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Full-length FE mock with insightful analytics, weakness diagnosis, and video explanations!
14 Jan 2021, 08:36
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D
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Difficulty:
95%
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Question Stats:
41% (02:32) correct
59%
(03:04)
wrong
based on 68
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If \(x^2 - 5x + 1 = 0\), what is the value of \(x^3 + \frac{1}{x^3}\) ?
A. 90
B. 100
C. 105
D. 110
E. 120
A. 90
B. 100
C. 105
D. 110
E. 120
14 Jan 2021, 09:20
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Bunuel
If \(x^2 - 5x + 1 = 0\), what is the value of \(x^3 + \frac{1}{x^3}\) ?
A. 90
B. 100
C. 105
D. 110
E. 120
A. 90
B. 100
C. 105
D. 110
E. 120
\((a+b)^3=a^3+b^3+3ab(a+b)\)
If a=x and b=1/x
\((x + \frac{1}{x})^3=x^3 + \frac{1}{x^3}+3(x + \frac{1}{x})\)
\(( \frac{x^2+1}{x})^3=x^3 + \frac{1}{x^3}+3( \frac{x^2+1}{x})\)...(I)
Now \(x^2 - 5x + 1 = 0........x^2+1=5x\)
Substitute the value in (I)
\(( \frac{5x}{x})^3=x^3 + \frac{1}{x^3}+3( \frac{5x}{x})\)
\(( 5)^3=x^3 + \frac{1}{x^3}+3(5)\)
\(125=x^3 + \frac{1}{x^3}+15\)
\(x^3 + \frac{1}{x^3}=125-15=110\)
D
20 Jan 2021, 01:08
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x^2 - 5x + 1 = 0
x^2+1 = 5x
Dividing by x on both sides
x + 1/x = 5 .............. (1)
Cubing both sides : (a+b)^3 = a^3 + b^3 + 3ab(a+b)
x^3 + (1/x)^3 + 3(x * 1/x)(x + 1/x) = (5)^3
x^3 + (1/x)^3 + 3(5) = 125 ......... ( Taking value of x + 1/x = 5 from 1)
x^3 + (1/x)^3 + 15 = 125
x^3 + (1/x)^3 = 125 - 15
x^3 + (1/x)^3 = 110
Answer choice D
x^2+1 = 5x
Dividing by x on both sides
x + 1/x = 5 .............. (1)
Cubing both sides : (a+b)^3 = a^3 + b^3 + 3ab(a+b)
x^3 + (1/x)^3 + 3(x * 1/x)(x + 1/x) = (5)^3
x^3 + (1/x)^3 + 3(5) = 125 ......... ( Taking value of x + 1/x = 5 from 1)
x^3 + (1/x)^3 + 15 = 125
x^3 + (1/x)^3 = 125 - 15
x^3 + (1/x)^3 = 110
Answer choice D
