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Solutions 4. Determinants - Exercise 4.2 | Class 12 Mathematics-I - Toppers Study
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Solutions 4. Determinants - Exercise 4.2 | Class 12 Mathematics-I - Toppers Study
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Chapter 4 Mathematics-I class 12
Exercise 4.2 class 12 Mathematics-I Chapter 4. Determinants
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4. Determinants
| Exercise 4.2 |
Solutions 4. Determinants - Exercise 4.2 | Class 12 Mathematics-I - Toppers Study
Exercise - 4.2
Using the properties of determinants and without expanding in Exercise 1 to 5, prove that:
Ques.1. 
Ans. Given:
Operating 
0 = 0 [
C1 and C2 are identical]
L.H.S. = R.H.S.
hance proved.
Ques.2. 
Ans. On 
Operating 
= 
= 
= 0 = R.H.S.
[ All entries of one column here first are zero]
hance proved.
Ques.3. 
Ans. On 
operating 
= 
= 
= 9 x 0 = 0 [ two columns are identical]
hance Proved.
Ques.4. 
Ans. Given: 
= 
Operating 
= 
= 
= 0 [ two columns are identical]
hance Proved.
Ques.5. 
Ans. L.H.S. = 
operating 
= 
= 
= 
= 
[operating 
]
= 
[operating 
]
= 
[operating 
]
= 
[Interchanging 
and 
]
= 
[Interchanging and ]
= 
= R.H.S.
hance proved.
Using the properties of determinants and with expanding in Exercise 6 to 14, prove that:
Ques.6. 
Ans. Let 
= 
[Taking 
common from each row]
Interchanging rows and columns.,
hance Proved.
Ques.7. 
Ans. L.H.S. = 
Taking common 
from 
respectively,
= 
= 
[operating 
]
= 
= 
= 
= 
= R.H.S.
hance proved.
Ques.8. (i) 
(ii) 
Ans. (i) L.H.S. = 
operating 
and ,
= 
= 
[Expanding along 1st column]
= 
= 
= 
= 
= R.H.S.
hance Proved.
(ii) L.H.S. = 
operating 
and
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance proved.
Ques.9. 
Ans.
L.H.S. = 
= 
[Multiplying 
by 
respectively]
= 
= 
= 
[operating 
]
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance proved.
Ques.10. (i) 
(ii) 
Ans. (i) L.H.S. = 
= 
= 
= 
[operating and ]
= 
= 
= R.H.S.
hance proved.
(ii) L.H.S. = 
= 
= 
= 
[operating and ]
= 
= 
= 
= R.H.S.
hance Proved.
Ques.11. (i) 
(ii) 
Ans. (i) L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance Proved.
(ii) L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance Proved.
Ques.12. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance Proved.
Ques.13. 
Ans. L.H.S. = 
= 
= 
= 
= 
= 
= 
= R.H.S.
Ques.14. 
Ans. L.H.S. = 
Multiplying 
by respectively and then dividing the determinant by 
= 
= 
= 
= 
= 
= 
= 
= R.H.S.
hance, proved.
Choose the correct answer in Exercises 15 and 16.
Ques.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)]
hance, option (C) is correct.
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