of $a$ and $b$.\\
\hfill \mbox{\textit{CAIE P3 2021 Q6 [6]}}