## EXERCISE 4.7

#### 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.