Question

If the mean of \[x\] and \[\dfrac{1}{x}\] is \[M\] , then the mean of \[{x^2}\] and \[\dfrac{1}{{{x^2}}}\] is
A.\[{M^2}\]
B.\[\dfrac{{{M^2}}}{4}\]
C.\[2{M^2} - 1\]
D.\[2{M^2} + 1\]

Answer 1

Hint: Here, we have to find the value of mean of \[{x^2}\] and \[\dfrac{1}{{{x^2}}}\]. The mean is defined as the average of a given set of numbers. Here we will first use the given mean and form an equation. Then we will simplify the equation and square the terms so that we get an equation containing the terms \[{x^2}\] and \[\dfrac{1}{{{x^2}}}\] . We will then simplify the equation to get the required mean.

Formula Used:
 We will use the following formulas:
1. \[\overline x = \dfrac{1}{n}\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) = \dfrac{{{x_1} + {x_2} + {x_3} + ...... + {x_n}}}{n}\], where \[{x_1},{x_2},{x_3},.......,{x_n}\] be the samples of data.
2.The square of the sum of two numbers is given by the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
Complete step-by-step answer:
We know that the mean of \[x\] and \[\dfrac{1}{x}\] is \[M\].
Therefore,
\[\overline x = M\]
Substituting \[\overline x = \dfrac{{x + \dfrac{1}{x}}}{2}\] in the above equation, we get
\[ \Rightarrow \dfrac{{x + \dfrac{1}{x}}}{2} = M\]
On cross multiplication, we get
\[ \Rightarrow x + \dfrac{1}{x} = 2M\]
Squaring on both the sides, we get
\[ \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^2} = {\left( {2M} \right)^2}\]
\[ \Rightarrow \left( {{x^2} + \dfrac{1}{{{x^2}}} + 2 \cdot x \cdot \dfrac{1}{x}} \right) = 4{M^2}\]
Simplifying the equation, we have
\[ \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} + 2 = 4{M^2}\]
Rewriting the equation, we have
\[ \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} = 4{M^2} - 2\]
Dividing by 2 on both the sides, we get
\[ \Rightarrow \dfrac{{{x^2} + \dfrac{1}{{{x^2}}}}}{2} = \dfrac{{4{M^2} - 2}}{2}\]
\[ \Rightarrow \dfrac{{{x^2} + \dfrac{1}{{{x^2}}}}}{2} = 2{M^2} - 1\]
Therefore, the mean of \[{x^2}\] and \[\dfrac{1}{{{x^2}}}\] is \[2{M^2} - 1\].

Note: We should note that there are three types of mean namely arithmetic mean (A.M.), weighted Arithmetic mean, Geometric mean (G.M.) and Harmonic mean (H.M) respectively. When some of the values are not important, then we assign them certain numerical values to show their relative importance. These numerical values are called weights. Thus the mean calculated is called weighted arithmetic mean. The geometric mean, G, of a set of n positive values \[{X_1},{\rm{ }}{X_2},{\rm{ }} \ldots \ldots ,{\rm{ }}{X_n}\] is the nth root of the product of the values. The harmonic mean, H, of a set of \[n\] values\[{X_1},{\rm{ }}{X_2},{\rm{ }} \ldots \ldots ,{\rm{ }}{X_n}\] is the reciprocal of the arithmetic mean of the reciprocals of the values.