| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simultaneous equations |
| Type | Line intersecting general conic |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring standard techniques: finding a circle equation from diameter endpoints (midpoint and distance formula), solving simultaneous equations (substitution into circle equation gives quadratic), and calculating triangle area (using coordinate formula or base×height). All steps are routine A-level methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.02c Simultaneous equations: two variables by elimination and substitution1.02q Use intersection points: of graphs to solve equations1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Midpoint of AB is \((3, 1)\); Centre C of the circle is \((3, 1)\) | B1 | soi |
| Radius \(\sqrt{(7-3)^2 + (-2-1)^2} = 5\) | M1 | Attempt to find length of AB, AC or BC |
| So circle is \((x-3)^2 + (y-1)^2 = 25\) | M1 A1 [4] | Uses their midpoint and radius (do not allow diameter used). Need not be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Crosses \(y = 2x + 5\) where \((x-3)^2 + (2x+5-1)^2 = 25\) | M1 | Substituting \(y = 2x+5\) and attempting to collect terms oe |
| \(5x^2 + 10x = 0\) giving \(x = -2, 0\) | A1 A1 | Both values correct. Allow for quadratic solved BC providing it is seen in form \(ax^2 + bx = 0\) or \(ay^2 + by + c = 0\) |
| So points are \((-2, 1)\) and \((0, 5)\) | [3] | Correct \(y\) coordinates FT their \(x\)-coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(AQ = \sqrt{2}\) and \(BQ = \sqrt{7^2 + 7^2} = 7\sqrt{2}\); Triangle ABQ has a right angle at Q (angle in a semicircle) | M1 | Attempt to find two lengths to be used in area calculation (excluding AB). Note QAB=81.9° and QBA=8.1° |
| So area of triangle is \(\frac{1}{2} \times AQ \times BQ\) | M1 | Correct method for finding the area |
| Area \(= 7\) | A1 [3] | FT their Q |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Midpoint of AB is $(3, 1)$; Centre C of the circle is $(3, 1)$ | B1 | soi |
| Radius $\sqrt{(7-3)^2 + (-2-1)^2} = 5$ | M1 | Attempt to find length of AB, AC or BC |
| So circle is $(x-3)^2 + (y-1)^2 = 25$ | M1 A1 [4] | Uses their midpoint and radius (do not allow diameter used). Need not be simplified |
---
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Crosses $y = 2x + 5$ where $(x-3)^2 + (2x+5-1)^2 = 25$ | M1 | Substituting $y = 2x+5$ and attempting to collect terms oe |
| $5x^2 + 10x = 0$ giving $x = -2, 0$ | A1 A1 | Both values correct. Allow for quadratic solved BC providing it is seen in form $ax^2 + bx = 0$ or $ay^2 + by + c = 0$ |
| So points are $(-2, 1)$ and $(0, 5)$ | [3] | Correct $y$ coordinates FT their $x$-coordinates |
---
## Question 7(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $AQ = \sqrt{2}$ and $BQ = \sqrt{7^2 + 7^2} = 7\sqrt{2}$; Triangle ABQ has a right angle at Q (angle in a semicircle) | M1 | Attempt to find two lengths to be used in area calculation (excluding AB). Note QAB=81.9° and QBA=8.1° |
| So area of triangle is $\frac{1}{2} \times AQ \times BQ$ | M1 | Correct method for finding the area |
| Area $= 7$ | A1 [3] | FT their Q |
---7 In this question you must show detailed reasoning.\\
The points $\mathrm { A } ( - 1,4 )$ and $\mathrm { B } ( 7 , - 2 )$ are at opposite ends of a diameter of a circle.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the circle.
\item Find the coordinates of the points of intersection of the circle and the line $y = 2 x + 5$.
\item Q is the point of intersection with the larger $y$-coordinate.
Calculate the area of the triangle ABQ .
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2021 Q7 [10]}}