Question

$\sf{If\:the\:mean\:of\:x_1\:and\:x_2\:is\:M_1\:and\:that\:of\:x_1,x_2,x_3,x_4\:is\:M_2, } \\\sf{then\:mean\:of\:ax_1,ax_2,\dfrac{x^{3}}{a},\dfrac{x^{4}}{a}\: is }$

OPTIONS :


1. $\sf{\dfrac{M_1+M_2}{2}$



2. $\sf{\dfrac{aM_1+\dfrac {M_2}{2})}{2}}$



3. $\sf{\dfrac{1}{2a}[(a^{2}-1)M_1 +2M_2]}$



4. $\sf{\dfrac{1}{2a}[2(a^{2}-1)M_1 +2M_2]}$
​

Answer 1

$M_1=\dfrac{x_1+x_2}{2}$

$\implies x_1+x_2=2M_1$

$M_2=\dfrac{x_1+x_2+x_3+x_4}{4}\\\\\implies x_1+x_2+x_3+x_4=4M_2\\\\\implies 2M_1+x_3+x_4=4M_2\\\\\implies x_3+x_4=4M_2-2M_1$

Now we have to find the mean of $ax_1,ax_2,\dfrac{x^3}{a},\dfrac{x^4}{a}$

$ax_1+ax_2=a(x_1+x_2)\\\\\implies 2aM_1$

$\dfrac{x^3}{a}+\dfrac{x^4}{a}\\\\\implies \dfrac{1}{a}(x^3+x^4)\\\\\implies \dfrac{4m_2-2M_1}{a}\\\\\implies 2(\dfrac{2M_2-M_1}{a})$

Mean = sum of observation / number of terms ( which is 4 )

$\implies \dfrac{2aM_1+\dfrac{2(2M_2-M_1)}{a}}{4}\\\\\implies \dfrac{aM_1}{2}+\dfrac{2M_2-M_1}{2a}\\\\\implies \dfrac{a^2M_1-M_1+2M_2}{2a}\\\\\implies \dfrac{1}{2a}[M_1(a^2-1)+2M_2]$

The answer is Option 3 .

Answer 2

.hope it helps....................