2025 CHS PRELIMS P2 Q7

\(ABC\) is a triangle. \(\overrightarrow{AB}=\mathbf p\) and \(\overrightarrow{AC}=\mathbf q\). \(X\) is the midpoint of \(AB\). \(Z\) and \(Y\) are points on \(XC\) and \(AC\) respectively such that \(\overrightarrow{XZ}=\dfrac23\overrightarrow{XC}\) and \(\overrightarrow{YC}=k\overrightarrow{AC}\).

1. Express the following vectors in terms of \(\mathbf p\) and/or \(\mathbf q\), giving your answers in the simplest terms.
   1. \(\overrightarrow{AX}\),[1]
   2. \(\overrightarrow{CX}\).[2]
2. Express \(\overrightarrow{YZ}\) in terms of \(k\), \(\mathbf p\) and \(\mathbf q\).[2]
3. Given that \(YZ\) is parallel to \(AX\), find the numerical value of
   1. \(k\),[1]
   2. \(\dfrac{YZ}{AB}\),[1]
   3. \(\dfrac{\text{Area of}\hspace{0.5em}\triangle XYC}{\text{Area of}\hspace{0.5em}\triangle ABC}\).[1]