1. **Expressing $Z$ in exponential form given $\frac{Z + 1}{Z + 2 - j} = j$** We start with the equation: $$\frac{Z + 1}{Z + 2 - j} = j$$ Multiply both sides by the denominator: $$Z + 1 = j(Z + 2 - j)$$ Expand the right side: $$Z + 1 = jZ + 2j - j^2$$ Recall that $j^2 = -1$, so: $$Z + 1 = jZ + 2j + 1$$ Bring all terms to one side: $$Z - jZ + 1 - 1 - 2j = 0$$ $$Z(1 - j) - 2j = 0$$ Solve for $Z$: $$Z = \frac{2j}{1 - j}$$ Multiply numerator and denominator by the conjugate of the denominator $(1 + j)$: $$Z = \frac{2j(1 + j)}{(1 - j)(1 + j)} = \frac{2j + 2j^2}{1 + 1} = \frac{2j - 2}{2} = -1 + j$$ Now express $Z = -1 + j$ in exponential form: Calculate magnitude: $$|Z| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$$ Calculate argument (angle): $$\theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1)$$ Since real part is negative and imaginary part positive, $Z$ lies in the second quadrant, so: $$\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$ Therefore, $$Z = \sqrt{2} e^{j \frac{3\pi}{4}}$$ 2. **Finding $V_o$ in exponential form given:** $$V_o = V_2 + \frac{Z_2}{Z_1 + Z_2}(V_1 - V_2)$$ Given: $$V_1 = 2 \angle 0 = 2 e^{j0}$$ $$V_2 = e^{j \pi/4}$$ $$Z_1 = 1 + j2$$ $$Z_2 = 2 + j3$$ Calculate $Z_1 + Z_2$: $$Z_1 + Z_2 = (1 + 2) + j(2 + 3) = 3 + j5$$ Calculate the fraction: $$\frac{Z_2}{Z_1 + Z_2} = \frac{2 + j3}{3 + j5}$$ Multiply numerator and denominator by conjugate of denominator: $$= \frac{(2 + j3)(3 - j5)}{(3 + j5)(3 - j5)} = \frac{6 - 10j + 9j - 15j^2}{9 + 25} = \frac{6 - j + 15}{34} = \frac{21 - j}{34}$$ Simplify: $$= \frac{21}{34} - j \frac{1}{34}$$ Calculate $V_1 - V_2$: $$2 e^{j0} - e^{j \pi/4} = 2 - \left(\cos \frac{\pi}{4} + j \sin \frac{\pi}{4}\right) = 2 - \frac{\sqrt{2}}{2} - j \frac{\sqrt{2}}{2}$$ Multiply the fraction by $V_1 - V_2$: $$\left(\frac{21}{34} - j \frac{1}{34}\right) \left(2 - \frac{\sqrt{2}}{2} - j \frac{\sqrt{2}}{2}\right)$$ Expand: Real part: $$\frac{21}{34} \left(2 - \frac{\sqrt{2}}{2}\right) - \frac{1}{34} \left(\frac{\sqrt{2}}{2}\right)$$ Imaginary part: $$- \frac{1}{34} \left(2 - \frac{\sqrt{2}}{2}\right) - \frac{21}{34} \left(\frac{\sqrt{2}}{2}\right)$$ Calculate numerically: $$2 - \frac{\sqrt{2}}{2} \approx 2 - 0.707 = 1.293$$ Real part: $$\frac{21}{34} \times 1.293 - \frac{1}{34} \times 0.707 \approx 0.799 - 0.021 = 0.778$$ Imaginary part: $$- \frac{1}{34} \times 1.293 - \frac{21}{34} \times 0.707 \approx -0.038 - 0.437 = -0.475$$ So the product is approximately: $$0.778 - j0.475$$ Add $V_2 = e^{j \pi/4} = 0.707 + j0.707$: Real part: $$0.707 + 0.778 = 1.485$$ Imaginary part: $$0.707 - 0.475 = 0.232$$ Calculate magnitude: $$|V_o| = \sqrt{1.485^2 + 0.232^2} \approx \sqrt{2.206 + 0.054} = \sqrt{2.26} = 1.503$$ Calculate angle: $$\theta = \tan^{-1}\left(\frac{0.232}{1.485}\right) = 0.155 \text{ radians}$$ Therefore, $$V_o \approx 1.503 e^{j 0.155}$$ 3. **Solving for $a$ and $b$ in:** $$2 e^{-j \pi/2} - 3 e^{j \pi} = a + j b$$ Calculate each term: $$e^{-j \pi/2} = \cos(-\pi/2) + j \sin(-\pi/2) = 0 - j1 = -j$$ $$2 e^{-j \pi/2} = 2(-j) = -2j$$ $$e^{j \pi} = \cos \pi + j \sin \pi = -1 + j0 = -1$$ $$-3 e^{j \pi} = -3(-1) = 3$$ Sum: $$-2j + 3 = 3 - 2j$$ So, $$a = 3, \quad b = -2$$ 4. **Expressing** $$\frac{3 (e^{j 4 \omega \theta} + e^{-2 j \omega \theta})}{e^{j \omega \theta}}$$ Divide numerator terms by denominator: $$3 \left(e^{j (4 \omega \theta - \omega \theta)} + e^{j (-2 \omega \theta - \omega \theta)}\right) = 3 \left(e^{j 3 \omega \theta} + e^{-j 3 \omega \theta}\right)$$ Recall Euler's formula: $$e^{j x} + e^{-j x} = 2 \cos x$$ So, $$3 \times 2 \cos(3 \omega \theta) = 6 \cos(3 \omega \theta)$$ **Final answers:** 1. $$Z = \sqrt{2} e^{j \frac{3\pi}{4}}$$ 2. $$V_o \approx 1.503 e^{j 0.155}$$ 3. $$a = 3, \quad b = -2$$ 4. $$6 \cos(3 \omega \theta)$$