# Class 12 NCERT Solutions- Mathematics Part I – Chapter 4 Determinants – Exercise 4.2 | Set 1

### Using the properties of determinants and without expanding in Exercise 1 to 7, prove that:

### Question 1.

**Solution:**

L.H.S.=

C

_{1}→C_{1}+C_{2}=

According to Properties of Determinant

=0 [∵ C

_{1 }& C_{3}are identical]Now, L.H.S.=R.H.S.

Hence Proved

### Question 2.

**Solution:**

L.H.S.=

=0 [∵ Every element of C

_{1}are 0]Now, L.H.S.=R.H.S.

Hence Proved

### Question 3.

**Solution:**

L.H.S.=

C

_{3}→C_{3}-C_{1 }=

=

=9 ×0=0 [∵C

_{2}& C_{3 }are identical]Now, L.H.S.=R.H.S.

Hence Proved

### Question 4.

**Solution:**

L.H.S.=

=

Now, L.H.S.=R.H.S.

Hence Proved

### Question 5.

**Solution:**

L.H.S.=

Now, L.H.S.=R.H.S.

Hence Proved

### Question 6.

**Solution:**

Let Δ=

Taking (-1) common from every row

Δ=(-1)

^{3}Interchange rows and columns

Δ=-

Now, Δ=-Δ

Δ+Δ=0

2Δ=0

Δ=0

Now, L.H.S.=R.H.S.

Hence Proved

### Question 7.

**Solution:**

L.H.S.=

Taking common a from Row 1,

b from Row 2,

c from Row 3, we have

Now, L.H.S.=R.H.S.

Hence Proved

### By using properties of determinants, in Exercises 8 to 14, show that:

### Question 8(i).

### (ii)

**Solution:**

(i)L.H.S.=Now, L.H.S.=R.H.S.

Hence Proved

(ii)L.H.S.=Now, L.H.S.=R.H.S.

Hence Proved

### Question 9.

**Solution:**

L.H.S.=

Now, L.H.S.=R.H.S.

Hence Proved

### Question 10.(i)

### (ii)

**Solution:**

(i)L.H.S.=Now, L.H.S.=R.H.S.

Hence Proved

(ii)L.H.S.=Now, L.H.S.=R.H.S.

Hence Proved

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