RBSE Solutions for Class 12 Maths Chapter 4 सारणिक Ex 4.1

Rajasthan Board RBSE Solutions for Class 12 Maths Chapter 4 सारणिक Ex 4.1 Textbook Exercise Questions and Answers.

RBSE Class 12 Maths Solutions Chapter 4 सारणिक Ex 4.1

प्रश्न 1.
\(\left|\begin{array}{cc} 2 & 4 \\ -5 & -1 \end{array}\right|\)
हल:
\(\left|\begin{array}{cc} 2 & 4 \\ -5 & -1 \end{array}\right|\) = 2 × (- 1) - 4 × (- 5)
= - 2 + 20 = 18

प्रश्न 2.
(i) \(\left|\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right|\)
हल:
\(\left|\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right|\) = cos θ × cos θ - sin θ (- sin θ)
= cos² + sin²θ

(ii) \(\left|\begin{array}{cc} x^2-x+1 & x-1 \\ x+1 & x+1 \end{array}\right|\)
हल:
\(\left|\begin{array}{cc} x^2-x+1 & x-1 \\ x+1 & x+1 \end{array}\right|\) = (x² - x + 1) (x + 1) - (x - 1)
(x + 1) = (x³ + 1) - (x² - 1)
= x³ + 1 - x² + 1
= x³ - x² + 2

प्रश्न 3.
यदि A = \(\left[\begin{array}{ll} 1 & 2 \\ 4 & 2 \end{array}\right]\), तो दिखाइए |2A| = 4 |A|
हल:

प्रश्न 4.
यदि A = \(\left[\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{array}\right]\) हो, तो दिखाइए |3A| = 27 |A|
हल:

प्रत्येक पंक्ति से 3 निकालने पर
|3A| = 3 × 3 × 3 × \(\left|\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{array}\right|\) = 27|A|
⇒ |3A| = 27 |A|

प्रश्न 5.
निम्नलिखित सारणिकों का मान ज्ञात कीजिए:
(i)
\(\left|\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{array}\right|\)
हल:
|A| = \(\left|\begin{array}{ccc} 3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0 \end{array}\right|\)
दूसरी पंक्ति के अनुदिश |A| का प्रसरण

= 3 × - 5 - (- 1) × 3
= - 15 + 3 = - 12

(ii)
\(\left|\begin{array}{ccc} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{array}\right|\)
हल:
\(\left|\begin{array}{ccc} 3 & -4 & 5 \\ 1 & 1 & -2 \\ 2 & 3 & 1 \end{array}\right|\) पहली पंक्ति से प्रसरण करते हुए

= 3 (1 + 6) + 4 (1 + 4) + 5 (3 - 2)
= 21 + 20 + 5 = 46

(iii)
\(\left|\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{array}\right|\)
हल:
|A| = \(\left|\begin{array}{ccc} 0 & 1 & 2 \\ -1 & 0 & -3 \\ -2 & 3 & 0 \end{array}\right|\) पहली पंक्ति से प्रसरण करते हुए

= - (0 - 6) + 2 (- 3 + 0) = 6 - 6 = 0

(iv)
\(\left|\begin{array}{ccc} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{array}\right|\)
हल:
\(\left|\begin{array}{ccc} 2 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{array}\right|\) पहली पंक्ति से प्रसरण करते हुए

= 2 (0 - 5) + (0 + 3) - 2 (0 - 6)
= - 10 + 3 + 12 = 5

प्रश्न 6.
यदि A = \(\left[\begin{array}{lll} 1 & 1 & -2 \\ 2 & 1 & -3 \\ 5 & 4 & -9 \end{array}\right]\) हो, तो |A| ज्ञात कीजिए।
हल:

= (- 9 + 12) - (- 18 + 15) - 2 (8 - 5)
= 3 + 3 - 6 = 0

प्रश्न 7.
x के मान ज्ञात कीजिए यदि
(i) \(\left|\begin{array}{ll} 2 & 4 \\ 5 & 1 \end{array}\right|\) = \(\left|\begin{array}{cc} 2 x & 4 \\ 6 & x \end{array}\right|\)
हल:
बायाँ पक्ष = \(\left|\begin{array}{ll} 2 & 4 \\ 5 & 1 \end{array}\right|\) = 2 - 20 = - 18
दायाँ पक्ष = \(\left|\begin{array}{cc} 2 x & 4 \\ 6 & x \end{array}\right|\) = (2x² - 24)
अतः - 18 = 2x² - 24 या 2x² = 24 - 18 = 6
∴ x² = 3, x = ± √3

(ii) \(\left|\begin{array}{ll} 2 & 3 \\ 4 & 5 \end{array}\right|\) = \(\left|\begin{array}{cc} x & 3 \\ 2 x & 5 \end{array}\right|\)
हल:
\(\left|\begin{array}{ll} 2 & 3 \\ 4 & 5 \end{array}\right|\) = \(\left|\begin{array}{cc} x & 3 \\ 2 x & 5 \end{array}\right|\)
दोनों ओर प्रसरण करते हुए या 10 - 12 = 5x - 6x
⇒ - 2 = - x
⇒ x = 2

प्रश्न 8.
याद \(\left|\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right|\) = \(\left|\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right|\) हो तो x बराबर है
(A) 6
(B) ± 6
(C) - 6
(D) 0
उत्तर:
(B) ± 6

हल:
\(\left|\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right|\) = \(\left|\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right|\)
⇒ x² - 36 = 36 - 36
या x² - 36 = 0 ∴ x² = 36
अतः x = ± 6
अतः सही विकल्प (B) है।