Standard +0.3 This is a standard two-part binomial expansion question requiring students to first expand (2-3x)^{-2}, then use coefficient conditions to find two unknowns. While it involves multiple steps and algebraic manipulation, the techniques are routine for C3/C4 students: apply the binomial theorem formula, multiply by a linear numerator, and solve simultaneous equations from coefficients. The problem-solving is straightforward with no novel insights required.
4. (a) Use the binomial theorem to expand
$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$
in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
$$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$
In the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), the coefficient of \(x\) is 0 and the coefficient of \(x ^ { 2 }\) is \(\frac { 9 } { 16 }\)
Find
(b) the value of \(a\) and the value of \(b\),
(c) the coefficient of \(x ^ { 3 }\), giving your answer as a simplified fraction.
4. (a) Use the binomial theorem to expand
$$( 2 - 3 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 3 }$$
in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient as a simplified fraction.
$$\mathrm { f } ( x ) = \frac { a + b x } { ( 2 - 3 x ) ^ { 2 } } , \quad | x | < \frac { 2 } { 3 } , \quad \text { where } a \text { and } b \text { are constants. }$$
In the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, the coefficient of $x$ is 0 and the coefficient of $x ^ { 2 }$ is $\frac { 9 } { 16 }$
Find\\
(b) the value of $a$ and the value of $b$,\\
(c) the coefficient of $x ^ { 3 }$, giving your answer as a simplified fraction.\\
\hfill \mbox{\textit{Edexcel C34 Q4 [13]}}