Solve the following problems:

1. Given the equations:

[tex]\[
x + y + z = 11
\][/tex]

[tex]\[
x y + y z + z x = 36 \quad \text{when} \quad x \ \textgreater \ y
\][/tex]

(a) Simplify the expression: [tex]\[ 2a^4 - a^2 - 1 \][/tex].

(b) Evaluate the expression: [tex]\[ (2x + 3y)^2 + (2y + 3z)^2 + (2z + 3x)^2 \][/tex].

(c) If [tex]\[ z = 6 \][/tex], solve for [tex]\[ x \][/tex] and [tex]\[ y \][/tex], then find [tex]\[ x^5 - y^5 = 211 \][/tex].

Question

26-06-2024
Mathematics

Answered

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Solve the following problems:

1. Given the equations:

[tex]\[
x + y + z = 11
\][/tex]

[tex]\[
x y + y z + z x = 36 \quad \text{when} \quad x \ \textgreater \ y
\][/tex]

(a) Simplify the expression: [tex]\[ 2a^4 - a^2 - 1 \][/tex].

(b) Evaluate the expression: [tex]\[ (2x + 3y)^2 + (2y + 3z)^2 + (2z + 3x)^2 \][/tex].

(c) If [tex]\[ z = 6 \][/tex], solve for [tex]\[ x \][/tex] and [tex]\[ y \][/tex], then find [tex]\[ x^5 - y^5 = 211 \][/tex].

Sagot :

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GinnyAnswer GinnyAnswer · Answer 1 · 2024-06-26T20:55:49-04:00

चलिए दिए गए सवाल के सभी भागों को चरणबद्ध तरीके से हल करते हैं।

(क) दोघटकों को (a और b) समीकरण $ (2a^4 - a^2 - 1) $ में पहचानें:

गुणांक:
[tex]\[ 2a^4 - a^2 - 1 = 0 \][/tex]

इस समीकरण को जड़कर:
[tex]\[ z = 6 \][/tex]

(घ) $ (2x + 3y)^2 + (2y + 3z)^2 + (2z + 3x)^2 $:

दी गई समीकरणों $ x + y + z = 11 $ और $ xy + yz + zx = 36 $ का उपयोग करके अनुमान लगाएं यदि $ z = 6 $, तब:

$ x + y + 6 = 11 $
इससे:
[tex]\[ x + y = 5 \][/tex]
अब:
$ xy + 6(x + y) = 36 $
इसे $ x + y $ से बदलते हुए:
[tex]\[ xy + 6 \cdot 5 = 36 \][/tex]
इससे:
[tex]\[ xy + 30 = 36 \][/tex]
और:
[tex]\[ xy = 6 \][/tex]
समीकरण:
[tex]\[ (2x + 3y)^2 + (2y + 3z)^2 + (2z + 3x)^2 = (2x + 3y)^2 + (2y + 18)^2 + (12 + 3x)^2 \][/tex]
जे संज्ञात्मक समीकरण $ x $ और $ y $ में रखकर $ y = 1 $ और $ x = 5 $:

क्लीयरण:
[tex]\[ (2x + 3y)^2 + (2y + 18)^2 + (12 + 3x)^2 \][/tex]

(ग) $ x^5 - y^5 = 211 $:

जड़:
$ x = 5 $, $ y = 1 $, तब $ x^5 - y^5 $ की गणना करके:
[tex]\[ 5^5 - 1^5 = 211 \][/tex]

इसलिए सभी दिए गए समीकरणों की उपयोगी समझ बनाकर हल किया गया।

ध्यान दें कि यहां [tex]$ x > y $[/tex] की स्थिति में इस तरह की जड़कर चर्चायोग्य गणनाएं लागू होते हैं।

Sagot :

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