Math - Others Questions

6 A Give the graphical representation of the following deterministic finite state automata with its transition function K 90 91 92 E 0 1 90 A 9 Transition function Present State 70 71 72 Next State for Input 0 90 92 72 Start symbol S P S AB Aa Bb B b Next State for Input I B Give the languages and their natures descriptions generated by the following grammars i N S A B a b ii N S A B a b Start symbol S P SAB AaAla BbBlb 9 9

5 A SJHS student swam 56m with the current and 48m against the current in the same length of time What is the swimmer s speed in still water if the current is 4km h 313x 15 10 T

Use Warshall s algorithm to find the shortest paths of the following weighted graph I 4 4 9 1 2 3 8 1 2

B Give the characterizing property of the following sets i C 7 9 11 13 14 19 ii D 7 11 13 17 19 23

5 A The Bursaries Committee of a certain country offers scholarships to only 4 students from a particular school who come out first second third and fourth in their final year examinations i Find the number of ordered ways the students can qualify for a scholarship if the school has 10 students sitting for their final examinations ii Solve for the problem i above if order is not important B i In a group of 100 people each person picks a number from 97 to 115 What is the minimum number of people you need to be sure two of them have the same number iii At a certain university 525 of the seniors are history majors or math majors or both There are 102 senior math majors and 35 seniors are majoring in both history and math How many seniors are majoring in history 10 Marks

1 A List the elements of each of the following sets i A x x is a perfect square and 0 x 30 ii B x x is an integer and 1 x 5

population and the land area of a particular country for selected years from 1800 through 2000 Use the information to find the population density of a certain country for the year 1900 to the nearest tenth expressed in people per square kilometer Click here to view the bar graph The population density in 1900 was approximately Round to the nearest tenth as needed Bar Graph and Data pulation millions 350 300 250 200 150 1 in 10 1 yd 1 m 1 acre Resident Population and Land Area Population Land Area 40 35 3 0 25 lan OU 6 5 cm 0 09 m 08m2 2 6 km 0 4 hectare ha abs to suog A

3 A For a function f IR R defined by f x x 4 find the following using images and inverse images given that A 1 1 2 and B 5 4 12 23 60 i f B NA ii f A uf B B Show if the expression f x x 4 defined in A above has an inverse by first finding out if it is bijective Write its inverse if it has 10 Marks

2 Given the following two polynomials P x 4x 12 and P 3x 6 Carm out the following operations a P P2 P P2 2P1 P x P2 and P P divide using algebraic long division method b Factorization of P and P2 c Simplification of using factorization done in b P 120 Marksl

Give the common difference of the sequence 10 9 8 7 Note the three dotes mean the sequence of numbers does not end at the value 7 it keeps on going to infinity Answer Question 12 Not yet answered Marked out of 10 00 The closed formula for the partial sum 1 2 3 4 n is O a n n 1 2 O b n n 1 3 O c n n 1 3

The expression x a means that we have x as a base and a as an exponent Simplifying 2x 3 2x 3 we have Answer

Metric Equivalents for Capacity along with dimensional analysis to convert the given unit to the unit indicated 13 mL to tsp Click to check the table of English and Metric Equivalents for Capacity 3 mL tsp Round to two decimal places as needed English and Metric Equivalents for Cap English and Metric Equivalents for Capacit 1 teaspoon tsp 5 milliliters mL 15 milliliters mL 30 milliliters ml 1 tablespoon tbsp 1 fluid ounce fl oz

The average rainfall amounts for two of the world s wettest places are given average rainfall and by how many inches Round to the nearest inch Lloro Colombia 13 300 millimeters and Cherrapunji India 464 inches Click the icon to view a table of English and Metric Equivalents The average rainfall is greater in Data table The average rainfall is greater by approximately English and Metric Equivalents 1 in 2 54 cm 1 ft 30 48 cm inches X

evaluate the left Riemann sum with n 6 taking the sample points to be left endpoints Round your answer to six decimal places 3 f x 3 cos x 0 x 4 What does the Riemann sum represent The Riemann sum represents the net area of the rectangles with respect to the x axis Illustrate with a diagram y 3 0 3 3jp 5A 8 15 3 4 y 3 0 3 k 100 R 8 714 3A 8 KIN 2 5 8 3r y 3 0 3 1 8 I 4 3 A 8 5 pr 8 y 3 O 3 4 3jpr 8

Find the Riemann sum for f x 3x 1 6 x 4 with five equal subintervals taking the sample points to be right endpoints Explain with the aid of a diagram what the Riemann sum represents 11 Need Help SESE 9 7 5 3 53 9 11 13 17 191 11 9 7 5 3 5 9 11 13 15 17 19 The Riemann sum represents the net area of the rectangles with respect to the x axis 6 O 4 2 y 11 9 7 5 3 3 5 9 11 13 15 17 191 114 9 7 5 3 S 7 y 9 11 13 15 17 19 2

9 Implied interest rate and period Consider the case of the following annuities and the need to compute either their expected rate of return or duration Matthew needed money for some unexpected expenses so he borrowed 3 900 55 from a friend and agreed to repay the loan in four equal installments of 1 100 at the end of each year The agreement is offering an implied interest rate of Matthew s friend Gregory has hired a financial planner for advice on retirement Considering Gregory s current expenses and expected future lifest changes the financial planner has stated that once Gregory crosses a threshold of 4 136 860 in savings he will have enough money for retirement Gregory has nothing saved for his retirement yet so he plans to start depositing 40 000 in a retirement fund at a fixed rate of 5 00 at the end of each year It will take for Gregory to reach his retirement goal

Mary Jarvis is a single individual who is working on filing her tax return for the previous year She has assembled the following relevant information She received 77 000 in salary She received 17 500 of dividend income She received 3 700 of interest income on Home Depot bonds She received 22 500 from the sale of Disney stock that was purchased 2 years prior to the sale at a cost of 6 500 She received 9 000 from the sale of Google stock that was purchased 6 months prior to the sale at a cost of 5 300 Mary receives one exemption 4 000 and she has allowable itemized deductions of 7 500 These amounts will be deducted from her gross income to determine her taxable income Assume that her tax rates are based on Table 3 5 The data has been collected in the Microsoft Excel Online file below Open the spreadsheet and perform the required analysis to answer the questions below X Open spreadsheet a What is Mary s federal tax liability Round your answer to the nearest cent Do not round intermediate calculations b What is her marginal tax rate Round your answer to 1 decimal place c What is her average tax rate Round your answer to 2 decimal places

8 Present value of annuities and annuity payments The present value of an annuity is the sum of the discounted value of all future cash flows You have the opportunity to invest in several annuities Which of the following 10 year annuities has the greatest present value PV Assume that all annuities earn the same positive interest rate An annuity that pays 500 at the end of every six months An annuity that pays 500 at the beginning of every six months An annuity that pays 1 000 at the beginning of each year An annuity that pays 1 000 at the end of each year An ordinary annuity selling at 10 538 38 today promises to make equal payments at the end of each year for the next twelve years N If the annuity s appropriate interest rate I remains at 6 50 during this time the annual annuity payment PMT will be You just won the lottery Congratulations The jackpot is 35 000 000 paid in twelve equal annual payments The first payment on the lottery jackpot will be made today In present value terms you really won assuming annual interest rate of 6 50

10 Perpetuities Perpetuities are also called annuities with an extended or unlimited life Based on your understanding of perpetuities answer the following questions Which of the following are characteristics of a perpetuity Check all that apply A perpetuity is a series of regularly timed equal cash flows that is assumed to continue indefinitely into the future The present value of a perpetuity is calculated by dividing the amount of the payment by the investor s opportunity interest rate In a perpetuity returns in the form of a series of identical cash flows are earned A perpetuity continues for a fixed time period Your grandfather wants to establish a scholarship in his father s name at a local university and has stipulated that you will administer it As you ve committed to fund a 15 000 scholarship every year beginning one year from tomorrow you ll want to set aside the money for the scholarship immediately At tomorrow s meeting with your grandfather and the bank s representative you will need to deposit rounded to the nearest whole dollar so that you can fund the scholarship forever assuming that the account will earn 4 50 per annum every year Oops The bank representative just reported that he misquoted the available interest rate on the scholarship s account Your account should earn 7 00 The amount of your required deposit should be revised to relationship between the This suggests there is interest rate earned on the account and the present value of the perpetuity

The number of compounding periods in one year is called compounding frequency The compounding frequency affects both the present and future values of cash flows An investor can invest money with a particular bank and earn a stated interest rate of 4 40 however interest will be compounded quarterly What are the nominal periodic and effective interest rates for this investment opportunity Interest Rates Nominal rate Periodic rate Effective annual rate Rahul needs a loan and is speaking to several lending agencies about the interest rates they would charge and the terms they offer He particularly likes his local bank because he is being offered a nominal rate of 4 But the bank is compounding bimonthly every two months What is the effective interest rate that Rahul would pay for the loan 4 067 3 926 4 244 4 381 Another bank is also offering favorable terms so Rahul decides to take a loan of 12 000 from this bank He signs the loan contract at 5 compounded daily for 12 months Based on a 365 day year what is the total amount that Rahul owes the bank at the end of the loan s term Hint To calculate the number of days divide the number of months by 12 and multiply by 365 12 236 75 O 13 372 12 O 13 615 21

7 Future value of annuities There are two categories of cash flows single cash flows referred to as lump sums and annuities Based on your understanding of annuities answer the following questions Which of the following statements about annuities are true Check all that apply A perpetuity is a series of equal payments made at fixed intervals that continue infinitely and can be thought of as an infinite annuity When equal payments are made at the end of each period for a certain time period they are treated as ordinary annuities An ordinary annuity of equal time earns less interest than an annuity due When equal payments are made at the end of each period for a certain time period they are treated as an annuity due Which of the following is an example of an annuity An investment in a certificate of deposit CD A lump sum ayment made to a life insurance company that promises to make a series of equal payments later for some Ashley has a large and growing collection of animated movies She wants to replace her old television with a new LCD model so she has started saving for it At the end of each year she deposits 1 060 in her bank account which pays her 7 interest annually Ashley wants to keep saving for two years and then buy the newest LCD model that is available Ashley s savings are an example of an annuity How much money will Ashley have to buy a new LCD TV at the end of two years 1 865 07 of time O 2 194 20 O 2 347 79 O 1 916 50 by the end of two

14 Loan amortization and capital recovery After Shipra got a job the first thing she bought was a new car She took out an amortized loan for 45 000 with no 0 down payment She agreed to pay off the loan by making annual payments for the next four years at the end of each year Her bank is charging her an interest rate of 10 per year Yesterday she called to ask that you help her compute the annual payments necessary to repay her loan Calculate the annual payment and complete the following loan amortization table Year Beginning Amount 1 2 3 4 45 000 00 Payment Interest Paid 4444 Principal Paid Ending Balance 0 02

Lloyd is a divorce attorney who practices law in Florida He wants to join the American Divorce Lawyers Association ADLA a professional organization for divorce attorneys The membership dues for the ADLA are 750 per year and must be paid at the beginning of each year For instance membership dues for the first year are paid today and dues for the second year are payable one year from today However the ADLA also has an option for members to buy a lifetime membership today for 8 000 and never have to pay annual membership dues Obviously the lifetime membership isn t a good deal if you only remain a member for a couple of years but if you remain a member for 40 years it s a great deal Suppose that the appropriate annual interest rate is 7 9 What is the minimum number of years that Lloyd must remain a member of the ADLA so that the lifetime membership is cheaper on a present value basis than paying 750 in annual membership dues Note Round your answer up to the nearest year 17 years 18 years 14 years O 20 years In 1626 Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe Historians estimate that the price he paid for the island was about 24 worth of goods including beads trinkets cloth kettles and axe heads Many people find it laughable that Manhattan Island would be sold for 24 but you need to consider the future value FV of that price in more current times If the 24 purchase price could have been invested at a 4 25 annual interest rate what is its value as of 2012 386 years later 193 638 065 25 227 809 488 53 O 261 980 911 81

Open spreadsheet a Complete an amortization schedule for a 45 000 loan to be repaid in equal installments at the end of each of the next three years The interest rate is 10 compounded annually Round all answers to the nearest cent Year 1 2 3 Year 1 Year 2 Year 3 Beginning Balance Interest Payment Principal b What percentage of the payment represents interest and what percentage represents principal for each of the three years Round all answers to two decimal places Interest Repayment of Principal Ending Balance c Why do these percentages change over time I These percentages change over time because even though the total payment is constant the amount of interest paid each year is declining as the remaining or outstanding balance declines II These percentages change over time because even though the total payment is constant the amount of interest paid each year is increasing as the remaining or outstanding balance declines III These percentages change over time because even though the total payment is constant the amount of interest paid each year is declining as the remaining or outstanding balance increases IV These percentages change over time because even though the total payment is constant the amount of interest paid each year is increasing as the remaining or outstanding balance increases V These percentages do not change over time interest and principal are each a constant percentage of the total payment

y 2 June 2023 0 0 Maximize subject to 16 04 0 p 2x y x y2 35 x 2y 60 2x y 60 X 25 0 y 0

Suppose that the future price p 1 of a certain item is given by the following exponential function In this function p is measured in dollars and is the number of years from today p t 1800 1 019 Find the initial price of the item Does the function represent growth or decay O growth O decay By what percent does the price change each year

1 Show that I 2 1 2 Show that 2 1 2 n n 1 2 1 341 WIN 3 113

Incorrect Your answer is incorrect Use the ALEKS calculator to evaluate each expression Round your answers to the nearest thousandth Do not round any intermediate computations log 10 13 0 log 0

Given the indicated maturities listed in the following table assume the following yields for US Treasury securities Maturity Years 1 5 Yield 3 6 On the following graph plot the yield curve implied by these interest rates Place a blue point circle symbol at each maturity and interest rate in the table and the yield curve will draw itself INTEREST RATE Percent 10 9 8 7 0 10 20 30 5 5 5 5 4 2 4 0 10 15 20 25 30 MATURITY Years Yield Curve

Suppose the Federal Reserve the Fed decides to tighten credit by contracting the money supply Use the following graph by moving the black X to show what happens to the equilibrium level of borrowing and the new equilibrium interest rate INTEREST RATE r Percent 16 D CAPITAL Billions of dollars S2 Short term interest rates Which tend to be more volatile short or long term interest rates O Long term interest rates S1 Equilibrium If the inflation rate was 3 40 and the nominal interest rate was 5 60 over the last year what was the real rate of interest over the last year Faith etis average Round intermediate calculations to four decimal places

S Treasury yield curve is plotted in the following graph INTEREST RATE 5 7 3 2 0 5 10 15 25 30 20 YEARS TO MATURITY Based on an upward sloping normal yield curve as shown which of the following statements is correct O There is a positive maturity risk premium O Inflation must be expected to increase in the future O If the pure expectations theory is correct future short term rates are expected to be higher than cu thoopt must be correct

The following graph shows the supply of and demand for capital in a market over the last year You can see that the demand for capital has increased over the last year the demand curve shifted to the right Place the black X at the equilibrium interest rate and the quantity of capital INTEREST RATE r Percent 20 18 16 14 12 10 8 2 D2 D1 0 0 2 4 14 16 18 20 6 10 12 6 8 CAPITAL Billions of dollars Equilibrium The market interest rate increased by 2 0 and the amount of capital borrowed Which of the following factors could be responsible for the change in demand shown in the graph by The Federal Reserve the Fed decided to relax Its monetary policy and expanded the money supply O Expected inflation decreased billion

Everyone uses money and it is important to understand what factors affect the cost of money Consider the following scenario A friend comes to you and asks you to invest in his business instead of investing in Treasury bonds You think he has a good business model so you tell him you are willing to invest as long as the expected return on the investment is at least four times the return you would have received on the Treasury bonds Determine which of these fundamental factors is affecting the cost of money in the scenario described Time preferences for consumption Inflation Risk

For the given cost function C x 250 give at least 3 decimal places a The cost when producing 1850 items x 3375 b The average cost when producing 1850 items c The marginal cost when producing 1850 items e The minimal average cost find the following For average and marginal costs d The production level that will minimize the average cost per item per item per item items L

A company makes color television sets It produces a bargain set that sells for 330 profit and a deluxe set that sells for 590 profit On the assembly line the bargain set requires 5 hours while the deluxe set takes 7 hours The cabinet shop spends 1 hour on the cabinet for the bargain and 2 hours on the cabinet for the deluxe set Both sets require 4 hours for testing and packing On a particular production run the company has available 3600 work hours on the assembly line 900 work hours in the cabinet shop and 2600 work hours in the testing and packing department Complete parts a through c a How many sets of each type should the company produce to maximize profit What is the maximum profit The company should produce deluxe sets and bargain sets to maximize profit Tyne whole numbers

At the beginning of this year a manufacturer set its predetermined overhead rate using these estimates overhead costs 800 and direct materials costs 400 000 Actual costs for the year are 791 450 for overhead and 400 000 for direct materials Complete this question by entering your answers in the tabs below Required 1 Required 2 Required 3 Compute the predetermined overhead rate using estimated direct materials costs viremators Estimated overhead costs 800 0001 77321 400 000 Overhead rate 200

A company produces two types of can openers manual and electric Each requires in its manufacture the use of two machines A and B Each manual can opener require the use of machine A for 2 hours and machine B for 1 hour An electric can opene requires 1 hour on A and 2 hours on B Furthermore suppose the maximur numbers of hours available per month for the use of machines B are 90 6 respectively The profit on a manual can opener is 1 5 and on an electric ca opener it is 2 If the company can sell all the can openers it can produce how mar of each type should it make in order to maximize the monthly profit n order to solve this linear programming problem answer the following questions

8 Identify the conic solve for y and graph using your calculator Find the angle of rotation write the unrotated equation in general form 2x 3 xy y 2x 3y 4 0 a Type Unrotated eqn Graph of rotated eqn y

7 Determine if the equation below is an example of exponential growth or decay 719 0 81x exponential growth O exponential decay O This is not an exponential equation This equation is exponential but shows neither growth nor decay

A pilot looked out his window and saw an airport He estimated it to be at a 7 degree angle of depression while his plane was at an altitude of 1 855 miles How far in miles was the plane from the airport at that moment see distance to the airport in figures Airport distance to airport 7 Altitude

Suppose that the growth rate of a tree is 0 514 feet per year What would be the growth rate in mm per day use 1 foot 304 8 mm and 1 day 365 25 days Your Answer

with replacement A fair die is rolled 6 times What is the probability of having no 1 and no 3 among the rolls Round your answer to three decimal places 0 X 00

Complete the following chart to evaluate Marca s current budget then adjust monthly amounts to make a new budget such that each expense falls within the guidelines Expense Category Housing utilities Food Clothing Health Personal care Transportation Recreation Education Savings Miscellaneous Total Recommended of Net Monthly Income 27 33 20 26 3 5 12 14 6 8 6 10 12 18 Monthly Monthly Current Low High New Amount OK Amount 1000 985 125 500 250 100 300 100 New

6 Since birth Hailee has had a savings account that started at 1000 and has been growing at a rate of 4 per year The amount of money in the account can be modeled by the equation y P 1 04 where y is the value of the account x is the number of years and P was the original deposit amount Is it possible for Hailee s account to grow to 14972 71 within Hailee s lifetime Hint try to figure out the bounds of the parameter no O yes None of these is correct

24 G 1 25 26 A bag contains ten tiles labeled B C D E F G H I J and K One tile will be randomly picked What is the probability of picking a vowel Write your answer as a fraction 00 20 5 3

24 25 26 27 28 When Tom had 2 years left in college he took out a student loan for 13 525 The loan has an annual interest rate of 5 7 Tom graduated 2 years after acquiring the loan and began repaying the loan immediately upon graduation According to the terms of the loan Tom will make monthly payments for 5 years after graduation During the 2 years he was in school and not making payments the loan accrued simple interest Answer each part Do not round intermediate computations and round your answers to the nearest cent If necessary refer to the list of financial formulas a If Tom s loan is subsidized find his monthly payment Subsidized loan monthly payment S b If Tom s loan is unsubsidized find his monthly payment Unsubsidized loan monthly payment S X 3

Elsa has a new aquarium The tank is in the shape of a rectangular prism 60 in long 24 in wide and 36 in deep She plans to raise fish and sell them to a local pet store The pet store will inspect the aquarium to make sure that the fish are being kept in a healthful environment One of the main concerns is that each fish has enough water Elsa looks up the amount of water needed for the species of fish she plans to raise She finds the answer but it is given in cubic feet a Find the volume of water that the tank holds in cubic feet Use the table of conversion facts as needed b The particular species Elsa plans to raise requires at least 0 42 ft of water per fish If there is any amount less than this no matter how much less the pet store will not accept Elsa s fish What is the largest whole number of fish that she can keep in the aquarium fish c The pet store will pay 1 79 for each fish If she fully populates her aquarium and sells all the fish how much money will she collect Conversion facts for length 12 inches in 3 feet ft 36 inches in I foot ft 1 yard yd I yard yd F B

24 25 26 27 X 28 8 X 0 0 3 29 8 a Rachel entered a raffle to win a movie ticket The probability that she wins a movie ticket is Find the odds in favor of her winning a movie ticket 11 0 to 5 30 31 b Eric is watching his favorite soccer team playing a match The odds against his favorite tearn winning are 10 tearn winning 0 32 33 34 35 What is the probability of his favorite

A runner s racecourse has a total distance of 2 5 mi Race officials want to mark the course every 200 yd with a flag including one at the finish line The starting line does not need a flag How many flags are needed First fill in the blanks on the left side of the equation using three of the ratios shown Then write your answer on the right side of the equation Ratios 200 yd 1 flag 2 5 mi 1 1 flag 200 yd 5280 ft 1 mi 1 mi 5280 ft 1 3 ft 1 yd 1 yd 3 ft 1 B flags ft yd in flag E

4 Evaluate express final answer in scientific notation 0 00012 0 000000013 G