Let sets A and B have 5 elements each. Let mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 from each element of A and adding 2 to each element of B. Then the sum of the mean and variance of the elements of C is:
1
36
2
40
3
32
4
38
Official Solution
Correct Option: (4)
Let sets A and B be as follows:
Given:
and the variances:
First, calculate the sum of the elements of sets A and B:
Next, calculate the sum of squares for A and B using the formula for variance:
Similarly, for set B:
Now, set C is formed by subtracting 3 from each element of set A and adding 2 to each element of set B:
where:
and
Now, we calculate the mean of C:
Next, calculate the variance of C:
Using the identity and similarly for :
Thus, the sum of the mean and variance of C is:
02
PYQ 2024
medium
mathematicsID: jee-main
Let . Let the mean and the variance of 6 observations be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:
1
2
3
4
Official Solution
Correct Option: (1)
To solve this problem, we need to find the mean deviation about the mean for the 6 observations given. Let's break down the solution in a step-by-step manner:
Understanding Mean and Variance: The formula for the mean (average) of observations is:
Given, the mean of observations is 2:
Simplifying, we have:
Solving for :
Calculating Variance: The formula for variance is:
Given that the variance is 23:
Simplifying individual terms:
Plug in values:
Solving Simultaneous Equations: Now, solve for and using:
Substitute in equation 2:
Solving this:
Solving quadratic :
So, . If , then . If , then . Either way, and take the values 4 and 6.
Calculating Mean Deviation: Mean deviation about a mean is given by:
Compute mean deviation:
03
PYQ 2024
easy
mathematicsID: jee-main
The variance of the data Is _______.
Official Solution
Correct Option: (1)
Calculate sums:
Calculate the mean :
Calculate the variance :
where .
Plugging in values:
Compute:
Thus, the variance is: 29.09090909
04
PYQ 2024
medium
mathematicsID: jee-main
If the mean and variance of the data where are and respectively, then is equal to
Official Solution
Correct Option: (1)
Step 1: Calculate the Mean
Given that the mean .
Step 2: Calculate the Variance
Given that the variance .
Step 3: Set up the Equations for and
Using the formula for variance with mean and variance values:
Step 4: Solve for
Rearranging, we find:
So, the correct answer is: 6344
05
PYQ 2025
medium
mathematicsID: jee-main
For a statistical data of 10 values, a student obtained the mean as 5.5 andHe later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is:
1
7
2
4
3
9
4
5
Official Solution
Correct Option: (1)
We are given a statistical data set with the mean and the sum of squares as follows:
Mean of the data: 5.5
The student initially noted two values incorrectly as 4 and 5 instead of 6 and 8. We need to find the variance of the corrected data.
Step-by-Step Solution:
Calculate the initial sum of data from the given mean:
The incorrect data values are 4 and 5. Their sum is 9.
The correct data values are 6 and 8. Their sum is 14.
Correct the sum of the data:
Calculate the corrected mean:
Adjust the sum of squares:
Subtract the squares of the incorrect values:
Add the squares of the correct values:
Calculate the corrected sum of squares:
Calculate the corrected variance:
Therefore, the variance of the corrected data is 7.
06
PYQ 2025
medium
mathematicsID: jee-main
Let be ten observations such that and their variance is . If and are respectively the mean and the variance of then is equal to:
1
90
2
110
3
100
4
120
Official Solution
Correct Option: (3)
The problem requires us to determine the value of , where and are the mean and variance of the transformed observations given by . We are provided with:
, where
Variance of is
First, we find the mean of using the first condition:
