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

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

Exercise 4.2

Set 1

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

Question 1Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

                C1→C1+C2

=Solutions Class 12 maths Chapter-4 (Determinants)

According to Properties of Determinant   

=0                                [∵  C& C3 are identical]

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

Hence Proved

Question 2Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

        Solutions Class 12 maths Chapter-4 (Determinants)

=0                        [∵ Every element of C1 are 0]

Now,  L.H.S.=R.H.S.

Hence Proved

Question 3Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

            C3→C3-C

        =Solutions Class 12 maths Chapter-4 (Determinants)

      =Solutions Class 12 maths Chapter-4 (Determinants)

=9 ×0=0          [∵C2 & C are identical]

Now,  L.H.S.=R.H.S.

Hence Proved

Question 4Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

        =Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

Question 5Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

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

Hence Proved

Question 6Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


Let Δ=Solutions Class 12 maths Chapter-4 (Determinants)

Taking (-1) common from every row

  Δ=(-1)3 Solutions Class 12 maths Chapter-4 (Determinants)

Interchange rows and columns

Δ=-Solutions Class 12 maths Chapter-4 (Determinants)

Now, Δ=-Δ

       Δ+Δ=0

       2Δ=0

        Δ=0

Now,  L.H.S.=R.H.S.

Hence Proved

Question 7Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Taking common a from Row 1,

                          b from Row 2,

                          c from Row 3, we have

Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

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

Question 8(i). Solutions Class 12 maths Chapter-4 (Determinants)

(ii)Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


(i) L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

(ii)  L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

Question 9.

 Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

R_{1} \rightarrow R_{1}-R_{2}

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

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

Hence Proved

Question 10.

(i)Solutions Class 12 maths Chapter-4 (Determinants)

(ii)Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


(i) L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

(ii) L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Now,  L.H.S.=R.H.S.

Hence Proved

Exercise 4.2

Set 2

Question 11. 

(i)Solutions Class 12 maths Chapter-4 (Determinants)

(ii)Solutions Class 12 maths Chapter-4 (Determinants)

Solution:


(i) L.H.S.=Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

∵ L.H.S.=R.H.S

Hence proved

(ii) Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

∵ L.H.S.=R.H.S

Hence proved

Question 12Solutions Class 12 maths Chapter-4 (Determinants)

Solution:

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

∵ L.H.S.=R.H.S

Hence proved

Question 13Solutions Class 12 maths Chapter-4 (Determinants)

Solution:

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

∵ L.H.S.=R.H.S

Hence proved

Question 14Solutions Class 12 maths Chapter-4 (Determinants)

Solution:

Solutions Class 12 maths Chapter-4 (Determinants)

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

Solutions Class 12 maths Chapter-4 (Determinants)

Solutions Class 12 maths Chapter-4 (Determinants)

∵ 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) k2|A|

(C) k3|A|

(D) 3k|A|

Solution:

 Let A =Solutions Class 12 maths Chapter-4 (Determinants) be a square matrix of order 3 x 3   ….(1)

Now, kA=Solutions Class 12 maths Chapter-4 (Determinants)

⇒|kA|=Solutions Class 12 maths Chapter-4 (Determinants)

⇒|kA|=k3 Solutions Class 12 maths Chapter-4 (Determinants)

∴ |kA|= k3|A|            [ from eq.(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.

Chapter-4 (Determinants)

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