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Exercise 4.6 - Chapter 4 Determinants class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Examine the consistency of the system of equations.

+ 2= 2

2x + 3= 3

Answer:

The given system of equations is:

+ 2= 2

2x + 3= 3

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6584/Chapter%204_html_30b0a9c2.gif

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.

Question 2:

Examine the consistency of the system of equations.

2− y = 5

x + = 4

Answer:

The given system of equations is:

2− y = 5

x + = 4

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6589/Chapter%204_html_625c2816.gif

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.

Question 3:

Examine the consistency of the system of equations.

x + 3y = 5

2x + 6y = 8

Answer:

The given system of equations is:

x + 3y = 5

2x + 6y = 8

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6594/Chapter%204_html_2c1ee03a.gif

∴ A is a singular matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6594/Chapter%204_html_10044d87.gif

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

Question 4:

Examine the consistency of the system of equations.

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

Answer:

The given system of equations is:

x + y z = 1

2x + 3y + 2z = 2

ax + ay + 2az = 4

This system of equations can be written in the form AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6601/Chapter%204_html_m4b4d1993.gif

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.

Question 5:

Examine the consistency of the system of equations.

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

Answer:

The given system of equations is:

3x − y − 2z = 2

2y − z = −1

3x − 5y = 3

This system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6606/Chapter%204_html_m54650ebf.gif

∴ A is a singular matrix.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6606/Chapter%204_html_m1e5b7c2a.gif

Thus, the solution of the given system of equations does not exist. Hence, the system of equations is inconsistent.

Question 6:

Examine the consistency of the system of equations.

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

Answer:

The given system of equations is:

5x − y + 4z = 5

2x + 3y + 5z = 2

5x − 2y + 6z = −1

This system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6608/Chapter%204_html_m631ef4b3.gif

∴ A is non-singular.

Therefore, A−1 exists.

Hence, the given system of equations is consistent.

Question 7:

Solve system of linear equations, using matrix method.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_m724cf69e.gif

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_m33c86301.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6613/Chapter%204_html_39a6a942.gif

Question 8:

Solve system of linear equations, using matrix method.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_1d13fa43.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_m2be58a95.gif

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_59b8eaa8.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6621/Chapter%204_html_m72cd731.gif

Question 9:

Solve system of linear equations, using matrix method.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_m326506ee.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_m5e3784fb.gif

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_5b6948cf.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6627/Chapter%204_html_324d0cda.gif

Question 10:

Solve system of linear equations, using matrix method.

5x + 2y = 3

3x + 2y = 5

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6632/Chapter%204_html_m2810aa19.gif

Thus, A is non-singular. Therefore, its inverse exists.

Question 11:

Solve system of linear equations, using matrix method.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_54e2ee67.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_mcb0a148.gif

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_m4ae283eb.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_m71cac2ce.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6638/Chapter%204_html_bf8a970.gif

Question 12:

Solve system of linear equations, using matrix method.

x − y + z = 4

2x + y − 3z = 0

x + y + z = 2

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_fffddc7.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_59c7d826.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6644/Chapter%204_html_2398fa92.gif

Question 13:

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

Answer:

The given system of equations can be written in the form AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_m6e5aee51.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_2a364aa6.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6647/Chapter%204_html_m480412b3.gif

Question 14:

Solve system of linear equations, using matrix method.

x − y + 2z = 7

3x + 4y − 5z = −5

2x − y + 3z = 12

Answer:

The given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_6f61bdbd.gif

Thus, A is non-singular. Therefore, its inverse exists.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_214d0473.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6652/Chapter%204_html_m76863815.gif

Question 15:

Ifhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_89d4e0a.gif , find A−1. Using A−1 solve the system of equations

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m4a65d85.gif

Answer:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m1d26ecb7.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_m4a89e4df.gif

Now, the given system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_1236668a.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6661/Chapter%204_html_24ed4cc2.gif

Question 16:

The cost of 4 kg onion, 3 kg wheat and 2 kg rice is Rs 60. The cost of 2 kg onion, 4 kg

wheat and 6 kg rice is Rs 90. The cost of 6 kg onion 2 kg wheat and 3 kg rice is Rs 70.

Find cost of each item per kg by matrix method.

Answer:

Let the cost of onions, wheat, and rice per kg be Rs x, Rs y,and Rs z respectively.

Then, the given situation can be represented by a system of equations as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_25a36f4.gif

This system of equations can be written in the form of AX = B, where

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_74fc5d5f.gif

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_5c8fdc20.gif

Now,

X = A−1 B

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/233/6666/Chapter%204_html_m5d62b9ab.gif

Hence, the cost of onions is Rs 5 per kg, the cost of wheat is Rs 8 per kg, and the cost of rice is Rs 8 per kg.

Also Read : Miscellaneous-Exercise-Chapter-4-Determinants-class-12-ncert-solutions-Maths

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