| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Easy -1.2 This is a straightforward completing the square question with standard follow-up parts. Part (a) requires routine algebraic manipulation, part (b) is a direct application of the completed square form, and part (c) asks for a basic sketch using the vertex. All techniques are standard textbook exercises with no problem-solving insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = 4\left(x + \frac{5}{2}\right)^2 - 19\) | ||
| \(a = 4\) | B1 | |
| \(b = \frac{5}{2}\) OE | B1 | |
| \(c = -19\) | B1 | There is no requirement for the candidate to list \(a\), \(b\) and \(c\). Look at values in their final expression, condone omission of \({}^2\), and award marks as follows. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\left(\text{Their } 4\left(x+\frac{5}{2}\right)^2 - 19\right) = 45 \Rightarrow \left(x + \frac{5}{2}\right)^2 = 16\) | *M1 | Equate their quadratic completed square form from 6(a) to 45 or re-start and use completing the square. |
| Solve as far as \(x =\) | DM1 | Any valid method leading to two answers. |
| \(\left[x =\right] \frac{3}{2},\ -\frac{13}{2}\) | A1 | SC: If M0 or M1 DM0 awarded, B1 available for correct final answers. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Quadratic curve that is the right way up (must be seen either side of stationary point) | B1 | No axes required, ignore any axes even if incorrect. |
| Stationary point stated using any valid method or correctly labelled on their diagram. | B1 FT | FT *their* values from 6(a) as long as *their* expression is of the form \(p(qx+r)^2 + s\). Expect \(\left(-\frac{5}{2}, -19\right)\). Condone if stated correctly but plotted incorrectly. |
| B1 FT |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 4\left(x + \frac{5}{2}\right)^2 - 19$ | | |
| $a = 4$ | B1 | |
| $b = \frac{5}{2}$ OE | B1 | |
| $c = -19$ | B1 | There is no requirement for the candidate to list $a$, $b$ and $c$. Look at values in their final expression, condone omission of ${}^2$, and award marks as follows. |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left(\text{Their } 4\left(x+\frac{5}{2}\right)^2 - 19\right) = 45 \Rightarrow \left(x + \frac{5}{2}\right)^2 = 16$ | *M1 | Equate their quadratic completed square form from **6(a)** to 45 or re-start and use completing the square. |
| Solve as far as $x =$ | DM1 | Any valid method leading to two answers. |
| $\left[x =\right] \frac{3}{2},\ -\frac{13}{2}$ | A1 | **SC:** If M0 or M1 DM0 awarded, B1 available for correct final answers. |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Quadratic **curve** that is the right way up (must be seen either side of stationary point) | B1 | No axes required, ignore any axes even if incorrect. |
| Stationary point stated using any valid method or correctly labelled on their diagram. | B1 FT | FT *their* values from **6(a)** as long as *their* expression is of the form $p(qx+r)^2 + s$. Expect $\left(-\frac{5}{2}, -19\right)$. Condone if stated correctly but plotted incorrectly. |
| | B1 FT | |
---6 The equation of a curve is $y = 4 x ^ { 2 } + 20 x + 6$.
\begin{enumerate}[label=(\alph*)]
\item Express the equation in the form $y = a ( x + b ) ^ { 2 } + c$, where $a$, $b$ and $c$ are constants.
\item Hence solve the equation $4 x ^ { 2 } + 20 x + 6 = 45$.
\item Sketch the graph of $y = 4 x ^ { 2 } + 20 x + 6$ showing the coordinates of the stationary point. You are not required to indicate where the curve crosses the $x$ - and $y$-axes.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q6 [9]}}