Standard +0.3 This is a straightforward Further Maths question requiring students to write inequalities for a circle and a half-line from a given diagram. While it involves Further Maths content (complex loci), it only requires direct translation of geometric features into standard forms |z - (1+3i)| ≤ r and arg(z) ≤ π/4, with minimal calculation needed to find r = √10 from the given information.
5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\).
\includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242}
State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.
Accept alternatives, e.g. Re (z) ≥ Im(z), or │z −1│≤ │z −i│
6
Answer
Marks
Guidance
`
(a)
n n n
r ( r + 2 ) = r 2 + 2 r
=r 1 =r 1 =r 1
1
= n ( n + 1 ) ( 2 n + 1 ) + n ( n + 1 )
6
1
= n(n+1)(2n+1+6)
6
1
= n ( n + 1 ) ( 2 n + 7 )
Answer
Marks
6
M1
A1
M1
A1
Answer
Marks
[4]
2.1
1.1
1.1
Answer
Marks
2.3
Factoring n or n +1
NB AG must show previous step
Question 5:
5 | z − 1 − 3 i 1 0
arg(z) ≤ /4 | M1
B1
A1
M1
A1
[5] | 2.5
1.1
1.1
1.1
1.1 | circle of form z − a = b used
radius 10 soi
all correct (must be )
half line arg(z) = a used
all correct, condone arg(z) ≤ 45
Accept alternatives, e.g. Re (z) ≥ Im(z), or │z −1│≤ │z −i│
6
` | (a) | n n n
r ( r + 2 ) = r 2 + 2 r
=r 1 =r 1 =r 1
1
= n ( n + 1 ) ( 2 n + 1 ) + n ( n + 1 )
6
1
= n(n+1)(2n+1+6)
6
1
= n ( n + 1 ) ( 2 n + 7 )
6 | M1
A1
M1
A1
[4] | 2.1
1.1
1.1
2.3 | Factoring n or n +1
NB AG must show previous step
5 An Argand diagram is shown below. The circle has centre at the point representing $1 + 3 i$, and the half line intersects the circle at the origin and at the point representing $4 + 4 \mathrm { i }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242}
State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2022 Q5 [5]}}