Permutations and Combinations Questions and Answers

The procedure is as follows Step 1 Start with the cell 1 1 at the north west corner of the transportation table and allocate it maximum possible amount x 1 min a b so that either the capacity of the origin O or the requirement for the destination D satisfies it or both Step 2 If X a cross off the first row of the transportation table and decrease b by a and go to step 3 If X 1 b cross off the first column of the transportation table and decrease a by by and go to step 3 If x a b cross off either the first row or the first column but not both Step 3 Repeat steps 1 and 2 for the resulting reduced transportation table until all the requirements are satisfied Note The N W corner rule of allocating values of the variables eliminates either a row or column of the table from further consideration and the last allocation eliminates both a row and column Hence a feasible solution obtained by this method has m n 1 basic variables Problem Find an initial basic feasible solution to the following TP using North West Comer Rule Solution 1 2 3 b a 1 2 3 F 1 10 0 2 Step 1 First allocate in 1 1 cell X 1 min 15 5 5 12 7 3 2 1 3 4 a 15 20 11 0 14 16 18 5 5 15 15 10 45 9 20 25 b 5 2 3 4 10 0 20 11 15 18 12 7 9 20 25 Step 2 Second allocate in 1 2 cell X 2 min 10 15 10 2 3 4 0 2011 10 0 14 16 18 5 15 15 1 45 7 9 20 25 14 16 18 5 5 15 10 Step 3 Third allocate in 2 2 cell X min 25 5 5

The minimum processing time among these jobs is 6 which corresponds to the job J4 on machine M and job J5 on machine M Hence J4 is placed in the first available position in the sequence as shown below J2 J4 J5 Finally the job J3 is placed as shown below J2 J4 J3 Thus J2J4J3 two machines Sequencing n Jobs on 3 Machines Processing n Jobs in m Machines Machines M M M3 J5 J J5J is the optimal sequence of the five jobs on the Let there be n jobs J IzIn each of which is to be processed in m machines say M M Mm in the order M M Mm The list of job numbers 1 2 n with their processing time is given in the following table Job numbers 1 t11 t21 t31 3 t13 t22 t23 t32 33 2 t12 J En tin tzn t3n Mm tmn tm1 tm3 An optimum solution to this problem can be obtained if either or both of the following conditions are satisfied a min t max ty for i 2 3 k 1 b min tj 2 max ty for i 2 3 k 1 35 If the above condition is satisfied then the problem can be converted to an equivalent two machines and n jobs problem Algorithm for optimum sequence for n jobs in m machines The iterative procedure for determining the optimal sequence for n jobs in m machines is given below Step 1 Find min tij and mintmj Also find the maximum of each of t2j t3j t for all j 1 2 n Step 2 Check the following a min ti 2 max tij for i 2 3 k 1 b min tm max tij for i 2 3 k 1 Step 3 If both the inequalities of step 2 are not satisfied this method fails

sin 42 cos 48 cos 51 sin 39 X N Y sin Y cos cos Y

ale formula to find the value of the annuity Find the interest Periodic Deposit 70 at the end of each month Rate 7 compounded monthly Click the icon to view some finance formulas a The value of the annuity is Do not round until the final answer Then round to the nearest dollar as needed Formulas A Time 30 years In the provided formulas P is the deposit made at the end of each compounding period r is the annual interest rate of the annuity in decimal form n is the number of compounding periods per year and A is the value of the annuity after t years P 1 r 1 nt P 1 2 1 n A P A 9 nt 1

b Find the interest Periodic Deposit 100 at the end of every six months Rate Time 4 5 compounded semiannually 25 years Click the icon to view some finance formulas a The value of the annuity is Do not round until the final answer Then round to the nearest dollar as needed Formulas A In the provided formulas P is the deposit made at the end of each compounding period r is the annual interest rate of the annuity in decimal form n is the number of compounding periods per year and A is the value of the annuity after t years A P 1 r 1 nt 1 1 P A CLA P F nt A

If the boys of a class are arrnged in rows each consistings of 24 36 and 40 boys 18 boys fall short in each case But they can be arranged in rows each containing 27 boys Find the least number of boys of the class

Based on the graph find the range of y f x 7 f x 10 x if x 0 if x 0 Ay 10 5 10 0 7 5 10 OA 10 10 OB 0 U 0 00 OC 0 0 U 0 U 0 0 OD 0 00 W

About the sequence x kEZ we have the following 1 EB x b for each t to b t b a b x 1 1 5 0 MK te to b te bk20 k 1 c x keZ is uniformly Cauchy on t b t b verification of these properties is left for the reader We consider the uniform limit of the sequence x 1 8 to 8B x b C which is given by x t lim x 1 1E1 b to b Thus X 1 lim x f s x s ds 6 X 1 1 da kix x lim ff s x s ds x flim f s x s ds Finally the validity of this integral equation has the following two implications X t x s x s ds x 0 x 1 x f f s lim x s ds x ff s x s ds Thus the function 1 x 1 satisfying the integral equation x 1 x ff s x s ds for all t t b t b 2 dx 0 aff s x s ds f 1 X 1 dt d This now proves that the curve X To 8 to 85 thus obtained is a solution of the initial value problem

About the sequence keZ we have the following a x 1 EB xo b for each 1 to b t b MKk b 1 x 1 te to b t b k20 k 1 x keZ is uniformly Cauchy on t b to b verification of these properties is left for the reader We consider the uniform limit of the sequence x 10 8 to 8 B x b C which is given by 1 l b to b Thus X 1 lim x x t lim x 1 1 2 f s X s ds Finally the validity of this integral equation has the following two implications X t x f s x s ds x 0 X x lim ff s x s ds x f lim f s x s ds x f f s lim x s ds x ff s x s ds Thus the function rx 1 satisfying the integral equation x 1 x f s x s ds for all 1 18 b t b dx 0 ff s x s ds 1 X 1 dt This now proves that the curve X to 8 4 82 thus obtained is a solution of the initial value problem

Simplify the expression below when x 1 and y 3 2y X 2

Solve the absolute value inequality Graph the solution set 3t 2 4 Select the correct choice below and fill in any answer boxes within your choice OA The solution set in interval notation is Type your answer in interval notation Use integers or fractions for any numbers in the expression OB The solution is the empty set Graph the solution set Choose the correct answer below OA O C 4 4 2 2 0 0 2 4 O B O D 4 2 2 0 0 2 LIN 4

5 7 3 8 2y x 4 Find the slope of each line 7 y x 3 4 y 2x 2 0 Find the slope of the line through each pair of points 5 13 9 9 4 6 6 2 12 14 0

Find the reciprocal of the fraction 9 10 9

Divide and simplify 2n 87 16 6 2n 2n 8n7 16n 3 2n6 Simplify your answ

2i 2 8i 2i 2 8i Type your answer in the form a biUse integers or fractions for any numbers in the exp

Without graphing determine the number of solutions of the system of equations State whether the system is consisten or inconsistent If the system is consistent determine whether the equations are dependent or independent y 5x 8 y 5x 8 Determine the number of solutions of the system of equations Choose the correct answer below O A There is one solution OB There is no solution OC There are infinitely many solutions Which answer below applies to the system of equations O A Consistent and dependent

Simplify the expression 26 4 6 4

Simplify 33 5 9 33 5 9

Which of the following represents 7 in exponential form O 9 X7 81 7 X9 O 9x7

01 07 MC Simplify the expression Write your answer as a complex number 8 49 36 O 56 6i O 56 6i O 56 18 56 18 Jhy

Simplify x5 O O O X N 2 1 3 X5 J

4x 10 6 X X

f x if x 5 if x 5 4x 6 2 of 3 Evaluate this function at x 9 Express your answer as an integer or simplified fraction If the function is undefined at the given value inde efined

Now Therefore a 12 11 3 1 1 a11 1 11 2 i 3j i 1 2 3 and j 1 2 12 3 11 12 3 21 2 213 3 11 0 a32 12 13 3 21 1 12 Hence the required matrix is given by A 0 3 For example A 1 1 2 5 92 111 3 21 2 3 3 Types of Matrices In this section we shall discuss different types of matrices i Column matrix A matrix is said to be a column matrix if it has only one column iii Square matrix 1 Rationalised 2023 24 In general A a mx is a column matrix of order m 1 1 Row matrix For example B 5 2 3 2 For example A A matrix is said to be a row matrix if it has only one row is a column matrix of order 4 1 3 3 2 In general B b 1 x is a row matrix of order is a row matr 2 Ban MATRICES n A matrix in which the number of rows equal to the number of columns said to be a square matrix Thus an m x n matrix is said to be a square matrix m n and is known as a square matrix of order n 1 0 3 2 1 3 oblished is a square matrix of order 3 3

2 Show steps to solve by Completing the Square Use fractions not decimals when applicable Other methods will not earn credit a x 13x 36 0 b x 4x 13 0

Suppose P x 2x 2x x 2 is divided by x 2 a Identify the quotient Quotient b Identify the remainder Remainder c Express P x in the form quotient divisor remainder Pfr

Check 81x 36x 4 Check

ws online is included in the membership fee but there is an additional fee of 1 50 to rent each movie that embership fee for an online television and movie service is 9 80 The cost of viewing television For one month Jill s membership and movie rental fees were 12 80 How many movies did Jill rent online that month a 1 b 2 c 3 d 4 19 If a 3 pound pizza is sliced in half and each half is sliced into thirds what is the weight in ounces of each of the slices 1 pound 16 ounces a 4 b 6 c 8 d 16 20 Nate walks 25 meters in 13 7 seconds If he walks at this same rate which of the following is closest to the distance he will walk in 4 minutes e 150 meters f 450 meters g 700 meters h 1 4000 meters

Use synthetic division to rewrite f x in the form f x x k q x r for the given value of k 3 f x 2x x x 5 k 1 Show the synthetic division in the following table What is f 1

Find the product of the polynomials x 2 x 5x 8

Divide using synthetic division 3 6x 17x 21x 3 6x 17x 21x 3 1 I X 1 6 16

Divide using synthetic division 4 2y 13y 25y 29y 50 y 4 2y 13y 25y 29y 50

Divide using synthetic division 2 x 8 x 3 x x 8 Y 3 11

Divide 4x4 26x 13x 7x 6 3 x 6 Write the answer to this problem below 4 v 13v 3 2 Av 26x 7x 6

Given that the polynomial function has the given zero find the other zeros f x x 10x 41x 50 2 The other zero s is are Do not factor Use a comma to separate answers as needed Express complex numbers in terms of i Type each solution only once

potential rational zeros 5 7x 22 C Which of the following choices lists all the potential r OA 1 2 11 22 OB 0 11 12 11 22 OC 11 22 OD 1 2

Consider the equation 13 2a 6a 3 Which two of the following show a correct equation after all variables are moved to one side of the equation Remember Tr A B Choose one of the variable terms Move it to the other side of the equation by adding its additive in both sides of the equation NOTE There are two correct answers to this problem 13 8a 3 13 8a 3 C 13 4a 3 13 4a 3

Multiply 5v 2f 5v 2f 5v 2f 5v 2f Simplify your answer

Find the following special product 2 8x y 2 0 Ou 16

Divide using synthetic division 3 y 10y 15 y 4 3 y 10y 15 v 4

following que Click here to watch the video Use synthetic division to find the quotient 5x 3x 5x 5 x 2 3 2 5x 3x 5x 5

Morbidity and mortality statistics of a fictitious town in North Carolina 2019 Population size Total population in 2019 Age and or gender specific data All persons 55 years of age and older Women 15 44 years of age Births Total live births in 2019 Deaths Total deaths in 2019 Age and or cause specific death data Fetal deaths Infant deaths Deaths of persons 55 years and older Morbidity Deaths from cancer Deaths from stroke Deaths from heart disease Deaths from cancer among persons 55 years and older Total persons with heart disease Persons newly diagnosed with heart disease in 2019 This table is also viewable in PDF here Calculate 2 measures of morbidity and show your work 2a The overall risk population 167 000 51 000 36 000 2 800 1 985 24 40 722 88 39 121 35 4 381 289 cumulative incidence of heart disease in 2019 per 1 000 2b The prevalence of heart disease in this town reported as a percent per 100 population

A 12 year cohort study was conducted to examine the association between periodontitis gum disease and myocardial infarctions heart attacks in a representative sample of older adults Among 7 500 adults with periodontitis 174 suffered a myocardial infarction Among the 37 500 adults free of periodontitis 344 new cases of myocardial infarction were observed over the study period Show your work Use per 1 000 where appropriate Remember to include proper UNITS cases per population per study period What is the incidence of myocardial infarction among adults free of periodontitis

Find the exact value of the following expression Do not use a calculator to enter a decimal approximation sin RIN sin Type an integer or a fraction Type an exact answer using radicals as needed Rationalize all denominators

5 through 5 4 perp to y x A y 4 x 5 B y 4 x 5 C y 4 x 5 D y 4 x 5 6 6 through 5 5 perp to y x 3 A y 5 5 C y 5 x 3 D y 5 x 3

5 6x 18 1 x 1 4x 14x 6

1 B Writing Equations of Lines Write the slope intercept form of the equation of each line 1 2 2 4 3 12 A y x 2 Ay 7 y x 2 C y A y 4 x x 5 5 7 B y x 2 B y 4 x 5 23 x 5 rights reserved D y x 2 5 5 through 5 4 perp to y x 3 6 C y 4 x 5 23 x 5 D y 4 x 5 4 3 2 6 A y 4x C y p x 4 Write the point slope form of the equation of the line described perp perpendicular 3 through 2 4 parallel to y 3x 3 44 through 1 4 parallel to x 0 A y 4 3 x 2 B y 4 3 x 2 A y 4 3 x 1 B y 1 3 x 4 C y 2 3 x 4 C 0 x 1 D y 2 3 x 4 D y 4 0 3 2 B y x 4 A y 5 x B y 5 5 6 through 5 5 perp to y x 3 8 x 5 x C y 5 x 5 D y 5 x 5 2 3 D y 4x x 5

1 Determine the slope of the line with the following equation a y 3x 1 b 2y x 3 c 2x y 3 0

O O 6 5 7 to f x x 2 x 4 x 5 f x x 2 x 4 x 5 O f x x 2 x 4 x 5 O f x x 2 x 4 x 5 5 7 6 5 4 3 2 1 2 3 5 6 7

O 6 O 12 O f x x 2 x 4 x 5 O f x x 2 x 4 x 5 to f x x 2 x 4 x 5 2 f x x 2 x 4 x 5 5 7 6 5 4 3 2 1 1 2 3 5 6 7 5