(10 points total) A small community organization consists of 20 families, of which 4 have one child, 8 have two children., 5 have three children, 2 have four children and 1 has five children. (a) (5 points)If one of these families is chosen at random, what is the probability that it has i children i = 1, 2, 3, 4, 5? (b) (5 points)If one of the children is randomly chosen, what is the probability that they came from a family of i children i = 1,2, 3, 4, 5?

5. (9 points) Each student at some college has a mathematics requirement M (to take at least one mathematics course) and a science requirement S (to take at least one science course). A poll of 150 sophomore students shows that:60 completed M, 45 completed S, and 25 completed both M and S. Find the number of students who have completed: (a) (3 points) At least one of the two requirements (b) (3 points) Exactly one of the two requirements (c) (3 points) Neither requirement.

4. Assume continuous intensity values, and suppose that the intensity values of an image have the PDF pr(r) = 2r/(L-1)² for 0 ≤ r ≤ L-1 and pr(r) = 0 for other values of r. (a). Find the transformation function that will map the input intensity values, r, into values, s, of a histogram-equalized image. (b). Find the transformation function that (when applied to the histogram-equalized intensities, s) will produce an image whose intensity PDF is p₂(z) = 3z²/(L-1)³ for 0 ≤ z ≤ L-1 and for other values of z.

1. The effectiveness of some solar energy heating units depends on the amount of radiation available from the sun. During a typical October, daily total solar radiation in Tampa, Florida, approximately follows the following probability density function (units are hundreds of calories): Ax) = { 3 32 (x- 2)(6 - x) 2 <xs 6 0 elsewhere a) Find the cumulative distribution function. b) Find the probability that solar radiation exceeds 300 calories on atypical October day. c) What amount of solar radiation is exceeded on exactly 50% of the d) Find the expected daily radiation for October.

Determine the points of intersection between the circle, x2 +y2 = 3, and the hyperbola, xy = 1. As shown in the plot, there are four roots. However, it is enough to find only one root because the others can be deduced from symmetry. a) Solve the system of equations using the Successive Substitution method,starting with the initial guess xo = 0.5 and yo = 1.5. Show two complete iterations. b) Solve the system of equations using Newton-Raphson method, starting with the initial guess xo= -0.5 and yo =-1.5. Show two complete iterations. Evaluate &s for the second iteration. c) Solve the system of nonlinear equations by calling the MATLAB function newtmult. Yoursolution should achieve an accuracy of 6 significant figures. Report the solution, number ofiterations and errors.

3. (5 points) Four independent flips of a fair coin are made. Let X denote the number of heads obtained. Calculate the probability mass function of the random variable Y = X – 3.

Use method of undetermined coefficients to solve the following non-homogeneous differentialequations. y^{\prime \prime}+4 y^{\prime}+3 y=65 \cos (2 x) x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x+1 \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+3 y=10 e^{-2 x} \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+y=x+e^{x}

Activity:Part 1:Your manager believes load power factor is poor and hence improvements are required. He has asked you to analyse the related engineering problem and produce a brief report to be presented to the colleagues with diverse backgrounds within the organization. Structure of the report is as follows: Title: Reducing the Line Currents to Improve the Electrical System and Maximum Demand Abstract : A brief of the content of the report 1 Problem Definition, Aims, and Objectives: * A brief of the current unsatisfactory situation with regard to line currents justified with calculations * Emphasize the importance of improvements with suitable circuit diagrams 2 Methodology: * Include an equivalent circuit showing the line impedance, power factor correction capacitor, and the load. * Vector diagram showing the line current before p.f. correction, Ib, line current after p.f.correction, Ia, capacitor current, Ic, and load current, IL. * Calculation of the capacitance required to be connected. * Calculation and comparison of line current before p.f. correction, Ib, and the line current after p.f. correction, Ia. * Express supply voltage in the form, v(t) = Vm sin wt Where Vm is the peak voltage. * Express the line current before and after in the forms is(t) = Ibm Sin(wt + ¢b) and ia(t) = Iam sin(wt + pa) * Express the capacitor current in the form and ic(t) = Icm Sin(wt + 4) * Analytically solve iL(t) + ic(t) using the theories of trigonometry and hence obtain the supply line current in the form, ta(t) = iL(t) + ic(t) = Iam Sin(wt + pa) * Hand-sketch i(t) and ic(t) on the same graph to a scale. Graphically sum up i(t) and ic(t) to obtain the sketch of ia(t) = in(t) + ic(t). Compare this result obtained graphically with the analytical result obtained above. 3 Results: * Plot is(t) and ia(t) on the same graph using appropriate software and discuss * Calculation of the maximum demand cost saving 4 Discussion: Provide a brief discussion about how far the aims and objectives are achieved

(a) A given number in base x can be converted to any other base y. According to the expansion method, if abc.de is any given number in base x, then write its value in base 10.[31 (b) Convert the following numbers using number system conversions, show your answer in details:[6] \text { i. }(723)_{8} \text { to hexa decimal system } \text { ii. }(0 . A B D F)_{16} \text { to decimal system } iii. Convert 0.375 to binary system iv. Which digits from (0,1,2,3,4,5) are not allowed in Quinary system (base5) representation. v. (11010.1011)2 to hexadecimal. vi. (257)10 to the binary system. Consider the binary number 10.0011 i. Convert the above number to the decimal system ii. What are the place values of the digits 1 in the number 0.00112 iii. what is the sum of (1+1+1+1) in binary system iv. calculate 101 divided by 10 using long division. Which one is the correct representation of a binary number from the following?[2] i. 1101ii. (214)2ii. (0000)2iv. (11)²

A flagpole is held in the vertical position by cables attached to three nearby structures. The tension in one of the cables is found to be 20 N. The figure below is a sketch of the situation as seen from above. All of the measurements given are correct, however the sketch is not to scale and poorly represents dimensions and angles. (a) Assuming the measurements given are all correct, use them to determine the angles 0, and 02 in degrees, to 2 significant figures. (b) Express the tension forces, T1 and T2, in component form to 2 significantfigures. Bby resolving the forces along x and y directions, find two equations involving the magnitudes T, and T2 of the tensions on the wires. Show that the matrix A (defined below) has an inverse. A=\left[\begin{array}{cc} 0.85 & -0.65 \\ 0.52 & 0.76 \end{array}\right] =) Evaluate the inverse of A. Express the equations found in part (c) in matrix form, and hence solve the equations to determine the magnitudes T, and T2 to 2 significant figures.