Use the exponential definitions of sine and cosine toshow that \text { (i) } \quad \cos 2 x=\cos ^{2} x-\sin ^{2} x \text { (ii) } \quad \sin 2 x=2 \sin x \cos x b)Use de Moivre's theorem to show that 8 \cos ^{3} \theta=2 \cos 3 \theta+6 \cos \theta

1. Consider a three-phase power system with one-line diagram shown in Figure 1. The three-phase trans- former between CBs 1 and 2 (CB: circuit breaker) nameplate ratings are listed: 5MVA, 13.8A-138.0YkV, the transformer reactance X₁1 = 3.80 (viewed from low voltage side 13.8kV, resistance is negligible). The impedance of the transmission line between CBs 3 and 4 is ZL1 = (10+j100). -(50 pts) (a) Pick up SB = 100MVA for the entire three-phase system, and rated voltage VB = 138.0kV, calculate the per-unit line L1 and transformer impedance values. (b) If an SLG fault occurs at the midpoint of the line (L2) between CBs 5 and 6, which breaker(s) should operate? If the CB 5 or CB 6 does not operate, which breaker(s) will provide the backup protection? (c) List the operating CB(s) for different zones, which are listed in Figure 2. (d) If the second generator is connected at bus 3, the system (generators, buses, and transmission lines) is protected by overcurrent relays R1 to R12. Assuming the directional overcurrent relays are used for three transmission lines, what is the remote backup relay(s) for R7? And why? G Generator mm www Transformer - GSU Bus 1 depending on which breakfas Bus 3 Transmission line L1 Shunt Reactor L3 Shunt Capacitor Figure 1: A three-phase power system. Bus 2 Distribution Transformer Feeder

Q1. Calculate capillary rise/fall in a glass tube 2 mm diameter when immersed in (a) water (b) mercury.Both the liquids are at 20°C and the surface tension values at this temperature for water and mercury are 0.072 N/m and 0.052 N/m respectively. The specific gravity of mercury is 13.6. The contact angle of water and mercury are 0° and 130° respectively.

The switch in the circuit has been closed for a long time, and it is opened at t=0. Find v(t) for t>= 0. Calculate the initial energy stored in the capacitor. (a). When the switch is closed, calculate the value of Vc. (b). When the switch is opened, find the time constant. (c). Find v(t) for t>= 0. (d). Find p(t) for t>= 0. (e). Calculate the initial energy stored in the capacitor.

• 2-40 Water is being heated in a closed pan on top of a range while being stirred by a paddle wheel. During the process, 30 kJ of heat is transferred to the water, and 5 kj of heat is lost to the surrounding air. The paddle-wheel work amounts to 500 N:m. Determine the final energy of the system if its initial energy is 12.5 kJ.

A rigid 10-L vessel initially contains a mixture of liquid water and vapor at 100° C with 12.3 percent quality. The mixture is then heated until its temperature is 150° C. Calculate the heat transfer required for this process in kJ.

3.19. An ideal gas initially at 600 K and 10 bar undergoes a four-step mechanically reversible cycle in a closed system. In step 12, pressure decreases isothermally to 3 bar; instep 23. pressure decreases at constant volume to 2 bar; in step 34, volume decreases at constant pressure; and in step 41, the gas returns adiabatically to its initial state.Take Cp = (7/2)R and Cy = (5/2)R. (a) Sketch the cycle on a PV diagram. (b) Determine (where unknown) both T and P for states 1, 2, 3, and 4. \text { (c) Calculate } Q, W, \Delta U, \text { and } \Delta H \text { for each step of the cycle. }

4) Refrigerant-134a enters the condenser of a residential heat pump at 800 kPa and 35°C at a rate of 0.018 kg/s and leaves atS00 kPa as a saturated liquid. If the compressor consumes 1.2 kW of power, determine (a) the COP of the heat pump and(b) the rate of heat absorption from the outside air.

E2A.6(a) A sample of 4.50g of methane occupies 12.7 dm3 at 310 K. (i) Calculate the work done when the gas expands isothermally against a constant external pressure of 200 Torr until its volume has increased by 3.3 dm². (ii) Calculate the work that would be done if the same expansion occurred reversibly. E2A.6(b) A sample of argon of mass 6.56g occupies 18.5 dm3 at 305 K.(i) Calculate the work done when the gas expands isothermally against a constant external pressure of 7.7kPa until its volume has increased by 2.5 dm3.(ii) Calculate the work that would be done if the same expansion occurred reversibly. F=\frac{k T}{2 l} \ln \left(\frac{1+v}{1-v}\right) \quad v=\frac{n}{N} where k is Boltzmann's constant, N is the total number of units, and l= 45 nm for DNA. (a) What is the magnitude of the force that must be applied to extend a DNA molecule with N=200 by 90 nm? (b) Plot the restoring force against v, noting that v can be either positive or negative. How is the variation of the restoring force with end-to-end distance different from that predicted by Hooke's law? (c) Keeping in mind that the difference in end-to-end distance from an equilibrium value is x = nl and, consequently, dx = ldn= Nldv,write an expression for the work of extending a DNA molecule. Hint: You must integrate the expression for w. The task can be accomplished best with mathematical software.

2. (1-62) A mercury manometer (p = 13,600 kg/m') is connected to an air duct to measure the pressure inside. The difference in the manometer levels is 15 mm, and the atmospheric pressure is 100kPa. a. Answer in one sentence: Judging from figure below,determine if the pressure in the duct is above or below the atmospheric pressure. (Points 2) b. Determine the absolute pressure in the duct. (Points 7)

Q3.1. Yeast have mitochondria and can perform cellular respiration. What would you expect to be consumed and produced during the process of cellular respiration in yeast? a.Glucose and O2 consumed; CO2 H20, and energy produced. b.Glucose, H2O, CO2, and energy consumed; O2 produced. c. CO2 and H2O consumed; glucose, O2, and energy produced. d. CO2 and energy consumed; H20, 02, and energy produced.