NCERT Solutions Class 12

NCERT Solutions Class 12

Determinants : NCERT Solutions – Class 12 Maths (Ex 7)

Miscellaneous

1. Prove that the determinant  is independent of  

Ans. Let 

Expanding along first row,

 

 

  =  =  which is independent of 


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.


 

3. Evaluate:  

Ans. Let 

Expanding along first row,

=

= 1


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 


5. Solve the equation:  

Ans. Given: 

  

 

Either 

  ……….(i)

Or 

  

 

 

 

But this is contrary as given that .

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


6. Prove that:  

Ans. L.H.S. = 

 

 =  

 =  = R.H.S.    Proved.


7. If  and B =  find  

Ans. Given:  and B = 

Since,   [Reversal law] ……….(i)

Now 

 = 

Therefore,  exists.

  and  and 

 adj. B =  = 

 

From eq. (i), 

 


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) 

  = 


9. Evaluate:  

Ans. Let 

 

 


10. Evaluate:  

Ans. Let 

 

 =  = 


 

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

11.  

Ans. L.H.S. =  =  

=

 

Expanding along third column, 

 = R.H.S.


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.


13.  

Ans. L.H.S. = 

 

 

 = R.H.S.


 

14.  

Ans. L.H.S. = 

 

 = 1 = R.H.S.


15.  = 0

Ans. L.H.S. = 

 

 [ C2 and C3 have become identical]

= 0 = R.H.S.


 

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),

 

 

 


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.

 


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.


 

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.

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