Edexcel PURE 2024 October — Question 1

Exam BoardEdexcel
ModulePURE
Year2024
SessionOctober
PaperDownload PDF ↗
TopicGeneralised Binomial Theorem
TypeFactoring out constants before expansion
DifficultyModerate -0.3 This is a straightforward application of the binomial expansion formula for negative/fractional powers, requiring substitution into the standard formula and simplification of fractions. Part (b) is a routine numerical approximation exercise. While it involves more algebraic manipulation than basic index laws, it's a standard textbook question with no problem-solving insight required, making it slightly easier than average.
Spec1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions
  1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of
$$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
(b) Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)
Question 1:
AnswerMarks
1(a)1
−
(8−3x)− 1 3 = 1 1− 3 x   3
AnswerMarks
2 8 B1
1
 1− 3 x   − 3 =1+(−1 )   − 3 x   + −1 3 (−1 3 −1 )   − 3 x   2 + −1 3 (−1 3 −1 )(−1 3 −2 )   − 3 x   3 +...
AnswerMarks
 8  3  8  2!  8  3!  8 M1
(8−3x)− 1 3 = 1 + 1 x+ 1 x2 + 7 x3+...
AnswerMarks
2 16 64 1536A1 A1
(4)
AnswerMarks
(b)2 3
1 1 2 1 2 7 2 2851 −1
 2851
+   +   +   +...=  3 6 =   =...
AnswerMarks
2 163 643 15363 5184 5184M1
5184 2 3 3 3
= or 1
AnswerMarks
2851 2 8 5 1A1
(2)

Total 6

1 1 1 7
A1: + x+ x2 + x3. Isw following a correct answer.
2 16 64 1536
1
Condone the position of xn provided it is not clearly on the denominator. e.g. condone
16 x
1 1
but not . Ignore terms with higher powers of x. Do not accept x1
16x 16
(b)
M1: Attempts to substitute x =
2
3
into their expansion from part (a) to achieve a fraction and
attempts to find the reciprocal of that fraction. May be implied by their fraction. You may need
to check this on your calculator. They may attempt to find an additional term which is fine. Do
not be concerned as to whether the fraction is correct for their binomial expansion if they have
shown the substitution. Condone slips including losing one of their terms.
"
1
2
+
1
1
6
 2
3

+
1
6 4
 2
3

2
+
1 5
7
3 6
 2
3

3
" =
A
B
B
A
scores M1
1 
2
5
8
1
5
8
1
4
scores M1
"
1
2
+
1
1
6
 2
3

+
1
6 4
1
 2
3

2
+
1 5
7
3 6
 2
3

3
"
on its own scores M0
A1:
5
2
1
8
8
5
4
1
or 1
2
2
3
8
3
5
3
1
PMT
. Isw once a correct answer is seen. Correct answer scores M1A1.
Question 1:
--- 1(a) ---
1(a) | 1
−
(8−3x)− 1 3 = 1 1− 3 x   3
2 8  | B1
1
 1− 3 x   − 3 =1+(−1 )   − 3 x   + −1 3 (−1 3 −1 )   − 3 x   2 + −1 3 (−1 3 −1 )(−1 3 −2 )   − 3 x   3 +...
 8  3  8  2!  8  3!  8  | M1
(8−3x)− 1 3 = 1 + 1 x+ 1 x2 + 7 x3+...
2 16 64 1536 | A1 A1
(4)
(b) | 2 3
1 1 2 1 2 7 2 2851 −1
 2851
+   +   +   +...=  3 6 =   =...
2 163 643 15363 5184 5184 | M1
5184 2 3 3 3
= or 1
2851 2 8 5 1 | A1
(2)
Total 6
1 1 1 7
A1: + x+ x2 + x3. Isw following a correct answer.
2 16 64 1536
1
Condone the position of xn provided it is not clearly on the denominator. e.g. condone
16 x
1 1
but not . Ignore terms with higher powers of x. Do not accept x1
16x 16
(b)
M1: Attempts to substitute x =
2
3
into their expansion from part (a) to achieve a fraction and
attempts to find the reciprocal of that fraction. May be implied by their fraction. You may need
to check this on your calculator. They may attempt to find an additional term which is fine. Do
not be concerned as to whether the fraction is correct for their binomial expansion if they have
shown the substitution. Condone slips including losing one of their terms.
"
1
2
+
1
1
6
 2
3

+
1
6 4
 2
3

2
+
1 5
7
3 6
 2
3

3
" =
A
B

B
A
scores M1
1 
2
5
8
1
5
8
1
4
scores M1
"
1
2
+
1
1
6
 2
3

+
1
6 4
1
 2
3

2
+
1 5
7
3 6
 2
3

3
"
on its own scores M0
A1:
5
2
1
8
8
5
4
1
or 1
2
2
3
8
3
5
3
1
PMT
. Isw once a correct answer is seen. Correct answer scores M1A1.
\begin{enumerate}
  \item (a) Find the first 4 terms of the binomial expansion, in ascending powers of $x$, of
\end{enumerate}

$$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$

giving each coefficient as a simplified fraction.\\
(b) Use the answer from part (a) with $x = \frac { 2 } { 3 }$ to find a rational approximation to $\sqrt [ 3 ] { 6 }$

\hfill \mbox{\textit{Edexcel PURE 2024 Q1}}
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