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

**2. Without expanding the determinants, prove that: **

**Ans. **L.H.S. =

Multiplying R_{1} by R_{2} by and R_{3} by , =

= = [Interchanging C_{1} and C_{3}]

= = [Interchanging C_{2} and C_{3}]

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

adj. B = =

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

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: **** **

**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. = =

=

=

##### Expanding along third column,

=

=

=

=

=

= = 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

**14. **** **

**Ans. **L.H.S. =

=

=

=

= = 1 = R.H.S.

**15. **** = 0**

**Ans. **L.H.S. =

=

=

=

=

= [ C_{2} and C_{3} 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

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) **** **

**Ans. **According to question, ……….(i)

Let

=

=

[From eq. (i)] = 0 [ R_{2} and R_{3} 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

adj. A = =

And

=

=

=

Therefore, option (A) is correct.

### 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

**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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