Question
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\(\begin{bmatrix} x & 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} = \begin{bmatrix} 45 \end{bmatrix}\)
तो निम्नलिखित में से कौन-सा x का एक मान है?
Answer (Detailed Solution Below)
Detailed Solution
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आव्यूह 1 और आव्यूह 2 को गुणा करें:
\(\begin{bmatrix} x & 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} = \begin{bmatrix} x+4+7 & 2x+5+8 & 3x+6+9 \end{bmatrix}\)
⇒ \(\begin{bmatrix} x+11 & 2x+13 & 3x+15 \end{bmatrix}\)
परिणामी आव्यूह को आव्यूह 3 से गुणा करें:
\(\begin{bmatrix} x+11 & 2x+13 & 3x+15 \end{bmatrix} \times \begin{bmatrix} 1 \\ 1 \\ x \end{bmatrix} \)
\(= (x+11) \cdot 1 + (2x+13) \cdot 1 + (3x+15) \cdot x\)
⇒ \((x+11) + (2x+13) + (3x^2+15x)\)
⇒ \((3x^2 + 18x + 24)\)
परिणाम को 45 के बराबर करें:
\((3x^2 + 18x + 24 = 45)\)
⇒ \((3x^2 + 18x - 21 = 0)\)
\((x^2 + 6x - 7 = 0)\)
\((x^2 + 7x - x - 7 = 0)\)
⇒ \((x = 1 \text{ या } x = -7)\)
चरण 5: सत्यापित करें:
\(x = 1\) के लिए, वापस प्रतिस्थापित करें:
\((3(1)^2 + 18(1) + 24 = 45)\)
45 =45
∴ x का सही मान 1 है।
इसलिए, सही उत्तर विकल्प 4 है।
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