# RD Sharma Solution CLass 12 Mathematics Chapter 6 Determinants

## Chapter 6 – Determinants Exercise Ex. 6.1

Question 1

Solution 1

Question 2
Solution 2
Question 3

Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6

Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10

Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19

Question 20

Solution 20

Question 21

Find the values of x, if

Solution 21

Question 22
Solution 22
Question 23

Solution 23

Question 24

Solution 24

## Chapter 6 – Determinants Exercise Ex. 6.2

Question 1
Solution 1
Question 2

Solution 2

Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Question 6
Solution 6
Question 7
Solution 7
Question 8

Solution 8

Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20

Without expanding, show that the value of the following determinants is
zero:

Solution 20

Question 21

Solution 21

Question 22

Without expanding, show that the value of the following determinants is
zero:

Solution 22

Question 23

Without expanding, show that the value of the following determinants is
zero:

Solution 23

Question 24

Without expanding, show that the value of the following determinants is
zero:

Solution 24

Question 25

Without expanding, show that the value of the following determinants is
zero:

Solution 25

Question 26
Solution 26
Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Evaluate the following:

Solution 31

Question 32

Evaluate the following:

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Prove the following identity:

Solution 60

Question 61

Solution 61

Question 62

Prove the following identities:

Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

Prove the following identity:

Solution 65

Question 66

Solution 66

Question 67

Prove the following identity:

Solution 67

Question 68

Prove the following identities:

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

Question 80

Solution 80

Question 81

Solution 81

Question 82

Solution 82

Question 83

Solve the following determinant equation:

Solution 83

## Chapter 6 – Determinants Exercise Ex. 6.3

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4

find the area of the triangle with vertices at the points:

(0, 0), (6, 0), (4, 3)

Solution 4

Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21

Solution 21

## Chapter 6 – Determinants Exercise Ex. 6.4

Question 1
Solution 1
Question 2

Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7

Solve the following systems of linear equation by Cramer’s rule:

2x + 3y = 10

x + 6y = 4

Solution 7

Question 8
Solution 8
Question 9
Solution 9
Question 10

Solve the following systems of linear equations by Cramer’s rule

x + 2y = 1

3x + y = 4

Solution 10

Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Solve the following system of the linear equations by
Cramer’s rule:

2x – 3y – 4z = 29

-2x + 5y – z = -15

3x – y + 5z = – 11

Solution 17

Question 18
Solution 18
Question 19

Solution 19

Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24
Question 25
Solution 25
Question 26
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
Solution 29
Question 30
Solution 30
Question 31
Solution 31
Question 32
Solution 32

## Chapter 6 – Determinants Exercise Ex. 6.5

Question 1
Solution 1
Question 2

Solve the following system of homogeneous linear equations:

2x + 3y + 4z = 0

X + y + z = 0

2x + 5y – 2z = 0

Solution 2

Question 3
Solution 3
Question 4

Find the real values of λ for which the following system of linear equations has non – trivial solutions.

Also, find the non – trivial solutions

2λx – 2y + 3z = 0

x + λy + 2z = 0

2x + λz = 0

Solution 4

Question 5
Solution 5

## Chapter 6 – Determinants Exercise Ex. 6VSAQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Write the value of the determinant:

Solution 51

Question 52

Solution 52

## Chapter 6 – Determinants Exercise MCQ

Question 1

If A and B are square matrices of order 2, then det (A + B) = 0 is possible only when

a. det (A) = 0 or det (B) = 0

b. det (A) + det (B) = 0

c. det (A) = 0 and det (B) = 0

d. A + B = O

Solution 1

Correct option: (d)

Question 2

Which of the following is not correct?

a. |A| = |AT|, where A = [aij]3×3

b. |kA| = k3 |A|, where A = [aij]3×3

c. If A is a skew-symmetric matrix of odd order, then |A| = 0

d.

Solution 2

Correct option: (d)

Question 3

a. a11C31+a12C32+a13C33

b. a11C11+a12C21+a13C31

c. a21C11+a22C12+a23C13

d. a11C11+a21C21+a31C31

Solution 3

Correct option: (d)

If A is a square matrix of order n then det(A) = a11C11+a21C21+a31C31

Question 4

Which of the following is not correct in a given determinant of A, where A = [aij]3×3.

a. Order of minor is less than order of the det (A).

b. Minor of an element can never be equal to cofactor of the same element

c. Value of a determinant is obtained by multiplying elements of a row or column by corresponding cofactors

d. Order of minors and cofactors of elements of A is same

Solution 4

Correct option: (b)

Minor of an element can never be equal to cofactor of the same element.

Cij=(-1)i+jMij

Question 5

a. 0

b. -16

c. 16

d. 1

e. None of these

Solution 5

Correct option: (e)

Question 6

a. n

b. a

c. x

d. none of these

Solution 6

Correct option: (a)

Question 7

a. Δ1 + Δ2 = 0

b. Δ1 + 2Δ2 = 0

c. Δ1 = Δ2

d. none of these

Solution 7

Correct option: (a)

Question 8

and

a. 4

b. 6

c. 8

d. none of these

Solution 8

Question 9

be an identity in x, where a, b, c, d, e, are independent of x. then the value of e is

a. 4

b. 0

c. 1

d. none of these

Solution 9

Correct option: (a)

Question 10

Using the factor theorem it is found that a + b, b + c and c + a are three factors of the determinant  The other factor in the value of the determinant is

a. 4

b. 2

c. a + b + c

d. none of these

Solution 10

Correct option: (a)

Question 11

If a, b, c are distinct, then the value of x satisfying

a. c

b. a

c. b

d. 0

Solution 11

Correct option: (d)

Question 12

If the determinant

a. A, b, c are in H.P.

b. α is a root of 4ax2 + 12 bx + 9c = 0 or, a, b, c are in G.P.

c. a, b, c are in G.P. only

d. a, b, c are in A.P.

Solution 12

Correct option: (b)

Question 13

If ω is a non-real cube root of unity and n is not a multiple of 3, then   is equal to

a. 0

b. ω

c. ω2

d. 1

Solution 13

Correct option: (a)

Question 14

a. n

b. 2n

c. -2n

d. n2

Solution 14

Correct option: (c)

Question 15

a. positive

b. (ac – b2)(ax2 + 2bx + c)

c. Negative

d. 0

Solution 15

Correct option: (c)

Question 16

a. 52

b. 0

c. 513

d. 59

Solution 16

Correct option: (b)

Question 17

a. 7

b. 10

c. 13

d. 17

Solution 17

Correct option: (b)

Question 18

a. 0

b. 1

c. x

d. 2x

Solution 18

Correct option:(a)

Question 19

a. 0

b. 1

c. 2 sin B tan A cos C

d. none of these

Solution 19

Correct option: (a)

Question 20

The number of distinct real roots of  lies in the interval

a. 1

b. 2

c. 3

d. 0

Solution 20

Correct option: (b)

Question 21

a. Det (A) = 0

b. Det (A) ∊ (2, ∞)

c. Det (A) ∊ (2, 4)

d. Det (A) ∊ [2, 4]

Solution 21

Correct option: (d)

Question 22

a. 3

b. ±3

c. ±6

d. 6

Solution 22

Correct option: (c)

Question 23

a. f(a) = 0

b. f(b) = 0

c. f(0) = 0

d. f(1) = 0

Solution 23

Correct option: (a)

Question 24

a. a3 + b3 + c3

b. 3bc

c. a3 + b3 + c3 – 3abc

d. none of these

Solution 24

Correct option: (c )

Question 25

a. xyz

b. x-1y-1z-1

c. -x-y-z

d. -1

Solution 25

Correct option: (d)

Question 26

a. abc(b – c) (c – a)(a – b)

b. (b – c) (c – a)(a – b)

c. (a + b + c)(b – c)(c – a)(a – b)

d. none of these

Solution 26

Correct option: (d)

Question 27

a.

b. [-1, 1]

c.

d.

Solution 27

Correct option: (a)

Question 28

a.

b.

c.

d.

Solution 28

Correct option: (a)

Question 29

a. 9x2(x + y)

b. 9y2 (x + y)

c. 3y2(x + y)

d. 7x2(x + y)

Solution 29

Correct option: (b)

Question 30

a. 0

b. -1

c. 2

d. 3

Solution 30

Correct option: (a)

Question 31

There are two values of a which makes the determinant  equal to 86. The sum of these two values is

a. 4

b. 5

c. -4

d. 9

Solution 31

Correct option: (c)

Question 32

a. 4

b. 8

c. 16

d. 32

Solution 32

Correct option: (d)

Question 33

a. 2

b. 4

c. 8

d. n2

Solution 33

Correct option: (c)

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