Vectors in problem solvingAQA A-Level Maths: Subtopic test

1. 1
   A particle of mass $2$ kg is acted on by two forces, $\mathbf F_1=(3\mathbf i+2\mathbf j)$ N and $\mathbf F_2=(-5\mathbf i+4\mathbf j)$ N, where $\mathbf i$ and $\mathbf j$ are perpendicular unit vectors.

   (a)

   Find the resultant of $\mathbf F_1$ and $\mathbf F_2$.

   [1 mark]

   • A$(-2\mathbf i+6\mathbf j)$ N
   • B$(8\mathbf i-2\mathbf j)$ N
   • C$(-2\mathbf i-2\mathbf j)$ N
   • D$(2\mathbf i-6\mathbf j)$ N

   (b)

   A third force $\mathbf F_3$ is now applied so that the particle is in equilibrium. Find $\mathbf F_3$.

   [1 mark]

   • A$(-2\mathbf i+6\mathbf j)$ N
   • B$(2\mathbf i+6\mathbf j)$ N
   • C$(2\mathbf i-6\mathbf j)$ N
   • D$\mathbf 0$ N

   (c)

   The force $\mathbf F_3$ is removed, so that only $\mathbf F_1$ and $\mathbf F_2$ act. Find the acceleration of the particle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The points $P$, $Q$ and $R$ have position vectors $\mathbf p=\mathbf i+2\mathbf j$, $\mathbf q=3\mathbf i+8\mathbf j$ and $\mathbf r=k\mathbf i+14\mathbf j$, where $k$ is a constant.

   (a)

   Find $\overrightarrow{PQ}$.

   [1 mark]

   • A$-2\mathbf i-6\mathbf j$
   • B$2\mathbf i+6\mathbf j$
   • C$4\mathbf i+10\mathbf j$
   • D$2\mathbf i+10\mathbf j$

   (b)

   Which statement must be true if $P$, $Q$ and $R$ lie on a straight line?

   [1 mark]

   • A$|\overrightarrow{PQ}|=|\overrightarrow{QR}|$
   • B$\overrightarrow{PQ}$ is perpendicular to $\overrightarrow{PR}$
   • C$\overrightarrow{PR}=\overrightarrow{PQ}$
   • D$\overrightarrow{PR}=\lambda\overrightarrow{PQ}$ for some scalar $\lambda$

   (c)

   Given that $P$, $Q$ and $R$ are collinear, find the value of $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Relative to a port $O$, the unit vectors $\mathbf i$ and $\mathbf j$ point due east and due north, and distances are in kilometres. At noon a ship is at the point with position vector $(8\mathbf i+6\mathbf j)$ km. It moves with constant velocity $(-2\mathbf i+\mathbf j)$ km h⁻¹.

   (a)

   Find the position vector of the ship at 15:00 and its distance from $O$ at that time.

   [3 marks]

   (b)

   At time $T$ hours after noon the ship is due north of the port. Find $T$ and the distance of the ship from the port at that time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In triangle $OAB$, $\overrightarrow{OA}=\mathbf a$ and $\overrightarrow{OB}=\mathbf b$, where $\mathbf a$ and $\mathbf b$ are not parallel. The point $M$ lies on $AB$ so that $AM:MB=1:3$, and $C$ is the midpoint of $OA$.

   (a)

   (i) Find $\overrightarrow{AB}$. (ii) Show that $\overrightarrow{OM}=\frac34\mathbf a+\frac14\mathbf b$. (iii) The point $N$ is such that $\overrightarrow{ON}=3\mathbf a+\mathbf b$. Show that $O$, $M$ and $N$ lie on a straight line.

   [6 marks]

   (b)

   The lines $OM$ and $CB$ meet at the point $X$. Find $\overrightarrow{OX}$ in terms of $\mathbf a$ and $\mathbf b$.

   [6 marks]

   Total for question 4: 12 marks

End of questions