The following table gives the literacy rate (in percentage) o f 35 cities. Find the mean literacy rate.

Question : Literacy rate (in %) 45 – 55 55 – 65 65 – 75 75 – 85 85 – 95 Number of cities 3 10 11 8 3 Solution : Let the assumed mean, A = 70 and class size, h = 10 So, \(u_i\) = \(x_i – 70\over 10\) Literacy rate (in %) …

The following table gives the literacy rate (in percentage) o f 35 cities. Find the mean literacy rate. Read More »

A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.

Question : Number of days 0 – 6 6 – 10 10 – 14 14 – 20 20 – 28 28 – 38 38 – 40 Number of students 11 10 7 4 4 3 1 Solution : Here, the class size varies, and \(x_i\)’s are small, Number of days Mid-Value (\(x_i\)) Number Students (\(f_i\)) …

A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent. Read More »

To find out concentration of \(SO_2\) in the air (in parts per million, i.e. ppm), the data was collected for 30 localities in a certain city and is presented below. Find the mean concentration of \(SO_2\) in the air.

Question : Concentration of \(SO_2\) (in ppm) Frequency 0.00 – 0.04 4 0.04 – 0.08 9 0.08 – 0.12 9 0.12 – 0.16 2 0.16 – 0.20 4 0.20 – 0.24 2 Solution : Let the assumed mean a = 0.10 Concentration of \(SO_2\) (in ppm) Frequency (\(f_i\)) Mid – Value (\(x_i\)) \(u_i\) = \(x_i …

To find out concentration of \(SO_2\) in the air (in parts per million, i.e. ppm), the data was collected for 30 localities in a certain city and is presented below. Find the mean concentration of \(SO_2\) in the air. Read More »

The table below shows expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by a suitable method.

Question : Daily expenditure (in Rupees) 100 – 150 150 – 200 200 – 250 250 – 300 300 – 350 Number of households 4 5 12 2 2 Solution : Let the assumed mean, A = 225, class size, h = 50 So, \(u_i\) = \(x_i – A\over h\) = \(x_i – 225\over 50\) …

The table below shows expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by a suitable method. Read More »

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose ?

Question : Number of Mangoes 50 – 52 53 – 55 56 – 58 59 – 61 62 – 64 Number of Boxes 15 110 135 115 25 Solution : Here, the class intervals are formed by the exclusive method. If we make the data an inclusive one, the mid-values remain same. So there is …

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose ? Read More »

Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarised as follows. Find the mean heart beats per minute for these women, choosing a suitable method.

Question : Number of heart beats per minute 65 – 68 68 – 71 71 – 74 74 – 77 77 – 80 80 – 83 83 – 86 Number of women 2 4 3 8 7 4 2 Solution : Let assumed mean a = 75.5 Number of heart beats No. of women \(f_i\) …

Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarised as follows. Find the mean heart beats per minute for these women, choosing a suitable method. Read More »

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is 18 rupees. Find the missing frequency f.

Question : Class Interval 11 – 13 13 – 15 15 – 17 17 – 19 19 – 21 21 – 23 23 – 25 Frequency 7 6 9 13 f 5 4 Solution : We have, Class Interval Mid-value (\(x_i\)) Frequency (\(f_i\)) \(f_ix_i\) 11 – 13 12 7 84 13 – 15 14 6 …

The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is 18 rupees. Find the missing frequency f. Read More »

Consider the following distribution of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method.

Question : Daily Wages in Rupees 0 – 120 120 – 140 14 – 160 160 – 180 180 – 200 Number of Workers 12 14 8 6 10 Solution : Let the assumed mean A = 150 and class size h = 50 So, \(u_i\) = \(x_i – A\over h\) = \(x_i – 150\over …

Consider the following distribution of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method. Read More »

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Question : Number of plants 0 – 2 2 – 4 4 – 6 6 – 8 8 – 10 10 – 12 12 – 14 No. of houses 1 2 1 5 6 2 3 Which method do you use for finding the mean and why ? Solution : Number of Plants Number of …

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Read More »

Prove the following identities, where the angles involved are acute angles for which the expressions are defined :

Question : (i)  \(1 – cos\theta\over 1 + cos\theta\) = \(({cosec\theta – cot\theta})^2\) (ii)  \(1 + sin\theta\over cos\theta\) + \(cos\theta\over 1 + sin\theta\) = \(2sec\theta\) (iii)  \(tan\theta\over 1 – cot\theta\) + \(cot\theta\over 1 – tan\theta\) = \(sec\theta + cosec\theta\) + 1 (iv)  \(1 + sec\theta\over sec\theta\) = \(sin^2\theta\over 1 – cos\theta\) (v)  \(cos A – …

Prove the following identities, where the angles involved are acute angles for which the expressions are defined : Read More »