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Solutions ⇒ Class 12th ⇒ Mathematics-I ⇒ 4. Determinants

# Solutions 4. Determinants - Miscellaneous Exercise on Chapter - 4 | Class 12 Mathematics-I - Toppers Study

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## Solutions 4. Determinants - Miscellaneous Exercise on Chapter - 4 | Class 12 Mathematics-I - Toppers Study

CBSE board students who preparing for class 12 ncert solutions maths and Mathematics-I solved exercise chapter 4. Determinants available and this helps in upcoming exams 2023-2024.

### You can Find Mathematics-I solution Class 12 Chapter 4. Determinants

• All Chapter review quick revision notes for chapter 4. Determinants Class 12
• NCERT Solutions And Textual questions Answers Class 12 Mathematics-I
• Extra NCERT Book questions Answers Class 12 Mathematics-I
• Importatnt key points with additional Assignment and questions bank solved.

Chapter 4 Mathematics-I class 12

### Miscellaneous Exercise on Chapter - 4 class 12 Mathematics-I Chapter 4. Determinants

• Solutions 4. Determinants - Miscellaneous Exercise On Chapter - 4 | Class 12 Mathematics-I - Toppers Study
• Class 12 Ncert Solutions
• Solution Chapter 4. Determinants Class 12
• Solutions Class 12
• Chapter 4. Determinants Miscellaneous Exercise On Chapter - 4 Class 12

## 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

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

….(ii)

Now to find  (say), where

C =

C =

C =  =  =

hance,  exists.

and

and

……….(iii)

Again

= A (given)

(i)

=

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

(ii)

=

Ans. Let

### Ques.10. Evaluate:

Ans. Let

=  =

Ques.11.

Ans. L.H.S. =  =

=

= 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

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

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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## Study Materials List:

##### Solutions ⇒ Class 12th ⇒ Mathematics-I
1. Relations and Functions
2. Inverse Trigonometric Functions
3. Matrices
4. Determinants
5. Continuity And Differentiability
6. Application of Derivatives

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