Standard +0.3 This is a straightforward partial differentiation problem requiring finding where ∂z/∂x = 0 and ∂z/∂y = 0, then solving the resulting simultaneous equations. While it involves two variables and cubic terms, it's a standard textbook exercise in multivariable calculus with no novel insight required—slightly easier than average A-level difficulty.
9 x ( x − 2 ) = 0 ( a n d x 0 ) x = 2 y = − 6
Answer
Marks
So (0, 0, 0) and (2, –6, –12) and no others
B1
B1
B1
M1
M1
A1
B1
Answer
Marks
[7]
1.1a
1.1
1.1
1.1
1.1
1.1
Answer
Marks
1.1
Condone poor notation for B1B1
provided intention clear.
No justification required. Must
be paired. Must follow from their
derivatives.
Can be implied by seeing both
their derivatives set to 0 and an
attempt to solve simultaneously
Setting both derivatives equal to
0 and eliminating one unknown
Finding the x and y coordinates of
the “non-zero” SP. Must be
paired.
Do not ISW. Do not condone
position vectors. Accept x = , y
Answer
Marks
=, z =.if clearly in triplets.
Allow correctly embedded in
grad
o r e g 6 x + 2 y = 0 3 x = − y
9 x 2 + 6 y = 0 y 2 + 6 y = 0
Correct answer, no working is
A0.
Correct answer, no working is
B0.
Question 1:
1 | z
=9x2 +6y
x
z
=6x+2y
y
SP where x = 0 and y = 0
z z
SPs where both = 0 and =0
x y
6 x + 2 y = 0 y = − 3 x
9 x 2 + 6 y = 0 9 x 2 − 1 8 x = 0
9 x ( x − 2 ) = 0 ( a n d x 0 ) x = 2 y = − 6
So (0, 0, 0) and (2, –6, –12) and no others | B1
B1
B1
M1
M1
A1
B1
[7] | 1.1a
1.1
1.1
1.1
1.1
1.1
1.1 | Condone poor notation for B1B1
provided intention clear.
No justification required. Must
be paired. Must follow from their
derivatives.
Can be implied by seeing both
their derivatives set to 0 and an
attempt to solve simultaneously
Setting both derivatives equal to
0 and eliminating one unknown
Finding the x and y coordinates of
the “non-zero” SP. Must be
paired.
Do not ISW. Do not condone
position vectors. Accept x = , y
=, z =.if clearly in triplets. | Allow correctly embedded in
grad
o r e g 6 x + 2 y = 0 3 x = − y
9 x 2 + 6 y = 0 y 2 + 6 y = 0
Correct answer, no working is
A0.
Correct answer, no working is
B0.
1 A surface is defined in 3-D by $z = 3 x ^ { 3 } + 6 x y + y ^ { 2 }$.\\
Determine the coordinates of any stationary points on the surface.
\hfill \mbox{\textit{OCR MEI Further Extra Pure 2023 Q1 [7]}}