IB AA HL May 2026 P2 TZA Q11

A child’s toy is made from a solid right cone and a solid hemisphere of equal radius. The circular faces of each solid are joined to form a single circle \(C\). The point \(P\) lies on the circumference of \(C\), and the point \(V\) is the vertex of the cone as shown in the following diagram.

In this question, all units are given in cm.

The toy is positioned so that the circle \(C\) lies completely within the plane defined by \(x+2y-z=5\).

The point \(P\) has coordinates \((-3,6,4)\) and the point \(V\) has coordinates \((4,16,-5)\).

1. Find a vector equation for the line passing through \(V\) and the centre of \(C\).[3]
2. Find the coordinates of the centre of \(C\).[3]
3. 1. Show that the radius of \(C\) is \(\sqrt{14}\text{ cm}\).
   2. Find the total surface area of the toy.[6]
4. The toy is rotated so that the position of the vertex is fixed at \(V\) and the image \(P\prime\) of the point \(P\) has coordinates \((-9,11,1)\).
   Find angle \(P\hat{V}P\prime\), giving your answer in degrees.[3]
5. Find an equation of the plane containing the points \(P\), \(V\) and \(P\prime\), giving your answer in the form \(ax+by+cz=d\), where \(a,b,c,d\in\mathbb{Z}\).[4]