The number of values of a ∈ N such that the variance of 3 , 7 , 12 , a , 43 - a is a natural number is:

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MathematicsStatisticsMeasures of DispersionMedium2 minPYQ_2022

MathematicsMediumsingle choice

The number of values of $a \in N$ such that the variance of $3, 7, 12, a, 43 - a$ is a natural number is:

Answer:

Solution:

Calculating Mean $= \frac{3 + 7 + 12 + a + 43 - a}{5} = 13$

Variance $= \frac{9 + 49 + 144 + a^{2} + (43 - a)^{2}}{5} - 13^{2} \in N$

$\Rightarrow \frac{2 a^{2} - a + 1}{5} = n$ (where $n \in N$ )

$\Rightarrow 2 a^{2} - a + 1 - 5 n = 0$ must have solution as natural numbers

So, $D = 40 n - 7$ always has $3$ at unit place for any value of $n$

$\Rightarrow D$ can't be perfect square

So, $a$ can't be integer.

Stream:JEESubject:MathematicsTopic:StatisticsSubtopic:Measures of Dispersion

⏱ 2mℹ️ Source: PYQ_2022

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