OCR MEI Further Extra Pure 2019 June — Question 2 11 marks

Exam BoardOCR MEI
ModuleFurther Extra Pure (Further Extra Pure)
Year2019
SessionJune
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
Topic3x3 Matrices
TypeDeterminant calculation and singularity
DifficultyChallenging +1.2 This is a Further Maths question on multivariable calculus requiring partial derivatives, completing the square, and geometric interpretation. While the techniques are standard (finding stationary points, sketching contours, analyzing sections), the multivariable context and the need to connect algebraic and geometric reasoning across multiple parts elevates it above typical A-level questions. The individual steps are routine for Further Maths students, but the synthesis required makes it moderately challenging.
Spec8.05c Sections and contours: sketch and relate to surface8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero8.05f Nature of stationary points: classify using Hessian matrix
A surface \(S\) is defined by \(z = 4x^2 + 4y^2 - 4x + 8y + 11\).
  1. Show that the point P\((0.5, -1, 6)\) is the only stationary point on \(S\). [2]
    1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\). [2]
    2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\). [3]
  3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\). Find all possible values of \(a\) and \(c\). [4]
Question 2:
AnswerMarks Guidance
2(a) ¶z ¶z
=8x-4 and =8y+8
¶x ¶y
8x  4 = 0  x = 0.5 and 8y + 8 = 0  y = 1 
z=4(0.5)2+4(-1)2-4(0.5)+8(-1)+11 = 6
AnswerMarks
[so P is the only stationary point]M1
A1
AnswerMarks
[2]1.1
2.2aBoth
z calculation required for A1
NB AG
AnswerMarks Guidance
2(b) (i)
Þ(x-0.5)2 +(y+1)2 =9
Sketch of circle
AnswerMarks
with centre (0.5, 1) and radius 3B1
B1
AnswerMarks
[2]1.1
2.2asoi but must be evident from
sketch or description in words
Sketch can be ‘by hand’ but must
be clearly intended to be a circle
(i.e. closed single curve that is at
least roughly circular)
AnswerMarks Guidance
2(b) (ii)
contour
P has z = 6 so is below the contour where z = 42
Moving from P to the contour in every direction
must be upwards [since there is no other stationary
AnswerMarks
point,] hence P must be a minimumB1
B1
B1
AnswerMarks
[3]2.1
2.1
AnswerMarks
2.1Must mention:
 P ‘inside’ contour
B1
 P is below contour
B1
 Upwards from P
B1
Allow good sketch
with complete
argument
AnswerMarks Guidance
2(c) z=4x2+4c2-4x+8c+11
dz
=8x-4=0Þx=0.5 [so a = 0.5]
dx
é 2
æ 1ö
êz =4 x- +4c2 +8c+10Þx=0.5 so a=0.5]
ç ÷
ê è 2ø
ë
22=4´0.52+4c2-4´0.5+8c+11
AnswerMarks
hence c = 1 or 3M1
ù A1
ú
ú
û
M1
A1
AnswerMarks
[4]3.1a
1.1
3.1a
AnswerMarks
1.1Must be ‘z =’
Condone ‘= 22’
Correct value of a identified
Substituting their value of a and
setting z = 22
AnswerMarks
bothDo not award M1 if
expression contains y
or by completing the
sq. with correct
argument
(c2+2c-3=0)
AnswerMarks Guidance
21 2 
Question 2:
2 | (a) | ¶z ¶z
=8x-4 and =8y+8
¶x ¶y
8x  4 = 0  x = 0.5 and 8y + 8 = 0  y = 1 
z=4(0.5)2+4(-1)2-4(0.5)+8(-1)+11 = 6
[so P is the only stationary point] | M1
A1
[2] | 1.1
2.2a | Both
z calculation required for A1
NB AG
2 | (b) | (i) | 42=4x2+4y2 -4x+8y+11
Þ(x-0.5)2 +(y+1)2 =9
Sketch of circle
with centre (0.5, 1) and radius 3 | B1
B1
[2] | 1.1
2.2a | soi but must be evident from
sketch or description in words
Sketch can be ‘by hand’ but must
be clearly intended to be a circle
(i.e. closed single curve that is at
least roughly circular)
2 | (b) | (ii) | (The vertical projection of) P lies inside the
contour
P has z = 6 so is below the contour where z = 42
Moving from P to the contour in every direction
must be upwards [since there is no other stationary
point,] hence P must be a minimum | B1
B1
B1
[3] | 2.1
2.1
2.1 | Must mention:
 P ‘inside’ contour
B1
 P is below contour
B1
 Upwards from P
B1
Allow good sketch
with complete
argument
2 | (c) | z=4x2+4c2-4x+8c+11
dz
=8x-4=0Þx=0.5 [so a = 0.5]
dx
é 2
æ 1ö
êz =4 x- +4c2 +8c+10Þx=0.5 so a=0.5]
ç ÷
ê è 2ø
ë
22=4´0.52+4c2-4´0.5+8c+11
hence c = 1 or 3 | M1
ù A1
ú
ú
û
M1
A1
[4] | 3.1a
1.1
3.1a
1.1 | Must be ‘z =’
Condone ‘= 22’
Correct value of a identified
Substituting their value of a and
setting z = 22
both | Do not award M1 if
expression contains y
or by completing the
sq. with correct
argument
(c2+2c-3=0)
2 | 1 | 2 | 3 | 4
A surface $S$ is defined by $z = 4x^2 + 4y^2 - 4x + 8y + 11$.

\begin{enumerate}[label=(\alph*)]
\item Show that the point P$(0.5, -1, 6)$ is the only stationary point on $S$. [2]
\item \begin{enumerate}[label=(\roman*)]
\item On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to $z = 42$. [2]
\item By using the sketch in part (b)(i), deduce that P must be a minimum point on $S$. [3]
\end{enumerate}
\item In the section of $S$ corresponding to $y = c$, the minimum value of $z$ occurs at the point where $x = a$ and $z = 22$.
Find all possible values of $a$ and $c$. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Extra Pure 2019 Q2 [11]}}
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