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

St. Britto Hr. Sec. School - Madurai

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

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

Cramer’s rule is applicable only to get an unique solution when

\({ \triangle }_{ z }\neq 0\)
\({ \triangle }_{ x }\neq 0\)
\({ \triangle }_\neq 0\)
\({ \triangle }_{ y }\neq 0\)

\(\left| { A }_{ n\times n } \right| \)=3 \(\left| adjA \right| \) =243 then the value n is

If A= \(\begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}\),then \(\rho (A)\) is

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

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

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

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

The rank of the unit matrix of order n is

n −1
n
n +1
n²

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

non- singular matrix
singular matrix
zero matrix
none of these

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

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

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
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.

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

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, 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.

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

Find the rank of the following matrices.

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

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.

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?

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

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?

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.

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?

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).

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

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

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

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

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

St. Britto Hr. Sec. School - Madurai

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

Part - A

Multiple Choice Question - Answer All Question

Cramer’s rule is applicable only to get an unique solution when

\(\left| { A }_{ n\times n } \right| \)=3 \(\left| adjA \right| \) =243 then the value n is

If A= \(\begin{pmatrix} 2 & 0 \\ 0 & 8 \end{pmatrix}\),then \(\rho (A)\) is

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

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

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

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

The rank of the unit matrix of order n is

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

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

Solve the equations x+2y+z=7,2x−y+2z=4,x+y−2z = −1 by using 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 equations by using Cramer’s rule 2x + 3y = 7; 3x + 5y = 9

Find the rank of the following matrices. \(\left( \begin{matrix} 5 & 6 \\ 7 & 8 \end{matrix} \right) \)

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

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.

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

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

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

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

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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
2x + y −z = 3, x + y + z =1, x− 2y− 3z = 4

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

Find the rank of the following matrices.

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

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.

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.

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

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