# NCERT solution class 12 chapter 4 Determinants exercise 4.7 mathematics part 1

## EXERCISE 4.7

#### Question 1:

Prove that the determinant is independent of θ.

Hence, Î” is independent of Î¸.

#### Question 2:

Without expanding the determinant, prove that

Hence, the given result is proved.

#### Question 3:

Evaluate

Expanding along C3, we have:

#### Question 4:

If ab and are real numbers, and,

Show that either a + b + c = 0 or a = b = c.

Expanding along R1, we have:

Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.

#### Question 5:

Solve the equations

#### Question 6:

Prove that

Expanding along R3, we have:

Hence, the given result is proved.

If

We know that.

Let verify that

(i)

(ii)

(i)

We have,

(ii)

#### Question 9:

Evaluate

Expanding along R1, we have:

#### Question 10:

Evaluate

Expanding along C1, we have:

#### Question 11:

Using properties of determinants, prove that:

Expanding along R3, we have:

Hence, the given result is proved.

#### Question 12:

Using properties of determinants, prove that:

Expanding along R3, we have:

Hence, the given result is proved.

#### Question 13:

Using properties of determinants, prove that:

Expanding along C1, we have:

Hence, the given result is proved.

#### Question 14:

Using properties of determinants, prove that:

Expanding along C1, we have:

Hence, the given result is proved.

#### Question 15:

Using properties of determinants, prove that:

Hence, the given result is proved.

#### Question 16:

Solve the system of the following equations

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,

A11 = 75, A12 = 110, A13 = 72

A21 = 150, A22 = −100, A23 = 0

A31 = 75, A32 = 30, A33 = − 24

#### Question 17:

If abc, are in A.P., then the determinant

A. 0 B. 1 C. D. 2x

Here, all the elements of the first row (R1) are zero.

Hence, we have Δ = 0.

#### Question 18:

If xyz are nonzero real numbers, then the inverse of matrix is

A.  B.

C.  D.

#### Question 19:

Let, where 0 ≤ θ≤ 2π, then

A. Det (A) = 0

B. Det (A) ∈ (2, ∞)

C. Det (A) ∈ (2, 4)

D. Det (A)∈ [2, 4]