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Which of the following is not true about the congruent angles in triangles, created by bisecting a parallelogram? a. The two triangles are congruent. b. The two triangle are congruent because of the SAS Postulate. c. None of the above. d. The two triangles are congruent because of the SSS Postulate. e. Angle x and angle y are congruent because of the Opposite Interior Postulate. f. Opposite sides of a parallelogram are congruent.

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Exercise 8.1 Consider the following Penalty game. It gives the probability of scoring a goal when the row player (the striker) adopts one of the strategies $L$ (shoot left), $R$ (shoot right), and the column player (the goalkeeper) uses the strategies $l$ (jump left), $w$ (wait then jump), $r$ (jump right). The row player is interested in maximizing and the column player in minimizing the probability of scoring a goal. min $l$ Max $w$ $r$ $L$ 0.6 0.7 1.0 $R$ 1.0 0.8 0.7 (a) Find all equilibria of this game in mixed (including pure) strategies, and their equilibrium payoffs. Why is the payoff unique? (b) Now suppose that player I has an additional strategy $M$ (shoot down the middle), so that the payoff matrix is min $l$ Max $w$ $r$ $L$ 0.6 0.7 1.0 $R$ 1.0 0.8 0.7 $M$ 1.0 0.0 1.0 Find an equilibrium of this game, and the equilibrium payoff. [Hint: The result from (a) will be useful.]

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What is the IEEE754 32-bit single precision representation of the hexadecimal code (AB)$_{16}$? 0 1000 0110 01010110000000000000000 0 1000 0110 01010100000000000000000 0 1000 0110 01110110000000000000000 0 1000 0110 01110100000000000000000

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Question 7 Consider the short run production for a firm. If marginal cost is decreasing, we know that: Omarginal product must be at its minimum Omarginal product must be increasing O average product must be decreasing O average cost must be increasing

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Choosing a jury is an important aspect of litigation strategy. ? True ? False

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Using the table below: Initial Concentration and Initial Rate Experiment [NO] [H2] Rate (M/s) 1 0.0050 0.0020 1.25 x 10^-5 2 0.010 0.0020 5.00 x 10^-5 3 0.010 0.0040 1.00 x 10^-4 a. Determine the rate law for the reaction: 2NO(g) + 2H2(g) -> 2H2O(g) + N2(g) b. What is the overall order of the reaction? c. Determine the rate constant for the reaction.

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Dr. Jack, Dr. Jill and Dr. Bell are working with small town of Nigeria. It is 07 found that there is no other doctor in the town and 45% of population of town prefer Dr. Jill, 25% prefer Dr. Jack and the rest prefer Dr. Bell. Also it is noted that Dr. Jack advices 10% of his patient for taking pain killer, while Dr. Jill advices 12% and Dr. Bell advices 16% of their patients for taking pain killer. What is the probability that a randomly selected patient is found to be advices for pain killer, advices by (a) Dr. Jill (b) Dr. Jack or Dr. Jill?

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8. Determine the linear time-invariant recursive system described by the given difference equation is stable. y(n) = ay(n-1) + x(n)

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Find actual solution of the following system $x' + 3x - y = 0$ $y' + 4x - 2y = 0$ that satisfies the initial conditions: $x(0) = 0$ $y(0) = 3$ a. $x = e^t - e^{-t}, y = e^t - e^{-2t}$ b. $x = e^t - e^{-2t}, y = 4e^t - e^{-2t}$ c. $x = e^t + 2e^{-2t}, y = 4e^t + 2e^{-2t}$ d. $x = 2e^t - e^{-2t}, y = 4e^t - 2e^{-2t}$ e. $x = e^{2t} - e^{-2t}, y = 4e^{2t} - e^{-2t}$

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1-2 Problem 1.2 (37.5 points) (Semiconductor laser) The schematics of a semiconductor GaAs/AlGaAs based laser is shown in the figure below. The table on the right specifies the performance parameters of this laser. AlGaAs GaAs Corrugated dielectric Structure (Bragg reflector) Active layer Caussian beam Wavelength Power Far field divergence Beam quality Line width $\lambda$ = 850 nm P = 10 mW $\theta$ = 1 mrad M$^2$ < 1.2 $\Delta\lambda$ = 0.02 nm (a) What is the diameter of the beam immediately after the output of the laser. What about (i) 1 m, (ii) 10 m, and (iii) 100 m away from the laser? (b) A small spherical lens is mounted on the front end of the laser to couple light into a single mode fiber with a 10 µm core. What should the focal distance of this lens be, in order to achieve efficient coupling into the fiber? (c) What is the expected coherence time and coherence length for this laser? (d) As shown in the figure the required high reflectivity at the back of the laser is achieved using a corrugated dielectric geometry, a.k.a. Bragg reflector, that consists of a periodic structure of high and low index of refraction materials. In this particular case the structure is made out of GaAs and AlGaAs with indexes of refraction 3.631 and 3.3 respectively. Calculate what should the pitch $d$ of the Bragg grating be? (e) (Bonus: 10 points) How wide should the Bragg reflector be for 99% reflectivity? (Hint: from homework #3, for a dielectric mirror the reflectance is: $R_{Nlayers} = \frac{(n_2^2N - n_1^2N)^2}{(n_2^2N + n_1^2N)^2}$ where N is the number of layers and for the Bragg reflector considered: $n_0 = n_1 =$ n$_{GaAs}$; $n_2 = n_3 = n_{AlGaAs}$).

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kelly h.