| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question testing standard techniques: (i) parallel planes share normal vectors, (ii) perpendicular distance formula between planes, (iii) finding a line perpendicular to a given vector within a plane. All parts are routine applications of formulas with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04h Shortest distances: between parallel lines and between skew lines4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute coordinates \((5, 2, -2)\) in \(x + 4y - 8z = d\) | M1 | |
| Obtain plane equation \(x + 4y - 8z = 29\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt to use perpendicular formula to find perpendicular from \((5, 2, -2)\) to \(m\) | M1 | |
| Obtain a correct unsimplified expression, e.g. \(\dfrac{5+8+16-2}{\sqrt{1+16+64}}\) | A1 | |
| Obtain answer 3 | A1 | |
| Alternative method 1: | ||
| State or imply perpendicular from \(O\) to \(m\) is \(\frac{2}{9}\) or from \(O\) to \(n\) is \(\frac{29}{9}\) | B1 | |
| Find difference in perpendiculars | M1 | |
| Obtain answer 3 | A1 | |
| Alternative method 2: | ||
| Obtain correct parameter value, or position vector or coordinates of the foot of the perpendicular from \((5,2,-2)\) to \(m\), e.g. \(\mu = \pm\frac{1}{3}\); \(\left(\frac{14}{3}, \frac{2}{3}, \frac{2}{3}\right)\) | B1 | |
| Calculate the length of the perpendicular | M1 | |
| Obtain answer 3 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Calling the direction vector \(a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\), use a scalar product to form a relevant equation in \(a\), \(b\) and \(c\), e.g. \(a+4b-8c=0\) or \(5a+2b-2z=0\) | B1 | |
| Solve two relevant equations for the ratio \(a:b:c\) | M1 | |
| Obtain \(a:b:c = 4:-19:-9\) | A1 | OE |
| State answer \(\mathbf{r} = 5\mathbf{i}+2\mathbf{j}-2\mathbf{k}+\lambda(4\mathbf{i}-19\mathbf{j}-9\mathbf{k})\) | A1 | OE |
| Alternative method: | ||
| Attempt to calculate vector product of two relevant vectors, e.g. \((\mathbf{i}+4\mathbf{j}-8\mathbf{k})\times(5\mathbf{i}+2\mathbf{j}-2\mathbf{k})\) | M1 | |
| Obtain two correct components | A1 | |
| Obtain \(8\mathbf{i}-38\mathbf{j}-18\mathbf{k}\) | A1 | OE |
| State answer \(\mathbf{r} = 5\mathbf{i}+2\mathbf{j}-2\mathbf{k}+\lambda(4\mathbf{i}-19\mathbf{j}-9\mathbf{k})\) | A1 | OE |
## Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute coordinates $(5, 2, -2)$ in $x + 4y - 8z = d$ | M1 | |
| Obtain plane equation $x + 4y - 8z = 29$, or equivalent | A1 | |
## Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to use perpendicular formula to find perpendicular from $(5, 2, -2)$ to $m$ | M1 | |
| Obtain a correct unsimplified expression, e.g. $\dfrac{5+8+16-2}{\sqrt{1+16+64}}$ | A1 | |
| Obtain answer 3 | A1 | |
| **Alternative method 1:** | | |
| State or imply perpendicular from $O$ to $m$ is $\frac{2}{9}$ or from $O$ to $n$ is $\frac{29}{9}$ | B1 | |
| Find difference in perpendiculars | M1 | |
| Obtain answer 3 | A1 | |
| **Alternative method 2:** | | |
| Obtain correct parameter value, or position vector or coordinates of the foot of the perpendicular from $(5,2,-2)$ to $m$, e.g. $\mu = \pm\frac{1}{3}$; $\left(\frac{14}{3}, \frac{2}{3}, \frac{2}{3}\right)$ | B1 | |
| Calculate the length of the perpendicular | M1 | |
| Obtain answer 3 | B1 | |
## Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Calling the direction vector $a\mathbf{i}+b\mathbf{j}+c\mathbf{k}$, use a scalar product to form a relevant equation in $a$, $b$ and $c$, e.g. $a+4b-8c=0$ or $5a+2b-2z=0$ | B1 | |
| Solve two relevant equations for the ratio $a:b:c$ | M1 | |
| Obtain $a:b:c = 4:-19:-9$ | A1 | OE |
| State answer $\mathbf{r} = 5\mathbf{i}+2\mathbf{j}-2\mathbf{k}+\lambda(4\mathbf{i}-19\mathbf{j}-9\mathbf{k})$ | A1 | OE |
| **Alternative method:** | | |
| Attempt to calculate vector product of two relevant vectors, e.g. $(\mathbf{i}+4\mathbf{j}-8\mathbf{k})\times(5\mathbf{i}+2\mathbf{j}-2\mathbf{k})$ | M1 | |
| Obtain two correct components | A1 | |
| Obtain $8\mathbf{i}-38\mathbf{j}-18\mathbf{k}$ | A1 | OE |
| State answer $\mathbf{r} = 5\mathbf{i}+2\mathbf{j}-2\mathbf{k}+\lambda(4\mathbf{i}-19\mathbf{j}-9\mathbf{k})$ | A1 | OE |7 The plane $m$ has equation $x + 4 y - 8 z = 2$. The plane $n$ is parallel to $m$ and passes through the point $P$ with coordinates $( 5,2 , - 2 )$.\\
(i) Find the equation of $n$, giving your answer in the form $a x + b y + c z = d$.\\
(ii) Calculate the perpendicular distance between $m$ and $n$.\\
(iii) The line $l$ lies in the plane $n$, passes through the point $P$ and is perpendicular to $O P$, where $O$ is the origin. Find a vector equation for $l$.\\
\hfill \mbox{\textit{CAIE P3 2019 Q7 [9]}}