Further Pure 1: Further vectorsEdexcel A-Level Further Maths: Topic test

1. 1
   The vectors $\mathbf{a}$ and $\mathbf{b}$ are given by $\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}$.

   (a)

   What is $\mathbf{a}\times\mathbf{b}$?

   [1 mark]

   • A$-\mathbf{i}-5\mathbf{j}+7\mathbf{k}$
   • B$\mathbf{i}-5\mathbf{j}-7\mathbf{k}$
   • C$-\mathbf{i}+5\mathbf{j}-7\mathbf{k}$
   • D$-\mathbf{i}+5\mathbf{j}+7\mathbf{k}$

   (b)

   What is the area of the parallelogram with adjacent sides $\mathbf{a}$ and $\mathbf{b}$?

   [1 mark]

   • A$75$
   • B$5\sqrt3$
   • C$\frac{5\sqrt3}{2}$
   • D$11$

   (c)

   Find a unit vector that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l$ has vector equation $\mathbf{r}=(3\mathbf{i}-\mathbf{j}+2\mathbf{k})+\lambda(2\mathbf{i}+3\mathbf{j}-6\mathbf{k})$.

   (a)

   What are the direction cosines of $l$?

   [1 mark]

   • A$\frac27,\ \frac37,\ -\frac67$
   • B$\frac{2}{49},\ \frac{3}{49},\ -\frac{6}{49}$
   • C$\frac{2}{\sqrt7},\ \frac{3}{\sqrt7},\ -\frac{6}{\sqrt7}$
   • D$\frac{2}{11},\ \frac{3}{11},\ -\frac{6}{11}$

   (b)

   Which is a Cartesian equation of $l$?

   [1 mark]

   • A$\frac{x+3}{2}=\frac{y-1}{3}=\frac{z+2}{-6}$
   • B$\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-2}{6}$
   • C$\frac{x-3}{2}=\frac{y+1}{3}=\frac{z-2}{-6}$
   • D$\frac{x-2}{3}=\frac{y-3}{-1}=\frac{z+6}{2}$

   (c)

   Find the acute angle between $l$ and the $x$-axis, giving your answer to the nearest $0.1^\circ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The points $A$, $B$, $C$ and $D$ have coordinates $A(1,2,0)$, $B(3,1,2)$, $C(2,4,3)$ and $D(4,2,5)$.

   (a)

   Find the area of triangle $ABC$.

   [3 marks]

   (b)

   Find the volume of the tetrahedron $ABCD$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $l$ has equation $\left(\mathbf{r}-(2\mathbf{i}+\mathbf{j}-3\mathbf{k})\right)\times(2\mathbf{i}-2\mathbf{j}+\mathbf{k})=\mathbf{0}$ and the point $P$ has coordinates $(5,4,0)$.

   (a)

   Find the perpendicular distance from $P$ to $l$.

   [6 marks]

   (b)

   The plane $\Pi$ contains $l$ and the point $P$. Find a Cartesian equation of $\Pi$, and hence find the perpendicular distance from the origin to $\Pi$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The lines $l_1$ and $l_2$ have direction vectors $\mathbf{d}_1=\mathbf{i}-2\mathbf{j}+2\mathbf{k}$ and $\mathbf{d}_2=4\mathbf{i}-3\mathbf{j}+12\mathbf{k}$ respectively.

   (a)

   What is the value of $\mathbf{d}_1\cdot\mathbf{d}_2$?

   [1 mark]

   • A$22$
   • B$34$
   • C$39$
   • D$4\mathbf{i}+6\mathbf{j}+24\mathbf{k}$

   (b)

   What is the cosine of the acute angle between $l_1$ and $l_2$?

   [1 mark]

   • A$\frac{34}{39}$
   • B$\frac{34}{507}$
   • C$\frac{34}{16}$
   • D$\frac{34}{13}$

   (c)

   Find the acute angle between $l_1$ and $l_2$, giving your answer to the nearest $0.1^\circ$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A parallelepiped has edges represented by the vectors $\mathbf{a}=\mathbf{i}+2\mathbf{k}$, $\mathbf{b}=3\mathbf{j}+\mathbf{k}$ and $\mathbf{c}=2\mathbf{i}+\mathbf{j}$.

   (a)

   What is $\mathbf{b}\times\mathbf{c}$?

   [1 mark]

   • A$\mathbf{i}-2\mathbf{j}+6\mathbf{k}$
   • B$-\mathbf{i}-2\mathbf{j}-6\mathbf{k}$
   • C$-\mathbf{i}+2\mathbf{j}-6\mathbf{k}$
   • D$\mathbf{i}+2\mathbf{j}-6\mathbf{k}$

   (b)

   What is the volume of the parallelepiped?

   [1 mark]

   • A$-13$
   • B$26$
   • C$\frac{13}{6}$
   • D$13$

   (c)

   Taking the face with edges $\mathbf{b}$ and $\mathbf{c}$ as the base, find the perpendicular height of the parallelepiped, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The line $l$ passes through the points $A(1,-2,3)$ and $B(2,2,11)$.

   (a)

   Find the direction ratios and the direction cosines of $l$.

   [3 marks]

   (b)

   Write the equation of $l$ in the form $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$ and hence find the perpendicular distance from the origin to $l$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The lines $l_1$ and $l_2$ have vector equations $\mathbf{r}=(2\mathbf{i}+\mathbf{j})+s(\mathbf{i}+2\mathbf{j}+2\mathbf{k})$ and $\mathbf{r}=(\mathbf{i}+3\mathbf{j}+\mathbf{k})+t(2\mathbf{i}-\mathbf{j}+2\mathbf{k})$.

   (a)

   Show that $l_1$ and $l_2$ are skew lines.

   [6 marks]

   (b)

   Find the shortest distance between $l_1$ and $l_2$, giving your answer to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions