Question

If the mean and the standard deviation of the data $3,5,7,a,b$ are $5$ and $2$ respectively, then $a$ and $b$ are the roots of the equation
1. ${x^2} - 20x + 18 = 0$
2. ${x^2} - 10x + 19 = 0$
3. $2{x^2} - 20x + 19 = 0$
4. ${x^2} - 10x + 18 = 0$

Answer 1

Hint: I In this question, the mean and standard deviation are $5$ and $2$ of the given data ($3,5,7,a,b$). We have to find the quadratic equation whose roots are given as $a$ and $b$. Calculate the sum and product of the roots using standard deviation formula.\

Formula used:
Mean $\left( {\overline x } \right) = \dfrac{{\sum {{x_i}} }}{n}$
Standard deviation $\left( {S.D.} \right) = \sqrt {\dfrac{{\sum {{x_i}^2} }}{n} - {{\left( {\overline x } \right)}^2}} $, $\overline x $(mean)
Quadratic equation –
${x^2} - Px + Q = 0$
Where, $P = $ sum of the roots
$Q = $ Product of the roots

Complete step by step solution: 
Given that,
Mean$\left( {\overline x } \right) = 5$
$ \Rightarrow \dfrac{{3 + 5 + 7 + a + b}}{5} = 5$
$a + b = 10 - - - - - \left( 1 \right)$
Standard deviation$\left( {S.D.} \right) = 2$
Given data is $3,5,7,a,b$
Therefore,
$\sum {{x_i}^2} = {\left( 3 \right)^2} + {\left( 5 \right)^2} + {\left( 7 \right)^2} + {\left( a \right)^2} + {\left( b \right)^2}$
$\sum {{x_i}^2} = 83 + {a^2} + {b^2}$
Using Standard deviation formula,
$2 = \sqrt {\dfrac{{83 + {a^2} + {b^2}}}{5} - {{\left( 5 \right)}^2}} $
$4 = \dfrac{{83 + {a^2} + {b^2}}}{5} - {\left( 5 \right)^2}$
$4 = \dfrac{{83 + {a^2} + {b^2}}}{5} - 25$
$4 + 25 = \dfrac{{83 + {a^2} + {b^2}}}{5}$
$29 \times 5 - 83 = {a^2} + {b^2}$
$ \Rightarrow {a^2} + {b^2} = 62$
Apply ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
${\left( {a + b} \right)^2} - 2ab = 62$
${\left( {10} \right)^2} - 2ab = 62\left( {equation\left( 1 \right)} \right)$
$2ab = 38$
$ab = 19$
Quadratic equation will be,
${x^2} - \left( {a + b} \right)x + ab = 0$
${x^2} - 10x + 19 = 0$
Hence, Option (2) is the correct answer.

Note: The key concept involved in solving this problem is the good knowledge of quadratic equations. Students must know that if roots are given then we can directly find the equation using ${x^2} - Px + Q = 0$where $P$ and $Q$ are the sum and product of the roots respectively. Likewise, if the equation is given as $a{x^2} + bx + c = 0$and we have to find the sum and product of roots we can find directly using Sum of roots $ = \dfrac{{ - b}}{a}$ and product $ = \dfrac{c}{a}$.