Easy -1.8 This is a very straightforward algebraic manipulation question requiring only substitution of basic circle formulae (C = πd, A = πr² = πd²/4) and simplification to show k = 4. It involves no problem-solving, just routine algebraic verification of a given result using well-known formulae.
4 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).
obtaining a correct relationship in any 3 of \(C\), \(d\), \(r\) and \(A\)
M2
may substitute into given relationship; e.g. M2 for \(Cd = 4\pi r^2\) or \(\pi d^2 = k\pi r^2\) seen/obtained
or obtaining a correct relationship in \(k\) and no more than 2 other variables
M2
or M1 for at least two of \(A = \pi r^2\), \(C = \pi d\), \(C = 2\pi r\), \(d = 2r\) or \(r = \frac{d}{2}\) seen or used; condone e.g. Area \(= \pi r^2\); allow \(A = \pi\left(\frac{d}{2}\right)^2\) to imply \(A = \pi r^2\) and \(r = \frac{d}{2}\) and so earn M1, if M2 not earned
convincing argument leading to \(k = 4\)
A1
must be from general argument, not just substituting values for \(r\) or \(d\); may start from given relationship and derive \(k = 4\); e.g. M1 only for e.g. \(A = \pi r^2\) and \(C = \pi d\) and so \(k = 4\) with no further evidence
Total: [3]
## Question 4:
obtaining a correct relationship in any 3 of $C$, $d$, $r$ and $A$ | M2 | may substitute into given relationship; e.g. M2 for $Cd = 4\pi r^2$ or $\pi d^2 = k\pi r^2$ seen/obtained
or obtaining a correct relationship in $k$ and no more than 2 other variables | M2 | or M1 for at least two of $A = \pi r^2$, $C = \pi d$, $C = 2\pi r$, $d = 2r$ or $r = \frac{d}{2}$ seen or used; condone e.g. Area $= \pi r^2$; allow $A = \pi\left(\frac{d}{2}\right)^2$ to imply $A = \pi r^2$ and $r = \frac{d}{2}$ and so earn M1, if M2 not earned
convincing argument leading to $k = 4$ | A1 | must be from general argument, not just substituting values for $r$ or $d$; may start from given relationship and derive $k = 4$; e.g. M1 only for e.g. $A = \pi r^2$ and $C = \pi d$ and so $k = 4$ with no further evidence
**Total: [3]**
---
4 A circle has diameter $d$, circumference $C$, and area $A$. Starting with the standard formulae for a circle, show that $C d = k A$, finding the numerical value of $k$.
\hfill \mbox{\textit{OCR MEI C1 Q4 [3]}}