# NCERT Solutions Class 12 maths Chapter-4 (Determinants)Exercise 4.2

**NCERT Solutions Class 12 Maths**from class

**12th**Students will get the answers of

**Chapter-4 (Determinants)Exercise 4.2**This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.

We have given the answers of all the questions of

**NCERT Board Mathematics Textbook**in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

### 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.=

C1→C1+C2

=

According to Properties of Determinant

=0 [∵ C1 & C3 are identical]

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

Hence Proved

**Question 2. **

**Solution:**

L.H.S.=

=0 [∵ Every element of C1 are 0]

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

Hence Proved

**Question 3. **

**Solution:**

L.H.S.=

C3→C3-C1

=

=

=9 ×0=0 [∵C2 & C3 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

### Exercise 4.2

### Set 2

**Question 11. **

**(i)**

**(ii)**

**Solution:**

**(i)** L.H.S.=

∵ L.H.S.=R.H.S

Hence proved

**(ii)**

∵ L.H.S.=R.H.S

Hence proved

**Question 12. **

**Solution:**

∵ L.H.S.=R.H.S

Hence proved

**Question 13****. **

**Solution:**

∵ L.H.S.=R.H.S

Hence proved

**Question 14. **

**Solution:**

Multiplying column1,column2,column3 by a,b,c respectively and then dividing the determinant by a,b,c .

∵ L.H.S.=R.H.S

Hence proved

**Choose the correct answer in Exercises 15 and 16.**

**Question 15. Let A be a square matrix of order 3 x 3, then|A| is equal to:**

**(A) k|A|**

**(B) k**^{2}**|A|**

**(C) k**^{3}**|A|**

**(D) 3k|A|**

**Solution:**

Let A = be a square matrix of order 3 x 3 ….(1)

Now, kA=

⇒|kA|=

⇒|kA|=k3

∴ |kA|= k3|A| [ from eqn .(1) ]

∴ option (C) is correct.

**Question 16.Which of the following is correct**

**(A) Determinant is a square matrix.**

**(B) Determinant is a number associated to a matrix.**

**(C) Determinant is a number associated to a square matrix.**

**(D) None of these**

**Solution:**

Since, Determinant is a number which is always associated to a square matrix.

∴ option (C) is correct.

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