Question

If \[x + \dfrac{1}{x} = \sqrt 3 \], then the value of \[{x^{18}} + {x^{12}} + {x^6} + 1\] is
A.\[0\]
B.\[1\]
C.\[2\]
D.\[3\]

Answer 1

Hint: To solve this question first we have to find the relation by \[x + \dfrac{1}{x} = \sqrt 3 \] equation that will eliminate some powers from the equation \[{x^{18}} + {x^{12}} + {x^6} + 1\]. So in particular the equation first cube both sides and eliminate some power and find the relation in higher powers.

Complete step-by-step answer:
Given,
Equation given in question \[x + \dfrac{1}{x} = \sqrt 3 \]
To find,
The value of \[{x^{18}} + {x^{12}} + {x^6} + 1\]
So, the given equation.
\[x + \dfrac{1}{x} = \sqrt 3 \]
On cubing both sides.
\[{\left( {x + \dfrac{1}{x}} \right)^3} = {\left( {\sqrt 3 } \right)^3}\]
Using the property of \[{(a + b)^3} = {a^3} + {b^3} + 3(a + b)\]
\[{(x + \dfrac{1}{x})^3} = {x^3} + \dfrac{1}{{{x^3}}} + 3(x + \dfrac{1}{x})\]
On putting the value
\[{x^3} + \dfrac{1}{{{x^3}}} + 3(x + \dfrac{1}{x}) = 3\sqrt 3 \]
Now putting the value of given equation in question
On putting the value \[x + \dfrac{1}{x} = \sqrt 3 \]
\[{x^3} + \dfrac{1}{{{x^3}}} + 3\sqrt 3 = 3\sqrt 3 \]
On further solving
\[{x^3} + \dfrac{1}{{{x^3}}} = 0\]
Now taking LCM in denominator
\[\dfrac{{{x^6} + 1}}{{{x^3}}} = 0\]
On further solving
\[{x^6} + 1 = 0\] (i)
Now, on putting the value in \[{x^{18}} + {x^{12}} + {x^6} + 1\]
Let \[i = {x^{18}} + {x^{12}} + {x^6} + 1\]
Taking \[{x^6}\] common from the equation
\[i = {x^{12}}({x^6} + 1) + \left( {{x^6} + 1} \right)\]
Now putting the value from equation (i)
\[i = {x^{12}}(0) + \left( 0 \right)\]
On further solving
\[i = 0\]
The value of the expression \[{x^{18}} + {x^{12}} + {x^6} + 1\] is the value of \[x + \dfrac{1}{x} = \sqrt 3 \]
\[ \Rightarrow {x^{18}} + {x^{12}} + {x^6} + 1 = 0\]
So, the correct answer is “Option A”.

Note: To solve this type of question first we have to make some relation such that the relation must simplify the power and come to the perfect value that is used to simplify the given expression. You may commit a mistake in finding the relationship and applying that relationship to the expression that’s the value we have to find. We have to take commonly from the expression in such a way that the value of the obtained equation is used to simplify that relationship and we are able to find the value of that given expression.