| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a straightforward completing the square exercise with direct application to circle equations. Parts (i) and (ii) are routine algebraic manipulation requiring only the standard formula, while parts (iii) and (iv) involve simple recognition of circle centre and radius from completed square form. No problem-solving or novel insight required—purely procedural with clear scaffolding. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x^2 + 3x = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\) | B1, B1, 2 | \(a = \frac{3}{2}\); \(b = -\frac{9}{4}\) o.e. |
| (ii) \(y^2 - 4y - \frac{11}{4} = (y-2)^2 - \frac{27}{4}\) | B1, B1, 2 | \(p = -2\); \(q = -\frac{27}{4}\) o.e. |
| (iii) Centre \(\left(-\frac{3}{2}, 2\right)\) | B1√, 1 | \(\left(-\frac{3}{2}, 2\right)\) N.B. If question is restarted in this part, ft from part (iii) working only |
| (iv) Radius \(= \sqrt{\frac{27}{4} + \frac{9}{4}} = \sqrt{9} = 3\) | M1, A1, 2 | \(\sqrt{\text{-their'} b^2 \text{-their'} q'}\) or use \(\sqrt{(f^2 + g^2 - c)}\); \(3\) (\(\pm 3\) scores A0) |
**(i)** $x^2 + 3x = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4}$ | B1, B1, 2 | $a = \frac{3}{2}$; $b = -\frac{9}{4}$ o.e.
**(ii)** $y^2 - 4y - \frac{11}{4} = (y-2)^2 - \frac{27}{4}$ | B1, B1, 2 | $p = -2$; $q = -\frac{27}{4}$ o.e.
**(iii)** Centre $\left(-\frac{3}{2}, 2\right)$ | B1√, 1 | $\left(-\frac{3}{2}, 2\right)$ **N.B. If question is restarted in this part, ft from part (iii) working only**
**(iv)** Radius $= \sqrt{\frac{27}{4} + \frac{9}{4}} = \sqrt{9} = 3$ | M1, A1, 2 | $\sqrt{\text{-their'} b^2 \text{-their'} q'}$ or use $\sqrt{(f^2 + g^2 - c)}$; $3$ ($\pm 3$ scores A0)5 (i) Express $x ^ { 2 } + 3 x$ in the form $( x + a ) ^ { 2 } + b$.\\
(ii) Express $y ^ { 2 } - 4 y - \frac { 11 } { 4 }$ in the form $( y + p ) ^ { 2 } + q$.
A circle has equation $x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0$.\\
(iii) Write down the coordinates of the centre of the circle.\\
(iv) Find the radius of the circle.
\hfill \mbox{\textit{OCR C1 2006 Q5 [7]}}