1. 1
   The points $A$ and $B$ have coordinates $A(2,\,-1,\,5)$ and $B(8,\,3,\,-7)$. The point $M$ is the midpoint of $[AB]$.

   (a)

   Find the distance $AB$.

   [1 mark]

   • A$22$
   • B$\sqrt{108}$
   • C$196$
   • D$14$

   (b)

   Find the coordinates of $M$.

   [1 mark]

   • A$(3,\,2,\,-6)$
   • B$(10,\,2,\,-2)$
   • C$(5,\,1,\,-1)$
   • D$(5,\,1,\,6)$

   (c)

   The point $D(p,\,1,\,3)$, where $p>5$, is such that $DM=5$. Find the value of $p$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A solid ornament is made from a right circular cone joined to a hemisphere. The flat face of the cone is the same circle as the flat face of the hemisphere, and this circle has radius 3 cm. The perpendicular height of the cone is 4 cm.

   (a)

   Find the slant height of the cone.

   [1 mark]

   • A$5$ cm
   • B$7$ cm
   • C$\sqrt{7}$ cm
   • D$25$ cm

   (b)

   Find the volume of the ornament, in cm$^3$.

   [1 mark]

   • A$18\pi$
   • B$30\pi$
   • C$48\pi$
   • D$54\pi$

   (c)

   Find the total surface area of the ornament. Give your answer as an exact multiple of $\pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A right pyramid $VABCD$ has a square base $ABCD$ with sides of length 10 cm. The apex $V$ is vertically above $O$, the centre of the base, and $VO = 12$ cm. The point $N$ is the midpoint of $[BC]$.

   (a)

   Find the angle between the edge $[VA]$ and the base $ABCD$. A calculator may be used.

   [3 marks]

   (b)

   Find the total surface area of the pyramid.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A warehouse is in the shape of a cuboid 20 m long, 12 m wide and 9 m high. Coordinates are set up in metres with one corner of the floor at the origin $O(0,0,0)$, so that the other floor corners are $P(20,0,0)$, $Q(20,12,0)$ and $R(0,12,0)$, and the $z$-axis is vertical. A security camera is fixed at $C(0,0,9)$, on the ceiling directly above $O$. A calculator may be used.

   (a)

   (i) Find the distance from the camera to the corner $Q$.
   (ii) Find the angle between the line $CQ$ and the floor of the warehouse.
   (iii) A light is to be hung at the midpoint of $[CQ]$. Write down its coordinates.

   [6 marks]

   (b)

   A straight cable is fixed from $C$ to a point $S$ on the floor edge $[PQ]$. The cable is 23 m long.
   (i) Find the distance $PS$.
   (ii) Find the angle between the cable and the floor, and hence determine whether the cable makes a steeper angle with the floor than the line $CQ$.

   [6 marks]

   Total for question 4: 12 marks

End of questions