An alternating voltage of 240V, 50Hz is connected across an impedance of (60-1100)Ω. Determine (a) the resistance (b) the capacitance (c) the magnitude of the impedance and its phase angle and (d) the current flowing.

The real part of the impedance is the resistance.

$$R = 60 \, \Omega$$

The imaginary part of the impedance is related to the capacitive reactance $X_C$.

$$X_C = -100 \, \Omega$$ $$X_C = \frac{1}{2\pi f C}$$ $$C = \frac{1}{2\pi f |X_C|} = \frac{1}{2\pi \times 50 \times 100}$$ $$C = \frac{1}{10000\pi} \, \text{F}$$

$$|Z| = \sqrt{R^2 + X_C^2} = \sqrt{60^2 + 100^2}$$ $$|Z| = \sqrt{3600 + 10000} = \sqrt{13600}$$ $$|Z| = 116.62 \, \Omega$$

$$\phi = \tan^{-1}\left(\frac{X_C}{R}\right) = \tan^{-1}\left(\frac{-100}{60}\right)$$ $$\phi = \tan^{-1}\left(-\frac{5}{3}\right) \approx -59.04^\circ$$

$$I = \frac{V}{|Z|} = \frac{240}{116.62}$$ $$I \approx 2.06 \, \text{A}$$

Final Answers:

• Resistance: $\boxed{60 \, \Omega}$
• Capacitance: $\boxed{\frac{1}{10000\pi} \, \text{F}}$
• Magnitude of Impedance: $\boxed{116.62 \, \Omega}$
• Phase Angle: $\boxed{-59.04^\circ}$
• Current: $\boxed{2.06 \, \text{A}}$

Q2: Lagrange Interpolation to find $f(1.25)$.

Given data points:
$(x_0, y_0) = (0, 1)$, $(x_1, y_1) = (1, 2.25)$, $(x_2, y_2) = (2, 3.75)$, $(x_3, y_3) = (3, 4.25)$.

The Lagrange interpolation formula is:

$$P(x) = \sum_{j=0}^{n} y_j L_j(x)$$

where

$$L_j(x) = \prod_{k=0, k \neq j}^{n} \frac{x - x_k}{x_j - x_k}$$

Calculate $L_j(1.25)$:

1. $L_0(1.25) = \frac{(1.25-1)(1.25-2)(1.25-3)}{(0-1)(0-2)(0-3)} = \frac{(-0.75)(-0.75)(-1.75)}{(-1)(-2)(-3)} = \frac{-0.984375}{6} = -0.1640625$

2. $L_1(1.25) = \frac{(1.25-0)(1.25-2)(1.25-3)}{(1-0)(1-2)(1-3)} = \frac{(1.25)(-0.75)(-1.75)}{(1)(-1)(-2)} = \frac{1.640625}{2} = 0.8203125$

3. $L_2(1.25) = \frac{(1.25-0)(1.25-1)(1.25-3)}{(2-0)(2-1)(2-3)} = \frac{(1.25)(0.25)(-1.75)}{(2)(1)(-1)} = \frac{-0.546875}{-2} = 0.2734375$

4. $L_3(1.25) = \frac{(1.25-0)(1.25-1)(1.25-2)}{(3-0)(3-1)(3-2)} = \frac{(1.25)(0.25)(-0.75)}{(3)(2)(1)} = \frac{-0.234375}{6} = -0.0390625$

Calculate $P(1.25)$:

$$P(1.25) = 1(-0.1640625) + 2.25(0.8203125) + 3.75(0.2734375) + 4.25(-0.0390625)$$ $$P(1.25) = -0.1640625 + 1.845703125 + 1.025390625 - 0.166015625$$ $$P(1.25) = 2.541015625$$

Final Answer:

$$\boxed{2.541}$$

Q3: Use Gauss elimination method to solve the system of equations.

The system of equations is:

1. $x_1 - x_2 + x_3 = 0$
2. $-x_1 + x_2 - x_3 = 0$
3. $10x_1 + 2x_2 - x_3 = 90$
4. $20x_2 + 10x_3 = 80$