Number of values of a ∈ N such that variance of 3,7,12 a, 43-a is a natural number is (Mean

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13.Statistics

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The number of values of $a \in N$ such that the variance of $3,7,12 a, 43-a$ is a natural number is (Mean $=13$)

A

$0$

B

$2$

C

$5$

D

infinite

(JEE MAIN-2022)

Solution

Mean $=13$

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

$\Rightarrow \frac{2 a^{2}-a+1}{5} \in N$

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

its $D=40 n-7$ always has $3$ at unit place

$\Rightarrow D$ can't be perfect square

So, a can't be integer.

Standard 11

Mathematics

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