NCERT Solutions Class 12

Exercise 4.6

Examine the consistency of the system of equations in Exercises 1 to 3.

1.  

Ans. Matrix form of given equations is AX = B

 

  A =  and B = 

  = 

Therefore, Unique solution and hence equations are consistent.

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2.  

Ans. Matrix form of given equations is AX = B

  

  A =  and B = 

  = 

Therefore, Unique solution and hence equations are consistent.

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3. 

Ans. Matrix form of given equations is AX = B

  

 A =  and B = 

  = 6 – 6 = 0

Now (adj. A) B =  = 

Therefore, given equations are inconsistent, i.e., have no common solution.

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Examine the consistency of the system of equations in Exercises 4 to 6.

4.  

Ans. Matrix form of given equations is AX = B

 

Here A = 

 

  =   0

Therefore, Unique solution and hence equations are consistent.

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5.  

Ans. Matrix form of given equations is AX = B

  

Here A = 

 

  =  = 

Now (adj. A) = 

And (adj. A) B =  =  = 

Therefore, given equations are inconsistent.

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6.  

Ans. Matrix form of given equations is AX = B

  

Here A = 

 

  = 

Therefore, Unique solution and hence equations are consistent.

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Solve the system of linear equations, using matrix method, in Exercise 7 to 10.

7.  

Ans. Matrix form of given equations is AX = B

  

Here    A = , X =  and B = 

   = 

Therefore, solution is unique and  = 

 

= 

Therefore,  and 

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8.  

Ans. Matrix form of given equations is AX = B

 

Here    A = , X =  and B = 

  = 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

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9.  

Ans. Matrix form of given equations is AX = B

 

Here    A = , X =  and B = 

  = 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

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10.  

Ans. Matrix form of given equations is AX = B

 

Here    A = , X =  and B = 

  = 

Therefore, solution is unique and  = 

  = 

Therefore,  and 

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Solve the system of linear equations, using matrix method, in Exercise 11 to 14.

11.  

Ans. Matrix form of given equations is AX = B

 

Here A =  , X =  and B = 

  =  = 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

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12.  

Ans. Matrix form of given equations is AX = B

 

Here    A =  , X =   and B = 

 

= 

= 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

 

 

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13.  

Ans. Matrix form of given equations is AX = B

 

Here A =  , X =   and B = 

 

= 

= 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

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14.  

Ans. Matrix form of given equations is AX = B

 

Here A =  , X =   and B = 

 

= 

= 

Therefore, solution is unique and  = 

 

= 

= 

Therefore,  and 

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15. If A =  find  Using  solve the system of equations    

Ans. Given: Matrix A = 

 

  = 

  exists and  ……….(i)

Now,  and  and 

 adj. A =  = 

 From eq. (i),

= 

Now, Matrix form of given equations is AX = B

 

Here A =  , X =   and B = 

Therefore, solution is unique and 

 

= 

= 

Therefore,  and 

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16. The cost of 4 kg onion, 3 kg wheat and 2 kg rice is ` 60. The cost of 2 kg onion, 4 kg wheat and 2 kg rice is ` 90. The cost of 6 kg onion, 2 k wheat and 3 kg rice is ` 70. Find cost of each item per kg by matrix method.

Ans. Let ` ` ` per kg be the prices of onion, wheat and rice respectively.

 According to given data, we have three equations,

Matrix form of given equations is AX = B

 

Here A =  , X =   and B = 

  =  = 

Therefore, solution is unique and  =  …….(i)

Now,   

 adj. A = 

 From eq. (i),

= 

= 

Therefore,  and 

Hence, the cost of onion, wheat and rice are ` 5, ` 8 and ` 8 per kg.