# Question 9:

## Part (i) EITHER method:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute coordinates of general point of $l$ in equation of plane and equate constant terms, obtaining an equation in $b$ and $c$ | M1* | |
| Obtain a correct equation, e.g. $8 + 2b - c = 1$ | A1 | |
| Equate the coefficient of $t$ to zero, obtaining an equation in $b$ and $c$ | M1* | |
| Obtain a correct equation, e.g. $4 - b - 2c = 0$ | A1 | |

## Part (i) OR method:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitute $(4, 2, -1)$ in the plane equation | M1* | |
| Obtain a correct equation in $b$ and $c$, e.g. $2b - c = -7$ | A1 | |
| **EITHER** Find a second point on $l$ and obtain an equation in $b$ and $c$ | M1* | |
| Obtain a correct equation in $b$ and $c$, e.g. $b + 2c = 4$ | A1 | |
| **OR** Calculate scalar product of a direction vector for $l$ and a vector normal for the plane and equate to zero | M1* | |
| Obtain a correct equation for $b$ and $c$ | A1 | |
| Solve for $b$ or for $c$ | M1(dep*) | |
| Obtain $b = -2$ and $c = 3$ | A1 | **[6 marks]** |

## Part (ii) EITHER method:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Find $\overrightarrow{PQ}$ for a point $Q$ on $l$ with parameter $t$, e.g. $4\mathbf{i} - 5\mathbf{k} + t(2\mathbf{i} - \mathbf{j} - 2\mathbf{k})$ | B1 | |
| Calculate scalar product of $\overrightarrow{PQ}$ and a direction vector for $l$ and equate to zero | M1 | |
| Solve and obtain $t = -2$ | A1 | |
| Carry out a complete method for finding the length of $\overrightarrow{PQ}$ | M1 | |
| Obtain the given answer $\sqrt{5}$ correctly | A1 | |

## Part (ii) OR 1:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Calling $(4, 2, -1)$ $A$, state $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) in component form, e.g. $4\mathbf{i} - 5\mathbf{k}$ | B1 | |
| Calculate vector product of $\overrightarrow{AP}$ and a direction vector for $l$, e.g. $(4\mathbf{i} - 5\mathbf{k}) \times (2\mathbf{i} - \mathbf{j} - 2\mathbf{k})$ | M1 | |
| Obtain correct answer, e.g. $-5\mathbf{i} - 2\mathbf{j} - 4\mathbf{k}$ | A1 | |
| Divide modulus of the product by that of the direction vector | M1 | |
| Obtain the given answer correctly | A1 | |

## Part (ii) OR 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) in component form | B1 | |
| Use a scalar product to find the projection of $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) on $l$ | M1 | |
| Obtain correct answer in any form, e.g. $\dfrac{18}{\sqrt{9}}$ | A1 | |
| Use Pythagoras to find the perpendicular | M1 | |
| Obtain the given answer correctly | A1 | |

## Part (ii) OR 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) in component form | B1 | |
| Use a scalar product to find the cosine of $PAQ$ | M1 | |
| Obtain correct answer in any form, e.g. $\dfrac{18}{\sqrt{41}\cdot\sqrt{9}}$ | A1 | |
| Use trig to find the perpendicular | M1 | |
| Obtain the given answer correctly | A1 | |

## Part (ii) OR 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| State $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) in component form | B1 | |
| Find a second point $B$ on $l$ and use the cosine rule in triangle $APB$ to find the cosine of $A$, $B$ or $P$, or use a vector product to find the area of $APB$ | M1 | |
| Obtain correct answer in any form | A1 | |
| Use trig or area formula to find the perpendicular | M1 | |
| Obtain the given answer correctly | A1 | |

## Part (ii) OR 5:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Find $\overrightarrow{PQ}$ for a point $Q$ on $l$ with parameter $t$, e.g. $4\mathbf{i} - 5\mathbf{k} + t(2\mathbf{i} - \mathbf{j} - 2\mathbf{k})$ | B1 | |
| Use correct method to express $PQ^2$ (or $PQ$) in terms of $t$ | M1 | |
| Obtain a correct expression in any form, e.g. $(4+2t)^2 + (-t)^2 + (-5-2t)^2$ | A1 | |
| Carry out a complete method for finding its minimum | M1 | |
| Obtain the given answer correctly | A1 | **[5 marks]** |

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