## NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Ex 4.6

#### Page No 136:

#### Question 1:

Examine the consistency of the system of equations.

*x*+ 2

*y*= 2

2

*x*+ 3*y*= 3#### Answer:

The given system of equations is:

*x*+ 2

*y*= 2

2

*x*+ 3*y*= 3
The given system of equations can be written in the form of

*AX*=*B*, where
∴

*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

*x*−*y*= 5*x*+

*y*= 4

#### Answer:

The given system of equations is:

2

*x*−*y*= 5*x*+

*y*= 4

The given system of equations can be written in the form of

*AX*=*B*, where
∴

*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*+ 3

*y*= 5

2

*x*+ 6*y*= 8#### Answer:

The given system of equations is:

*x*+ 3

*y*= 5

2

*x*+ 6*y*= 8
The given system of equations can be written in the form of

*AX*=*B*, where
∴

*A*is a singular matrix.
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

2

*x*+ 3*y*+ 2*z*= 2*ax*+

*ay*+ 2

*az*= 4

#### Answer:

The given system of equations is:

*x*+

*y*+

*z*= 1

2

*x*+ 3*y*+ 2*z*= 2*ax*+

*ay*+ 2

*az*= 4

This system of equations can be written in the form

*AX*=*B*, where
∴

*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.

3

*x*−*y*− 2z = 2
2

*y*−*z*= −1
3

*x*− 5*y*= 3#### Answer:

The given system of equations is:

3

*x*−*y*− 2z = 2
2

*y*−*z*= −1
3

*x*− 5*y*= 3
This system of equations can be written in the form of

*AX*=*B*, where
∴

*A*is a singular matrix.
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.

5

*x*−*y*+ 4*z*= 5
2

*x*+ 3*y*+ 5*z*= 2
5

*x*− 2*y*+ 6*z*= −1#### Answer:

The given system of equations is:

5

*x*−*y*+ 4*z*= 5
2

*x*+ 3*y*+ 5*z*= 2
5

*x*− 2*y*+ 6*z*= −1
This system of equations can be written in the form of

*AX*=*B*, where
∴

*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.

#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 8:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 9:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 10:

Solve system of linear equations, using matrix method.

5

*x*+ 2*y*= 3
3

*x*+ 2*y*= 5#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 11:

Solve system of linear equations, using matrix method.

#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 12:

Solve system of linear equations, using matrix method.

*x*−

*y*+

*z*= 4

2

*x*+*y*− 3*z*= 0*x*+

*y*+

*z*= 2

#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 13:

Solve system of linear equations, using matrix method.

2

*x*+ 3*y*+ 3*z*= 5*x*− 2

*y*+

*z*= −4

3

*x*−*y*− 2*z*= 3#### Answer:

The given system of equations can be written in the form

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Question 14:

Solve system of linear equations, using matrix method.

*x*−

*y*+ 2

*z*= 7

3

*x*+ 4*y*− 5*z*= −5
2

*x*−*y*+ 3*z*= 12#### Answer:

The given system of equations can be written in the form of

*AX*=*B*, where
Thus,

*A*is non-singular. Therefore, its inverse exists.#### Page No 137:

#### Question 15:

If, find

*A*^{−1}. Using A^{−1}solve the system of equations#### Answer:

Now, the given system of equations can be written in the form of

*AX*=*B*, where#### 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:

This system of equations can be written in the form of

*AX*=*B*, where
Now,

*X*=

*A*

^{−1}

*B*

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.

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