NCERT Solutions for Class 12 Maths Exercise Miscellaneous Chapter 4 Determinants – FREE PDF Download
NCERT Solutions for Class 12 Maths Chapter 4 – Determinants is a sure-shot way of obtaining the complete marks in the particular chapter for Board Exam 2019- 2020. CoolGyan provides you with Free PDF download of the same solved by Expert Teachers as per NCERT (CBSE) Book guidelines. They provide the students with precise and to the point answers which fetch very good marks in Board Exams. Download today the NCERT CBSE Solutions for Class 12 Maths Chapter 4 – Determinants to achieve your goal score. Class 12 Maths Chapter 4 – Determinants solved by Expert Teachers as per NCERT (CBSE) Book guidelines.
NCERT Solutions for Class 12 Maths Chapter 4 – Determinants
1. Prove that the determinant 
is independent of 
Expanding along first row,
= 
= 
which is independent of
2. Without expanding the determinants, prove that: 
Multiplying R1 by a, R2 by b and R3 by c.
= 
= 
= 
[Interchanging C1 and C3]
= 
[Interchanging C2 and C3]
= 
= 
=RHS
Proved.
3. Evaluate:
Expanding along first row,
=
= 
= 
= 
= 1
4. If and are real numbers and Show that either a+b+c=0 or a=b=c
and 
are real numbers and 
Show that either a+b+c=0 or a=b=c
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:
Either 
……….(i)
Or 
But this is contrary as given that 
.
Therefore, from eq. (i), is only the solution.
6. Prove that:
= 
= 
= 
= 
= 
= 
= 
= R.H.S. Proved.
7. If and B = find
and B = 
find 
and B = 
Since, 
[Reversal law] ……….(i)
Now 
= 
= 
Therefore, 
exists.
and 
and 
adj. B = 
= 
From eq. (i), 
=∣∣∣∣9−21−310502∣∣∣∣=|9−35−210102|
8. Let A = verify that:
verify that:
(i) 
(ii) 
= 
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:
= 
= 
= 
= 
= 
= 
= 
= 
= 
10. Evaluate:
= 
= 
= 
= 
Using properties of determinants in Exercises 11 to 15, prove that:
11. 
= 
=
= 
Expanding along third column,
= 
= 
= 
= 
= 
= 
= R.H.S.
12.
= 
= 
(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.
= 
= 
= 
= 
= 
(a+b+c)[4bc+2ab+2ac+a2−a2+ac+ab−bc](a+b+c)[4bc+2ab+2ac+a2−a2+ac+ab−bc]
= 
= 
= R.H.S.
14.
= 
= 
= 
= 
=> 1 = R.H.S.
15. = 0
= 0
= 
= 
= 
= 
= 
[
C2 and C3 have become identical]
= 0 = R.H.S.
16. Solve the system of the following equations: (Using matrices):
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) 
……….(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:
are non-zero real numbers, then the inverse of matrix A = 
is:
(A) 
(B) 
(C) 
(D) 
exists and unique solution is 
……….(i)
Now 
and 
and 
adj. A = 
= 
And 
= 
= 
= 
Therefore, option (A) is correct.
19. Let A = where Then:
where 
Then:
(A) Det (A) = 0
(B) Det (A) 
(C) Det (A) 
(D) Det (A) 
……….(i)
Since 
[
cannot be negative]
Therefore, option (D) is correct.