Moderate -0.8 This is a straightforward multi-part question testing completing the square (routine algebraic manipulation), sketching a parabola (basic graph work), and calculating a discriminant (direct formula application). All parts are standard textbook exercises requiring only recall and basic technique with no problem-solving insight needed.
4. (a) Show that \(x ^ { 2 } + 6 x + 11\) can be written as
$$( x + p ) ^ { 2 } + q$$
where \(p\) and \(q\) are integers to be found.
(b) In the space at the top of page 7 , sketch the curve with equation \(y = x ^ { 2 } + 6 x + 11\), showing clearly any intersections with the coordinate axes.
(c) Find the value of the discriminant of \(x ^ { 2 } + 6 x + 11\)
or \(p=3\) or \(\frac{6}{2}\). Ignore "= 0" so \((x+3)^2+2=0\) can score both marks
\(q = 2\)
B1
Part (b):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
U shape with min in 2nd quadrant (must be above \(x\)-axis and not on \(y\)-axis)
B1
Curve need not cross \(y\)-axis but min should NOT touch \(x\)-axis and should be left of (not on) \(y\)-axis
U shape crossing \(y\)-axis at \((0, 11)\) only; condone \((11, 0)\) marked on \(y\)-axis
B1
Just 11 marked on \(y\)-axis is fine. Condone stopping at \((0,11)\)
Part (c):
Answer
Marks
Guidance
Answer/Working
Marks
Guidance
\(b^2 - 4ac = 6^2 - 4 \times 11\)
M1
For some correct substitution into \(b^2 - 4ac\); substitution into \(b^2 < 4ac\) or \(b^2 = 4ac\) or \(b^2 > 4ac\) is M0
\(= -8\)
A1
For \(-8\) only. If \(-8 < 0\) treat the \(< 0\) as ISW and award A1. If \(-8 \geq 0\) then A0. Only award marks for discriminant in part (c)
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(x+3)^2 + 2$ | B1 | or $p=3$ or $\frac{6}{2}$. Ignore "= 0" so $(x+3)^2+2=0$ can score both marks |
| $q = 2$ | B1 | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| U shape with min in 2nd quadrant (must be above $x$-axis and not on $y$-axis) | B1 | Curve need not cross $y$-axis but min should NOT touch $x$-axis and should be left of (not on) $y$-axis |
| U shape crossing $y$-axis at $(0, 11)$ only; condone $(11, 0)$ marked on $y$-axis | B1 | Just 11 marked on $y$-axis is fine. Condone stopping at $(0,11)$ |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $b^2 - 4ac = 6^2 - 4 \times 11$ | M1 | For some correct substitution into $b^2 - 4ac$; substitution into $b^2 < 4ac$ or $b^2 = 4ac$ or $b^2 > 4ac$ is M0 |
| $= -8$ | A1 | For $-8$ only. If $-8 < 0$ treat the $< 0$ as ISW and award A1. If $-8 \geq 0$ then A0. Only award marks for discriminant in part (c) |
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4. (a) Show that $x ^ { 2 } + 6 x + 11$ can be written as
$$( x + p ) ^ { 2 } + q$$
where $p$ and $q$ are integers to be found.\\
(b) In the space at the top of page 7 , sketch the curve with equation $y = x ^ { 2 } + 6 x + 11$, showing clearly any intersections with the coordinate axes.\\
(c) Find the value of the discriminant of $x ^ { 2 } + 6 x + 11$\\
\hfill \mbox{\textit{Edexcel C1 2010 Q4 [6]}}