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# NCERT Solutions class 12 Maths Chapter 4 Miscellaneous

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NCERT solutions for Class 12 Maths Chapter 4

## NCERT Solutions class 12 Maths Determinants

### 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.

class 12 Maths Chapter 4 Miscellaneous

### 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

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

=

=  ….(ii)

Now to find  (say), where

C =

=

C =

C =  =  =

Therefore,  exists.

and

and

=  ……….(iii)

Again

=

=  = A (given)

(i)

=

[From eq. (ii) and (iii)]

(ii)

=

Ans. Let

=

=

=

=

=

=

=

=

=

### 10. Evaluate:

Ans. Let

=

=  =  =

NCERT Solutions class 12 Maths Miscellaneous

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

11.

Ans. L.H.S. =  =

=

=

=

=

=

=

=

=  = 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.

class 12 Maths Chapter 4 Miscellaneous

Ans. L.H.S. =

=

=

=

=  = 1 = R.H.S.

### 15.  = 0

Ans. L.H.S. =

=

=

=

=

=  [ C2 and C3 have become identical]

= 0 = R.H.S.

NCERT Solutions class 12 Maths Miscellaneous

### 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

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.

NCERT Solutions class 12 Maths Miscellaneous

### 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

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.

## NCERT Solutions class 12 Maths Chapter 4 Miscellaneous

NCERT Solutions class 12 Maths Chapter 4 Miscellaneous PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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### 4 thoughts on “NCERT Solutions class 12 Maths Chapter 4 Miscellaneous”

1. Awesome concept

2. Nice work

I please request you to check the questions 7th and rewrite the correct answer at last
Thank you

4. thanks a lot