| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Complete the square, then discriminant |
| Difficulty | Moderate -0.3 Part (i) is routine completing the square, part (ii) is direct application of discriminant/completed square form, and part (iii) requires explanation of a general principle but follows immediately from understanding part (ii). Slightly easier than average due to straightforward algebraic manipulation and standard reasoning about roots. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4\left[x^2 - 3x\right] + 11\) — no marks until attempt to complete the square | ||
| \(4\left[\left(x-\frac{3}{2}\right)^2 - \frac{9}{4}\right] + 11\), \(a=4\) | B1 | Must be of the form \(4\left(x \pm \alpha\right)^2 \pm \ldots\) |
| \((x - 3/2)^2\) | B1 | |
| \(4\left(x-\frac{3}{2}\right)^2 + 2\), \(c=2\) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| No real roots | B1 [1] | Zero, none, 0, … if not 'no real roots' must be consistent with their (i) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(r=0 \Rightarrow 1\) real root or 1 repeated root; \(r<0 \Rightarrow 2\) real roots; \(r>0 \Rightarrow\) no real roots | M1 | Attempt to relate the value of \(r\) to number of real roots (implied with at least one correct statement) |
| All three statements correct | A1 [2] |
# Question 4:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4\left[x^2 - 3x\right] + 11$ — no marks until attempt to complete the square | | |
| $4\left[\left(x-\frac{3}{2}\right)^2 - \frac{9}{4}\right] + 11$, $a=4$ | B1 | Must be of the form $4\left(x \pm \alpha\right)^2 \pm \ldots$ |
| $(x - 3/2)^2$ | B1 | |
| $4\left(x-\frac{3}{2}\right)^2 + 2$, $c=2$ | B1 [3] | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| No real roots | B1 [1] | Zero, none, 0, … if not 'no real roots' must be consistent with their (i) |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $r=0 \Rightarrow 1$ real root or 1 repeated root; $r<0 \Rightarrow 2$ real roots; $r>0 \Rightarrow$ no real roots | M1 | Attempt to relate the value of $r$ to number of real roots (implied with at least one correct statement) |
| All three statements correct | A1 [2] | |
---4 (i) Express $4 x ^ { 2 } - 12 x + 11$ in the form $a ( x + b ) ^ { 2 } + c$.\\
(ii) State the number of real roots of the equation $4 x ^ { 2 } - 12 x + 11 = 0$.\\
(iii) Explain fully how the value of $r$ is related to the number of real roots of the equation $p ( x + q ) ^ { 2 } + r = 0$ where $p , q$ and $r$ are real constants and $p > 0$.
\hfill \mbox{\textit{OCR PURE Q4 [6]}}