JEE Mathematics 2023 Question with Solution

Given: The mean of the $8$ numbers $x, y, 10, 12, 6, 12, 4, 8$ is $9$ and the variance is $9.25$.

Find: The value of $3x-2y$.

Using the mean,

So,

Hence,

Now use the variance formula:

The sum of squares of the known numbers is

Therefore,

So,

Hence,

Now,

Substitute $x+y=20$ and $x^2+y^2=218$:

Thus $x$ and $y$ are roots of

Factorizing,

So the two values are $13$ and $7$. Since $x>y$,

Therefore,

So the required value is $25$.

Given: Mean $=9$, variance $=9.25$ for the numbers $x, y, 10, 12, 6, 12, 4, 8$.

Find: $3x-2y$.

From the mean,

The known numbers add up to

Therefore,

From the variance,

So,

The sum of squares of the known numbers is

Hence,

Use

Then,

Now solve the pair with sum $20$ and product $91$. The numbers are $13$ and $7$. Since $x>y$, we take

Therefore,

Hence, the required numerical value is $25$.