## EXERCISE 4.1

#### Page No 108:

#### Question 1:

Evaluate the determinants in Exercises 1 and 2.

#### Answer:

= 2(−1) − 4(−5) = − 2 + 20 = 18

#### Question 2:

Evaluate the determinants in Exercises 1 and 2.

(i) (ii)

#### Answer:

(i) = (cos *θ*)(cos *θ*) − (−sin *θ*)(sin *θ*) = cos^{2} *θ*+ sin^{2} *θ* = 1

(ii)

= (*x*^{2} − *x* + 1)(*x* + 1) − (*x* − 1)(*x* + 1)

= *x*^{3} − *x*^{2} + *x* + *x*^{2} − *x* + 1 − (*x*^{2} − 1)

= *x*^{3} + 1 − *x*^{2} + 1

= *x*^{3} − *x*^{2} + 2

#### Question 3:

If, then show that

#### Answer:

The given matrix is.

#### Question 4:

If, then show that

#### Answer:

The given matrix is.

It can be observed that in the first column, two entries are zero. Thus, we expand along the first column (C_{1}) for easier calculation.

From equations (i) and (ii), we have:

Hence, the given result is proved.

#### Question 5:

Evaluate the determinants

(i) (iii)

(ii) (iv)

#### Answer:

(i) Let.

It can be observed that in the second row, two entries are zero. Thus, we expand along the second row for easier calculation.

(ii) Let.

By expanding along the first row, we have:

(iii) Let

By expanding along the first row, we have:

(iv) Let

By expanding along the first column, we have:

#### Page No 109:

#### Question 6:

If, find.

#### Answer:

Let

By expanding along the first row, we have:

#### Question 7:

Find values of *x*, if

(i)

2451=2x46x(ii)

2345=x32x5

#### Answer:

(i)

(ii)

#### Question 8:

If, then *x* is equal to

(A) 6 (B) ±6 (C) −6 (D) 0

#### Answer:

**Answer: B**

Hence, the correct answer is B.