| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Point on line satisfying condition |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question with routine techniques: (i) checking parallel by dot product of direction and normal vectors, (ii) angle between planes using normal vectors formula, (iii) using distance formula and solving for λ. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| *EITHER*: Expand scalar product of a normal to \(m\) and a direction vector of \(l\) | M1 | |
| Verify scalar product is zero | A1 | |
| Verify that one point of \(l\) does not lie in the plane | A1 | |
| *OR*: Substitute coordinates of a general point of \(l\) in the equation of the plane \(m\) | M1 | |
| Obtain correct equation in \(\lambda\) in any form | A1 | |
| Verify that the equation is not satisfied for any value of \(\lambda\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct method to evaluate a scalar product of normal vectors to \(m\) and \(n\) | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer \(74.5°\) or \(1.30\) radians | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| *EITHER*: Using the components of a general point \(P\) of \(l\) form an equation in \(\lambda\) by equating the perpendicular distance from \(n\) to \(2\) | M1 | |
| *OR*: Take a point \(Q\) on \(l\), e.g. \((5, 3, 3)\) and form an equation in \(\lambda\) by equating the length of the projection of \(QP\) onto a normal to plane \(n\) to \(2\) | M1 | |
| Obtain a correct modular or non-modular equation in any form | A1 | |
| Solve for \(\lambda\) and obtain a position vector for \(P\), e.g. \(7\mathbf{i} + 5\mathbf{j} + 7\mathbf{j}\) from \(\lambda = 3\) | A1 | |
| Obtain position vector of the second point, e.g. \(3\mathbf{i} + \mathbf{j} - \mathbf{k}\) from \(\lambda = -1\) | A1 | |
| Total: 4 |
## Question 10(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| *EITHER*: Expand scalar product of a normal to $m$ and a direction vector of $l$ | M1 | |
| Verify scalar product is zero | A1 | |
| Verify that one point of $l$ does not lie in the plane | A1 | |
| *OR*: Substitute coordinates of a general point of $l$ in the equation of the plane $m$ | M1 | |
| Obtain correct equation in $\lambda$ in any form | A1 | |
| Verify that the equation is not satisfied for any value of $\lambda$ | A1 | |
| **Total: 3** | | |
## Question 10(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct method to evaluate a scalar product of normal vectors to $m$ and $n$ | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer $74.5°$ or $1.30$ radians | A1 | |
| **Total: 3** | | |
## Question 10(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| *EITHER*: Using the components of a general point $P$ of $l$ form an equation in $\lambda$ by equating the perpendicular distance from $n$ to $2$ | M1 | |
| *OR*: Take a point $Q$ on $l$, e.g. $(5, 3, 3)$ and form an equation in $\lambda$ by equating the length of the projection of $QP$ onto a normal to plane $n$ to $2$ | M1 | |
| Obtain a correct modular or non-modular equation in any form | A1 | |
| Solve for $\lambda$ and obtain a position vector for $P$, e.g. $7\mathbf{i} + 5\mathbf{j} + 7\mathbf{j}$ from $\lambda = 3$ | A1 | |
| Obtain position vector of the second point, e.g. $3\mathbf{i} + \mathbf{j} - \mathbf{k}$ from $\lambda = -1$ | A1 | |
| **Total: 4** | | |10 The planes $m$ and $n$ have equations $3 x + y - 2 z = 10$ and $x - 2 y + 2 z = 5$ respectively. The line $l$ has equation $\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )$.\\
(i) Show that $l$ is parallel to $m$.\\
(ii) Calculate the acute angle between the planes $m$ and $n$.\\
(iii) A point $P$ lies on the line $l$. The perpendicular distance of $P$ from the plane $n$ is equal to 2 . Find the position vectors of the two possible positions of $P$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\hfill \mbox{\textit{CAIE P3 2018 Q10 [10]}}