If x - x-1 - -5- evaluate x2 - x-2 and x4 - x-4 -2 MARKS-

Substitution: 0.5 Mark each Calculation: 0.5 Mark each Given x + x^ 1 nbsp;= √(5) nbsp; ...(i) Squaring nbsp; on the both side of nbsp;equation (i) (x ...

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Standard X

Mathematics

Method of Substitution to Find the Solution of a Pair of Linear Equations

If x + x-1 = ...

Question

If $x + x^{- 1} = \sqrt{5}$ , evaluate $x^{2} + x^{- 2}$ and $x^{4} + x^{- 4}$ [2 MARKS]

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Solution

Substitution: 0.5 Mark each
Calculation: 0.5 Mark each
Given
$x + x^{- 1} = \sqrt{5}$ ...(i)
Squaring on the both side of equation (i)
$({x + x^{- 1})}^{2}$ = $(\sqrt{5})^{2}$
$\Rightarrow x^{2} + x^{- 2} + 2 = 5$
$\Rightarrow x^{2} + x^{- 2} = 5 - 2 = 3$ ...(ii)

Squaring on the both side of equation (ii)
$({x^{2} + x^{- 2})}^{2} = 3^{2}$
$\Rightarrow x^{4} + x^{- 4} + 2 = 9$
$\Rightarrow x^{4} + x^{- 4} = 9 - 2 = 7$

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If x+x−1=√5, evaluate x2+x−2 and x4+x−4 [2 MARKS]

Substitution: 0.5 Mark each Calculation: 0.5 Mark each Given x+x−1=√5 ...(i) Squaring on the both side of equation (i) (x+x−1)2 = (√5)2 ⇒ x2+x−2+2=5 ⇒ x2+x−2=5−2=3 ...(ii) Squaring on the both side of equation (ii) (x2+x−2)2=32 ⇒ x4+x−4+2=9 ⇒ x4+x−4=9−2=7

If $x + x^{- 1} = \sqrt{5}$ , evaluate $x^{2} + x^{- 2}$ and $x^{4} + x^{- 4}$ [2 MARKS]