12th Business Maths Monthly Test - 1 ( Applications of Matrices and Determinants )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Monthly Test - 1 ( Applications of Matrices and Determinants )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

In a transition probability matrix, all the entries are greater than or equal to

if \(\rho (A)\neq \rho (A,B),\) then the system is

Consistent and has infinitely many solutions
Consistent and has a unique solution
inconsistent
consistent

Rank of a null matrix is

0
-1
\(\infty \)
1

If the number of variables in a non- homogeneous system AX = B is n, then the system possesses a unique solution only when

\(\rho (A)=\rho (A,B)>n\)
\(\rho (A)=\rho (A,B)
\(\rho (A)=\rho (A,B)=n\)
none of these

IfA =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AA^Tis

if T=\(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right) \) is a transition probability matrix, then at equilibrium A is equal to

\(\frac { 1 }{ 4 } \)
\(\frac { 1 }{ 5 } \)
\(\frac { 1 }{ 6 } \)
\(\frac { 1 }{ 8 } \)

If A=(1 2 3), then the rank of AA^T is

if \(\rho (A)=\rho (A,B)\) then the system is

Consistent and has infinitely many solutions
Consistent and has a unique solution
Consistent
inconsistent

For the system of equations x+2y+3z=1, 2x+y+3z=25x+5y+9z =4

there is only one solution
there exists infinitely many solutions
there is no solution
None of these

If \(\rho (A)=\rho (A,B)\)= the number of unknowns, then the system is

Consistent and has infinitely many solutions
Consistent and has a unique solution
inconsistent
consistent

The rank of the matrix \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is

if \(\frac { { a }_{ 1 } }{ x } +\frac { { b }_{ 1 } }{ y } ={ c }_{ 1 },\frac { { a }_{ 2 } }{ x } +\frac { { b }_{ 2 } }{ y } ={ c }_{ 2 },{ \triangle }_{ 1= }\begin{vmatrix} { a }_{ 1 } & { b }_{ 1 } \\ { a }_{ 2 } & { b }_{ 2 } \end{vmatrix};\quad { \triangle }_{ 2 }=\begin{vmatrix} { b }_{ 1 } & { c }_{ 1 } \\ { b }_{ 2 } & { c }_{ 2 } \end{vmatrix}{ \triangle }_{ 3 }=\begin{vmatrix} { c }_{ 1 } & { a }_{ 1 } \\ { c }_{ 2 } & a_{ 2 } \end{vmatrix}\) then (x,y) is

\(\left( \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } \frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } } \right) \)
\(\left( \frac { { \triangle }_{ 3 } }{ { \triangle }_{ 1 } } \frac { { \triangle }_{ 2 } }{ { \triangle }_{ 1 } } \right) \)
\(\left( \frac { { \triangle }_{ 1 } }{ { \triangle }_{ 2 } } \frac { { \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right) \)
\(\left( \frac { { -\triangle }_{ 1 } }{ { \triangle }_{ 2 } } \frac { {- \triangle }_{ 1 } }{ { \triangle }_{ 3 } } \right) \)

The system of equations 4x+6y=5, 6x+9y=7 has

a unique solution
no solution
infinitely many solutions
none of these

Which of the following is not an elementary transformation?

\({ R }_{ i }\leftrightarrow { R }_{ j }\)
\({ R }_{ i }\rightarrow { 2R }_{ i }+{ 2c }_{ j }\)
\({ R }_{ i }\rightarrow { 2R }_{ i }-{ 4R }_{ i }\)
\({ C }_{ i }\rightarrow { C }_{ i }+{ 5C }_{ j }\)

If the rank of the matrix \(\left( \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right) \) is 2. Then \(\lambda \) is

1
2
3
only real number

if \(\left| A \right| \neq 0,\) then A is

non- singular matrix
singular matrix
zero matrix
none of these

The rank of m×n matrix whose elements are unity is

The system of linear equations x+y+z=2,2x+y−z=3,3x+2y+k =4 has unique solution, if k is not equal to

4
0
-4
1

If \(\rho (A)\) =r then which of the following is correct?

all the minors of order r which does not vanish
A has at least one minor of order r which does not vanish
A has at least one (r+1) order minor which vanishes
all (r+1) and higher order minors should not vanish

if T= \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.7 } & \overset { B }{ 0.3 } \\ 0.6 & x \end{matrix} \right) \) is a transition probability matrix, then the value of x is

Part - B

Generic 2 - Answer All Question

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

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

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).

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?

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

Show that the equations5x+3y+7z=4,3x+26y+2z=9,7x+2y+10z =5 are consistent and solve them by rank method.

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

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).

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?

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

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

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 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.

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?

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.

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

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).

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.

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

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.

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

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?

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

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

Name	Number of hours		Total amount spent (in Rs)
Name	Horse Riding	Quad Bike Riding	Total amount spent (in Rs)
Keren	3	4	780
Benita	2	3	560

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

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12th Standard English Medium

12th Standard Tamil Medium

11th Standard English Medium

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10th Standard English Medium

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12th Business Maths Monthly Test - 1 ( Applications of Matrices and Determinants )

St. Britto Hr. Sec. School - Madurai

12th Business Maths Monthly Test - 1 ( Applications of Matrices and Determinants )-Aug 2020

Part - A

Multiple Choice Question - Answer All Question

In a transition probability matrix, all the entries are greater than or equal to

if \(\rho (A)\neq \rho (A,B),\) then the system is

Rank of a null matrix is

If the number of variables in a non- homogeneous system AX = B is n, then the system possesses a unique solution only when

IfA =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AAT is

if T=\(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.4 } & \overset { B }{ 0.6 } \\ 0.2 & 0.8 \end{matrix} \right) \) is a transition probability matrix, then at equilibrium A is equal to

If A=(1 2 3), then the rank of AAT is

if \(\rho (A)=\rho (A,B)\) then the system is

For the system of equations x+2y+3z=1, 2x+y+3z=25x+5y+9z =4

If \(\rho (A)=\rho (A,B)\)= the number of unknowns, then the system is

The rank of the matrix \(\left( \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 4 & 9 \end{matrix} \right) \) is

The system of equations 4x+6y=5, 6x+9y=7 has

Which of the following is not an elementary transformation?

If the rank of the matrix \(\left( \begin{matrix} \lambda & -1 & 0 \\ 0 & \lambda & -1 \\ -1 & 0 & \lambda \end{matrix} \right) \) is 2. Then \(\lambda \) is

if \(\left| A \right| \neq 0,\) then A is

The rank of m×n matrix whose elements are unity is

The system of linear equations x+y+z=2,2x+y−z=3,3x+2y+k =4 has unique solution, if k is not equal to

If \(\rho (A)\) =r then which of the following is correct?

if T= \(_{ B }^{ A }\left( \begin{matrix} \overset { A }{ 0.7 } & \overset { B }{ 0.3 } \\ 0.6 & x \end{matrix} \right) \) is a transition probability matrix, then the value of x is

Part - B

Generic 2 - Answer All Question

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

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

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).

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

Show that the equations5x+3y+7z=4,3x+26y+2z=9,7x+2y+10z =5 are consistent and solve them by rank method.

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

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).

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

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

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 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.

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

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

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.

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

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.

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

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

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IfA =\(\left( \begin{matrix} 1 \\ 2 \\ 3 \end{matrix} \right) \) then the rank of AA^Tis

If A=(1 2 3), then the rank of AA^T is

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

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

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).

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
5x + 3y = 17; 3x + 7y = 31

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).

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 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.

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

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.

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

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.

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