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12th Business Maths Monthly Test - 2 ( Applications of Matrices and Determinants )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Monthly Test - 2 ( Applications of Matrices and Determinants )-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

Solve the following equation by using Cramer’s rule
5x + 3y = 17; 3x + 7y = 31

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

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).

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

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

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 `5,000/-, `2,000/- and
`5,500/- respectively Find the prices per unit of three commodities by rank method.

Show that the following system of equations have unique solution:
x+y+z=3,x+2y+3z=4,x+4y+9z = 6 by rank method.
Show that the equations5x+3y+7z=4,3x+26y+2z=9,7x+2y+10z =5 are consistent and solve them by rank method.

If A=\(\left( \begin{matrix} 1 & 1 & -1 \\ 2 & -3 & 4 \\ 3 & -2 & 3 \end{matrix} \right) \) and B=\(\left( \begin{matrix} 1 & -2 & 3 \\ -2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right) \), then find the rank of AB and the rank
of BA.

The subscription department of a magazine sends out a letter to a large mailing list inviting subscriptions for the magazine. Some of the people receiving this letter
already subscribe to the magazine while others do not. From this mailing list, 60% of those who already subscribe will subscribe again while 25% of those who do
not now subscribe will subscribe. On the last letter it was found that 40% of those receiving it ordered a subscription. What percent of those receiving the current
letter can be expected to order a subscription?

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?

Solve the following equations by using Cramer’s rule
2x + 3y = 7; 3x + 5y = 9

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.

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

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

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

The subscription department of a magazine sends out a letter to a large mailing list
inviting subscriptions for the magazine. Some of the people receiving this letter
already subscribe to the magazine while others do not. From this mailing list, 45%
of those who already subscribe will subscribe again while 30% of those who do
not now subscribe will subscribe. On the last letter, it was found that 40% of those
receiving it ordered a subscription. What percent of those receiving the current
letter can be expected to order a subscription?

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

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 the rank of the matrix A =\(\left( \begin{matrix} 4 & 5 & 2 \\ 3 & 2 & 1 \\ 4 & 4 & 8 \end{matrix}\begin{matrix} 2 \\ 6 \\ 0 \end{matrix} \right) \)

A commodity was produced by using 3 units of labour and 2 units of capital, the
total cost is Rs 62. If the commodity had been produced by using 4 units of labour
and one unit of capital, the cost is Rs 56. What is the cost per unit of labour and
capital? (Use determinant method).

Examine the consistency of the system of equations: x+y+z=7,x+2y+3z=18,y+2z=6 .

Find the rank of the matrix A =\(\left( \begin{matrix} 1 & -3 \\ 9 & 1 \end{matrix}\begin{matrix} 4 & 7 \\ 2 & 0 \end{matrix} \right) \)

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.

Find the rank of the matrix A =\(\left( \begin{matrix} -2 & 1 & 3 & 4 \\ 0 & 1 & 1 &2\\ 1 & 3 & 4&7 \end{matrix} \right) \)

Find the rank of the following matrices.

\(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

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

At marina two types of games viz., Horse riding and Quad Bikes riding are available
on hourly rent. Keren and Benita spent Rs 780 and Rs 560 during the month of May.

Name Number of hours Total amount spent
(in Rs)

Horse Riding Quad Bike Riding

Keren 3 4 780

Benita 2 3 560

Find the hourly charges for the two games (rides). (Use determinant method).

A total of Rs 8,600 was invested in two accounts. One account earned \(4\frac { 3 }{ 4 } %\)% annual
interest and the other earned \(6\frac { 1 }{ 2 } %\)% annual interest. If the total interest for one year
was Rs 431.25, how much was invested in each account? (Use determinant method).

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?

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

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