Easy -1.2 This is a straightforward computational exercise requiring direct expansion of a complex number cubed. While it's a Further Maths question, it only requires basic complex arithmetic (multiplication and addition) with no conceptual insight or problem-solving. The 'show that' format makes it even more routine since the answer is given. Significantly easier than average A-level questions.
Expands \((2+i)^3\) to four terms with no more than one incorrect term; or correctly expands \((2+i)^2\) to \(4+4i+i^2\) and multiplies by \(2+i\) to produce at least three terms with no more than one incorrect term. Terms may be unsimplified.
\(= 8 + 12i + 6(-1) + (-i)\)
B1
At least one instance of \(i^2\) replaced with \(-1\) or \(i^3\) replaced with \(-i\); PI by \(3+4i\)
\(= 2 + 11i\)
R1
Completes a reasoned argument to show that \((2+i)^3\) is \(2+11i\)
## Question 5:
$(2+i)^3 = 1\cdot2^3i^0 + 3\cdot2^2i^1 + 3\cdot2^1i^2 + 1\cdot2^0i^3$ | M1 | Expands $(2+i)^3$ to four terms with no more than one incorrect term; or correctly expands $(2+i)^2$ to $4+4i+i^2$ and multiplies by $2+i$ to produce at least three terms with no more than one incorrect term. Terms may be unsimplified.
$= 8 + 12i + 6(-1) + (-i)$ | B1 | At least one instance of $i^2$ replaced with $-1$ or $i^3$ replaced with $-i$; PI by $3+4i$
$= 2 + 11i$ | R1 | Completes a reasoned argument to show that $(2+i)^3$ is $2+11i$
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