Core Pure: Further vectorsEdexcel A-Level Further Maths: Topic test

1. 1
   The line $l$ passes through the points $A(3,-2,1)$ and $B(7,0,5)$.

   (a)

   Which is a Cartesian equation of $l$?

   [1 mark]

   • A$\frac{x+3}{2}=\frac{y-2}{1}=\frac{z+1}{2}$
   • B$\frac{x-3}{2}=\frac{y+2}{1}=\frac{z-1}{2}$
   • C$\frac{x-3}{4}=\frac{y+2}{2}=\frac{z-1}{5}$
   • D$\frac{x-7}{3}=\frac{y}{-2}=\frac{z-5}{1}$

   (b)

   Which point lies on $l$?

   [1 mark]

   • A$(-1,-4,-1)$
   • B$(1,-3,1)$
   • C$(-1,-3,-3)$
   • D$(-1,-4,-3)$

   (c)

   The line $m$ has vector equation $\mathbf{r}=\begin{pmatrix}2\\-3\\0\end{pmatrix}+\mu\begin{pmatrix}3\\2\\3\end{pmatrix}$. Given that $l$ and $m$ intersect, find the coordinates of their point of intersection.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The plane $\Pi_1$ has equation $x-2y+2z=5$ and the plane $\Pi_2$ has equation $2x+y-2z=3$.

   (a)

   What is the scalar product of a normal vector to $\Pi_1$ and a normal vector to $\Pi_2$?

   [1 mark]

   • A$-4$
   • B$4$
   • C$0$
   • D$-9$

   (b)

   What is the acute angle between $\Pi_1$ and $\Pi_2$?

   [1 mark]

   • A$26.4^\circ$
   • B$116.4^\circ$
   • C$63.6^\circ$
   • D$75.5^\circ$

   (c)

   Find a vector equation of the line through the point $(2,1,-1)$ that is perpendicular to $\Pi_1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l$ has equation $\mathbf{r}=\begin{pmatrix}2\\-1\\0\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\-2\end{pmatrix}$ and the plane $\Pi$ has equation $2x-y+3z=-9$.

   (a)

   Find the coordinates of the point where $l$ meets $\Pi$.

   [3 marks]

   (b)

   The point $A$ has coordinates $(1,1,2)$. Find the perpendicular distance from $A$ to $\Pi$, giving your answer in the form $k\sqrt{14}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lines $l_1$ and $l_2$ have equations $l_1$: $\mathbf{r}=\begin{pmatrix}1\\2\\-1\end{pmatrix}+s\begin{pmatrix}2\\0\\1\end{pmatrix}$ and $l_2$: $\mathbf{r}=\begin{pmatrix}0\\3\\1\end{pmatrix}+t\begin{pmatrix}1\\-1\\2\end{pmatrix}$.

   (a)

   Find the shortest distance between $l_1$ and $l_2$, and deduce that the lines are skew.

   [6 marks]

   (b)

   The plane $\Pi$ contains $l_1$ and is parallel to $l_2$. Find a Cartesian equation of $\Pi$. The line $l_3$ has direction $(2,1,1)$. Find the acute angle between $l_3$ and $\Pi$, giving your answer to 1 decimal place.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The line $l$ has Cartesian equation $\frac{x+1}{3}=\frac{2-y}{2}=\frac{z-4}{6}$.

   (a)

   Which is a direction vector of $l$?

   [1 mark]

   • A$\begin{pmatrix}3\\2\\6\end{pmatrix}$
   • B$\begin{pmatrix}-1\\2\\4\end{pmatrix}$
   • C$\begin{pmatrix}3\\-2\\6\end{pmatrix}$
   • D$\begin{pmatrix}-3\\-2\\-6\end{pmatrix}$

   (b)

   What is the acute angle between $l$ and the $x$-axis, to 1 decimal place?

   [1 mark]

   • A$64.6^\circ$
   • B$25.4^\circ$
   • C$115.4^\circ$
   • D$33.7^\circ$

   (c)

   Show that the point $(8,-4,22)$ lies on $l$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The plane $\Pi$ passes through the points $P(1,0,0)$, $Q(0,2,0)$ and $R(0,0,3)$.

   (a)

   Which vector is perpendicular to $\Pi$?

   [1 mark]

   • A$\begin{pmatrix}1\\2\\3\end{pmatrix}$
   • B$\begin{pmatrix}2\\3\\6\end{pmatrix}$
   • C$\begin{pmatrix}3\\2\\1\end{pmatrix}$
   • D$\begin{pmatrix}6\\3\\2\end{pmatrix}$

   (b)

   What is the acute angle between $\Pi$ and the $xy$-plane, to 1 decimal place?

   [1 mark]

   • A$16.6^\circ$
   • B$73.4^\circ$
   • C$106.6^\circ$
   • D$52.9^\circ$

   (c)

   The point $(k,1,1)$ lies on $\Pi$. Find the value of $k$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The line $l$ has equation $\mathbf{r}=\begin{pmatrix}0\\1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\2\end{pmatrix}$ and the point $P$ has coordinates $(4,3,7)$.

   (a)

   The point $F$ is the foot of the perpendicular from $P$ to $l$. Find the coordinates of $F$.

   [3 marks]

   (b)

   Hence find the perpendicular distance from $P$ to $l$, and the coordinates of the point $P'$, the reflection of $P$ in $l$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The line $l$ has Cartesian equation $\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z}{2}$ and the plane $\Pi$ has equation $x+4y-8z=7$.

   (a)

   Find the coordinates of the point where $l$ meets $\Pi$, and the acute angle between $l$ and $\Pi$, giving your answer to 1 decimal place.

   [6 marks]

   (b)

   The point $B(3,-4,2)$ lies on $l$. Find the perpendicular distance from $B$ to $\Pi$, and hence find the distance along $l$ from $B$ to the point where $l$ meets $\Pi$.

   [6 marks]

   Total for question 8: 12 marks

End of questions