Show that the equations 2x + y + z = 5, x + y + z = 4, x − y + 2z = 1 are consistent and hence solve them.

Show that the equations x + y + z = 6, x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and solve them.

Show that the equations are inconsistent x − 4y + 7z = 14, 3x + 8y − 2z = 13, 7x − 8y + 26z = 5 

Find k, if the equations x + 2y − 3z = −2, 3x − y − 2z = 1, 2x + 3y − 5z = k are consistent.

Find k, if the equations x + y + z = 7,  x + 2y + 3z = 18,  y + kz = 6 are inconsistent

Investigate for what values of ‘a’ and ‘b’ the following system of equations x + y + z = 6,x + 2y + 3z = 10, x + 2y + az = b have
(i) no solution
(ii) a unique solution
(iii) an infinite number of solutions.

The total number of units produced (P) is a linear function of amount of over times in labour (in hours) (l), amount of additional machine time (m) and fixed finishing time (a)
i.e, P = a + bl + cm
From the data given below, find the values of constants a, b and c

Day	Production (in Units P)	Labour (in Hrs l)	Additional Machine Time (in Hrs m)
Monday Tuesday Wednesday	6,950 6,725 7,100	40 35 40	10 9 12

Estimate the production when overtime in labour is 50 hrs and additional machine time is 15 hrs.

Solve the following system of equations by rank method
x + y + z = 9, 2x + 5y + 7z = 52, 2x − y − z = 0

Show that the equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 are consistent and solve them by rank method.

For what values of the parameter λ, will the following equations fail to have unique solution: 3x − y+λz = 1, 2x + y + z = 2, x + 2y − λz = −1 by rank method.

The price of three commodities X, Y and Z are x, y and z respectively Mr. Anand purchases 6 units of Z and sells 2 units of X and 3 units of Y. Mr. Amar purchases a unit of Y and sells 3 units of X and 2units of Z. Mr. Amit purchases a unit of X and sells 3 units of Y and a unit of Z. In the process they earn Rs. 5,000/-, Rs. 2,000/- and Rs. 5,500/- respectively. Find the prices per unit of three commodities by rank method.

An amount of Rs. 5,000/- is to be deposited in three different bonds bearing 6%, 7% and 8% per year respectively. Total annual income is Rs. 358/-. If the income from first two investments is Rs. 70/- more than the income from the third, then find the amount of investment in each bond by rank method.

Solve by Cramer’s rule x + y + z = 4, 2x − y + 3z = 1, 3x + 2y − z = 1

The price of 3 Business Mathematics books, 2 Accountancy books and one Commerce book is Rs. 840. The price of 2 Business Mathematics books, one Accountancy book and one Commerce book is Rs. 570. The price of one Business Mathematics book, one Accountancy book and 2 Commerce books is Rs. 630. Find the cost of each book by using Cramer’s rule.

An automobile company uses three types of Steel S₁, S₂ and S₃ for providing three different types of Cars C₁, C₂ and C₃. Steel requirement R (in tonnes) for each type of car and total available steel of all the three types are summarized in the following table.

Types of Steel	Types of Car			Total Steel available
	C1	C2	C3
S1	3	2	4	28
S2	1	1	2	13
S3	2	2	2	14

Determine the number of Cars of each type which can be produced by Cramer’s rule.

In a market survey three commodities A, B and C were considered. In finding out the index number some fixed weights were assigned to the three varieties in each of the commodities. The table below provides the information regarding the consumption of three commodities according to the three varieties and also the total weight received by the commodity

Commodity Variety	Variety			Total weight
	I	II	III
A	1	2	3	11
B	2	4	5	21
C	3	5	6	27

Find the weights assigned to the three varieties by using Cramer’s Rule.

A total of Rs. 8,500 was invested in three interest earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was Rs. 380 and the amount, invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule).

80% of students who do maths work during one study period, will do the maths work at the next study period. 30% of students who do english work during one study period, will do the english work at the next study period. Initially there were 60 students do maths work and 40 students do english work.
Calculate,
(i) The transition probability matrix
(ii) The number of students who do maths work, english work for the next subsequent 2 study periods.

A new transit system has just gone into operation in Chennai. Of those who use the transit system this year, 30% will switch over to using metro train next year and 70% will continue to use the transit system. Of those who use metro train this year, 70% will continue to use metro train next year and 30% will switch over to the transit system. Suppose the population of Chennai city remains constant and that 60% of the commuters use the transit system and 40% of the commuters use metro train this year.
(i) What percent of commuters will be using the transit system after one year?
(ii) What percent of commuters will be using the transit system in the long run?

Two types of soaps A and B are in the market. Their present market shares are 15% for A and 85% for B. Of those who bought A the previous year, 65% continue to buy it again while 35% switch over to B. Of those who bought B the previous year, 55% buy it again and 45% switch over to A. Find their market shares after one year and when is the equilibrium reached?

Two products A and B currently share the market with shares 50% and 50% each respectively. Each week some brand switching takes place. Of those who bought A the previous week, 60% buy it again whereas 40% switch over to B. Of those who bought B the previous week, 80% buy it again where as 20% switch over to A. Find their shares after one week and after two weeks. If the price war continues, when is the equilibrium reached?

Find k if the equations 2x + 3y − z = 5, 3x − y + 4z = 2, x + 7y − 6z = k are consistent.

Find k if the equations x + y + z = 1, 3x − y − z = 4, x+ 5y + 5z = k are inconsistent.

Solve the equations x + 2y + z = 7, 2x − y + 2z = 4, x + y − 2z = −1 by using Cramer’s rule

The cost of 2kg of wheat and 1kg of sugar is Rs. 100. The cost of 1kg of wheat and 1kg of rice is Rs. 80. The cost of 3kg of wheat, 2kg of sugar and 1kg of rice is Rs. 220. Find the cost of each per kg using Cramer’s rule.

A salesman has the following record of sales during three months for three items A, B and C, which have different rates of commission.

Months	Sales of units			Total commission drawn (in Rs)
	A	B	C
January	90	100	20	800
February	130	50	40	900
March	60	100	30	850

Find out the rate of commission on the items A, B and C by using Cramer’s rule

Solve the following equation by using Cramer’s rule
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

Solve the following equation by using Cramer’s rule
x + y + z = 6, 2x + 3y− z =5, 6x−2y− 3z = −7

Solve the following equation by using Cramer’s rule
x + 4y + 3z = 2, 2x−6y + 6z = −3, 5x− 2y + 3z = −5