Question 6 - A-Level Maths

Edexcel C12 2015 June — Question 6 6 marks

Exam Board	Edexcel
Module	C12 (Core Mathematics 1 & 2)
Year	2015
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Binomial Theorem (positive integer n)
Type	Ratio of coefficients condition
Difficulty	Moderate -0.8 This is a straightforward binomial expansion question requiring recall of the binomial theorem formula and basic algebraic manipulation. Part (a) is routine application of (a+b)^n expansion, and part (b) involves equating two coefficients to solve for a constant—a standard textbook exercise with no problem-solving insight required.
Spec	1.04a Binomial expansion: (a+b)^n for positive integer n

6. (a) Find the first 3 terms in ascending powers of $x$ of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where $a$ is a non-zero constant. Give each term in its simplest form. Given that, in the expansion, the coefficient of $x$ is equal to the coefficient of $x ^ { 2 }$ (b) find the value of $a$.

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Question 6:

Part (a):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$(2+ax)^6 = 2^6 + \binom{6}{1}2^5(ax) + \binom{6}{2}2^4(ax)^2 + \ldots$	M1	Attempt at binomial expansion; correct binomial coefficient × correct power of $x$; condone bracket errors or omissions in powers of 2
$= 64 + 192ax + 240a^2x^2 + \ldots$	B1, A1, A1	B1: simplified to 64; A1: $192ax$; A1: both $192ax$ and $240a^2x^2$ correct

Part (b):

Answer	Marks	Guidance
Answer/Working	Mark	Guidance
$192a = 240a^2$	M1	Setting coefficients of $x$ and $x^2$ terms equal; must reach equation not involving $x$
$a = \dfrac{192}{240} = 0.8$ or equivalent	A1	cso; any equivalent fraction or decimal to $0.8$; ignore any reference to $a=0$

## Question 6:

### Part (a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $(2+ax)^6 = 2^6 + \binom{6}{1}2^5(ax) + \binom{6}{2}2^4(ax)^2 + \ldots$ | M1 | Attempt at binomial expansion; correct binomial coefficient × correct power of $x$; condone bracket errors or omissions in powers of 2 |
| $= 64 + 192ax + 240a^2x^2 + \ldots$ | B1, A1, A1 | B1: simplified to 64; A1: $192ax$; A1: both $192ax$ and $240a^2x^2$ correct |

### Part (b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $192a = 240a^2$ | M1 | Setting coefficients of $x$ and $x^2$ terms equal; must reach equation not involving $x$ |
| $a = \dfrac{192}{240} = 0.8$ or equivalent | A1 | cso; any equivalent fraction or decimal to $0.8$; ignore any reference to $a=0$ |

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6. (a) Find the first 3 terms in ascending powers of $x$ of the binomial expansion of

$$( 2 + a x ) ^ { 6 }$$

where $a$ is a non-zero constant. Give each term in its simplest form.

Given that, in the expansion, the coefficient of $x$ is equal to the coefficient of $x ^ { 2 }$\\
(b) find the value of $a$.\\

\hfill \mbox{\textit{Edexcel C12 2015 Q6 [6]}}

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