Question 6 - A-Level Maths

CAIE P1 2023 March — Question 6 6 marks

Exam Board	CAIE
Module	P1 (Pure Mathematics 1)
Year	2023
Session	March
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Binomial Theorem (positive integer n)
Type	Ratio of coefficients condition
Difficulty	Standard +0.3 This is a straightforward binomial expansion problem requiring students to find the general term, identify which terms give x^4 and x, set up a ratio equation, and solve for a. While it involves multiple steps, each step uses standard techniques (binomial coefficients, index laws, algebraic manipulation) with no novel insight required. It's slightly easier than average because the approach is mechanical once you identify the relevant terms.
Spec	1.04a Binomial expansion: (a+b)^n for positive integer n

6 In the expansion of $\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }$, it is given that $$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$ Find the possible values of the constant $a$.

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Question 6:

Answer	Marks	Guidance
$7C1\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)$ or $7C6\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)^2$ $7C2\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2$ or $7C5\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2$	B1 B1	Coefficients $x^4$ & $x$. Can be seen in an expansion
$\dfrac{\left(\dfrac{7}{a^5}\right)}{\left(\dfrac{21}{a^3}\right)} = 3$	M1	OE. Allow extraneous $x^4$ and $x$ at this stage; numerator and denominator must be functions of $a$. Allow errors in evaluation of combinations
A1	Completely correct
$a^2 = \frac{1}{9}$	A1	SOI (implied by $a = \frac{1}{3}$)
$a = \pm\frac{1}{3}$	A1	Allow $\pm 0.333$
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## Question 6:

| $7C1\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)$ or $7C6\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)^2$ $7C2\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2$ or $7C5\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2$ | B1 B1 | Coefficients $x^4$ & $x$. Can be seen in an expansion |
|---|---|---|
| $\dfrac{\left(\dfrac{7}{a^5}\right)}{\left(\dfrac{21}{a^3}\right)} = 3$ | M1 | OE. Allow extraneous $x^4$ and $x$ at this stage; numerator and denominator must be functions of $a$. Allow errors in evaluation of combinations |
| | A1 | Completely correct |
| $a^2 = \frac{1}{9}$ | A1 | SOI (implied by $a = \frac{1}{3}$) |
| $a = \pm\frac{1}{3}$ | A1 | Allow $\pm 0.333$ |
| | **6** | |

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6 In the expansion of $\left( \frac { x } { a } + \frac { a } { x ^ { 2 } } \right) ^ { 7 }$, it is given that

$$\frac { \text { the coefficient of } x ^ { 4 } } { \text { the coefficient of } x } = 3 \text {. }$$

Find the possible values of the constant $a$.\\

\hfill \mbox{\textit{CAIE P1 2023 Q6 [6]}}

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Q1 4 Q2 4 Q3 4 Q4 5 Q5 6 Q6 6 Q7 8 Q8 8 Q9 9 Q10 10 Q11 11

Answer	Marks	Guidance
\(7C1\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)\) or \(7C6\left(\frac{x}{a}\right)^6\left(\frac{a}{x^2}\right)^2\) \(7C2\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2\) or \(7C5\left(\frac{x}{a}\right)^5\left(\frac{a}{x^2}\right)^2\)	B1 B1	Coefficients \(x^4\) & \(x\). Can be seen in an expansion
\(\dfrac{\left(\dfrac{7}{a^5}\right)}{\left(\dfrac{21}{a^3}\right)} = 3\)	M1	OE. Allow extraneous \(x^4\) and \(x\) at this stage; numerator and denominator must be functions of \(a\). Allow errors in evaluation of combinations
A1	Completely correct
\(a^2 = \frac{1}{9}\)	A1	SOI (implied by \(a = \frac{1}{3}\))
\(a = \pm\frac{1}{3}\)	A1	Allow \(\pm 0.333\)
6