| Exam Board | OCR MEI |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2014 |
| Session | June |
| Marks | 24 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vector Product and Surfaces |
| Type | Normal vector or normal line to surface |
| Difficulty | Challenging +1.2 This is a systematic Further Maths question on partial derivatives and surface normals requiring multiple standard techniques (computing gradient vector, finding normal line equation, using linear approximation, solving simultaneous equations). While it has five parts and involves Further Maths content, each part follows established procedures without requiring novel insight—the gradient gives the normal direction, parallel to y-axis means x and z components are zero, and tangent plane normal matches gradient. The length and Further Maths classification push it above average, but the mechanical nature keeps it from being truly challenging. |
| Spec | 8.05a 3D surfaces: z = f(x,y) and implicit form, partial derivatives8.05d Partial differentiation: first and second order, mixed derivatives8.05g Tangent planes: equation at a given point on surface |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{\partial g}{\partial x} = 2x + 6z - 4y\) | B1 | |
| \(\dfrac{\partial g}{\partial y} = 6y + 2z - 4x\) | B1 | |
| \(\dfrac{\partial g}{\partial z} = 4z + 2y + 6x\) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| At \(P\): \(\dfrac{\partial g}{\partial x} = -32\), \(\dfrac{\partial g}{\partial y} = 24\), \(\dfrac{\partial g}{\partial z} = 16\) | B1 | |
| Normal line is \(\mathbf{r} = \begin{pmatrix}2\\6\\-2\end{pmatrix} + \lambda\begin{pmatrix}-4\\3\\2\end{pmatrix}\) | M1 | Direction of normal line |
| A1 [3] | FT; Condone omission of '\(\mathbf{r} =\)' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(h \approx \delta g \approx\right) \dfrac{\partial g}{\partial x}\delta x + \dfrac{\partial g}{\partial y}\delta y + \dfrac{\partial g}{\partial z}\delta z\) | M1 | |
| \(h = (-32)(-4\lambda) + (24)(3\lambda) + (16)(2\lambda)\) \(\left(= 232\lambda\right)\) | A1 FT | |
| Approx distance is \( | \lambda | \sqrt{4^2+3^2+2^2}\) |
| \(= \sqrt{29} | \lambda | = \dfrac{\sqrt{29} |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Require \(\dfrac{\partial g}{\partial x} = \dfrac{\partial g}{\partial z} = 0\) | M1 | |
| \(2x + 6z - 4y = 0\) and \(4z + 2y + 6x = 0\) | ||
| \(y = -x\), \(z = -x\) | M1 | For (e.g.) \(y\) and \(z\) as multiples of \(x\) |
| \(x^2 + 3x^2 + 2x^2 + 2x^2 - 6x^2 + 4x^2 - 24 = 0\) | M1 | Quadratic in one variable |
| \(6x^2 - 24 = 0\) | A1 | In simplified form |
| Points \((2, -2, -2)\) and \((-2, 2, 2)\) | A1A1 [6] | If neither point correct, give A1 for any four correct coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\begin{pmatrix}2x+6z-4y\\6y+2z-4x\\4z+2y+6x\end{pmatrix} = \lambda\begin{pmatrix}10\\-1\\2\end{pmatrix}\) | M1 | Allow M1 even if \(\lambda = 1\) |
| A1 FT | ||
| \(y = 3x\), \(z = -5x\) | M1 | For (e.g.) \(y\) and \(z\) as multiples of \(x\); Or \(x = -\frac{1}{4}\lambda\), \(y = -\frac{3}{4}\lambda\), \(z = \frac{5}{4}\lambda\) |
| \(x^2 + 27x^2 + 50x^2 - 30x^2 - 30x^2 - 12x^2 - 24 = 0\) | M1 | Quadratic in one variable |
| \(6x^2 - 24 = 0\) | A1 | Or \(y^2 - 36 = 0\) or \(z^2 - 100 = 0\); Or \(\lambda^2 - 64 = 0\) |
| Possible points \((2, 6, -10)\) and \((-2, -6, 10)\) | A1 | For one correct point |
| At \((2, 6, -10)\): \(10x - y + 2z = -6\) | M1 | Checking at least one point |
| At \((-2, -6, 10)\): \(10x - y + 2z = 6\) | ||
| It is the tangent plane at \((-2, -6, 10)\) | A1 [8] | |
| OR: \(10x - (3x) + 2(-5x) = 6\) giving \(x = -2\) | M1 | Equation in one variable |
| A1 | Or \(y = -6\) or \(z = 10\) or \(\lambda = 8\) | |
| M1 | Using this value to obtain at least two coordinates | |
| It is the tangent plane at \((-2, -6, 10)\) | A2 | Give A1 for two coordinates correct |
# Question 2:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{\partial g}{\partial x} = 2x + 6z - 4y$ | B1 | |
| $\dfrac{\partial g}{\partial y} = 6y + 2z - 4x$ | B1 | |
| $\dfrac{\partial g}{\partial z} = 4z + 2y + 6x$ | B1 **[3]** | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| At $P$: $\dfrac{\partial g}{\partial x} = -32$, $\dfrac{\partial g}{\partial y} = 24$, $\dfrac{\partial g}{\partial z} = 16$ | B1 | |
| Normal line is $\mathbf{r} = \begin{pmatrix}2\\6\\-2\end{pmatrix} + \lambda\begin{pmatrix}-4\\3\\2\end{pmatrix}$ | M1 | Direction of normal line |
| | A1 **[3]** | FT; Condone omission of '$\mathbf{r} =$' |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(h \approx \delta g \approx\right) \dfrac{\partial g}{\partial x}\delta x + \dfrac{\partial g}{\partial y}\delta y + \dfrac{\partial g}{\partial z}\delta z$ | M1 | |
| $h = (-32)(-4\lambda) + (24)(3\lambda) + (16)(2\lambda)$ $\left(= 232\lambda\right)$ | A1 FT | |
| Approx distance is $|\lambda|\sqrt{4^2+3^2+2^2}$ | M1 | |
| $= \sqrt{29}|\lambda| = \dfrac{\sqrt{29}|h|}{232}$ | A1 **[4]** | Accept $\dfrac{h}{8\sqrt{29}}$, $\dfrac{h}{43.1}$, $0.023h$ etc |
## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Require $\dfrac{\partial g}{\partial x} = \dfrac{\partial g}{\partial z} = 0$ | M1 | |
| $2x + 6z - 4y = 0$ and $4z + 2y + 6x = 0$ | | |
| $y = -x$, $z = -x$ | M1 | For (e.g.) $y$ and $z$ as multiples of $x$ |
| $x^2 + 3x^2 + 2x^2 + 2x^2 - 6x^2 + 4x^2 - 24 = 0$ | M1 | Quadratic in one variable |
| $6x^2 - 24 = 0$ | A1 | In simplified form |
| Points $(2, -2, -2)$ and $(-2, 2, 2)$ | A1A1 **[6]** | If neither point correct, give A1 for any four correct coordinates |
## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\begin{pmatrix}2x+6z-4y\\6y+2z-4x\\4z+2y+6x\end{pmatrix} = \lambda\begin{pmatrix}10\\-1\\2\end{pmatrix}$ | M1 | Allow M1 even if $\lambda = 1$ |
| | A1 FT | |
| $y = 3x$, $z = -5x$ | M1 | For (e.g.) $y$ and $z$ as multiples of $x$; Or $x = -\frac{1}{4}\lambda$, $y = -\frac{3}{4}\lambda$, $z = \frac{5}{4}\lambda$ |
| $x^2 + 27x^2 + 50x^2 - 30x^2 - 30x^2 - 12x^2 - 24 = 0$ | M1 | Quadratic in one variable |
| $6x^2 - 24 = 0$ | A1 | Or $y^2 - 36 = 0$ or $z^2 - 100 = 0$; Or $\lambda^2 - 64 = 0$ |
| Possible points $(2, 6, -10)$ and $(-2, -6, 10)$ | A1 | For one correct point |
| At $(2, 6, -10)$: $10x - y + 2z = -6$ | M1 | Checking at least one point |
| At $(-2, -6, 10)$: $10x - y + 2z = 6$ | | |
| It is the tangent plane at $(-2, -6, 10)$ | A1 **[8]** | |
| **OR:** $10x - (3x) + 2(-5x) = 6$ giving $x = -2$ | M1 | Equation in one variable |
| | A1 | Or $y = -6$ or $z = 10$ or $\lambda = 8$ |
| | M1 | Using this value to obtain at least two coordinates |
| It is the tangent plane at $(-2, -6, 10)$ | A2 | Give A1 for two coordinates correct |2 A surface $S$ has equation $\mathrm { g } ( x , y , z ) = 0$, where $\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24$. $\mathrm { P } ( 2,6 , - 2 )$ is a point on the surface $S$.\\
(i) Find $\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }$ and $\frac { \partial \mathrm { g } } { \partial z }$.\\
(ii) Find the equation of the normal line to the surface $S$ at the point P .\\
(iii) The point Q is on this normal line and close to P . At $\mathrm { Q } , \mathrm { g } ( x , y , z ) = h$, where $h$ is small. Find, in terms of $h$, the approximate perpendicular distance from Q to the surface $S$.\\
(iv) Find the coordinates of the two points on the surface at which the normal line is parallel to the $y$-axis.\\
(v) Given that $10 x - y + 2 z = 6$ is the equation of a tangent plane to the surface $S$, find the coordinates of the point of contact.
\hfill \mbox{\textit{OCR MEI FP3 2014 Q2 [24]}}