Q120 / 180 · COMEDK 2024 (Morning Shift) · Mathematics
The mean of the numbers $a, b, 8,5,10$ is 6 and the variance is 6.80 , then which of the following gives possible values of $a$ & $b$
- A$a=5, b=2$
- Correct answerB$a=3, b=4$
- C$a=0, b=7$
- D$a=1, b=6$
Solution
The mean of the numbers $a, b, 8, 5, 10$ is given by $\dfrac{a + b + 8 + 5 + 10}{5} = 6$.
This simplifies to $a + b + 23 = 30$, so $a + b = 7$.
The variance is given by $\sigma^2 = \dfrac{\sum x_i^2}{n} - (\bar{x})^2 = 6.8$.
Substituting the values: $\dfrac{a^2 + b^2 + 8^2 + 5^2 + 10^2}{5} - 6^2 = 6.8$.
$\dfrac{a^2 + b^2 + 64 + 25 + 100}{5} - 36 = 6.8$.
$\dfrac{a^2 + b^2 + 189}{5} = 42.8$.
$a^2 + b^2 + 189 = 214$, which gives $a^2 + b^2 = 25$.
Since $a + b = 7$, we have $(a + b)^2 = a^2 + b^2 + 2ab = 49$.
$25 + 2ab = 49$, so $2ab = 24$, which means $ab = 12$.
The quadratic equation with roots $a$ and $b$ is $x^2 - (a+b)x + ab = 0$, which is $x^2 - 7x + 12 = 0$.
Factoring the equation: $(x - 3)(x - 4) = 0$.
Thus, the values are $a=3, b=4$ or $a=4, b=3$.
Answer: $a=3, b=4$
This simplifies to $a + b + 23 = 30$, so $a + b = 7$.
The variance is given by $\sigma^2 = \dfrac{\sum x_i^2}{n} - (\bar{x})^2 = 6.8$.
Substituting the values: $\dfrac{a^2 + b^2 + 8^2 + 5^2 + 10^2}{5} - 6^2 = 6.8$.
$\dfrac{a^2 + b^2 + 64 + 25 + 100}{5} - 36 = 6.8$.
$\dfrac{a^2 + b^2 + 189}{5} = 42.8$.
$a^2 + b^2 + 189 = 214$, which gives $a^2 + b^2 = 25$.
Since $a + b = 7$, we have $(a + b)^2 = a^2 + b^2 + 2ab = 49$.
$25 + 2ab = 49$, so $2ab = 24$, which means $ab = 12$.
The quadratic equation with roots $a$ and $b$ is $x^2 - (a+b)x + ab = 0$, which is $x^2 - 7x + 12 = 0$.
Factoring the equation: $(x - 3)(x - 4) = 0$.
Thus, the values are $a=3, b=4$ or $a=4, b=3$.
Answer: $a=3, b=4$
Hence, the correct answer is option: 'B'.