## NCERT Solutions for Class 12 Maths Chapter 4 – Determinants Miscellaneous Exercise

#### Page No 141:

#### Question 1:

Prove that the determinant is independent of

*θ*.#### Answer:

Hence, Î” is independent of

*Î¸*.#### Question 2:

Without expanding the determinant, prove that

#### Answer:

Hence, the given result is proved.

#### Question 3:

Evaluate

#### Answer:

Expanding along C

_{3}, we have:#### Question 4:

If

*a*,*b*and*c*are real numbers, and,
Show that either

*a*+*b*+*c*= 0 or*a*=*b*=*c*.#### Answer:

Expanding along R

_{1}, we have:
Hence, if Δ = 0, then either

*a*+*b*+*c*= 0 or*a*=*b*=*c*.#### Question 5:

Solve the equations

#### Answer:

#### Question 6:

Prove that

#### Answer:

Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Question 7:

If

#### Answer:

We know that.

#### Page No 142:

#### Question 8:

Let verify that

(i)

(ii)

#### Answer:

(i)

We have,

(ii)

#### Question 9:

Evaluate

#### Answer:

Expanding along R

_{1}, we have:#### Question 10:

Evaluate

#### Answer:

Expanding along C

_{1}, we have:#### Question 11:

Using properties of determinants, prove that:

#### Answer:

Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Question 12:

Using properties of determinants, prove that:

#### Answer:

Expanding along R

_{3}, we have:
Hence, the given result is proved.

#### Question 13:

Using properties of determinants, prove that:

#### Answer:

Expanding along C

_{1}, we have:
Hence, the given result is proved.

#### Question 14:

Using properties of determinants, prove that:

#### Answer:

Expanding along C

_{1}, we have:
Hence, the given result is proved.

#### Question 15:

Using properties of determinants, prove that:

#### Answer:

Hence, the given result is proved.

#### Question 16:

Solve the system of the following equations

#### Answer:

Let

Then the given system of equations is as follows:

This system can be written in the form of

*AX*=*B*, where
A

Thus,

*A*is non-singular. Therefore, its inverse exists.
Now,

*A*

_{11}= 75,

*A*

_{12}= 110,

*A*

_{13}= 72

*A*

_{21}= 150,

*A*

_{22}= −100,

*A*

_{23}= 0

*A*

_{31}= 75,

*A*

_{32}= 30,

*A*

_{33}= − 24

#### Page No 143:

#### Question 17:

Choose the correct answer.

If

*a*,*b*,*c*, are in A.P., then the determinant**A.**0

**B.**1

**C.**

*x*

**D.**2

*x*

#### Answer:

**Answer:**

**A**

Here, all the elements of the first row (R

_{1}) are zero.
Hence, we have Δ = 0.

The correct answer is A.

#### Question 18:

Choose the correct answer.

If

*x*,*y*,*z*are nonzero real numbers, then the inverse of matrix is**A.**

**B.**

**C.**

**D.**

#### Answer:

**Answer: A**

The correct answer is A.

#### Question 19:

Choose the correct answer.

Let, where 0 ≤

*θ*≤ 2π, then**A.**Det (A) = 0

**B.**Det (A) ∈ (2, ∞)

**C.**Det (A) ∈ (2, 4)

**D.**Det (A)∈ [2, 4]

#### Answer:

Answer: D

Now,

0≤θ≤2π

⇒-1≤sinθ≤1 The correct answer is D.

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