| Exam Board | OCR |
|---|---|
| Module | Further Additional Pure AS (Further Additional Pure AS) |
| Year | 2024 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Standard +0.8 This is a multi-part question on multivariable calculus requiring finding stationary points via partial derivatives, numerical evaluation of derivatives at specific points, and sketching cross-sections. While systematic, it involves several techniques (solving simultaneous equations, partial differentiation, interpreting derivative signs) and requires careful coordination across parts. More demanding than standard single-variable calculus but not requiring deep insight. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05d Partial differentiation: first and second order, mixed derivatives8.05e Stationary points: where partial derivatives are zero |
| Answer | Marks |
|---|---|
| 3 | z |
| Answer | Marks |
|---|---|
| = (−1, −2, −6) | B1 |
| Answer | Marks |
|---|---|
| [6] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | 3 y − 2 3 x − 1 |
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 A1A03 The surface $S$ has equation $z = f ( x , y )$, where $f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$ for all real values of $x$ and $y$. You are given that $S$ has a stationary point at the origin, $O$, and a second stationary point at the point $P ( a , b , c )$, where $\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )$.
\begin{enumerate}[label=(\alph*)]
\item Determine the values of $a , b$ and $c$.
\item Throughout this part, take the values of $a$ and $b$ to be those found in part (a).
\begin{enumerate}[label=(\roman*)]
\item Evaluate $\mathrm { f } _ { x }$ at the points $\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )$ and $\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )$.
\item Evaluate $\mathrm { f } _ { y }$ at the points $\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )$ and $\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )$.
\item Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of $S$, given by
\begin{itemize}
\end{enumerate}\item $z = f ( x , b )$, for $| x - a | \leqslant 0.1$,
\item $z = f ( a , y )$, for $| y - b | \leqslant 0.1$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{OCR Further Additional Pure AS 2024 Q3 [12]}}