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Solutions 4. Determinants - Miscellaneous Exercise on Chapter - 4 | Class 12 Mathematics-I - Toppers Study
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Chapter 4 Mathematics-I class 12
Miscellaneous Exercise on Chapter - 4 class 12 Mathematics-I Chapter 4. Determinants
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4. Determinants
| Miscellaneous Exercise on Chapter - 4 |
Solutions 4. Determinants - Miscellaneous Exercise on Chapter - 4 | Class 12 Mathematics-I - Toppers Study
Ques.1. Prove that the determinant
is independent of
Ans. Let
Expanding along R1,
=
=
which is independent of
Ques.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.
Ques.3. Evaluate:
Ans. Let
Expanding along R1,
=
=
=
=
= 1
Ques.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)
hance, from eq. (i) and (ii), either or
Ques.5. Solve the equation:
Ans. Given:
Either
……….(i)
Or
But this is contrary as given that .
hance, from eq. (i), is only the solution.
Ques.6. Prove that:
Ans. L.H.S. =
=
=
=
= =
= =
= R.H.S. Proved.
Ques.7. If
and B =
find
Ans. Given: and B =
Since, [Reversal law] ……….(i)
Now
= =
hance, exists.
and
and
adj. B =
=
From eq. (i),
=
Ques.8. Let A =
verify that:
(i)
(ii)
Ans. Given: Matrix A =
=
Therefore, exists.
and
and
adj. A =
= B (say)
=
………(i)
= =
hance, exists.
and
and
adj. B =
=
=
= ….(ii)
Now to find (say), where
C =
=
C =
C = =
=
hance, exists.
and
and
adj. A =
= ……….(iii)
Again
=
= = A (given)
(i)
=
[From eq. (ii) and (iii)]
(ii)
=
Ques.9. Evaluate:
Ans. Let
=
=
=
=
=
=
=
=
=
Ques.10. Evaluate:
Ans. Let
=
= =
=
Ques.11.
Ans. L.H.S. = =
=
=
Expanding along C3 
=
=
=
=
=
= = R.H.S.
Ques.12.
Ans. L.H.S. =
=
= (say) ……….(i)
Now
=
=
=
From eq. (i), L.H.S. =
……….(ii)
Now
=
Expanding along C1,
=
=
=
=
=
From eq. (i), L.H.S. =
= = R.H.S.
Ques.13.
Ans. L.H.S. =
=
=
=
=
=
=
=
= = R.H.S.
Ques.14.
Ans. L.H.S. =
=
=
=
= = 1 = R.H.S.
Ques.15.
= 0
Ans. L.H.S. =
=
=
=
=
= [
C2 and C3 have become identical]
= 0 = R.H.S
Ques.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),
=
=
Ques.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]
hance, option (A) is correct.
Ques.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
=
=
=
hance, option (A) is correct.
Ques.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]
hance, option (D) is correct.
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