M1: Vectors in mechanicsEdexcel International A Level Maths: Topic test

1. 1
   A force $\mathbf{F}=(8\mathbf{i}-6\mathbf{j})$ N acts on a particle, where $\mathbf{i}$ is a unit vector in the horizontal direction and $\mathbf{j}$ is a unit vector vertically upwards.

   (a)

   Find the magnitude of $\mathbf{F}$.

   [1 mark]

   • A$2$ N
   • B$14$ N
   • C$10$ N
   • D$100$ N

   (b)

   The direction of $\mathbf{F}$ is

   [1 mark]

   • A$36.9^\circ$ below the horizontal.
   • B$53.1^\circ$ below the horizontal.
   • C$36.9^\circ$ above the horizontal.
   • D$53.1^\circ$ above the horizontal.

   (c)

   Find, in the form $p\mathbf{i}+q\mathbf{j}$, the force of magnitude 5 N that acts in the same direction as $\mathbf{F}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two forces $\mathbf{P}=(2\mathbf{i}+5\mathbf{j})$ N and $\mathbf{Q}=(-6\mathbf{i}+3\mathbf{j})$ N act on a particle.

   (a)

   Find the resultant of $\mathbf{P}$ and $\mathbf{Q}$.

   [1 mark]

   • A$(4\mathbf{i}-8\mathbf{j})$ N
   • B$(8\mathbf{i}+2\mathbf{j})$ N
   • C$(-12\mathbf{i}+15\mathbf{j})$ N
   • D$(-4\mathbf{i}+8\mathbf{j})$ N

   (b)

   Find the magnitude of the resultant.

   [1 mark]

   • A$12$ N
   • B$8.94$ N
   • C$80$ N
   • D$8$ N

   (c)

   A third force $\mathbf{R}$ is added so that the resultant of $\mathbf{P}$, $\mathbf{Q}$ and $\mathbf{R}$ is $(3\mathbf{i}-\mathbf{j})$ N. Find $\mathbf{R}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A hiker walks from a camp $O$. For 2 hours she walks with constant velocity $(3\mathbf{i}+4\mathbf{j})$ km h$^{-1}$, and then for 3 hours she walks with constant velocity $(-4\mathbf{i}+6\mathbf{j})$ km h$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north respectively.

   (a)

   Find the position vector of the hiker relative to $O$ at the end of the 5 hours.

   [3 marks]

   (b)

   Find the distance of the hiker from $O$ at the end of the 5 hours, and the bearing of the hiker from $O$, giving your answers to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are directed due east and due north respectively. At 12:00 a ferry $F$ is at the point with position vector $(-3\mathbf{i}+2\mathbf{j})$ km relative to a fixed origin $O$, and moves with constant velocity $(4\mathbf{i}+5\mathbf{j})$ km h$^{-1}$. At 12:00 a lifeboat $L$ is at the point with position vector $(9\mathbf{i}-4\mathbf{j})$ km and moves with constant velocity $(-2\mathbf{i}+8\mathbf{j})$ km h$^{-1}$. The time $t$ is measured in hours after 12:00.

   (a)

   Show that the ferry and the lifeboat meet, and find the time at which they meet and the position vector of the point where they meet.

   [6 marks]

   (b)

   (i) Find the speed of the lifeboat, giving your answer to 3 significant figures.
   (ii) Find the distance between the ferry and the lifeboat at 13:00, giving your answer to 3 significant figures.
   (iii) Find the bearing of the ferry from the lifeboat at 13:00, giving your answer to the nearest degree.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A force $\mathbf{F}$ of magnitude 12 N acts on a particle in a direction $60^\circ$ above the direction of the unit vector $\mathbf{i}$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

   What is the component of $\mathbf{F}$ in the direction of $\mathbf{i}$?

   [1 mark]

   • A$10.4$ N
   • B$6$ N
   • C$12$ N
   • D$7.2$ N

   (b)

   A second force $\mathbf{G}=(-4\mathbf{i}+2\mathbf{j})$ N also acts on the particle. What is the magnitude of the resultant of $\mathbf{F}$ and $\mathbf{G}$?

   [1 mark]

   • A$16.5$ N
   • B$10.2$ N
   • C$8.25$ N
   • D$12.6$ N

   (c)

   Find the angle between the resultant of $\mathbf{F}$ and $\mathbf{G}$ and the direction of $\mathbf{i}$, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A cyclist's velocity changes from $(6\mathbf{i}+2\mathbf{j})$ m s$^{-1}$ to $(-2\mathbf{i}+10\mathbf{j})$ m s$^{-1}$ in 4 s, with constant acceleration. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are directed due east and due north respectively.

   (a)

   Find the acceleration of the cyclist.

   [1 mark]

   • A$(-2\mathbf{i}+2\mathbf{j})$ m s$^{-2}$
   • B$(-8\mathbf{i}+8\mathbf{j})$ m s$^{-2}$
   • C$(2\mathbf{i}-2\mathbf{j})$ m s$^{-2}$
   • D$(2\mathbf{i}+6\mathbf{j})$ m s$^{-2}$

   (b)

   Find the magnitude of the acceleration of the cyclist.

   [1 mark]

   • A$4$ m s$^{-2}$
   • B$2$ m s$^{-2}$
   • C$2.83$ m s$^{-2}$
   • D$11.3$ m s$^{-2}$

   (c)

   Find the speed of the cyclist 6 s after the velocity was $(6\mathbf{i}+2\mathbf{j})$ m s$^{-1}$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Two forces $(5\mathbf{i}-2\mathbf{j})$ N and $(\lambda\mathbf{i}+7\mathbf{j})$ N act on a particle, where $\lambda$ is a constant and $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors. The magnitude of the resultant of the two forces is 13 N.

   (a)

   Find the two possible values of $\lambda$.

   [3 marks]

   (b)

   Given that $\lambda>0$, find the unit vector in the direction of the resultant, and find the acute angle between the resultant and the direction of $\mathbf{j}$, giving your answer to 1 decimal place.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two tugs pull a barge. Tug $A$ exerts a force of magnitude 800 N on a bearing of $030^\circ$, and tug $B$ exerts a force of magnitude 600 N on a bearing of $120^\circ$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are directed due east and due north respectively. Water resistance acts on the barge, and the barge moves at a constant speed of 2.5 m s$^{-1}$ in the direction of the resultant of the two tug forces.

   (a)

   Express each tug force in the form $p\mathbf{i}+q\mathbf{j}$, and hence find the resultant of the two tug forces and its magnitude.

   [6 marks]

   (b)

   (i) Find the velocity of the barge in the form $p\mathbf{i}+q\mathbf{j}$, with $p$ and $q$ to 3 significant figures.
   (ii) The barge passes through $O$ at time $t=0$. Find its position vector 40 s later, with components to 3 significant figures.
   (iii) Find the bearing on which the barge is moving, giving your answer to 1 decimal place.

   [6 marks]

   Total for question 8: 12 marks

End of questions