NCERT Solutions Class 12

NCERT Solutions Class 12

Determinants : NCERT Solutions – Class 12 Maths (Ex 2)

Exercise 4.2

Using the properties of determinants and without expanding in Exercise 1 to 5, prove that:

1.  

Ans. Given: 

Operating  

 0 = 0 [ C1 and C2 are identical]

 L.H.S. = R.H.S.


2. 

Ans. On  Operating 

 = 0 = R.H.S.

[ All entries of one column here first are zero]


3.  

Ans. On  operating 

 = 

= 9 x 0 = 0 [ two columns are identical] Proved.


4.  

Ans. Given:      = 

Operating  

 = 0 [ two columns are identical] Proved.


5.  

Ans. L.H.S. =  operating 

 [operating ]

 [operating ]

 [operating ]

 [Interchanging  and ]

 [Interchanging  and ]

 = R.H.S.


6. 

Ans. Let 

 [Taking  common from each row]

Interchanging rows and columns in the determinants on R.H.S.,

 

 

  Proved.


7.  

Ans. L.H.S. = 

Taking common  from  respectively,

   [operating ]

 = 

 =  = R.H.S.


8. (i)  

(ii)  

Ans. (i) L.H.S. =  operating  and ,

 [Expanding along 1st column]

 =  = R.H.S. Proved.

(ii) L.H.S. =  operating  and 

=  

 = R.H.S.


9.  

Ans. 

L.H.S. =  =  [Multiplying  by  respectively]

 = 

 [operating ]

 = R.H.S.


10. (i)  

(ii)  

Ans. (i) L.H.S. = 

 

 [operating  and ]

 = R.H.S.

(ii) L.H.S. = 

 

 [operating  and ]

 = 

= R.H.S.  Proved.


11. (i)  

(ii)  

Ans. (i) L.H.S. = 

 

 

 = R.H.S. Proved.

(ii) L.H.S. = 

 

 

 = R.H.S. Proved.


12. 

Ans. L.H.S. = 

 

 

 = R.H.S.      Proved.


13. 

Ans. L.H.S. = 

 

 

 = R.H.S.


14.  

Ans. L.H.S. = 

Multiplying  by  respectively and then dividing the determinant by 

 

 

 = R.H.S.        Proved.


 

Choose the correct answer in Exercises 15 and 16.

15. Let A be a square matrix of order 3 x 3, then  is equal to:

(A) 

(B) 

(C) 

(D) 

Ans. Let A =  be a square matrix of order 3 x 3.   ……….(i)

  

  

  

 [From eq. (i)]

Therefore, option (C) is correct.


 

16. Which is the following is correct:

(A) Determinant is a square matrix.

(B) Determinant is a number associated to a matrix.

(C) Determinant is a number associated to a square matrix.

(D) None of these.

Ans. Since, Determinant is a number associated to a square matrix.

Therefore, option (C) is correct.

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