Answer: \(\lambda=\mu=\frac23\), and \(\overrightarrow{OZ}=\frac13(\mathbf a+\mathbf b)\).

Write each position or direction vector in terms of the given basis vectors, then compare coefficients to find the required constants or ratio.

(a) Since \(X\) is the midpoint of \(OB\),

\(\overrightarrow{OX}=\frac12\mathbf b\).

Therefore

\(\overrightarrow{AX}=\overrightarrow{OX}-\overrightarrow{OA}=\frac12\mathbf b-\mathbf a\).

Given \(\overrightarrow{AZ}=\lambda\overrightarrow{AX}\),

\(\overrightarrow{AZ}=\lambda\left(\frac12\mathbf b-\mathbf a\right)\).

Hence

\(\overrightarrow{OZ}=\overrightarrow{OA}+\overrightarrow{AZ}\).

So

\(\overrightarrow{OZ}=\mathbf a+\lambda\left(\frac12\mathbf b-\mathbf a\right)\).

(b) Since \(Y\) is the midpoint of \(AB\),

\(\overrightarrow{OY}=\frac12(\mathbf a+\mathbf b)\).

Given \(\overrightarrow{OZ}=\mu\overrightarrow{OY}\),

\(\overrightarrow{OZ}=\frac\mu2(\mathbf a+\mathbf b)\).

(c) The two expressions for \(\overrightarrow{OZ}\) are equal:

\(\mathbf a+\lambda\left(\frac12\mathbf b-\mathbf a\right)=\frac\mu2(\mathbf a+\mathbf b)\).

Expand the left-hand side:

\((1-\lambda)\mathbf a+\frac\lambda2\mathbf b=\frac\mu2\mathbf a+\frac\mu2\mathbf b\).

Compare coefficients of \(\mathbf a\) and \(\mathbf b\):

\(1-\lambda=\frac\mu2\)

and

\(\frac\lambda2=\frac\mu2\).

The second equation gives \(\lambda=\mu\).

Substitute \(\mu=\lambda\) into the first equation:

\(1-\lambda=\frac\lambda2\).

So

\(2-2\lambda=\lambda\),

and hence

\(\lambda=\frac23\).

Therefore \(\mu=\frac23\) as well.

(d) Using

\(\overrightarrow{OZ}=\frac\mu2(\mathbf a+\mathbf b)\)

and \(\mu=\frac23\),

\(\overrightarrow{OZ}=\frac13(\mathbf a+\mathbf b)\).