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13.Statistics
medium
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
Similar Questions
Find the mean and variance for the data
| ${x_i}$ | $92$ | $93$ | $97$ | $98$ | $102$ | $104$ | $109$ |
| ${f_i}$ | $3$ | $2$ | $3$ | $2$ | $6$ | $3$ | $3$ |
hard
