Question (1 point) Find the maximum and minimum values of the function f(x, y, z) = 3x – y — 3z subject to the constraints x² + 2z2 + 361 and x + y – z = 4. Maximum value is occuring at ). Minimum value is occuring at ). 2

Use implicit differentiation to find dx 3y = X+8 y-8 dy dx 0 Find the derivative of y= In (tan - ? (8x®)) with respect to x. The derivative of y = In (tan (8x")) with respect to x is -1

The conditions for the operation of a short circuit asynchronous motor controlled by three switches (S0, S1, S2) and connected to the energy source via contactor K1 are given as follows:  The S1 button is pressed and released, then if the S2 button is pressed within 5 seconds, the motor will be activated.  If the S0 button is pressed while the motor is on or if it is overloaded (F2F), the motor will be deactivated. a) Draw the appropriate state diagram to implement this control function with a relay. b) Write the logical functions related to this diagram and draw the electrical (relay) control circuit.

1. [7 marks] (a) (2 marks] Suppose that two distinct complex numbers u and v have the same modulus, and u+v+0. U – V (i) Show that the real part of is zero. u+v (ii) Show that u – v and u + v are orthogonal. a — с (b) [3 marks] Suppose that a and b are distinct points in C, that m is the midpoint of the line segment with endpoints a and b, and that c is a 1. [7 marks] (a) (2 marks] Suppose that two distinct complex numbers u and v have the same modulus, and u+v+0. U – V (i) Show that the real part of is zero. u+v (ii) Show that u – v and u + v are orthogonal. a — с (b) [3 marks] Suppose that a and b are distinct points in C, that m is the midpoint of the line segment with endpoints a and b, and that c is a point on the circle C(m, ž\b – al) with centre m and radius 316 – all such that c + a and c #b. Use the ratio and the fact that a - c= a - m + m - c and b-c=b- m + m - c to show that Zacb is a right angle. (c) [2 marks] Prove the assertion in (b) using the geometry of triangles in the Euclidean plane only. a - C

17.7 Bedeivable: A Sive Problem Nagano International has 420 accounts receivable in its subsidiary ledger. All accounts are due in 30 days. On December 31, an aging schedule was prepared. The results are summarized below: Not Yet Due 1-30 Days Past Due 31-60 Days Past Due 61-90 Days Past Due Over 90 Days Past Due Total Customer (418 names) Subtotals $863,125 $458,975 $236,700 $108,350 $22.500 $36.600 Two accounts receivable were accidentally omitted from this schedule. The following data are available regarding these accounts: 1. J. Ardis owes $10,625 from two invoices: invoice no. 218, dated September 14, in the amount of $7.450; and invoice no. 568, dated November 9, in the amount of $3.175. 2. N. Selstad owes 59,400 from two invoices: invoice no. 574, dated November 19, in the amount of $3,375; and invoice no. 641, dated December 5, in the amount of 6,025. Instructions a. Complete the aging schedule as of December 31 by adding to the column subtotals an aging of the accounts of Ardis and Selstad. b. Prepare a schedule to compute the estimated portion of each age group that will prove uncol- lectible and the required balance in the Allowance for Doubtful Accounts. Arrange your sched ule in the format illustrated on page 292. The following percentages of each age group are es- timated to be uncollectible: Not yet due, 1%; 1-30 days, 4%; 31-60 days, 10%; 61-90 days, 30%, over 90 days, 50%. c. Prepare the journal entry to bring the Allowance for Doubtful Accounts up to its required bal ance at December 31. Prior to making this adjustment, the account has a credit balance of $34,500. d. Show how accounts receivabıle would appear in the company's balance sheet at Decem ber 31 e. On January 7 of the following year, the credit manager of Nagano International learns that the $10,625 account receivable from J. Ardis is uncollectible because Ardis has declared bank ruptcy. Prepare the journal entry to write off this account.

A recent study reported that Americans spend an average of 270 minutes per day watching TV. Assume the distribution of minutes per day watching TV follows a normal distribution with a standard deviation of 23 minutes. a. What percent of the population watch more than 300 minutes per day? (Round your intermediate calculations and final answer to 2 decimal places.) b. What percent of the population watch more than 220 minutes per day? (Round your intermediate calculations and final answer to 2 decimal places.) c. What percent of the population watch between 220 hours and 300 hours? (Round your intermediate calculations and final answer to 2 decimal places.) d. Let’s define a “binge watcher” as someone in the upper 5% of the distribution of minutes watching TV. How many minutes does a “binge watcher” spend per day watching TV? (Round your intermediate calculations and final answer to 1 decimal place.)

Problem 3 (30 points) Consider the unity feedback system given below. a) Draw the root locus showing open-loop poles and zeros, and asymptotes. (20 pts) OUF b) Determine if a test point so = -4+j is on the root locus, where j² = -1. Justify your answer. (10 pts) R(S)+ Hint. The equations of the asymptotes of the Root Locus are given by: #aymptotes = #finite Problem 3 (30 points) Consider the unity feedback system given below. a) Draw the root locus showing open-loop poles and zeros, and asymptotes. (20 pts) OUF b) Determine if a test point so = -4+j is on the root locus, where j² = -1. Justify your answer. (10 pts) R(S)+ Hint. The equations of the asymptotes of the Root Locus are given by: #aymptotes = #finite poles - #finite zeros, - K(s+1) (s+2)(s+3)(s+4) Efinite poles-E finite zeros #aymptotes 0a= (2 #ay