Geometry and trigonometryIB Maths: Applications and Interpretation HL: Topic test

1. 1
   In a warehouse, with coordinates in metres, a sensor is at $S(1,-2,4)$ and a camera is at $C(5,2,2)$.

   (a)

   Find the distance $SC$.

   [1 mark]

   • A$36$ m
   • B$6$ m
   • C$\sqrt{68}$ m
   • D$3$ m

   (b)

   Find the midpoint of $[SC]$.

   [1 mark]

   • A$(2,2,-1)$
   • B$(6,0,6)$
   • C$(4,4,-2)$
   • D$(3,0,3)$

   (c)

   Find the unit vector in the direction of $\overrightarrow{SC}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A simple graph $G$ has six vertices, with degrees $1, 2, 2, 3, 4, 4$.

   (a)

   How many edges does $G$ have?

   [1 mark]

   • A$8$
   • B$16$
   • C$6$
   • D$15$

   (b)

   How many more edges would have to be added to $G$ to make it the complete graph $K_6$?

   [1 mark]

   • A$15$
   • B$8$
   • C$7$
   • D$5$

   (c)

   State, with a reason, whether $G$ could be a tree.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A council plans its emergency response using three ambulance stations $X(1,0)$, $Y(7,4)$ and $Z(1,6)$, where the units on the map are kilometres. A Voronoi diagram is drawn with the stations as its sites.

   (a)

   Find the equation of the perpendicular bisector of $[XY]$. Give your answer in the form $ax+by=c$, where $a,b,c\in\mathbb{Z}$.

   [3 marks]

   (b)

   The three cell boundaries of the diagram meet at a vertex $V$. Find the coordinates of $V$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two straight cables are modelled by the lines $L_1:\ \mathbf{r}=\begin{pmatrix}1 \\ 2 \\ 0\end{pmatrix}+s\begin{pmatrix}2 \\ 1 \\ 2\end{pmatrix}$ and $L_2:\ \mathbf{r}=\begin{pmatrix}7 \\ 6 \\ 5\end{pmatrix}+t\begin{pmatrix}2 \\ 2 \\ 1\end{pmatrix}$, where $s,t\in\mathbb{R}$ and the coordinates are in metres.

   (a)

   Show that the two cables meet, and find the coordinates of the point $X$ where they meet.

   [6 marks]

   (b)

   (i) Find the acute angle, in degrees, between the two cables. (ii) The cables are fixed to the ground at $P(1,2,0)$ on $L_1$ and $Q(7,6,5)$ on $L_2$. Use a vector product to find the area of triangle $PXQ$, where $X$ is the point of intersection.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A garden sprinkler at $O$ waters a sector of a circle of radius $5$ m. The angle of the sector at $O$ is $1.2$ radians.

   (a)

   Find the length of the curved edge of the watered sector.

   [1 mark]

   • A$4.17$ m
   • B$3$ m
   • C$6$ m
   • D$0.105$ m

   (b)

   Find the area of the watered sector.

   [1 mark]

   • A$15$ m$^2$
   • B$30$ m$^2$
   • C$6$ m$^2$
   • D$0.262$ m$^2$

   (c)

   The sprinkler is reset to water a sector of angle $80^\circ$ with the same radius. Find the area now watered.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The angle $\theta$ satisfies $\cos\theta=-\frac{5}{13}$, where $\frac{\pi}{2}<\theta<\pi$.

   (a)

   Find the value of $\sin\theta$.

   [1 mark]

   • A$-\frac{12}{13}$
   • B$\frac{5}{13}$
   • C$-\frac{5}{12}$
   • D$\frac{12}{13}$

   (b)

   Find the value of $\tan\theta$.

   [1 mark]

   • A$\frac{12}{5}$
   • B$-\frac{12}{5}$
   • C$-\frac{5}{12}$
   • D$\frac{5}{12}$

   (c)

   Find the other solution of $\cos x=-\frac{5}{13}$ in the interval $0\le x\le2\pi$. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   On a design app, a logo is a triangle. Reflection in the line $y=x$ is represented by $\mathbf{R}=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ and a stretch by factor $3$ parallel to the $x$-axis and factor $2$ parallel to the $y$-axis is represented by $\mathbf{S}=\begin{pmatrix}3 & 0 \\ 0 & 2\end{pmatrix}$. The logo is reflected and then stretched. This combined transformation is represented by the matrix $\mathbf{T}$.

   (a)

   Find $\mathbf{T}$, and hence find the image of the point $(4,-1)$.

   [3 marks]

   (b)

   (i) The logo has area $5$ cm$^2$. Find the area of its image under $\mathbf{T}$. (ii) The image of a point $P$ under $\mathbf{T}$ is $(6,-4)$. Find the coordinates of $P$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A park has five attractions $A$, $B$, $C$, $D$ and $E$ joined by two-way paths. The walking times in minutes are $AB=4$, $AC=6$, $BC=3$, $BD=6$, $CD=2$, $CE=7$ and $DE=4$.

   (a)

   A groundskeeper must walk along every path at least once, starting and ending at the same attraction. Use the Chinese postman algorithm to find the shortest possible walking time, and explain why some paths have to be walked twice.

   [6 marks]

   (b)

   Ignore the walking times. The adjacency matrix of the park, with the attractions in the order $A, B, C, D, E$, is $\mathbf{M}$. (i) Write down $\mathbf{M}$. (ii) Use your GDC to find the number of walks of length $3$ from $A$ to $E$. (iii) Find the number of walks from $A$ to $E$ of length at most $4$.

   [6 marks]

   Total for question 8: 12 marks

End of questions