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List of top Mathematics Questions

The following table gives the distribution of the life time of 400 neon lamps :

Life time (in hours)	Number of lamps
1500 - 2000	14
2000 - 2500	56
2500 - 3000	60
3000 - 3500	86
3500 - 4000	74
4000 - 4500	62
4500 - 5000	48

Find the median life time of a lamp.

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

Number of letters	1 - 4	4 - 7	7 - 10	10 - 13	13 - 16	16 - 19
Number of surnames	6	30	40	16	4	4

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

Weight (in kg)	40 - 45	45 - 50	50 - 55	65 - 60	70- 65	65 - 70	70 - 75
Number of students	2	3	8	6	6	3	2

A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data :

Number of cars	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 -70	70 - 80
Frequency	7	14	13	12	20	11	15	8

The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.

Runs scored	Number of batsmen
3000 - 4000	4
4000 - 5000	18
5000 - 6000	9
6000 - 7000	7
7000 - 8000	6
8000 - 9000	3
9000 - 10000	1
10000 - 11000	1

Find the mode of the data.

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.

Number of students per teacher	Number of states / U.T
15 - 20	3
20 - 25	8
25 -30	9
30 - 35	10
35 - 40	3
40 - 45	0
45 - 50	0
50 - 55	2

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

The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :

Lifetimes (in hours)	0 - 20	20 - 40	40 - 60	60 - 80	80 - 100	100 - 120
Frequency	10	35	52	61	28	29

Determine the modal lifetimes of the components.

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:\(\frac{(\text{1 + tan² A})}{(\text{1 + cot² A})} = (\frac{\text{1 - tan A }}{\text{ 1 - cot A}})^²= \text{tan² A}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\text{(cosec A - sin A)(sec A - cos A)} =\frac{ 1}{\text{(tan A + cot A)}}\) [Hint : Simplify LHS and RHS separately]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\text{(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{\text{(sinθ - 2sin³θ)}}{\text{(2cos³θ - cosθ)}} = \text{tan θ}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\sqrt{\frac{\text{1 + sin A}}{\text{1 - sin A }}}= \text{sec A+ tan A}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(\text{cos A - sin A + 1})}{\text{(cos A + sin A + 1)}} = \text{cosec A + cot A}\), using the identity cosec² A = 1 + cot² A.

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\(\frac{(1 + \text{sec A})}{\text{sec A}} = \frac{\text{sin² A}}{(\text{1 - cos A})}\) [Hint: Simplify LHS and RHS separately]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined: \(\frac{\text{tan θ}}{(\text{1 - cot θ})} + \frac{\text{cot θ}}{(\text{1 - tan θ})} = 1 + \text{secθ cosecθ}\) [Hint: Write the expression in terms of \(\text{sin θ}\) and \(\text{cos θ}\)]

Prove the following identities, where the angles involved are acute angles for which the expressions are defined: \(\frac{\text{cos A}}{(1 + \text{sin A})} + \frac{(1 +\text{ sin A})}{\text{cos A }}=\text{ 2 sec A}\)

Prove the following identities, where the angles involved are acute angles for which the expressions are defined:\( (\text{cosec θ}-\text{cot θ})²=\frac{(1-\text{cos θ})}{(1 +\text{cos θ})}\)

Choose the correct option. Justify your choice. \(\frac{(1 + \text{tan}^2 A)}{(1 +\text{ cot}^ 2 A)}\)

Choose the correct option. Justify your choice. (sec A + tan A) (1 - sin A)

Choose the correct option. Justify your choice. \((1+\text{tan θ+sec θ}) (1+\text{cot θ - cosec θ}) =\)