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RBSE Solutions for Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Ex 5.1

May 6, 2019 by Fazal 2 Comments

RBSE Solutions for Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Ex 5.1 is part of RBSE Solutions for Class 12 Maths. Here we have given Rajasthan Board RBSE Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Exercise 5.1.

Rajasthan Board RBSE Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Ex 5.1

प्रश्न 1.
x के किस मान के लिए आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 1
अव्युत्क्रमणीय है।
हल :
दिया है कि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 2
अव्युत्क्रमणीय है।
तब
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 3
⇒ 1(6 – 2) + 2(3 – x) + 3(2 – 2x) = 0
⇒ 4 + 6 – 2x + 6 – 6x = 0
⇒ – 8x = – 16
⇒ x = \(\frac { 16 }{ 8 }\)
अतः x = 2

प्रश्न 2.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 4
हो, तो adj•A ज्ञात कीजिए तथा सिद्ध कीजिए कि A(adj•A) = |A|I3 = (adj•A)A.
हल :
दिया गया आव्यूह,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 5
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 6
आव्यूह A के सहखण्डों से बना आव्यूह,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 7
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 8
अतः A.(adj.A) = |A|I3 …(ii)
(i) व (ii) से,
A(adj A) = |A|I3 = (adjA) A
इति सिद्धम्।

प्रश्न 3.
निम्नलिखित आव्यूह का व्युत्क्रमणीय आव्यूह ज्ञात कीजिए :
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 9
हल :
(i) दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 10
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 11
आव्यूह A के सहखण्डों से बना आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 12
(ii) दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 13
| A | = 1(16 – 9) – 3(4 – 3) + 3(3 – 4)
= 7 – 3 – 3
|A | = 1 ≠ 0
अतः A-1 का अस्तित्व है।
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 14
आव्यूह A सहखण्डों से बना आव्यूह B
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 15
(iii) दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 16
| A | = 0( – 12 + 12) – 1(16 – 12) – 1( – 12 + 9)
= 0 – 4 + 3
| A | = -1 ≠ 0
अतः A-1 का अस्तित्व है।
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 17
आव्यूह A के सहखण्डों से निर्मित आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 18

प्रश्न 4.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 19
हो, तो A-1 ज्ञात कीजिए तथा सिद्ध कीजिए कि :
(i) A-1 A = I3
(ii) A-1 = F( – α)
(iii) A(adjA) = |A|I = (adjA).A
हल :
दिया है आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 20
|A| = cos α (cos α – 0) + sin α (sin α – 0) + 0(0 – 0)
= cos² α + sin² α
| A | = 1 ≠ 0
अतः A-1 का अस्तित्व है।
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 21
आव्यूह A के सहखण्डों से बना आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 22
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 23
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 24
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 25
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 26

प्रश्न 5.
यदि
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 27
तो सिद्ध कीजिए कि A-1 = AT
हल :
माना कि
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 28
प्रथम पंक्ति के सापेक्ष प्रसार करने पर
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 29
= – 8(16 + 56) – (16 – 7) + 4( – 32 – 4)
= – 8.72 – 9 – 4.36
= – 9(64 + 1 + 16)
= – 9 × 81
= – 9³
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 31
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 32
अतः (iii) और (iv) से स्पष्ट है कि
A-1 = AT
इति सिद्धम्।

प्रश्न 6.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 33
हो, तो सिद्ध कीजिए कि A-1 = A3
हल :
दिया गया आव्यूह,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 34
अतः A-1 का अस्तित्व है।
आव्यूह A के सहखण्ड ज्ञात करने पर,
a11 = – 1, a12 = – 2, a21 = 1, a22 = 1
आव्यूह A के सहखण्डों के बना आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 35
समीकरण (i) व (ii) से,
A-1 = A3.
इति सिद्धम्।

प्रश्न 7.
यदि
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 36
तथा
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 37
हो, तो (AB)-1 ज्ञात कीजिए।
हल :
दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 38
तब |A| = 5(3 – 4) – 0(2 – 2) + 4(4 – 3)
= – 5 – 0 + 4
|A| = – 1 ≠ 0 .
अतः A-1 का अस्तित्व है।
आव्यूह A के सहखण्ड ज्ञात करने पर,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 39
आव्यूह A के सहखण्डों से निर्मित आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 40
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 41

प्रश्न 8.
यदि
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 42
हो, तो सिद्ध कीजिए कि
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 43
हल :
दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 44
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 45

प्रश्न 9.
सिद्ध कीजिए कि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 46
समीकरण A² – 6A + 17I = 0 को सन्तुष्ट करता है तथा A-1 भी ज्ञात कीजिए।
हल :
दिया गया आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 47
अत: दिया गया आव्यूह समीकरण A² – 6A + 17I = 0 को सन्तुष्ट करता है।
अब A² – 6A + 17I = 0
⇒ = A² – 6A = – 17I
⇒ A-1 (A² – 6A) = – 17A-1I
⇒ [A-1 का दोनों पक्षों में वाम गुणन करने पर]
⇒ A-1A² – 6A-1A = – 17A-1 [∴A-1I = A-1]
⇒ A – 6I = – 17A-1 [∴A-1A = I]
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 48
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 49

प्रश्न 10.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 50
हो, तो सिद्ध कीजिए कि A² + 4A – 42I = 0 तत्पश्चात् A-1 ज्ञात कीजिए।
हल :
दिया गया आव्यूह,
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 51
अब A² + 4A – 42I
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 52
अतः दिया गया आव्यूह समीकरण A² + 4A – 42I = 0 को सन्तुष्ट करता है।
अब A² + 4A – 42I= 0
A² + 4A = 42I
A-1 (A² + 4A) = 42A-1I
(A-1 का दोनों पक्षों में गुणा करने पर)।
RBSE Solutions for Class 12 Maths Chapter 5 Ex 5.1 53

We hope the given RBSE Solutions for Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Ex 5.1 will help you. If you have any query regarding RBSE Solutions for Class 12 Maths Chapter 5 व्युत्क्रम आव्यूह एवंरैरिवक समीकरण Ex 5.1, drop a comment below and we will get back to you at the earliest.

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