Binomial theorem Questions and Answers

K Find the number of combinations 8 objects taken 3 at a time There are combinations

A company estimates that it has a 50 chance of being successful in bidding on a 50 000 contract If it costs 6 300 in consultant fees to prepare the bid what is the expected gain or loss for the company if it decides to bid on this contract There is an expected Type an integer or a c for the company of gain loss

Consider the monthly payment for the following loan new car financing of 3 3 on a 30 month 12 550 loan Identify the following for the given loan where P is the amount of the loan in dollars r is the annual interest rate as a decimal n is the number of payments p year and it is the time in years P r n t Find the monthly payment for the loan Round your answer to the nearest cent

An insurance policy offers you the option of being paid 750 per month for 20 years or a lump sum of 100 000 Assume that the current rate of return is 4 5 compounded monthly and you expect to live for 20 years Identify the following for the annuity monthly payment option where m is the periodic payment in dollars r is the annual interest rate as a decimal n is the number of payments per year and t is the time in years m r n t Determine the present value of the annuity Round your answer to the nearest cent Which has more value O the annuity O the lump sum payment

You want to retire at age 65 You decide to make a deposit to yourself at the end of each year into an account paying 4 compounded annually Assume you a now 25 and can spare 1 900 per year Identify the following from the given information Let m represent the periodic payment in dollars r the annual interest rate as a decimal n the number of payments per year and t the time in years until retirement m r n t How much will you have when you retire at age 65 Round your answer to the nearest cent

2 points Write 2 2k 3 2 3k as a single summation and then evaluate the sum k 1 k 1 n n n 2 2k 3 2 3k

Carbon dioxide CO is sealed inside a piston cylinder assembly initially at 100 kPa and 27 C wit volume equal to 0 6 m An electrical resistor inside the cylinder heats up the gas until it temperature reaches 627 C with a final volume of 0 2 m During this process pressure and volume maintain a relation as PV2 constant while the gas loses 100 kJ heat to the surroundings Please answer the following questions Note In order to earn the full credit please start with the general governing equations followed by simplifications based on assumptions a 10 points Draw a schematic of the system including the boundary and label all the heat and work interactions between the system and the surroundings List at least three assumptions and approximations that you will use to generate and simplify the governing equations to be used for solving the following parts

Based on the slot machine shown to the right in how many ways can we obtain oranges on all three wheels There are ways to obtain oranges on all three wheels Wheel 1 Wheel 2 Wheel 3 HAR BAR BAR

As a general rule which of the following will not satisfy the UCC requirement for a record e mails and faxes merchant s confirmation memorandum letters between the parties phone conversations that someone else overheard

C 4 5 Let us consider an example of how to obtain the CNF of a Boolean expression Example 5 Obtain the CNF of the Boolean expression X X1 X2 X3 x1 AX2 x X3 Solution We have x A X2 x V x2 x V x VO x V x V x3 x3 x V x2 V x3 A x V x2 V x3 Similarly you can check that x A x3 x1 V x2 V x3 A x V x2 V x3 Thus the required CNF of the expression X X X2 X3 given here is x V x2 V x3 A x V x2 V x3 A x1 V x2 V x3 A x1 V x2 V x3 Try the following exercise now E3 Obtain the CNF of the Boolean expression X X1 X2 X3 x x V x A x3 As we have said earlier in the context of simplifying circuits we need to reduce Boolean expressions to simpler ones Simple means that the expression has fewer connectives and all the literals involved are distinct We illustrate this technique now Solution a Here we can write xi A x2 A xi A x 2 Example 6 Reduce the following Boolean expressions to a simpler form a X X1 X2 x AX2 x1 x b X X1 X2 X3 x AX V x1 Ax2 AX3 V X1 X3 xi A x2 x1 Ax2 x1 X2 X2 X1 A X2 A x x A O 0 b We can write xi A x2 V x1 A x2 A x3 V x1 A X3 X1 x V x2 AX3 x1 AX3 X1 x V x x2 V x3 x1 AX3 x1 IA x2 V x3 A X1 AX3 De Morgan s Law Identity law Complementation law Distributive law Thus in its simplified form the expression given in a above is O i e a null expression X1 A X2 V X3 X1 AX3 x1 AX2 V x1 AX3 X1 AX3 AD X1 A x2 AX3 V X3 x1 AX3 x1 A X2 AX3 V x1 X3 Complementation la Identity law Distributive law x1 AX2 A XI A X3 V x1 A X3 A X1 A x3 Distributive law Idemp assoc law Distributive law Associative law Absorption law Associative law Complementation law Identity law Distributive law Distributive law Absorption law Thus the simplified form of the expression given in b is x Ax3

b 9 A 11 L 01 6 00 B 20 i Find n An BnC ii Find n AUB UC iii Find n An B UC

Anderson s Fish House purchases a tract of land and an existing building for 900 000 The company plans to remove the old building and construct a new restaurant on the site In addition to the purchase price Anderson pays closing costs including title insurance of 2 000 The company also pays 12 000 in property taxes which includes 8 000 of back taxes unpaid taxes from previous years paid by Anderson on behalf of the seller and 4 000 due for the current fiscal year after the purchase date Shortly after closing the company pays a contractor 45 000 to tear down the old building and remove it from the site Anderson is able to sell salvaged materials from the old building for 3 000 and pays an additional 10 000 to level the land

Find an angle between 0 and 2 radians that is coterminal with an angle of 15 Give an exact answer Use symbolic notation and fractions where needed coterminal angle

This clearly shows that to meet the purchase order of F and F the raw material required is 335 units of R 467 units of R and 147 units of R which is much more than the available raw material Since the amount of raw material required to manufacture each unit of the three products is fixed we can either ask for an increase in the available raw material or we may ask the clients to reduce their orders Remark If we replace n place A in Example 3 by A given by i e if the clients agree to reduce their purchase orders then A B 10 A 9 12 6 10 20 0 Similarly we have 9 12 6 10 20 0 3 4 0 793 5 12 7 Rationalised 2023 24 141 216 78 170 220 60 MATHEMATICAL MODELLING This requires 311 units of R 436 units of R and 138 units of R which are well below the available raw materials i e 330 units of R 455 units of R and 140 units of R Thus if the revised purchase orders of the clients are given by A then the firm can easily supply the purchase orders of the two clients Note One may further modify A so as to make full use of the available raw material R R R P 3 4 0 B P 7 Query Can we make a mathematical model with a given B and with fixed quantities of the available raw material that can help the firm owner to ask the clients to modify their orders in such a way that the firm makes the full use of its available raw material The answer to this query is given in the following example 201 Example 4 Suppose P P P3 and R R R are as in Example 2 Let the firm has 330 units of R 455 units of R and 140 units of R available with it and let the amount of raw materials R R and R required to manufacture each unit of the three products is given by a d 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 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 4x 9y 12z 455 for raw material R 3y 7z 140 for raw material R and This system of equations can be expressed in matrix form as

tan Step 5 is not required in this situation as exact values of the are known PQ QB I 200 Example 2 Let a business firm produces three types products P P and P that uses three types of raw materials R R and R Let the firm has purchase orders from two clients F and F Considering the situation that the firm has a limited quantity of R R and R respectively prepare a model to determine the quantities of the raw material R R and R required to meet the purchase orders Solution Step 1 The physical situation is well identified in the problem Step 2 Let A be a matrix that represents purchase orders from the two clients F and F Then A is of the form MATHEMATICS A F ot to he Let B be the matrix that represents the amount of raw materials R R and R required to manufacture each unit of the products P P and P Then B is of the form A P B P or 1 h cot B AB Rationalised 2023 24 Step 3 Note that the product which in this case is well defined of matrices A and B is given by the following matrix R R R F F 10 15 6 10 20 0 AB R R R3 which in fact gives the desired quantities of the raw materials R R and R to fulfill the purchase orders of the two clients F and F Example 3 Interpret the model in Example 2 in case 3 40 neters h 1 a and B B 7 9 3 5 12 7 and the available raw materials are 330 units of R 455 units of R Solution Note that 10 15 6 10 20 0 r 793 5 12 7 lied R R3 F 165 247 87 R 140 units of R

Four students are measuring the temperature of a solution They each measure the temperature five times The actual temperature of the solution is 24 8 degrees Celsius Student 1 20 6 20 8 20 3 20 5 21 Student 2 25 2 14 5 24 6 25 1 24 8 Student 3 27 9 22 9 27 24 3 22 a The data collected by Student 1 is b The data collected by Student 2 is c The data collected by Student 3 is d The data collected by Student 4 is Question Help Message instructor Select an answer Select an answer Select an answer Select an answer Student 4 29 9 25 7 38 26 4 36 4

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

he position of a weight attached to a spring is s t 4 cos 3t What are the frequency and period of the syste

A technical support contracting firm hires people to work from home using their proprietary support scripting system The more employess they have the more contracts they can support and therefore the more revenue they can generate Suppose that the company s revenue and number of employees is related by R 64x where is R is the revenue in thousands of USD and z is the number of employees in hundreds If there is no shortage of work to be done the company currently has 1600 employees and the company wants to increase their revenue from 512 000 00 this year to 1 184 000 00 next year how many new employees should be hired in order to make that possible The company should hire new employees before next year

The editorial staff at a magazine company is reviewing 22 articles They wish to select six for the next issue of the magazine and must also decide in what order the stories will appear If it tak 1 minute to write a list of six articles selected for the magazine how many years would it take to write all possible lists of six articles Assume 60 minutes in an hour 24 hours in a day and 365 days in a year It would take year s to write all possible lists of six articles Round to the nearest whole number as needed

Determine the number of ways to perform the task described Four players are to be selected from a 18 player baseball team to visit schools to support a summer reading program In how many ways can this selection be made There are different ways 4 players can be selected from a 18 player baseball team Simplify your answer Type a whole number

24 1 lim zi 2 i

Calculate the length of an arc between two points on the Earth s surface given the radius of the Earth is 6 378 371 m and the angle subtended by the two points is 5 degrees O 105 993 65 m 658 781 49 m O405 712 89 m

Calculate the length of an arc between two points on a circle given the radius is 200 ft and the angle subtended by the two points is 15 degrees 52 36 ft 61 78 m 61 78 ft ro

e x 2x 3 f 10 19x 6x

A group of 7 musical notes is made into a 7 tone musical phrase consisting of 7 different notes How many different phrases are possible to construct There are different phrases possible to construct

Using the digits 0 1 2 8 9 determine how many 3 digit numbers can be constructed according to the following criteria The number must be odd and greater than 600 digits may be repeated The number of 3 digit numbers that can be constructed is

How many different three digit numbers can be formed using the digits 6 4 2 9 8 and 3 without repetition For example 229 is not allowed The number of different three digit numbers is

ivalent to a b C

promotions Price Promotions Amount versus Percentage Uff are commonly used by retailers to motivate consumers to make a purchase Both the amount off and percentage off are in widespread use and research on when to use each method has been mixed The current study evaluated intent to purchase for a lower and higher priced item using either a fixed amount or a percentage off For the lower priced item participants saw a promotion for balloons regularly selling for 48 pesos but on sale with either a 12 peso or a 25 discount The higher priced item was a 480 peso sweater on sale with either a 120 peso discount or a 25 discount One hundred and fifty one students were randomly assigned to the treatments and responded to two ques tions measuring value perceptions I would be saving a lot of money if I made my purchase at this store and This store is selling the advertised product at a considerable discount Participants answered each question on the scale 1 strongly disagree to 5 strongly agree

Question 5 Given positive integers n and m how many sequences a1 a2 an of n integers are there such that 1 a a an m Question 6 Given positive integers n and m how many sequences a1 a2 an of n integers are there such that 1 a a an m

olve 2xy 1 dx 2x y dy

students go to the same school Their family names are Smith Miller and Taylor and the first names are Joanna Bill and Mary However their first names and family names are not necessarily in the same order as written here Their ages are 19 21 and 22 Ms Smith is three years older than Mary The student whose family name is Taylor is 21 What is the name of the youngest student O Ms Mary Taylor O Ms Joanna Taylor Mr Bill Taylor Mr Bill Smith Mr Bill Miller Jh Ms Mary Smith

Simplify the expression and answer the following a What is the resulting coefficient of b b What is the resulting coefficient of a b 3a 3a b 6b 7a b 4a 3b 4a a Coefficient of b 342 0 C

Mary a human cannonball is shot from a cannon Her height in metres after launch is given by the equation h t 2 5t 27t 21 5 What is her height at t 8 s m Round to two decimal places h

Anwyn the Bold must rescue her fianc Marcin the Mighty along with his troop of soldiers from the evil clutches of the Baron of Oxmoore They are located on the top floor of the round tower To facilitate the rescue she is going for a bold move the use of a catapult to assault the castle that is 62 m away The Baron and his archers are ready to shoot from the top of the tower This position is 15 m above the release point of any projectile Anwyn the Bold sends their way when she aims the catapult at an angle of 60 above the horizontal Determine a how fast the projectile must be launched so that it hits the archers b the velocity of the projectile when it reaches the archers and c what direction the projectile is moving when it reaches the archers Place your final answers in the spaces provided

dy 7 9 Solve the differential equation 1 x 2y 3 by using an appropriate substitution Obtain an explicit solution dx

Evaluate the following expressions a l 101 x 1 21 b l 121 1 21

Number of bags 123456 2nd attempt Willingness to pay per bag 5 4 3 2 1 0 Part 1 2 points Itranscript If a bag of jelly beans costs 3 how many bags will the student buy 3 bags of jelly beans Itranscript How much consumer surplus will she enjoy Assume that the student will purchase bags up to but not including the point at which she incurs a loss of consumer surplus 6 in consumer surplus Part 2 2 points See Hint See Hint

3 Change 75 millimeters to decimeters A 7 500 dm OB 75 dm OC 7 5 dm OD 750 dm

Situation 1 Movie Tickets A new movie is coming out that you and your family really want to see You are going to buy tickets for opening day as soon as they go on sale and want to get extra tickets so friends could come too You have 224 50 and wants to get as many tickets as possible You want to get at least 5 adult tickets and 4 kids tickets Use the ticket prices from your local movie theater or look up the average prices of child and adult tickets for the movie theater In your discussion post share the equations that you used to model this situation and answer the following questions 1 Can you get at least 5 adult tickets and 4 kids tickets 2 What is the most tickets you could have bought and met the given conditions What number of each type of ticket is that 3 After doing more research you find out that you can rent out a whole theater for 190 If you rent the theater out you can have up to 24 people regardless of whether they are children or adults In what situations would it be a better deal to rent out the theater

2 A function f is continuous on 2 2 and some of the values of f are shown to the right x f x 2 3 If f has only one root r on the closed interval 2 2 and r 0 then a possible value of b is e 1 a 3 b 2 c 1 d 0 Explain your reasoning in full sentences Hint Draw a picture 06 b 24

Let S 1 Vn be a linearly independent set in F Which of the following statements are true Select all true statements If T F Fm be a linear O transformation then T v1 T vk with k n is linearly independent O 1 is linearly dependent for k n If T F Fm be an invertible linear O transformation with n m then T v T vn is linearly independent V Un Un V is linearly independent for 1 i n O None of the above

2 Solve the following equations a 9x 13 103 3 b n 8 003 21

The square shown has sides of length 8z7 decimeters Find its area The area of the square is equal to

Use the distributive property to write the expression without parentheses Then simplify the result if possible 1 1 2 12x 22 3 2 12x 22 Use integers or fractions for any numbers in the expression

2 Four students had the following amounts of money in their pockets 3 14 67 2 45 and 1 14 How much money did they have total O A 7 04 B 7 44 O C 7 00 O D 7 40

17 Change the fraction 93 1 000 to a decimal OA 0 00093 OB 0 093 OC 0 93 OD 0 0093

14 Find the value of 8 3 x 24 2 x 0 03 Round your answer to the nearest hundredth O A 60 26 OB 6 03 O C 79 86 O D 7 99

a If seven people shake hands with one another exactly once how many handshakes take place b Generalize the solution for n people a For seven people there will be Simplify your answer handshakes b For n people there will be handshakes Simplify your answer