NCERT Solutions class 12 Maths Chapter 4 Miscellaneous

NCERT Solutions class 12 Maths Chapter 4 Miscellaneous book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

NCERT solutions for Class 12 Maths Chapter 4 Download as PDF

NCERT Solutions class 12 Maths Determinants 

1. Prove that the determinant  is independent of  

Ans. Let

Expanding along first row,

 

 

  =  =  which is independent of

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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.

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class 12 Maths Chapter 4 Miscellaneous

3. Evaluate:  

Ans. Let

Expanding along first row,

=

=

=

=

= 1

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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

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5. Solve the equation:  

Ans. Given:

  

 

Either

  ……….(i)

Or

  

 

 

 

But this is contrary as given that .

Therefore, from eq. (i),  is only the solution.

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6. Prove that:  

Ans. L.H.S. =

=

=

=  

=  =  

=  =  = R.H.S.    Proved.

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7. If  and B =  find  

Ans. Given:  and B =

Since,   [Reversal law] ……….(i)

Now

=  =

Therefore,  exists.

  and  and

 adj. B =  =

 

From eq. (i),

 

=

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8. Let A =  verify that:

(i)    

(ii)  

Ans. Given: Matrix A =

 

  =

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)

  =

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9. Evaluate:  

Ans. Let

=  

=

=  

=

=

=

=

=

=

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10. Evaluate:  

Ans. Let

=  

=  =  =

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NCERT Solutions class 12 Maths Miscellaneous

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

11.  

Ans. L.H.S. =  =  

=

=  

Expanding along third column,

=

=

=

=

=

=  = R.H.S.

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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.

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13.  

Ans. L.H.S. =

=  

=

=  

=

=

=

=

=  = R.H.S.

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class 12 Maths Chapter 4 Miscellaneous

14.  

Ans. L.H.S. =

=  

=

=

=  = 1 = R.H.S.

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15.  = 0

Ans. L.H.S. =

=

=  

=

=

=  [ C₂ and C₃ have become identical]

= 0 = R.H.S.

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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

 adj. A =  =

And

 From eq. (i),

 

=

=

 

 

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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.

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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

 adj. A =  =

And

=

=

=

Therefore, option (A) is correct.

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NCERT Solutions class 12 Maths Miscellaneous

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

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