NCERT Solutions Class 12

Miscellaneous

1. Prove that the determinant  is independent of  

Ans. Let 

Expanding along first row,

 

 

  =  =  which is independent of 

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2. Without expanding the determinants, prove that: 

Ans. L.H.S. = 

Multiplying R₁ by  R₂ by  and R₃ by ,  = 

=  =  [Interchanging C₁ and C₃]

=  =  [Interchanging C₂ and C₃]

Proved.

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3. Evaluate:  

Ans. Let 

Expanding along first row,

=

= 

= 

= 

= 1

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4. If  and  are real numbers and  Show that either  or 

Ans. Given: 

  

 

Here,   Either 

 ……….(i)

Or 

  

  [Expanding along first row]

 

 

 

 

 

 

 

  and  and  

  and  and  ……….(ii)

Therefore, from eq. (i) and (ii), either  or 

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5. Solve the equation:  

Ans. Given: 

  

 

Either 

  ……….(i)

Or 

  

 

 

 

But this is contrary as given that .

Therefore, from eq. (i),  is only the solution.

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6. Prove that:  

Ans. L.H.S. = 

= 

= 

=  

=  =  

=  =  = R.H.S.    Proved.

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7. If  and B =  find  

Ans. Given:  and B = 

Since,   [Reversal law] ……….(i)

Now 

=  = 

Therefore,  exists.

  and  and 

 adj. B =  = 

 

From eq. (i), 

 

= 

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8. Let A =  verify that:

(i)    

(ii)  

Ans. Given: Matrix A = 

 

  = 

Therefore,  exists.

  and 

and 

 adj. A =  = B (say)

  =            ………(i)

 

=  = 

Therefore,  exists.

  and 

and 

 adj. B =  = 

 

= 

=  ….(ii)

Now to find  (say), where

C = 

= 

C = 

C =  =  =

Therefore,  exists.

  and 

and 

 adj. A = 

=  ……….(iii)

Again 

= 

=  = A (given)

(i) 

 = 

[From eq. (ii) and (iii)]

(ii) 

  = 

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9. Evaluate:  

Ans. Let 

=  

= 

=  

= 

= 

= 

= 

= 

= 

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10. Evaluate:  

Ans. Let 

=  

=  =  = 

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Using properties of determinants in Exercises 11 to 15, prove that:

11.  

Ans. L.H.S. =  =  

=

=  

Expanding along third column, 

= 

= 

= 

= 

= 

=  = R.H.S.

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12.  

Ans. L.H.S. = 

= 

=  (say) ……….(i)

Now 

= 

= 

= 

 From eq. (i), L.H.S. =  ……….(ii)

Now 

=  

Expanding along third column, 

= 

= 

= 

= 

= 

 From eq. (i), L.H.S. = 

=  = R.H.S.

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13.  

Ans. L.H.S. = 

=  

= 

=  

= 

= 

= 

= 

=  = R.H.S.

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14.  

Ans. L.H.S. = 

=  

= 

= 

=  = 1 = R.H.S.

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15.  = 0

Ans. L.H.S. = 

= 

=  

= 

= 

=  [ C₂ and C₃ have become identical]

= 0 = R.H.S.

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16. Solve the system of the following equations: (Using matrices):

 

Ans. Putting  and  in the given equations,

  

 the matrix form of given equations is  [AX= B]

Here,   A =  X =  and B = 

 

= 

= 

  exists and unique solution is  ……….(i)

Now     and 

and 

 adj. A =  = 

And 

 From eq. (i),

 

= 

= 

 

 

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Choose the correct answer in Exercise 17 to 19.

17. If  are in A.P., then the determinant  is:

(A) 0  

(B) 1  

(C)     

(D)  

Ans. According to question,  ……….(i)

Let 

=  

= 

[From eq. (i)] = 0 [ R₂ and R₃ have become identical]

Therefore, option (A) is correct.

 

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18. If  are non-zero real numbers, then the inverse of matrix A =  is:

(A) 

(B)  

(C) 

(D)  

Ans. Given: Matrix A = 

 

 

  exists and unique solution is  ……….(i)

Now     and  and 

 adj. A =  = 

And 

= 

= 

= 

Therefore, option (A) is correct.

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19. Let A =  where  Then:

(A) Det (A) = 0 

(B) Det (A)   

(C) Det (A)  

(D) Det (A) 

Ans. Given: Matrix A = 

 

 

  ……….(i)

Since   

  [  cannot be negative]

 

 

 

Therefore, option (D) is correct.