RD Sharma Solutions For Class 12 Maths Exercise 6.2 Chapter 6 Determinants

This exercise deals with the properties of determinants. Exercise 6.2 of Chapter 6 consists of problems based on the evaluation of determinants with proving the identities. The solutions prepared by our faculty are in an interactive manner to make it easy for the students to understand the concepts. Students can refer to the PDF of solutions as a major study material to improve their speed in solving problems accurately. Students can access the RD Sharma Solutions for Class 12 Maths Exercise 6.2 of Chapter 6 Determinants.

RD Sharma Solutions For Class 12 Chapter 6 Determinants Exercise 6.2Download PDF Here

RD Sharma Class 12 Maths Solutions Chapter 6 Determinants Exercise 6.2
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Access other exercises of RD Sharma Solutions For Class 12 Chapter 6 – Determinants

Exercise 6.1 Solutions

Exercise 6.3 Solutions

Exercise 6.4 Solutions

Exercise 6.5 Solutions

Access answers to Maths RD Sharma Solutions For Class 12 Chapter 6 – Determinants Exercise 6.2

Exercise 6.2 Page No: 6.57

1. Evaluate the following determinant:

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 33

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 34

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 35

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 36

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 37

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 38

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 39

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 40

Solution:

(i) Given

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 41
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 42

(ii) Given

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 43

= 1[(109) (12) – (119) (11)]

= 1308 – 1309

= – 1

So, Δ = – 1

(iii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 44

= a (bc – f2) – h (hc – fg) + g (hf – bg)

= abc – af2 – ch2 + fgh + fgh – bg2

= abc + 2fgh – af2 – bg2 – ch2

So, Δ = abc + 2fgh – af2 – bg2 – ch2

(iv) Given

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 45

= 2[1(24 – 4)] = 40

So, Δ = 40

(v) Given

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 46
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 47

= 1[(– 7) (– 36) – (– 20) (– 13)]

= 252 – 260

= – 8

So, Δ = – 8

(vi) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 48

(vii) Given

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 49
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 50
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 51

(viii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 52

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

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 53

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 54

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 55

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 56

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 57

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 58

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 59

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 60

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 61

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 62

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 63

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 64

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 65

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 66

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 67

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 68

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 69

Solution:

(i) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 70

(ii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 71
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 72

(iii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 73

(iv) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 74
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 75

(v) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 76

(vi) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 77
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 78

(vii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 79

(viii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 80

(ix) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 81

As, C1 = C2, hence determinant is zero

(x) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 82

(xi) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 83
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 84

(xii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 85

(xiii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 86

(xiv) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 87
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 88

(xv) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 89
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 90

(xvi) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 91
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 92

(xvii) Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 93

Hence proved.

Evaluate the following (3 – 9):

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 94

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 95

= (a + b + c) (b – a) (c – a) (b – c)

So, Δ = (a + b + c) (b – a) (c – a) (b – c)

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 96

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 97
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 98

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 99
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 100

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 101
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 102

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 103
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 104

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 105
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 106

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 107
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 108

= a [a (a + x + y) + az] + 0 + 0

= a2 (a + x + y + z)

So, Δ = a2 (a + x + y + z)

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 109

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 110

Prove the following identities (11 – 45):

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 111

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 112
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 113

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 114
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 115

= – (a + b + c) [(b – c) (a + b – 2c) – (c – a) (c + a – 2b)]

= 3abc – a3 – b3 – c3

Therefore, L.H.S = R.H.S,

Hence the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 116

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 117
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 118
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 119

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 120,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 121
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 122

Solution:

Consider,

L.H.S = 
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 123

Now by applying, R1→R1 + R2 + R3, we get,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 124
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 125

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 126
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 127
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 128

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 129
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 130
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 131

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 132
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 133

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 134

Solution:

Given,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 135

= – xyz(x – y) (z – y) [z2 + y2 + zy – x2 – y2 – xy]

= – xyz(x – y) (z – y) [(z – x) (z + x0 + y (z – x)]

= – xyz(x – y) (z – y) (z – x) (x + y + z)

= R.H.S

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 136

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 137
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 138

= (a2 + b2 + c2) (b – a) (c – a) [(b + a) (– b) – (– c) (c + a)]

= (a2 + b2 + c2) (a – b) (c – a) (b – c) (a + b + c)

= R.H.S

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 139

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 140

= [(2a + 4) (1) – (1) (2a + 6)]

= – 2

= R.H.S

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 141

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 142
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 143

= – (a2 + b2 + c2) (a – b) (c – a) [(– (b + a)) (– b) – (c) (c + a)]

= (a – b) (b – c) (c – a) (a + b + c) (a2 + b2 + c2)

= R.H.S

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 144

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 145

= R.H.S

Hence, the proof.

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 146

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 147
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 148
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 149

Solution:

Consider,

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 150
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 151
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 152

Solution:

RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 153
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 154
RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 155

Expanding the determinant along R1, we have

Δ = 1[(1) (7) – (3) (2)] – 0 + 0

∴ Δ = 7 – 6 = 1

Thus,RD Sharma Solutions for Class 12 Maths Chapter 6 Determinants Image 156

Hence the proof.

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