Standard +0.3 This is a straightforward binomial expansion question requiring students to identify which term has x^0 (the constant term) and calculate its coefficient. While it involves fractional powers of x and requires careful tracking of indices, it's a standard textbook exercise with a clear method and no conceptual difficulty beyond basic binomial theorem application.
Identifying term as \(20(2x)^3\left(\frac{5}{x}\right)^3\) oe
M3
Condone lack of brackets; \(x\)s may be omitted e.g. M3 for \(20 \times 8 \times 125\)
M1 for \([k](2x)^3\left(\frac{5}{x}\right)^3\) soi (e.g. in list or table), condoning lack of brackets
M1
First M1 not earned if elements added not multiplied; otherwise if in list or table, bod intent to multiply
and M1 for \(k = 20\) or e.g. \(\frac{6\times5\times4}{3\times2\times1}\) or for 1 6 15 20 15 6 1 seen (e.g. Pascal's triangle, even if no attempt at expansion)
M1
M0 for binomial coefficient if it still has factorial notation
and M1 for selecting the appropriate term (may be implied by use of only \(k=20\), but this M1 is not dependent on the correct \(k\) used)
M1
May be gained even if elements added
20 000
A1
Or B4 for 20 000 obtained from multiplying out \(\left(2x+\frac{5}{x}\right)^6\); allow SC3 for 20000 as part of an expansion
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Identifying term as $20(2x)^3\left(\frac{5}{x}\right)^3$ oe | M3 | Condone lack of brackets; $x$s may be omitted e.g. M3 for $20 \times 8 \times 125$ |
| M1 for $[k](2x)^3\left(\frac{5}{x}\right)^3$ soi (e.g. in list or table), condoning lack of brackets | M1 | First M1 not earned if elements added not multiplied; otherwise if in list or table, bod intent to multiply |
| and M1 for $k = 20$ or e.g. $\frac{6\times5\times4}{3\times2\times1}$ or for 1 6 15 20 15 6 1 seen (e.g. Pascal's triangle, even if no attempt at expansion) | M1 | M0 for binomial coefficient if it still has factorial notation |
| and M1 for selecting the appropriate term (may be implied by use of only $k=20$, but this M1 is not dependent on the correct $k$ used) | M1 | May be gained even if elements added |
| 20 000 | A1 | Or B4 for 20 000 obtained from multiplying out $\left(2x+\frac{5}{x}\right)^6$; allow SC3 for 20000 as part of an expansion |
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4 The binomial expansion of $\left( 2 x + \frac { 5 } { x } \right) ^ { 6 }$ has a term which is a constant. Find this term.
\hfill \mbox{\textit{OCR MEI C1 Q4 [4]}}