Notes

M1: For an attempt at the binomial expansion with \(n = \dfrac{1}{2}\) and obtains the correct structure for term 3 or term 4. Award for the correct coefficient with the correct power of \(x\).

e.g. \(\dfrac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)}{2!}(\lambda x)^2\) or \(\dfrac{\left(\frac{1}{2}\right)\left(-\frac{1}{2}\right)\left(-\frac{3}{2}\right)}{3!}(\lambda x)^3\) where \(\lambda \neq 1\)

Condone missing or incorrect brackets around the \(x\) terms but the binomial coefficients must be correct. Allow 2! and/or 3! or 2 and/or 6.

Do not allow notation such as \(\begin{pmatrix}\frac{1}{2}\\1\end{pmatrix}, \begin{pmatrix}\frac{1}{2}\\2\end{pmatrix}\) unless these are interpreted correctly.

A1: Correct unsimplified expression as shown but the bracketing must be correct unless any missing brackets are implied by subsequent work.
May be implied by a correct simplified expression.

OR allow this mark for at least 2 correct simplified terms from \(-\dfrac{9}{2}x, -\dfrac{81}{8}x^2\) and \(-\dfrac{729}{16}x^3\)

A1: \(1 - \dfrac{9}{2}x - \dfrac{81}{8}x^2 - \dfrac{729}{16}x^3\) or simplified equivalent. Correct answer with no working can score full marks. Ignore any extra terms and allow the terms to be listed or in a different order. Apply isw once a correct expansion is seen. Condone \(+-\) (equivalent to listing).
Allow recovery if applicable e.g. if an “\(x\)” is lost then “reappears”.

Allow decimal equivalents \(1 - 4.5x - 10.125x^2 - 45.5625x^3\) provided they are exact.

Allow mixed numbers \(1 - 4\tfrac{1}{2}x - 10\tfrac{1}{8}x^2 - 45\tfrac{9}{16}x^3\)

Note: You may see attempts via direct expansion, but these will be scored using the main scheme, ignoring absence of powers on the 1s. The below attempts both score first M1A1.
If you are unsure, send to review.

Notes

B1: Expansion is valid for \(|x| \lt \dfrac{1}{9}\) or \(|x| \leqslant \dfrac{1}{9}\) and \(x = -\dfrac{2}{9}\) is outside this range.

Requires:

• an acceptable range of validity given
• an acceptable comparison of \(-\dfrac{2}{9}\) or \(\dfrac{2}{9}\) with their range leading to e.g. “not valid”.

Examples of acceptable alternatives include:

• (Valid for) \(|9x| \lt 1\) or \(|9x| \leqslant 1\) and as \(9x = -2\) (the expansion is) not valid.
• (Valid for) \(|x| \lt \dfrac{1}{9}\) or \(|x| \leqslant \dfrac{1}{9}\) and as \(\dfrac{2}{9} \gt \dfrac{1}{9}\) \(\left(\text{or } -\dfrac{2}{9} \lt -\dfrac{1}{9}\right)\) the expansion is not valid.
• (Valid for) \(-\dfrac{1}{9} \lt x \lt \dfrac{1}{9}\) and \(x = -\dfrac{2}{9}\) is too small/big (condoned as minimally acceptable).
• (Series converges for) \(\lvert -9x \rvert \leqslant 1\) and as \(-9x = 2\) the series will diverge.
• (Valid for) \(|x| \lt \dfrac{1}{9}\) but \(\left|-\dfrac{2}{9}\right| \gt \dfrac{1}{9}\) so \(-\dfrac{2}{9}\) cannot be used.

Do not accept vague statements such as "it is too big", "it is outside the range" without any mention of what the range is. \(-\dfrac{2}{9} \lt -\dfrac{1}{9}\) alone is insufficient evidence (without any mention of what the range is) and scores B0.
An attempt to evaluate the expansion and compare with \(\sqrt{3}\) is not acceptable on its own.

Typical solution

Typical solution

Typical solution

\(|x| \lt \dfrac{1}{8}\)

Typical solution

\(-4\)

Typical solution

(i)

(ii)

\(\dfrac{1}{2}\), \(\dfrac{3}{4}x\) and \(\dfrac{9}{8}x^2\) form a geometric sequence with common ratio \(\dfrac{3}{2}x\)

Typical solution

Let \(x = -\dfrac{1}{3}\)

Typical solution

(i)

(ii)

Typical solution

Typical solution

Since \(k \gt 0\)

Typical solution

Typical solution

The expansion is valid for \(|x| \lt 1\)

Typical solution

Typical solution

\(128\)

Typical solution

Typical solution

Typical solution

Typical solution

Typical solution

Typical solution

Notes

B1: Correct first two terms. Possibly unsimplified

M1: Attempt third term. Must be expanding \(\left(1 + \frac{1}{2}x\right)^{-3}\). Allow \(\frac{1}{2}x^2\) for \(\left(\frac{1}{2}x\right)^2\)

A1: Obtain correct third term. Possibly unsimplified

B1FT: Multiply their three term expansion by \(\frac{1}{8}\). Bracket expanded and coefficients simplified
Terms could be listed or summed
If B1M1A1 awarded, but attempt to simplify then goes wrong, B1FT is not also awarded
ISW once correct expansion seen

Notes

M1: Attempt expansion of \((1 + 4x)^{\frac{1}{2}}\). To obtain \(1 + 2x + kx^2\) (allow unsimplified)

A1: Obtain correct, simplified, expansion. Ignore any terms beyond \(x^2\)

M1: Attempt product of their two expansions ie their answer to (a) and their attempt at \((1 + 4x)^{\frac{1}{2}}\)
‘Hence’ so M0 for division attempts
To obtain attempts at the six relevant terms (ie 1 constant term, 2 terms in \(x\) and 3 terms in \(x^2\), but not necessarily all correct)
Ignore higher powers

A1: Obtain correct three terms. Terms could be listed or summed

Notes

B1*: Both conditions correct oe eg \(-2 \lt x \lt 2\) and \(-\frac{1}{4} \lt x \lt \frac{1}{4}\)
Allow \(\left|\frac{1}{2}x\right| \lt 1\) and/or \(|4x| \lt 1\) oe ie conditions on \(|kx|\) not \(|x|\)
\(|x| \lt \frac{1}{4}\) could also be \(|x| \leqslant \frac{1}{4}\) oe, as \(n \gt 0\)

B1dep*: Correct conclusion; reason needed that identifies that \(|x| \lt \frac{1}{4}\) is contained within \(|x| \lt 2\)
eg \(|x| \lt \frac{1}{4}\) is a subset of \(|x| \lt 2\)
eg nested interval
eg \(\frac{1}{4} \lt 2\)

Must now be \(|x| \lt \frac{1}{4}\) not \(|4x| \lt 1\)
B0 for vague statements such as ‘so that both are valid’ without any further clarification
Or \(|x| \leqslant \frac{1}{4}\)

Notes

B1: For reference: \(\frac{1}{9}\left(1 - \frac{2}{3}x\right)^{-2}\) - soi
or for \(3^{-2}(1 + \ldots)^{-2}\)

B1FT: Correct first two terms follow through their \(k\) – allow un-simplified
\(k \ne \pm 1, \pm 2\) - if correct \(k = -\frac{2}{3}\)

B1FT: Correct third term following through their \(k\) – allow un-simplified but must imply that the third term contains their \(k^2\) - for correct \(k\) condone \(\frac{2 \times 3}{2!}\left(\frac{2}{3}x\right)^2\) (or similar for their \(k\) if negative)
\(k \ne \pm 1, \pm 2\)
Condone \(\frac{2 \times 3}{2!}(kx)^2\) and 2 for 2!

B1: Or correct equivalent e.g. \(\frac{1}{27}\left(3 + 4x + 4x^2\right)\), \(\frac{1}{9} + \frac{4}{27}x + \frac{4}{27}x^2\), etc.
ISW after correct expansion seen
Ignore higher order terms if found – a correct answer scores all 4 marks www

Notes

B1: oe, for example, \(-\frac{3}{2} \lt x \lt \frac{3}{2}\) - allow \(-\frac{3}{2} \leqslant x \lt \frac{3}{2}\) but not \(-\frac{3}{2} \leqslant x \leqslant \frac{3}{2}\) (or any inequality that includes the \(\frac{3}{2}\)) - ISW once correct inequality seen. Allow \(\left[-\frac{3}{2}, \frac{3}{2}\right)\) or \(\left(-\frac{3}{2}, \frac{3}{2}\right)\) oe but not \(\left[0, \frac{3}{2}\right)\) (or equivalents in set notation)
\(-\frac{3}{2} \lt |x| \lt \frac{3}{2}\) is B0 but \(0 \leqslant |x| \lt \frac{3}{2}\) is B1
Note that \(|2x| \lt 3\) only is B0 (must be in terms of \(x\))

Notes

B1FT: Finding correct coefficient of \(x\) or the \(x\) term for their \((p + qx + \ldots)(a + x)\) - FT their \(p\) and \(q\) from part (a) (so their \(x\)-coefficient must be \(p + aq\)). Allow embedded in an expansion e.g. \(= \frac{1}{9}\left(\ldots + \left(\frac{4}{3}a + 1\right)x + \ldots\right)\) or \(= \frac{1}{9}\left(\ldots + \frac{4}{3}ax + x + \ldots\right)\)
This mark can be implied by the correct answer for \(a\) (or on the FT as detailed in the next mark)

B1FT: Follow through \(-\dfrac{\textit{their}\text{ constant term}}{\textit{their}\text{ coefficient of }x}\) from part (a)

Notes

B1: Correct first two terms
Allow unsimplified
Expect \(1 + \frac{9}{8}x\)

M1: Attempt third term
Condone lack of brackets when attempting to square ie \(\frac{3}{4}x^2\)
Coefficient must be \(\dfrac{\left(\frac{3}{2}\right)\left(\frac{1}{2}\right)}{2}\) or equiv

A1: Obtain correct third term
Allow unsimplified
\(\frac{3}{4}x^2\) is A0 unless recovered by later work
Expect \(\frac{27}{128}x^2\)

B1FT: Multiply their 3 term expansion by 8
Bracket expanded and coefficients simplified
If B1M1A1 awarded, but attempt to simplify then goes wrong, B1FT is not also awarded
ISW once correct expansion seen

Notes

B1: Could also be \(|x| \leqslant \frac{4}{3}\) or \(-\frac{4}{3} \leqslant x \leqslant \frac{4}{3}\), as \(n \gt 0\)
Must be condition for \(x\), not \(kx\)

Notes

M1: Expand \((1 + ax)^2\) and attempt at least one coeff of \(x^2\)
Allow \(ax\) as middle term, and/or \(ax^2\) as third term
Attempt at \(x^2\) term could be part of a fuller expansion

M1: Attempt all three coeff of \(x^2\), and no others
If part of fuller expansion then M1 awarded when only three relevant terms used

A1: Equate to \(\frac{107}{16}\) to obtain correct quadratic
aef, including unsimplified
A0 if a mix of terms and coefficients, but can be recovered

A1: Solve quadratic, possibly BC, to obtain \(a = -\frac{5}{2}\) and \(a = \frac{1}{4}\)

Notes

B1: Obtain \(243 + 810x\)
Condone \(3^5 + 810x\)
Allow terms not written as a sum eg written separately, or linked with a comma

M1: Attempt at least one further term – product of correct binomial coeff, power of 3 and attempted power of \(2x\), with powers totalling 5
Binomial coeff must be numerical; \({}^5\mathrm{C}_2\) is not yet enough
Allow BOD if brackets missing when index is applied to \(2x\), even if never recovered eg \(540x^2\) or \(180x^3\)

A1: Obtain correct third term
Coefficient simplified
Terms separate, listed or summed

A1: Obtain correct fourth term
Coefficient simplified
Could be separate term, part of a list or part of a sum
If expanding brackets then mark as above, but all 5 sets of brackets must be considered (allow irrelevant terms to be discarded)

Alternative method: expanding \(\left[3\left(1 + \frac{2}{3}x\right)\right]^5\)

B1: First two terms correct
Allow with 243 still outside the bracket

M1: Attempt one further term
Condone just 3 not \(3^5\) being used, but must be the correct binomial coeff and an attempt at the correct power of \(\frac{2}{3}x\), but allow BOD if no brackets

A1: Either 3^rd or 4^th term correct
Allow with 243 still outside the bracket

A1: Fully correct expansion
With the 243 now multiplied into the expansion

Notes

B1: Identify correct substitution
Could be stated, or implied by use in their binomial expansion

M1: Attempt to use binomial from (a) with their 2 term substitution
Must substitute into at least the \(x^2\) and \(x^3\) terms from their (a)
Allow M1 if using \(2y + 4y^2\) as their substitution

M1: Attempt expansion to obtain the two relevant terms in \(y^3\)
M0 if any other \(y^3\) terms
Expect 4(their 1080) and (their 720)
Allow M1 if using \(2y + 4y^2\) as their substitution - expect 16(their 1080) and 8(their 720)

A1: Allow \(5040y^3\)
Ignore any other non-cubic terms

Alternative method 1: attempting binomial expansion of \((3 + (2y + 4y^2))^5\) or \(((3 + 2y) + 4y^2)^5\)

B1: Group into two expressions, and attempt to use them

M1: Use their groups to obtain the appropriate two elements of their binomial expansion (ie those that would give \(y^3\) terms)

M1: Expand to attempt the two \(y^3\) terms, and no others

A1: Obtain 5040

Alternative method 2: attempting to expand all 5 brackets

M1: Attempt to use all 5 brackets
An attempt to use all 5 is sufficient

M1: Attempt all products that would give a \(y\)-cubed term
Condone additional terms, even those that would give another \(y^3\) term
Irrelevant terms (ie powers greater than 3) may never be seen

A1: Obtain correct terms or coefficients, with no more than one incorrect
They must have attempted all of the expected \(y^3\) terms, and no more, with no more than one coefficient error
If \((3 + 2y + 4y^2)^4 \times (3 + 2y + 4y^2)\) then expect \(2880 + 1296 + 864\),
If \((3 + 2y + 4y^2)^3 \times (3 + 2y + 4y^2)^2\) then expect \(1368 + 1728 + 1512 + 432\)
If they have not yet combined like terms then this A mark can only be implied by a later correct answer or relevant correct combination of terms

A1: Obtain 5040

Notes

B1: Obtain correct first two terms
Allow unsimplified second term, including product of two fractions

M1: Attempt third term in expansion of \(\left(1 - \frac{3}{8}x\right)^{\frac{1}{3}}\)
Allow BOD if no brackets, even if never recovered
Allow BOD if no negative sign

A1: Correct third term
Allow unsimplified fraction as coefficient, but must be single term

B1FT: Correct expansion of \((8 - 3x)^{\frac{1}{3}}\)
FT as 2 x their expansion (at least two terms)
Bracket expanded and fractions simplified

Notes

B1: Allow any equivalent eg \(-\frac{8}{3} \lt x \lt \frac{8}{3}\)
Must be strict inequality
Must be condition for \(x\), so B0 for \(|3x| \lt 8\)

Notes

M1: Attempt first three terms of expansion
Must be expanding \((1 + 2x)^{-2}\)
Allow BOD if no brackets on \(2x\), even if never recovered

A1: Obtain correct first three terms
Allow unsimplified fraction for coeff of third term

M1: Attempt all 3 relevant products
Finding 3 appropriate terms from the product of two 3-term quadratics
If part of full expansion then M1 when reqd 3 products and no others are combined

A1: Any exact equivalent, including 24.96875
Condone \(x^2\) still present

Notes

M1: Correct application of cosine rule

A1: AG – at least one line of intermediate working (must have \(p^2 = \ldots\))
Any errors or missing brackets then A0

Notes

B1: For reference: \(4\left(1 + \frac{3}{16}h^2\right)^{\frac{1}{2}}\)
or for \(16^{\frac{1}{2}}(1 + \ldots)^{\frac{1}{2}}\)

M1: Correct first two terms for their \(k\)
\(k \neq 1\)

A1ft: Correct third term following through their \(k\)

A1: \(\lambda = \frac{3}{8}, \mu = -\frac{9}{512}\) (oe)

SC if candidates assume that \(p = 4 + \lambda h^2 + \mu h^4\) and then substitute into \(p^2 = 16 + 3h^2\) to find \(\lambda\) and \(\mu\) then B1 for correct \(\lambda\) and B1 for correct \(\mu\) (so 2/4 max.)

Notes

M1: may be embedded eg \({}_{12}\mathrm{C}_3\,(2x)^3\);

A1: if M0 allow SCB1 for 1760 seen; A0 for final answer of \(1760x^3\)

Notes

M1: Use of binomial expansion as far as \(1 + \frac{1}{2}(3x)\)
soi by correct answer, nfww

M1: Full expansion up to quadratic term.
Condone e.g. \(3x^2\) but not \(x^2\)
soi by correct answer

A1: Allow exact decimals: \(1 + 1.5x - 1.125x^2\)
Ignore cubic and higher order terms.
Mark final answer
isw writing terms as a list after correct answer seen, but A0 if correct answer spoilt by further manipulation

Notes

B1: OR \(-\frac{1}{3} < x < \frac{1}{3}\) OR \(x < \frac{1}{3} \cap x > -\frac{1}{3}\)
Condone \(|x| \leqslant \frac{1}{3}\) or \(-\frac{1}{3} \leqslant x \leqslant \frac{1}{3}\) (as \(n > 0\))
Must be \(x\) not e.g. \(3x\)
Mark final answer

Notes

M1: two of the first three terms correct; ignore terms in \(x^3\) and above; may be embedded; must see at least substitution for third term

A1: may be unsimplified; may be embedded

A1: all three terms correct; ignore exra terms

if M0 allow SCB1 for \(\left(1 + \frac{1}{2}x - \frac{1}{4}x^2\right)\) following the equivalent method with use of \(\frac{3x}{2}\); may see eg \(2 + x - \frac{1}{2}x^2\)

if M0 allow SCB2 for correct expansion not fully supported

if M1A0 allow SCB1 for correct expansion not fully supported

Notes

B1FT: allow \(|x| \leqslant \frac{8}{3}\) or \(-\frac{8}{3} \leqslant x \leqslant \frac{8}{3}\); mark the final answer
FT their \(\left(1 + \frac{a}{b}x\right)\)

Notes

M1: allow recovery from bracket error; may be implied by award of second M1

M1: allow M1 for \(495 \times 2 \times k^4 = 79\,200\,000\) oe;
“\(= 79\,200\,000\)” may be implied by \(k = 5\)

A1: not from wrong working; but allow recovery from \(x^8\) on one side of equation only
allow SC2 for \(k = 5\) unsupported

Notes

B1: Do not accept \(\dfrac{x^6}{8}\) is always positive (ie \(> 0\))

Notes

E1: O.E.
See exemplars

Exemplars for 7b

Accept (eg)
The 3-term expansion is further away than the one with 2 terms
The 3-term expansion moves away from the 2-term expansion
The 3-term expansion is on the wrong side of the graph of the 2-term expansion
The 3-term expansion goes up at the end
The 3-term expansion should be more accurate than a 2-term expansion [and it isn’t]
The 3-term expansion moves away from the other 2

Do not accept (eg)
They don’t follow the same shape

Notes

M1: Working for 3^rd or 4^th term correct
Must see the negative \(x^3\)

A2: A2 all terms correct or A1 for three terms correct

Notes

B1: Curve reaches between \((-1, 3)\) to \((-1, 4.5)\)

Notes

M1: For their expression from 7(c) in an integration. Condone \(\mathrm{d}x\) missing.
2.5 and limits needed but may be seen later

M1: For integrating their expression (3 terms or more) allow one error
Limits could be wrong or missing here

M1: For attempt at their limits substituted into their integrand.
Substitution must be seen.

A1: AWRT 3.74 from correct working. Must include units.
1.495 will probably get either M2 or M3
Annotate final page

Special Case
If trapezium rule used on original function allow M1 A1 maximum
Likely answers
If 2 strips used: \(3.71\ \mathrm{m}^2\)
If 3 strips used: \(3.72\ \mathrm{m}^2\)
If 6 strips used: \(3.73\ \mathrm{m}^2\)

Notes

B1: Dealing correctly with the 2. Need not use negative powers for this mark

M1: Allow for expanding \((1+kx)^{-2}\) even where the B mark is not awarded

M1: Attempt to multiply their expansion by the numerator

A1: Convincing argument

Note there is mathematically nothing wrong with the direct expansion
\((2+x)^{-2} = 2^{-2} - 2\times 2^{-3}x + 3\times 2^{-4}x^2\)
Award B1M1 if seen

Alternative method

M1: Using partial fractions – allow an arithmetic slip

B1: Dealing correctly with the 2. Need not use negative powers for this mark

M1: Allow for expanding both \((1+kx)^{-2}\) and \((1+kx)^{-1}\) even where the B mark is not awarded

A1: Adding terms to complete a convincing argument

Notes

M1: ignore extra terms, allow one error

A1: two of three terms correct

A1: all three terms correct, ignore extra terms

if M0 allow SC2 for 2 of first three terms correct

Notes

B1: or \(|x| \leqslant \frac{1}{4}\) oe