\( 74.25 \)
Step 1: Identify the data set
The first 10 natural numbers that are multiples of 3 are: \[ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \] This forms an arithmetic sequence where: - First term (\( a \)) = 3 - Common difference (\( d \)) = 3 - Number of terms (\( n \)) = 10
Step 2: Compute the mean (\( \mu \))
The mean of a sequence is given by: \[ \mu = \frac{\sum x_i}{n} \] Using the sum formula for an arithmetic sequence: \[ S_n = \frac{n}{2} (2a + (n-1)d) \] \[ S_{10} = \frac{10}{2} (2(3) + (10-1)3) \] \[ = 5(6 + 27) = 5(33) = 165 \] \[ \mu = \frac{165}{10} = 16.5 \]
Step 3: Compute the variance (\( \sigma^2 \))
Variance is given by: \[ \sigma^2 = \frac{1}{n} \sum (x_i - \mu)^2 \] Computing the squared differences: \[ (3 - 16.5)^2 = (-13.5)^2 = 182.25 \] \[ (6 - 16.5)^2 = (-10.5)^2 = 110.25 \] \[ (9 - 16.5)^2 = (-7.5)^2 = 56.25 \] \[ (12 - 16.5)^2 = (-4.5)^2 = 20.25 \] \[ (15 - 16.5)^2 = (-1.5)^2 = 2.25 \] \[ (18 - 16.5)^2 = (1.5)^2 = 2.25 \] \[ (21 - 16.5)^2 = (4.5)^2 = 20.25 \] \[ (24 - 16.5)^2 = (7.5)^2 = 56.25 \] \[ (27 - 16.5)^2 = (10.5)^2 = 110.25 \] \[ (30 - 16.5)^2 = (13.5)^2 = 182.25 \] Summing these: \[ 182.25 + 110.25 + 56.25 + 20.25 + 2.25 + 2.25 + 20.25 + 56.25 + 110.25 + 182.25 = 742.5 \] \[ \sigma^2 = \frac{742.5}{10} = 74.25 \] Thus, the correct answer is: \[ \mathbf{74.25} \]
The mean deviation about the mean for the following data is:
Given the vectors:
\[ \mathbf{a} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} \]
\[ \mathbf{b} = 3(\mathbf{i} - \mathbf{j} + \mathbf{k}) = 3\mathbf{i} - 3\mathbf{j} + 3\mathbf{k} \]
where
\[ \mathbf{a} \times \mathbf{c} = \mathbf{b} \]
\[ \mathbf{a} \cdot \mathbf{x} = 3 \]
Find:
\[ \mathbf{a} \cdot (\mathbf{x} \times \mathbf{b} - \mathbf{c}) \]
If three numbers are randomly selected from the set \( \{1,2,3,\dots,50\} \), then the probability that they are in arithmetic progression is:
A student has to write the words ABILITY, PROBABILITY, FACILITY, MOBILITY. He wrote one word and erased all the letters in it except two consecutive letters. If 'LI' is left after erasing then the probability that the boy wrote the word PROBABILITY is: \