Class12 ISC Maths Chapter Determinants Exercise 4.1

Class 12 ISC Maths Chapter Determinants Exercise 4.1

Explore our extensive collection of questions on the topic of "Determinants," designed to help you understand and apply this fundamental concept in linear algebra. Our question bank covers key topics such as properties of determinants, cofactor expansion, minors, applications in solving systems of linear equations, and finding the inverse of matrices. These practice questions are perfect for students aiming to build a strong foundation in determinants and excel in exams. Whether you're revising concepts or looking to enhance your problem-solving skills, our comprehensive collection provides the ideal resource for mastering "Determinants."

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Determinants

x²-x+1	x-1
x+1	x+1

cos 90º	-cos45º
sin90º	sin45º

2cosθ	-2sinθ
sinθ	cosθ

sin30º	cos30º
-sin60º	cos60º

Class 12 ISC Maths Chapter Determinants Exercise 4.1

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Q1 Evaluate the following Determinants :

(i) x2-x+1 x-1 x+1 x+1

(ii) cos 90º -cos45º sin90º sin45º

(iii) 2cosθ -2sinθ sinθ cosθ

Q2 Evaluate the following Determinants :

(i) sin30º cos30º -sin60º cos60º

(ii) -1 31 40 0 5 -432 0 0 20

Q3(i) Find the value of sin A -sinB cosA cosB ,where A=63º and B=27º

Q3(ii) Find the value of cos A sin A -sin B cos B ,where A=75º and B=45º

Q4(i) Write the cofactor of a12 in the determinant 2 -3 5 6 0 4 1 5 7

Q4(ii) If A= 5 6 -3 -4 3 2 -4 -7 3 ,then write the cofactor of the elenent a21 of its 2nd row.

Q4(iii) If Aij is the cofactor of the element aij of the determinant 2 -3 5 6 0 4 1 5 -7 ,then write the value of a32A32

Q5 Find the value of x if

(i) 2x+5 3 5x+2 9 =0

(ii) x 4 2 2x =0

(iii) x+2 3 x+5 4 =3

Q6 Find the value of x if

(i) 3x 7 -2 4 = 8 7 6 4

(ii) x 3 2x 5 = 2 3 4 5

Q7(i) If x+1 x-1 x-3 x+2 = 4 -1 1 3 ,then write the value of x.

Q7(ii) If √6 x √20 √24 = 6 2 5 2 ,then write the value of x.

Q8(i) If x ∈N and x 3 4 x = 4 -3 0 1 ,then write the value(s) of x.

Q8(ii) If x x 1 x = 3 4 1 2 ,write the positive value of x.

Q9 If x∈I and 2x 3 -1 x = 3 1 x 3 ,find the value(s) of x.

Q10 If A= 1 2 4 2 ,then show that |2A|=4|A|

Q11 Evaluate the following determinants:

(i) 2 4 1 8 5 2 -1 3 7

(ii) 3 -4 5 1 1 -2 2 3 1

(iii) 2 -1 -2 0 2 -1 3 -5 0

Q12 Prove that 1 a b -a 1 c -b -c 1 =1+a2+b2+c2

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(i)

x²-x+1 x-1

x+1 x+1

(ii)

cos 90º -cos45º

sin90º sin45º

(iii)

2cosθ -2sinθ

sinθ cosθ

(i)

sin30º cos30º

-sin60º cos60º

(ii)

-1 31 40

0 5 -432

0 0 20

Q3(i) Find the value of

sin A -sinB

cosA cosB

,where A=63º and B=27º

Q3(ii) Find the value of

cos A sin A

-sin B cos B

,where A=75º and B=45º

Q4(i) Write the cofactor of a₁₂ in the determinant

2 -3 5

6 0 4

1 5 7

Q4(ii) If

A=

5 6 -3

-4 3 2

-4 -7 3

,then write the cofactor of the elenent a₂₁ of its 2nd row.

Q4(iii) If A_ij is the cofactor of the element a_ij of the determinant

2 -3 5

6 0 4

1 5 -7

,then write the value of a₃₂A₃₂

(i)

2x+5 3

5x+2 9

=0

(ii)

x 4

2 2x

=0

(iii)

x+2 3

x+5 4

=3

(i)

3x 7

-2 4

=

8 7

6 4

(ii)

x 3

2x 5

=

2 3

4 5

Q7(i) If

x+1 x-1

x-3 x+2

=

4 -1

1 3

,then write the value of x.

Q7(ii) If

√6 x

√20 √24

=

6 2

5 2

,then write the value of x.

Q8(i) If x ∈N and

x 3

4 x

=

4 -3

0 1

,then write the value(s) of x.

Q8(ii) If

x x

1 x

=

3 4

1 2

,write the positive value of x.

Q9 If x∈I and

2x 3

-1 x

=

3 1

x 3

,find the value(s) of x.

Q10 If
A=

1 2

4 2

,then show that |2A|=4|A|

(i)

2 4 1

8 5 2

-1 3 7

(ii)

3 -4 5

1 1 -2

2 3 1

(iii)

2 -1 -2

0 2 -1

3 -5 0

Q12 Prove that

1 a b

-a 1 c

-b -c 1

=1+a²+b²+c²