Challenging +1.2 This is a standard partial differentiation problem requiring students to find ∂z/∂x = 0 and ∂z/∂y = 0, then solve the resulting simultaneous equations. While it involves Further Maths content (multivariable calculus), the technique is routine and the algebraic manipulation is straightforward, making it moderately above average difficulty but not requiring novel insight.
3 A surface has equation \(z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y\) for all real values of \(x\) and \(y\). Determine the coordinates of all stationary points of this surface.
Setting both first partial derivatives equal to zero and
attempt to eliminate one variable
Either y 3 − 3 y + 2 = 0 or 4 x 3 − 3 x + 1 = 0
i.e. (y – 1)2(y + 2) = 0 or (2x – 1)2(x + 1) = 0
(x, y, z) = ( 12 , 1, 34 )
Answer
Marks
= (−1, −2, −6)
B1
B1
M1
M1
A1
A1
Answer
Marks
[6]
1.1
1.1
1.1a
1.1
1.1
Answer
Marks
1.1
3 y − 2 3 x − 1
Either directly via x = or y =
2 y 2 2 x 2
3 y − 2 3 x − 1
OR indirectly via 2 x y = = y = 2x
y x
Any cubic equation in one variable
First SP correct BC www
Second SP correct BC www
SC1 for both pairs of (x, y) correct with z’s missing
or z incorrect
NB Extra SP A1A0
Question 3:
3 | z
= 2 x y 2 − 3 y + 2
x
z
= 2 x 2 y − 3 x + 1
y
Setting both first partial derivatives equal to zero and
attempt to eliminate one variable
Either y 3 − 3 y + 2 = 0 or 4 x 3 − 3 x + 1 = 0
i.e. (y – 1)2(y + 2) = 0 or (2x – 1)2(x + 1) = 0
(x, y, z) = ( 12 , 1, 34 )
= (−1, −2, −6) | B1
B1
M1
M1
A1
A1
[6] | 1.1
1.1
1.1a
1.1
1.1
1.1 | 3 y − 2 3 x − 1
Either directly via x = or y =
2 y 2 2 x 2
3 y − 2 3 x − 1
OR indirectly via 2 x y = = y = 2x
y x
Any cubic equation in one variable
First SP correct BC www
Second SP correct BC www
SC1 for both pairs of (x, y) correct with z’s missing
or z incorrect
NB Extra SP A1A0
3 A surface has equation $z = x ^ { 2 } y ^ { 2 } - 3 x y + 2 x + y$ for all real values of $x$ and $y$. Determine the coordinates of all stationary points of this surface.
\hfill \mbox{\textit{OCR Further Additional Pure AS 2023 Q3 [6]}}