Math - Others Questions

7 9 3 5 1 Solving Rational Equations Solve each equation Remember to check for extraneous solutions 312 45X11 4r 2 15r 4 55r 5r 5r 1 x 3 3r r 6 2 6 x 3 1 3 a 5 3a 15a 18 r 19 1 r 3 3 a 6 Simplify each expression 5 5 x 4 x 3 4 v 9v 20 v 9v 20 4 2 3x 9 3 1 2X17 x 2x 2x r a 4 a 5a 6 v 3 v 4 Name ro 4 1 2 6 2 b 1 8 k 6 2k 8k 6 k 6k Date 1 n 5 10 b 56 4 n 5n 6r 3r 9 4 3 Period 5b 20 b 56 4 n 10n 24 n 5n

10 IB QUESTION The following graph represents a function y f x where 3 x 5 The function has a maximum at 3 1 and a minimum at 1 1 2 3 2 0 F3 a The functions u and v are defined as u x x 3 v x 2x where x R i State the range of the function uo f

How many units of each product is to be made so as to utilise the full available raw material Solution Step 1 The situation is easily identifiable y fitne Step 2 Suppose the firm produces x units of P y units of P and z units of P Since product P requires 3 units of R P requires 7 units of R and P requires 5 units of R observe matrix B and the total number of units of R available is 330 we have 3x 7y 5z 330 for raw material R Similarly we have and 4x 9y 12z 455 for raw material R 3y 72 140 for raw material R This system of equations can be expressed in matrix form as 202 MATHEMATICS 375 4 9 12 037 330 y 455 140 X Rationalised 2023 24 Step 3 Using elementary row operations we obtain 100 X 010 00 Z This gives x 20 y 35 and z 5 Thus the firm can produce 20 units of P 35 units of P and 5 units of P to make full use of its available raw material 20 35 Remark One may observe that if the manufacturer decides to manufacture according to the available raw material and not according to the purchase orders of the two clients F and F as in Example 3 he she is unable to meet these purchase orders as F demanded 6 units of P where as the manufacturer can make only 5 units of P3 Example 5 A manufacturer of medicines is preparing a production plan of medicines M and M There are sufficient raw materials available to make 20000 bottles of M and 40000 bottles of M but there are only 45000 bottles into which either of the medicines can be put Further it takes 3 hours to prepare enough material to fill 1000 bottles of M it takes 1 hour to prepare enough material to fill 1000 bottles of M and there are 66 hours available for this operation The profit is Rs 8 per bottle for M and Rs 7 per bottle for M How should the manufacturer schedule his her production in order to maximise profit Solution Step 1 To find the number of bottles profit under the given hypotheses bed the and M in order to maximise the Step 2 Let x be the number of bottles of type M medicine and y be the number of bottles of type M medicine Since profit is Rs 8 per bottle for M and Rs 7 per bottle for M therefore the objective function which is to be maximised is given by Z Z x y 8x 7y The objective function is to be maximised subject to the constraints Refer Chapter 12 on I inear Programming

Fig A 2 3 The co ordinates of vertices O P Q R S and I are 0 0 20000 0 20000 6000 10500 34500 5000 40000 and 0 40000 respectively Note that 0 40 000 T X Rationalised 2023 24 P 20 000 0 x 20 000 MATHEMATICAL MODELLING S 5 000 40 000 y 40 000 R 10 500 34 500 NCH Q 20 000 6 000 x y 45 000 203 persinday Z at P 0 0 0 Z at P 20000 0 8 20000 160000 Z at Q 20000 6000 8 20000 7 6000 202000 Z at R 10500 34500 8 10500 7 34500 325500 Z at S 5000 40000 8 5000 7 40000 320000 Z at T 0 40000 7 40000 280000 Now observe that the profit is maximum at x 10500 and y 34500 and the maximum profit is 325500 Hence the manufacturer should produce 10500 bottles of M medicine and 34500 bottles of M medicine in order to get maximum profit of 325500 Example 6 Suppose a company plans to produce a new product that incur some costs fixed and variable and let the company plans to sell the product at a fixed price Prepare a mathematical model to examine the profitability Solution Step 1 Situation is clearly identifiable

7 Computational exercise Write a script to use Euler s method to compute the approximate solution to y cos 2t sin 3t for 0 t 1 y 0 1 and h 0 1 Please give the approximate solution w at t 0 h 2h Nh 1

1 a Use Taylor s theorem to show that if a 2 1 then 0 1 x e b Let m 0 Is the following statement True or false Explain why If 0 1 z e then 0 1 2 e

Revenue Cost of Revenue Gross Profit Operating Expenses Operating Income Loss Non Operating Income Loss Pretax Income Loss Adjusted Abnormal Losses Gains Pretax Income Loss GAAP Income Tax Expense Benefit Net Income Interest Expense In Millions of USD Cash Cash Equivalents STI Accounts Notes Receiv Inventories Other ST Assets Total Current Assets Property Plant Equip Net Property Plant Equip Accumulated Depreciation LT Investments Receivables Other LT Assets Total Noncurrent Assets Total Assets Liabilities Shareholders Equity Payables Accruals ST Debt Other ST Liabilities Total Current Liabilities LT Debt Other LT Liabilities Total Noncurrent Liabilities Total Liabilities Total Equity 274 515 365 817 394 328 169 559 212 981 223 546 104 956 152 836 170 782 43 887 51 345 108 949 119 437 26 334 119 103 0 38 668 66 288 721 67 009 82 67 091 9 680 57 411 2 873 42 296 15 229 47 867 105 392 107 049 108 923 284 109 207 119 103 14 527 19 300 94 680 99 803 FY 2020 FY 2021 09 26 2020 09 25 2021 90 943 62 639 16 120 26 278 4 061 6 580 32 589 39 339 143 713 134 836 45 336 49 527 112 096 119 810 66 760 70 283 100 887 127 877 33 952 38 762 180 175 216 166 323 888 351 002 46 108 153 157 258 549 65 339 2 645 54 763 17 141 53 577 FY 2022 09 24 2022 48 304 28 184 4 946 2 931 53 971 135 405 52 534 124 874 72 340 120 805 44 011 217 350 352 755 64 115 22 773 67 094 153 982 109 707 38 394 125 481 119 381 43 050 162 431 148 101 287 912 302 083 63 090 50 672

In Millions of USD except Per Share 12 Months Ending Revenue Cost of Revenue Gross Profit Operating Expenses Operating Income Loss Pretax Income Loss Adjusted Abnormal Losses Gains Pretax Income Loss GAAP Income Tax Expense Benefit Net Income Interest Expense In Millions of USD Total Assets Cash Cash Equivalents STI Accounts Notes Receiv Inventories Other ST Assets Total Current Assets Property Plant Equip Net Property Plant Equip Accumulated Depreciation LT Investments Receivables Other LT Assets Microsoft Corp MSFT US Standardized Total Noncurrent Assets Total Assets Liabilities Shareholders Equity Payables Accruals ST Debt Other ST Liabilities Total Current Liabilities LT Debt Other LT Liabilities Total Noncurrent Liabilities Total Liabilities FY 2020 FY 2021 FY 2022 06 30 2022 06 30 2020 06 30 2021 143 015 0 168 088 0 198 270 0 46 078 0 52 232 0 62 650 0 96 937 0 115 856 0 135 620 0 43 528 0 45 940 0 52 237 0 53 409 0 69 916 0 83 383 0 53 420 0 69 870 0 83 255 0 384 0 1 232 0 461 0 53 036 0 71 102 0 8 755 0 9 831 0 44 281 0 61 271 0 Total Equity Total liabilities Equity 2 591 0 FY 2020 06 30 2020 2 346 0 FY 2021 06 30 2021 136 527 0 130 334 0 32 011 0 38 043 0 1 895 0 2 636 0 11 482 0 13 393 0 181 915 0 184 406 0 52 904 0 70 803 0 96 101 0 122 154 0 43 197 0 51 351 0 2 965 0 5 984 0 63 527 0 72 586 0 119 396 0 149 373 0 301 311 0 333 779 0 118 304 0 301 3110 27 394 0 10 825 0 50 438 0 88 657 0 71 453 0 83 716 0 10 978 0 72 738 0 141 988 0 333 779 0 2 063 0 22 534 0 33 728 0 5 905 0 6 037 0 43 871 0 55 317 0 72 310 0 95 082 0 76 205 0 72 363 0 34 492 0 31 681 0 30 853 0 110 697 0 103 134 0 103 216 0 183 007 0 191 791 0 198 298 0 FY 2022 06 30 2022 104 757 0 44 261 0 3 742 0 16 924 0 169 684 0 87 546 0 147 206 0 59 660 0 6 891 0 100 719 0 195 156 0 364 840 0 166 542 0 264 940 al

a Let f y t y sin t and let D R Show that f is a Lipschitz function in y on D and state the Lipschitz constant b Let f y t 1 12 and let D y t MSys M teR for some M 0 Show that f is Lipschitz continuous in y on D and state the Lipschitz constant Hint it will be useful to remember the triangle inequality 121 221 121 22

Answer the following Poisson exercises using the Excel function Poisson in your equation Enter your Excel equations in the highlighted cells 1 a b C d e A textile manufacturer process finds that on average four flaws occurs per every 60 yards of material produced Write the Excel equation using an Excel function that will calculate the probability of exactly three flaws in a 60 yard piece of material Write the Excel equation using an Excel function that will calculate the probability of three or less flaws in a 60 yard piece of material Write the Excel equation using an Excel function that will calculate the probability of five or more flaws in a 60 yard piece of material Write the Excel equation using an Excel function that will calculate the probability of two or less flaws in a 30 yard piece of material Write the Excel equation using an Excel function that will calculate the probability of two or less flaws in a 15 yard piece of material

Income Statement The following account balances were taken from the adjusted trial balance for Shaw Messenger Service a delivery service firm for the fiscal year ended April 30 20Y1 Depreciation Expense Fees Earned Insurance Expense Miscellaneous Expense Rent Expense Salaries Expense Supplies Expense Utilities Expense Prepare an income statement Fees earned Expenses 12 000 637 200 2 280 Shaw Messenger Service Income Statement For the Year Ended April 30 20Y1 Amount Line Item Description 4 800 91 200 320 400 4 080 34 800 Salacies expense Rent expense Utilities expense Depreciation expense Supplies expense Insurance expense Miscellaneous expense Net income Total expenses V 320 400 4 080 x 12 000 X 4 800 x 2 280 x 91 200 x 34 800 x Amount 637 200 469 560 V 167 640

Adjusting entries for unearned and accrued fees The balance in the unearned fees account before adjustment at the end of the year is 113 370 Of these fees 70 290 have been earned In addition 13 600 of fees have been earned but have not been billed a Journalize the adjusting entry to adjust the unearned fees account If an amount box does not require an entry leave it blank Unearned Fees Fees Earned Feedback Check My Work Consider the accounts involved in the recording of this transaction Recall that unearned revenues are liabilities until they become earned revenues over time or during normal operations Unearned fees decrease for the amount of the revenue that has been earned b Journalize the adjusting entry to record the accrued fees If an amount box does not require an entry leave it blank Accounts Receivable Fees Earned Feedback Check My Work

Closing Entries After the accounts have been adjusted at March 31 the end of the fiscal year the following balances were taken from the ledger of Cabriolet Services Co Retained Earnings 783 900 29 000 Dividends 612 350 402 000 96 500 36 180 12 060 Fees Earned Wages Expense Rent Expense Supplies Expense Miscellaneous Expense Journalize the two entries required to close the accounts If an amount box does not require an entry leave it blank Mar 31 Fees Earned Wages Expense Rent Expense Supplies Expense Miscellaneous Expense Retained Earnings Mar 31 Retained Earnings V

Adjustment for prepaid expense The prepaid insurance account had a beginning balance of 7 340 and was debited for 3 230 of premiums paid during the year Journalize the adjusting entry required at the end of the year assuming the amount of unexpired insurance related to future periods is 4 120 If an amount box does not require an entry leave it blank Insurance Expense

The prepaid insurance account had a balance of 5 900 at the beginning of the year The account was debited for 6 500 for premiums on policies purchased during the year Journalize the adjusting entry required under each of the following alternatives for determining the amount of the adjustment a The amount of unexpired insurance applicable to future periods is 1 100 If an amount box does not require an entry leave it blank Insurance Expense Prepaid Insurance Feedback Check My Work b The amount of insurance expired during the year is 11 300 If an amount box does not require an entry leave it blank Insurance Expense Prepaid Insurance

The balance in the prepaid insurance account before adjustment at the end of the year is 15 090 Journalize the adjusting entry required under each of the following alternatives for determining the amount of the adjustmer a The amount of insurance expired during the year is 11 470 If an amount box does not require an entry leave it blank Insurance Expense Prepaid Insurance Feedback Check My Work Insurance Expense Prepaid Insurance 38 b The amount of unexpired insurance applicable to future periods is 3 620 If an amount box does not require an entry leave it blank

djustment for Unearned Revenue On June 1 20Y2 Monarch Co received 43 790 for the rent of land for 12 months ournalize the adjusting entry required for unearned rent on December 31 20Y2 Assume no previous adjustment has been made to Unearned Revenue during the year Round your answers to the nearest dollar amount If an amount box does not require an entry leave it blank Dec 31 Unearned Rent Rent Revenue

Adjustment for accrued expense City Realty Co pays weekly salaries of 13 375 on Monday for a six day workweek ending the preceding Saturday Journalize the necessary adjusting entry at the end of the accounting period assuming that the period ends on Friday Round your answers to nearest whole dollar If an amount box does not require an entry leave it blar 88 Salaries Expense Salaries Payable

Adjustment for accrued revenues At the end of the current year 14 290 of fees have been earned but have not been billed to clients Journalize the adjusting entry to record the accrued fees If an amount box does not require an entry leave it blank Accounts Receivable Fees Earned 8 8

The price of a certain ice cream cone is increasing at the rate of 15e0 06t cents per year where t is measured in years and t corresponds to 2014 Find the total change in price between the years 2014 and 2027 Round your answer to the nearest cen

Next question Suppose that you decide to buy a car for 57 000 including taxes and license fees You saved 10 000 for a down payment The dealer is offering you a choice between two incentives Incentive A is 6000 off the price of the car followed by a four year loan at 7 08 Incentive B does not have a cash rebate but provides free financing no interest over four years What is the difference in monthly payments between the two offers Which incentive is the better deal Use PMT PA 9 The difference in monthly payments between the two offers is Round to the nearest cent as needed Save

The price of a condominium is 188 000 The bank requires a 5 down payment and one point at the time of closing The cost of the condominium is financed with a 30 year fixed rate mortgage at 9 5 Use the following formula to determine the regular payment amount Complete parts a through e below PMT P a Find the required down payment b Find the amount of the mortgage c How much must be paid for the one point at closing Round to the nearest dollar as needed d Find the monthly payment excluding escrowed taxes and insurance Round to the nearest dollar as needed e Find the total cost of interest over 30 years

Use PMT P nt to determine the regular payment amount rounded to the nearest dollar The price of a home is 125 000 The bank requires a 20 down payment and three points at the time of closing The cost of the home is financed with a 30 year fixed rate mortgage at 10 Complete parts a through e below a Find the required down payment b Find the amount of the mortgage c How much must be paid for the three points at closing Round to the nearest dollar as needed d Find the monthly payment excluding escrowed taxes and insurance Round to the nearest dollar as needed e Find the total cost of interest over 30 years Round to the nearest dollar as needed

Suppose that you decide to borrow 13 000 for a new car You can select one of the following loans each requiring regular monthly payments Installment Loan A three year loan at 6 3 Installment Loan B five year loan at 5 2 P Use PMT to complete parts a through c below a Find the monthly payments and the total interest for Loan A The monthly payment for Loan A is Do not round until the final answer Then round to the nearest cent as needed The total interest for Loan A is Round to the nearest cent as needed b Find the monthly payments and the total interest for Loan B The monthly payment for Loan B is Do not round until the final answer Then round to the nearest cent as needed The total interest for Loan B is Round to the nearest cent as needed c Compare the monthly payments and the total interest for the two loans

Suppose that you borrow 12 000 for three years at 6 toward the purchase of a car Use PMT find the monthly payments and the total interest for the loan The monthly payment is Do not round until the final answer Then round to the nearest cent as needed The total interest for the loan is Use the answer from part a to find this answer Round to the nearest cent as needed P A nt to

Question 8 5 pts Write the equation for the stated transformation on the following function Translated down seven units f x x 3

At the age of 27 to save for retirement you decide to deposit 90 at the end of each month in an IRA that pays 4 5 compounded monthly a Use the following formula to determine how much you will have in the IRA when you retire at age 65 A A P 1 r 1 r A b Find the interest or I question 1 point s possible Submit quiz a You will have approximately when you retire Do not round until the final answer Then round to the nearest dollar as needed in the IRA b The interest is approximately Use the answer from part a to find this answer Round to the nearest dollar as needed

One safe investment pays 7 per year and a more risky investment pays 9 per year a How much must be invested in each account if an investor of 96 000 would like a return of 7 240 per year b Why might the investor use two accounts rather than put all the money in the 9 investment a is invested in the 7 account is invested in the 9 account b Why might the investor use two accounts OA Investing in two accounts always yields more money than investing in just one account OB Although the 9 account earns more interest the 7 account has no hidden fees OC The 9 investment is riskier than the 7 investment There is a greater chance the investor will lose money if it is all placed in the 99 OD The 9 account does not earn as much interest as the 7 account

Write the equation for the stated transformation on the following function A reflection in the x axis f x x 6

Find the inverse of the following function f x 3x 5

Find the domain of the following rational expression x 3 x 7x 30

Perform the stated operation on the following functions f x 2x g x x 1 2 f g 2

Question 13 Perform the stated operations on the following functions f x 2x g x x g f 1

Find the domain of the following function f x 1 4x

Perform the stated operation on the following functions x 2 f x x g x 2x 4 f x g x

Professor L Question 3 Completely simplify the following expression 6x y z4 z

The following table gives the number of women age 16 years and older in millions in a country s Kovillan workforce for selected years from 1950 and projected to 2050 Complete parts a and b below Women in the Year Workforce millions Year 1950 2010 1960 2015 1970 2020 1980 2030 1990 2040 2000 2050 a Use x as the number of years past January 1st 1950 to create a cubic model y using these data 0x 0x 0x0 Type integers or decimals rounded to five decimal places b During what year does the model indicate that the number of women in the work force will reach its maximum 10 Round down to the nearest year 18 6 21 1 30 6 45 8 56 4 65 9 Women in the Workforce millio 75 3 77 2 79 8 80 8 86 1 91 9

Write the line that is parallel to the following equation and passes through the point 3 4 Use slope intercept form y x 2 3

Name the property of real numbers illustrated by the equations 5 6 65 Associative Property O Multiplicative Inverse O Commutative Property

A car rental agency rents compact midsize and luxury cars Its goal is to purchase 70 cars with a total of 1 200 000 and to earn a daily rental of 3200 from all the cars The compact cars cost 10 000 and earn 40 per day in rental the midsize cars cost 20 000 and earn 50 per day and the luxury cars cost 40 000 and earn 60 per day To find the number of each type of car the agency should purchase solve the system of equations below where x y and z are the numbers of compact cars midsize cars and luxury cars respectively y 70 X 1 10 000x20 000y 40 000z 1 200 000 2 40x 50y 60z 3200 3 2 Select the correct choice below and fill in any answer boxes within your choice OA There is one solution The agency should purchase compact cars Simplify your answers OC There is no solution midsize cars and luxury cars OB There are infinitely many solutions If the agency purchases z luxury cars then the agency should purchase Type expressions using z as the variable compact cars and midsize cars

12 Months Ending Revenue Cost of Revenue Gross Profit Operating Expenses Operating Income Loss Non Operating Income Loss Pretax Income Loss Adjusted Abnormal Losses Gains Pretax Income Loss GAAP Income Tax Expense Benefit Net Income Interest Expense In Millions of USD Cash Cash Equivalents STI Accounts Notes Receiv Inventories Other ST Assets Total Current Assets Property Plant Equip Net Property Plant Equip Accumulated Depreciation LT Investments Receivables Other LT Assets Total Noncurrent Assets Total Assets Liabilities Shareholders Equity Payables Accruals ST Debt Other ST Liabilities Total Current Liabilities LT Debt Other LT Liabilities Total Noncurrent Liabilities Total Liabilities 09 26 2020 09 25 2021 09 24 2022 274 515 365 817 394 328 169 559 212 981 223 546 104 956 152 836 170 782 38 668 43 887 51 345 66 288 108 949 119 437 721 26 334 67 009 119 103 82 0 67 091 9 680 57 411 2 873 FY 2020 09 26 2020 90 943 16 120 4 061 32 589 143 713 45 336 112 096 42 296 15 229 47 867 108 923 284 109 207 14 527 94 680 FY 2021 09 25 2021 62 639 26 278 6 580 39 339 134 836 49 527 119 810 66 760 70 283 100 887 127 877 33 952 38 762 180 175 216 166 323 888 351 002 105 392 107 049 46 108 2 645 54 763 17 141 53 577 125 481 119 381 43 050 153 157 162 431 258 549 287 912 119 103 19 300 99 803 2 931 FY 2022 09 24 2022 48 304 28 184 4 946 53 971 135 405 52 534 124 874 72 340 120 805 44 011 217 350 352 755 64 115 22 773 67 094 153 982 109 707 38 394 148 101 302 083

A concert promoter needs to make 108 000 from the sale of 1900 tickets The promoter charges 60 for some tickets and 40 for the others Complete parts a through e a If there are x of the 60 tickets and y of the 40 tickets Write an addition equation that states that the total number of the tickets sold is 1900 Type an equation Do not simplify b How much money is made from the sale of x tickets at 60 each c How much money is made from the sale of y tickets at 40 each d Write an equation that states that the total amount made from the sale is 108 000 Type an equation Do not simplify e Solve the equations simultaneously to find how many tickets of each type must be sold to yield the 108 000 Select the correct choice below and if necessary fill in the answer box to complete your choice OA The promoter should sell tickets at 60 each and tickets at 40 each OB There are infinitely many solutions If the promoter sells y tickets at 40 each then Type an expression using y as the variable OC There is no solution tickets at 60 each should be sold

The matrix associated with the solution to a system of linear equations in x y and z is given Write the solution to the system if it exists 10 COD Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA xa y and z Simplify your answers OB There are infinitely many solutions If z is allowed to be any real number then x Type expressions using z as the variable OC There is no solution and y

The size of the civilian labor force in millions for selected years from 1950 and projected to 2050 can be modeled by y 0 0001x 0 0058x 1 43x 56 7 where the number of years after 1950 a What does this model project the civilian labor force to be in 2016 b In what year does the model project the civilian labor force to be 154 million a In 2016 the model projects the civilian labor force to be million Type an integer or decimal rounded to one decimal place as needed b The model projects the civilian labor force to be 154 million in

7 OPEN RESPONSE Use the figure If AB 3 15 what is BD Write the answer in simplest radical form Lesson 4 4 A B 1 D

Use the graph of the polynomial function f x a to solve f x 0 and b find the factorization of f x x 2x 6x 10x 6x 8 a Choose the correct solution for f x 0 OA x 1 andx 4 B x 1 x 1 and x 4 OC x 1 x 1 and x 4 OD x 1 andx 4 b The factorization of the polynomial function f x is Type your answer in factored form CIT 60 40 20 20 60 80 Ay

K The total number of a particular chain store in the world can be modeled by the cubic function y 10 4x 402x 5595x 8698 where x is the number of years aft 2000 and y is the total number of stores Complete parts a through c below a Graph the function on the window 2 20 2 by 8000 23000 1000 OA B O No OYes GILD b What does the model predict the number of stores will be in 2025 The model predicts that there will be A Round to the nearest whole number as needed c Does the number of stores ever decrease over this period of years ending in 2020 according to the model stores in 2025 O C OD

2 Express 1 414 1 414414414 as the ratio of two integers

21 Find x 7 45 201 60

Consider a cost function of the form Calculate MC 1 000 20 000 980x x for 0 x 1 000 for x 1 000 In an attempt to calculate the marginal cost at x 1 000 economists have devised the formula MC 1 000 lim C x lim C x 2 x 1 000 x 1 000 Provide your answer below C x 0