Updated By SaraNextGen

On March 11, 2024, 11:35 AM

Question 1:

Prove that the determinant  is independent of θ.

Answer:

Hence, Δ 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 C₃, 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 R₁, 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 R₃, we have:

Hence, the given result is proved.

Question 7:

If 

Answer:

We know that .

Question 8:

Let  verify that

(i) 

(ii) 

Answer:

(i)

We have,

(ii)

Question 9:

Evaluate 

Answer:

Expanding along R₁, we have:

Question 10:

Evaluate 

Answer:

Expanding along C₁, we have:

Question 11:

Using properties of determinants, prove that:

Answer:

Expanding along R₃, we have:

Hence, the given result is proved.

Question 12:

Using properties of determinants, prove that:

Answer:

Expanding along R₃, we have:

Hence, the given result is proved.

Question 13:

Using properties of determinants, prove that:

Answer:

Expanding along C₁, we have:

Hence, the given result is proved.

Question 14:

Using properties of determinants, prove that:

Answer:

Expanding along C₁, 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,

A₁₁ = 75, A₁₂ = 110, A₁₃ = 72

A₂₁ = 150, A₂₂ = −100, A₂₃ = 0

A₃₁ = 75, A₃₂ = 30, A₃₃ = − 24

Question 17:

Choose the correct Answer.

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

A. 0 B. 1 C. x D. 2x

Answer:

Answer: A

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