The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret the two measures.
The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure :
Expenditure (in Rs)
Number of families
1000 - 1500
24
1500 - 2000
40
2000 - 2500
33
2500 - 3000
28
3000 - 3500
30
3500 - 4000
22
4000 - 4500
16
4500 - 5000
7
Lifetimes (in hours)
0 - 20
20 - 40
40 - 60
60 - 80
80 - 100
100 - 120
Frequency
10
35
52
61
29
The following table shows the ages of tha year:
Age (in years)
5 - 15
15 - 25
25 - 35
35 - 45
45 - 55
55 - 65
Number of patients
6
11
21
23
14
5
Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)