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RBSE Solutions for Class 12 Maths Chapter 3 आव्यूह Miscellaneous Exercise

May 4, 2019 by Fazal Leave a Comment

Rajasthan Board RBSE Class 12 Maths Chapter 3 आव्यूह Miscellaneous Exercise

प्रश्न 1.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 1
हो, तो A² ज्ञात कीजिए।
हलः
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 2

प्रश्न 2.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 3
हो, तो (A – 2I). (A – 3I) ज्ञात कीजिए।
हलः
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 4

प्रश्न 3.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 5
हो, तो AB ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 6
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 7

प्रश्न 4.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 8
हो, तो BA ज्ञात कीजिए जहाँ i = √-1
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 9

प्रश्न 5.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 10
तथा
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 11
हो, तो आव्यूह A तथा B ज्ञात कीजिए।
हलः
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 12
(i) और (ii) को जोड़ने पर
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 13
(i) और (ii) को घटाने पर
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 14

प्रश्न 6.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 15
हो, तो x तथा y ज्ञात कीजिए।
हल :
हल : संगत अवयवों की तुलना करने पर
x + 2 = – 2
∴ x = – 4
– y – x = 5 ⇒ y = – x – 5 = – (-4) – 5
= 4 – 5 = -1
अतः = – 4, y = – 1

प्रश्न 7.
आव्यूह A का क्रम 3 x 4 है तथा B इस प्रकार का आव्यूह है कि AT B एवं ABT दोनों ही परिभाषित है तो B का क्रम लिखिए।
हल :
∴ A का क्रम 3 x 4 है।
∴ AT का क्रम 4 x 3 होगा
परन्तु ATB तथा ABT परिभाषित है
अत: B का क्रम भी 3 x 4 ही होगा।

प्रश्न 8.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 16
एक सममित आव्यूह है तो x का मान ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 17
एक सममित आव्यूह है,
अत: aij = aji, की तुलना करने पर
a32 = a23 ⇒ – x = 4
∴ x = – 4

प्रश्न 9.
एक 3 x 3 क्रम को आव्यूह B = [bij] लिखिए जिनके अवयव bij = (i) (j) हैं।
हल :
B11 = 1 x 1 = 1
B12 = 1 x 2 = 2
B13 = 1 x 3 = 3
B21 = 2 x 1 = 2.
B22 = 2 x 1 = 4.
B23 = 2 x 3 = 6
B31 = 3 x 1 = 3
B32 = 3 x 2 = 6
B33 = 3 x 3 = 9
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 18

प्रश्न 10.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 19
तथा
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 20
तो A + BT ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 21

प्रश्न 11.
आव्यूह A को सममित व विषम सममित आव्यूह के योग के रूप में व्यक्त कीजिए, जहाँ
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 22
है।
हल :
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 23
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 24

प्रश्न 12.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 25
हो तो सिद्ध कीजिए ।
(i) (AT)T = A
(ii) A + AT एक सममित आव्यूह है।
(ii) A – AT एक विषम सममित आव्यूह है।
(iv) AAT तथा ATA सममित आव्यूह है।
हल :
(i) (AT)T = A
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 26
इति सिद्धम्।

(ii) A + AT एक सममित आव्यूह है।
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 27
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 28
अत: A + AT एक सममित आव्यूह है।
इति सिद्धम्।

(iii)
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 29
अत: A – AT सममित आव्यूह है।
इति सिद्धम

(iv)
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 30
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 31
अत: AT A सममित आव्यूह है।
इति सिद्धम

प्रश्न 13.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 32
तथा 3A – 2B + C एक अशून्य आव्यूह है तो आव्यूह C लिखिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 33
अतः 3A – 2B + C = 0
∴ C = 2B – 3A + 0
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 34

प्रश्न 14.
एक 2×3 क्रम का आव्यूह B= |bij| लिखिए जिसके अवयव है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 35
हल :
दिया है, B = [bij] जिसके अवयव हैं।
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 36
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 37

प्रश्न 15.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 38
हों, तो ABC का प्रथम पंक्ति के अवयव ज्ञात कीजिए।
हल :
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 39
अतः पहली पंक्ति का अवयव 8 है।

प्रश्न 16.
यदि आव्यूह
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 40
हो तो AAT ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 41
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 42

प्रश्न 17.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 43
तो x का मान ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 44

प्रश्न 18.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 45
हो तो सिद्ध कीजिए कि
B² – (a + d)B = (bc – ad)I2, जहाँ
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 46
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 47
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 48
= (bc – ad)I2
= R.H.S.
इति सिद्धम्।

प्रश्न 19.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 49
हो तो (aA + bB) (aA – bB) को आव्यूह के रूप में ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 50

प्रश्न 20.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 51
हो, तो सिद्ध कीजिए कि
(A – B)² ≠ A² – 2AB + B².
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 52
A² – 2AB + B²
= A . A – 2 . A . B + B . B
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 53
समीकरण (i) और (ii) से
(A – B)² ≠ A² – 2AB + B².
इति सिद्धम्।

प्रश्न 21.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 54
तो k का मान ज्ञात कीजिए, जहाँ A² = kA – 2I2
हल : दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 55

प्रश्न 22.
यदि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 56
तथा i = √-1 निम्नलिखित सम्बन्धों का सत्यापन कीजिए
(i) A² = B² = C² = I2
(ii) AB = – BA = – C
हल :
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 57
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 58
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 59

प्रश्न 23.
याद
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 60
तथा f(A) = A² – 5A + 7I हो, तो f(A) ज्ञात कीजिए।
हल :
दिया है,
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 61

प्रश्न 24.
सिद्ध कीजिए कि
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 62
जबकि α – β = (2m – 1)\(\frac { \pi }{ 2 } \), m∈N
हल :
RBSE Solutions for Class 12 Maths Chapter 3 Additional Questions 63
= cos (α – β) [cos α cos β sin α sin β – cos α cos β sin α sin β]
= cos (α – β) x 0
= 0
= R.H.S.

RBSE Solutions for Class 12 Maths

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