Moderate -0.3 This is a straightforward binomial expansion question requiring application of the formula with a simple algebraic term, followed by equating two coefficients to solve for a constant. While it involves multiple steps and algebraic manipulation, it's a standard textbook exercise with no novel insight required—slightly easier than average due to the routine nature of the techniques involved.
5. (a) Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$
where \(a\) is a positive constant. Give each term in its simplest form.
Given that, in the expansion, the coefficient of \(x\) is equal to the coefficient of \(x ^ { 3 }\),
(b) find the exact value of \(a\) in its simplest form.
B1: first term 243; A1: two correct simplified terms from \(-\frac{405}{2}ax\), \(+\frac{135}{2}a^2x^2\), \(-\frac{45}{4}a^3x^3\); A1: all terms correct and simplified
Part (b):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\(\frac{405}{2}a = \frac{45}{4}a^3\)
M1
Set coefficient of \(x\) equal to coefficient of \(x^3\)
\(a^2 = \frac{810}{45} = 18\) or equivalent
A1
Correctly obtain \(a^2\) or \(a\) (may be unsimplified)
\(a = 3\sqrt{2}\)
A1
Condone \(a = \pm3\sqrt{2}\)
## Question 5:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(3-\frac{ax}{2}\right)^5 = 3^5 + \binom{5}{1}3^4\left(-\frac{ax}{2}\right) + \binom{5}{2}3^3\left(-\frac{ax}{2}\right)^2 + \binom{5}{3}3^2\left(-\frac{ax}{2}\right)^3...$ | M1 | Attempt at Binomial to get second and/or third and/or fourth term; correct binomial coefficient combined with correct power of $x$ |
| $= 243, -\frac{405}{2}ax + \frac{135}{2}a^2x^2 - \frac{45}{4}a^3x^3...$ | B1, A1, A1 | B1: first term 243; A1: two correct simplified terms from $-\frac{405}{2}ax$, $+\frac{135}{2}a^2x^2$, $-\frac{45}{4}a^3x^3$; A1: all terms correct and simplified |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{405}{2}a = \frac{45}{4}a^3$ | M1 | Set coefficient of $x$ equal to coefficient of $x^3$ |
| $a^2 = \frac{810}{45} = 18$ or equivalent | A1 | Correctly obtain $a^2$ or $a$ (may be unsimplified) |
| $a = 3\sqrt{2}$ | A1 | Condone $a = \pm3\sqrt{2}$ |
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5. (a) Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of
$$\left( 3 - \frac { a x } { 2 } \right) ^ { 5 }$$
where $a$ is a positive constant. Give each term in its simplest form.
Given that, in the expansion, the coefficient of $x$ is equal to the coefficient of $x ^ { 3 }$,\\
(b) find the exact value of $a$ in its simplest form.\\
\hfill \mbox{\textit{Edexcel C12 2016 Q5 [7]}}