| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Prove quadratic always positive/negative |
| Difficulty | Moderate -0.3 This is a standard C1 completing-the-square question with routine follow-up parts. Part (i) is textbook completing the square, part (ii) is basic sketching, part (iii) is solving a quadratic inequality using the completed square form, and part (iv) uses the discriminant. All techniques are core C1 with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-part structure. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable1.02m Graphs of functions: difference between plotting and sketching |
| Answer | Marks |
|---|---|
| 5 | ii |
| Answer | Marks |
|---|---|
| iv | (x − 2.5)2 o.e. |
| Answer | Marks |
|---|---|
| so yes, crosses | M1 |
| Answer | Marks |
|---|---|
| A1 | for clear attempt at −2.52 |
| Answer | Marks |
|---|---|
| conclusion | 4 |
Question 5:
5 | ii
iii
iv | (x − 2.5)2 o.e.
− 2.52 + 8
(x − 2.5)2 + 7/4 o.e.
min y = 7/4 o.e. [so above x axis]
or commenting (x − 2.5)2 ≥ 0
correct symmetrical quadratic
shape
8 marked as intercept on y axis
tp (5/2, 7/4) o.e. or ft from (i)
x2 − 5x − 6 seen or used
−1 and 6 obtained
x < − 1 and x > 6 isw or ft their
solns
min = (2.5, − 8.25) or ft from (i)
so yes, crosses | M1
M1
A1
B1
G1
G1
G1
M1
M1
M1
M1
A1 | for clear attempt at −2.52
allow M2A0 for (x − 2.5) + 7/4 o.e.
with no (x − 2.5)2 seen
ft, dep on (x − a)2 + b with b positive;
condone starting again, showing b2 −
4ac < 0 or using calculus
or (0, 8) seen in table
or (x − 2.5)2 [> or =] 12.25 or ft 14 − b
also implies first M1
if M0, allow B1 for one of x < − 1 and
x > 6
or M1 for other clear comment re
translated 10 down and A1 for
referring to min in (i) or graph in (ii);
or M1 for correct method for solving
x2 −5x −2 = 0 or using b2 − 4ac with
this and A1 for showing real solns eg
b2 − 4ac = 33; allow M1A0 for valid
comment but error in −8.25 ft; allow
M1 for showing y can be neg eg (0,
−2) found and A1 for correct
conclusion | 4
3
3
2\begin{enumerate}[label=(\roman*)]
\item Write $x^2 - 5x + 8$ in the form $(x - a)^2 + b$ and hence show that $x^2 - 5x + 8 > 0$ for all values of $x$. [4]
\item Sketch the graph of $y = x^2 - 5x + 8$, showing the coordinates of the turning point. [3]
\item Find the set of values of $x$ for which $x^2 - 5x + 8 > 14$. [3]
\item If $f(x) = x^2 - 5x + 8$, does the graph of $y = f(x) - 10$ cross the $x$-axis? Show how you decide. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 Q5 [12]}}