| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Solve quadratic by substitution |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard techniques: quadratic formula/completing the square for part (i), basic curve sketching for part (ii), and substitution (u = y^(1/2)) for part (iii). While part (iii) requires recognizing the substitution and squaring solutions, all steps are routine for C1 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(x = \frac{8 \pm \sqrt{64-44}}{2} = \frac{8 \pm \sqrt{20}}{2} = 4 \pm \sqrt{5}\) | M1, A1, B1, A1, 4 | Correct use of formula; \(\frac{8 \pm \sqrt{20}}{2}\) aef; \(\sqrt{20} = 2\sqrt{5}\) soi; \(4 \pm \sqrt{5}\) Alternative method: \((x-4)^2 - 16 + 11 = 0\) M1; \((x-4)^2 = 5\) A1; \(x = 4 + \sqrt{5}\) A1; or \(4 - \sqrt{5}\) A1 |
| (ii) +ve parabola; Root(s) in correct places; Completely correct curve with roots and (0, 11) labelled or referenced | B1, B1√, B1, 3 | |
| (iii) \(y = x^2 = \left(4 \pm \sqrt{5}\right)^2 = 16 + 5 \pm 8\sqrt{5} = 21 \pm 8\sqrt{5}\) | M1, M1, A1√, A1, 4 | \(y = x^2\) soi; Attempt to square at least one answer from part (i); Correct evaluation of \((a + b\sqrt{c})^2\) (a, b, c \(\ne\) 0); \(21 \pm 8\sqrt{5}\) |
**(i)** $x = \frac{8 \pm \sqrt{64-44}}{2} = \frac{8 \pm \sqrt{20}}{2} = 4 \pm \sqrt{5}$ | M1, A1, B1, A1, 4 | Correct use of formula; $\frac{8 \pm \sqrt{20}}{2}$ aef; $\sqrt{20} = 2\sqrt{5}$ soi; $4 \pm \sqrt{5}$ **Alternative method:** $(x-4)^2 - 16 + 11 = 0$ M1; $(x-4)^2 = 5$ A1; $x = 4 + \sqrt{5}$ A1; or $4 - \sqrt{5}$ A1
**(ii)** +ve parabola; Root(s) in correct places; Completely correct curve with roots and (0, 11) labelled or referenced | B1, B1√, B1, 3 |
**(iii)** $y = x^2 = \left(4 \pm \sqrt{5}\right)^2 = 16 + 5 \pm 8\sqrt{5} = 21 \pm 8\sqrt{5}$ | M1, M1, A1√, A1, 4 | $y = x^2$ soi; Attempt to square at least one answer from part (i); Correct evaluation of $(a + b\sqrt{c})^2$ (a, b, c $\ne$ 0); $21 \pm 8\sqrt{5}$7 (i) Solve the equation $x ^ { 2 } - 8 x + 11 = 0$, giving your answers in simplified surd form.\\
(ii) Hence sketch the curve $y = x ^ { 2 } - 8 x + 11$, labelling the points where the curve crosses the axes.\\
(iii) Solve the equation $y - 8 y ^ { \frac { 1 } { 2 } } + 11 = 0$, giving your answers in the form $p \pm q \sqrt { 5 }$.
\hfill \mbox{\textit{OCR C1 2006 Q7 [11]}}