Find the z-transform

Question

I have these expressions for which I need the z-transformed functions. Please help. The Question Expressions

My attempt for the first question

My attempt for the second question

My attempt for the third question

Answer 1

As the OP has included their efforts, I'm posting this answer.
For $|z|>|r|$, $$\mathcal Z \{r^n\cos(\omega_0n)u(n)\}=\frac{1-rz^{-1}\cos\omega_0}{1-2rz^{-1}\cos\omega_0+r^2z^{-2}}$$

and

$$\mathcal Z \{r^n\sin(\omega_0n)u(n)\}=\frac{rz^{-1}\sin\omega_0}{1-2rz^{-1}\cos\omega_0+r^2z^{-2}}$$

So for the question d, let $r = (ae^{-b})$

For the question e, $$x(n) = (n-1)a^{n-2} = a^{-2}(na^n-a^n)\\ \Rightarrow \mathcal Z \{x(n)\} = a^{-2}\left[-z\frac d{dz}\left(\frac{z}{z-a}\right)-\frac{z}{z-a}\right]=\frac{z(2a-z)}{a^2(a-z)^2}$$

For $f$, first simplify it.

$$\begin{align}x(n) &= (n-3)\left(\frac14\right)^{n-2}\cos\left(\frac{\pi n}2 - \frac\pi2\right)u(n) = 16(n-3)\left(\frac14\right)^n\sin\left(\frac{\pi n}2\right)u(n)\\& = 16\left[n\left(\frac14\right)^n\sin\left(\frac{\pi n}2\right)u(n)-3\left(\frac14\right)^n\sin\left(\frac{\pi n}2\right)u(n)\right]\end{align} $$

Now, for $\omega_0 = \frac{\pi}{2}$

$$\mathcal Z \left\{\left(\frac14\right)^n\sin\left(\frac{\pi n}2\right)\right\}=\frac{\frac{z^{-1}}{4}\cdot1}{1-2\cdot 0 + \frac{z^{-2}}{16}}=\frac{4z^{-1}}{16+z^{-2}}$$

Then use, $\mathcal Z \{nx(n)\} = -z\frac{d}{dz}X(z)$

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Your answers for a and b are correct. The answer of c is incorrect.

$$\begin{align}\mathcal Z \left\{2^nu(-n)+\left(\frac14\right)^nu(n-2)\right\}&=\sum_{n=-\infty}^\infty2^nu(-n)z^{-n} + \left(\frac14\right)^2\mathcal Z\left\{\left(\frac14\right)^{n-2}u(n-2)\right\}\\ &=\sum_{n=-\infty}^\infty2^nu(-n)z^{-n} + \left(\frac14\right)^2z^{-2}\mathcal Z\left\{\left(\frac14\right)^{n}u(n)\right\}\\ &=\sum_{n=-\infty}^02^nz^{-n} + \frac1{16}z^{-2}\frac{z}{z-\frac14} \\ & =\sum_{n=0}^\infty \left(\frac z2\right)^n+\frac{1}{4z(4z-1)} \\ &=\frac{1}{1-\frac z2}+\frac{1}{4z(4z-1)} \\ &= \frac{2}{2-z}+\frac{1}{4z(4z-1)} \end{align}$$

and ROC : $\frac14 < |z| <2$