Moderate -0.8 This is a straightforward application of the binomial theorem requiring identification of the correct term and calculation using the binomial coefficient formula. It's simpler than average as it involves only one specific term extraction with no algebraic manipulation beyond the standard formula, though the negative coefficient requires careful arithmetic.
Ignore terms for other powers; condone \(x^3\) included
B3 for 2560 from correct term (NB coefficient of \(x^4\) is 2560)
B3
or B3 for neg answer following \(10 \times 4 \times -64\) and then an error in multiplication
B3
e.g. \(10 \times 4 \times -64 = 40 - 64 = -24\) gets M2 only
or M2 for \(10 \times 2^2 \times (-4)^3\) oe; must have multiplication signs or be followed by a clear attempt at multiplication
M2
Condone missing brackets e.g. allow M2 for \(10 \times 2^2 \times -4x^3\); \(^5C_3\) or factorial notation not sufficient but accept \(\frac{5\times4\times3\times2\times1}{2\times1\times3\times2\times1}\) oe
or M1 for \(2^2 \times (-4)^3\) oe (condone missing brackets) or for 10 used or for 1 5 10 10 5 1 seen
M1
10 may be unsimplified; M1 only for e.g. 10, \(2^2\) and \(-4x^3\) seen in table with no multiplication signs or evidence of attempt at multiplication
For those finding coeff of \(x^2\) instead: allow M1 for 10 or 1 5 10 10 5 1 seen; further SC1 if they get 1280; similarly for finding coefficient of \(x^4\) as 2560
SC
Lack of neg sign in \(x^2\) or \(x^4\) terms means these are easier so not eligible for just a 1 mark MR penalty
## Question 3:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-2560$ | 4 marks total (www) | Ignore terms for other powers; condone $x^3$ included |
| B3 for 2560 from correct term (NB coefficient of $x^4$ is 2560) | B3 | |
| or B3 for neg answer following $10 \times 4 \times -64$ and then an error in multiplication | B3 | e.g. $10 \times 4 \times -64 = 40 - 64 = -24$ gets M2 only |
| or M2 for $10 \times 2^2 \times (-4)^3$ oe; must have multiplication signs or be followed by a clear attempt at multiplication | M2 | Condone missing brackets e.g. allow M2 for $10 \times 2^2 \times -4x^3$; $^5C_3$ or factorial notation not sufficient but accept $\frac{5\times4\times3\times2\times1}{2\times1\times3\times2\times1}$ oe |
| or M1 for $2^2 \times (-4)^3$ oe (condone missing brackets) or for 10 used or for 1 5 10 10 5 1 seen | M1 | 10 may be unsimplified; M1 only for e.g. 10, $2^2$ and $-4x^3$ seen in table with no multiplication signs or evidence of attempt at multiplication |
| For those finding coeff of $x^2$ instead: allow M1 for 10 or 1 5 10 10 5 1 seen; further SC1 if they get 1280; similarly for finding coefficient of $x^4$ as 2560 | SC | Lack of neg sign in $x^2$ or $x^4$ terms means these are easier so not eligible for just a 1 mark MR penalty |
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