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Given :

ABCD is a square P and Q are the mid points of AB and CD respectively. 

AB = 8cm 

PQ and BD intersect at O 

Now,

AP = BP = \(\frac{1}{2}\) AB

AP = BP = \(\frac{1}{2}\) x 8

= 4 cm 

AB = AD = 8 cm

QP ‖ AD 

Then, 

AD = QP 

So, 

OP = AD 

OP = \(\frac{1}{2}\) x 8 

= 4 cm 

Now,

Area (ΔOPB) = \(\frac{1}{2}\) x BP x PO

 = \(\frac{1}{2}\) x 4 x 4

= 8 cm²

Hence, 

Area (ΔOPB) is 8 cm²

Correct option is A) 72

Given that, 

D, E, F are the mid-points of BC, DC, AE respectively 

Let, 

AD is median of triangle ABC

Area (ΔADC) = \(\frac{1}{2}\)Area (ΔABC)

= \(\frac{1}{2}\) x 16

= 8 cm² 

Now, 

AE is a median of ΔADC

Area (ΔAED) = \(\frac{1}{2}\)Area (ΔADC)

= \(\frac{1}{2}\) x 8

= 4 cm²

Again, 

DE is the median of ΔAED 

Area (ΔDEF) = \(\frac{1}{2}\)Area (ΔAED)

= \(\frac{1}{2}\) x 4

= 2 cm²

ΔABC and ΔBDE are equilateral triangles

We know that,

Area of equilateral triangle = \(\frac{\sqrt3}{4}a^2\) 

D is the mid-point of BC then,

Area (ΔBDE) = \(\frac{\sqrt3}{4}\) x \((\frac{a}{2})^2\)

 = \(\frac{\sqrt3}{4}\) x \(\frac{a\times a}{2}\)

Now, 

Area (ΔABC) : Area (ΔBDE)

\(\frac{\sqrt3}{4}\) x a² : \(\frac{\sqrt3}{4}\) x \(\frac{a\times a}{2}\)

1 : \(\frac{1}{4}\)

4: 1 

Hence,

Area (ΔABC) : Area (ΔBDE) is 4: 1

Correct option is (d) P₅₀

Correct option is (b) There are open-ended class intervals,

• If the number of observations and mean of two or more groups are given, then the mean of the combined group is called the combined mean. It is denoted by x̄_c.
• If the number of observations and mean of two or more groups are given, then the mean of the combined group is called the combined mean. It is denoted by x̄_c.

Suppose, k groups having the observations n₁, n₂,…,n_k have means respectively x₁, x₂,…. x_k, then their combined mean is obtained as follows :
x̄_c = \(\frac{n_1x¯_1+n_2x¯_2+…+n_kx¯_k}{n_1+n_2+…+n_k}\)

The advantages of mode are as follows :

• Mode is easy to understand and calculate.
• Its value can be found merely by inspection.
• Its value is not affected by too large or too small values.
• It can be found graphically.

 When the importance of all the observations of the data is not equal, then by assigning weightage to each observation in proposition to its importance, the mean obtained is called weighted mean. It is denoted by x̄w.

Suppose the weights assigned to observations x₁, x₂,…. x_n are respectively w₁, w₂, …, w_n then
weighted mean x̄_w = \(\frac{w_1x_1+w_2x_2+…+w_nx_n}{w_1+w_2+…+w_n}\)
= \(\frac{Σwx}{Σw}\)

If n observations x₁, x₂, …, x_n of the data are positive and non-zero, then nth root of the product of these observations is called geometric mean. It is denoted by G. Its formula Is as follows :
G = \(n\sqrt{x_1⋅x_2⋅…⋅x_n}\)

Advantages of Geometric mean:

• It is rigidly defined.
• It is based on all observations of the data.
• It is a stable measure.
• It is suitable for algebric process.
• It balances the effect of too large or too small observations of the data and its value is less than mean.

Correct option is (c) x̄ ≥ G

In day to day practice, many times we state like, on an average the daily wage of workers of a factory is ₹ 110. On an average the daily production of a company is 50 units, the average life of a person in our country is 70 years, etc. In all these statements the context of the measure of an average is to the observation that occurs maximum number of time. It indicates a general value of observations. Such general value representing the average indicates the value of mode.

Thus, the value of observation occuring the maximum number of time in the data is called mode. It is denoted by Mc.

• In classified data the values of variable are cancentrated around a certain central value. This characteristic of data is called as Central Tendency.
• The central value around which the values of the variable are concentrated is called as measure of central tendency.

The value that occurs maximum number of times in the given data is called mode. It is denoted by M₀.

Correct option is (c) Value of 25.5th observation

Correct option is (a) 30

The noted statistician Karl Pearson gave the empirical formula between mean, median and mode.

The empirical relation between mean, median and mode is M₀ = 3M – 23c. Where, M₀ = Mode; M = Median; 3c – Mean.

While choosing an appropriate average the following factors are to be considered:

• The nature of data
• The nature of variable involved
• The purpose of study
• The type of classification used and

n = 10, M = 55. The largest observation increases from 100 to 110. The value of median is free from the effect of extreme values. Therefore, the new value of median remains same as 55.

Correct option is (c) x̄ = 9

Correct option is (d) 9

Correct option is (b) 28

x̄ = 5.5, M₀ = 6.4, M = ?
Using the empirical formula,
M₀ = 3M – 2x̄
∴6.4 = 3M – 2(5.5)
∴ 6.4 + 11.0 = 3M
∴ M = 17.4 / 3 = 5.8

Q₃ = 25.75
∴ P₇₅ = Q₃ = 25.75

x̄ = 17, y = x – 4
∴ y = x̄ – 4
= 17 – 4 = 13

Suppose, another number is x²
x̄ = \(\frac{6+x_2}{2}\)
∴ 5 =   \(\frac{6+x_2}{2}\)
∴ 10 = 6 + x²
∴ x² = 10 – 6 = 4
Hence, another number = 4.

Correct option is (c) 16

Arranging the observations in ascending order: 2, 4, 7, 7, 13, 15, 20 , Here, n = 7.

First quartile Q₁ = Value of \((\frac{n+1}{4})\)th observation

= Value of \((\frac{7+1}{4})\)

= 2nd observation

∴ Q₁ = 4.

The value of mean is based on all the observations of the data.

Mode M₀= Observation having a maximum frequency 48
∴ M₀ = 10

Seventh decile:

D₇ = Value of (n+110)th observation
= Value of 7(6110)
= Value of 7 (6.1)
= 42.7th observation Referring to column cf,
D₇ = 8 mobile phones

Fifteenth Percentile:

P₁₅ = Value of 15\(\left(\frac{n+1}{100}\right)\)th observation
= Value of 15 \(\left(\frac{60+1}{100}\right)\)
= Value of 15(0.61)
= 9.15th observation

Referring to column cf,
P₁₅ = 6 mobile phones

Interpretation:

• D₇ = 8 phones. Therefore, the sale of mobile phones of 70 % of days will be 8 or less phones.
• P₁₅ = 6 phones. Therefore, the sale of mobile phones of 15 % of days will be 6 or less phones.

Correct option is (d) Mode cannot be found.

Correct option is (b) The mean divides the data in 2 equal parts.

Median of number of vehicles:
M = Value of (n+1 / 2)th observation
= Value of (25+1 / 2)
= 13th observation,
Referring to cf column.
M = 2 vehicles
Hence, median of number of vehicles = 2 vehicles.

Mean of production (x) x = 81 Production cost y = 3x + 50
∴ Mean of production cost y = 3x + 50 Putting x̄ = 81,
ȳ = 3 (81) + 50 = 243 + 50 = 293

Concept : Factor Pricing. 

Explanation : Theory of factor pricing refers to determining the factor rewards for land, labour, capital and entrepreneur in the form of rent, wages, interest and profit respectively.

Shabana is an entrepreneur who has paid wages to its worker in a factory for production of goods and also paid interest on her bank loan in form of rewards to the factors of production.

Correct option: (3) a, c and d

Features of Micro Economics :

• Based on certain ssumption : Micro economics is based on ‘ceteris paribus’ assumption i.e., other things remaining constant like full employment, laissez faire policy, perfect competition, pure capitalism, etc.
• Study of Individual units : Micro economics deals with the study of behaviour of small individual units of the economy such as individual units of the economy such as individual consumer, individual firm, individual industries, individual prices, etc.
• Slicing Method : It divides or slices the economy into small units and studies each unit in detail e.g. study of a particular household demand in detail.
• Analysis of Market Structures : Micro economics analyses different market structures such as perfect competition, monopoly, monopolistic competition, oligopoly, etc.
• Use of Marginalism Principle : The term ‘marginal’ means change brought in total by an additional unit. Marginal analysis helps to study a variable through the changes by which producers and consumers take economic decisions using this principle.
• Price Theory : Micro economics is known as price theory because it determines the prices of goods and services as well as prices of factors of production.
• Limited Scope : The study of micro economics is limited to individual economic unit only. It does not deal with macro problems like unemployment, inflation, deflation, poverty, unemployment, population, etc.
• Partial I quilibrium : Micro economics analysis deals with partial equilibrium which analyses equilibrium position of an individual economic unit i.e. individual consumer, individual firm, etc. 

(1)-(c), (2)-(a), (3)-(b), (4)-(e).

It is true that measure of price elasticity of demand for a normal good carrier minus sign because there is an inverse relationship between P and Q. Whereas there is a positive relationship between price and quantity supplied. Therefore, price elasticity of supply carries plus sign.

Concept: Slicing method. 

Explanation : Micro economics splits or divides the whole economy into small individual units and then studies each unit separately in detail. Mukesh has divided its business into two separate divisions to understand the detail cost of operations and profitability separately for food business as well as garment business. Government collect whole information about population through census.

Concept : Lumping method / Macro economics.

Explanation : Macro economics use lumping method to study the whole economy rather than its part.

Concept : Free market economy. 

Explanation : A free market economy is that economy where the economic decisions regarding production of goods are taken at individual level.

Eg : What to produce? How much to produce? How to produce? etc. decisions are taken by producers.

With the help of free market economy Ramesh has taken decision related to production such as What to produce? and How to produce?

Concept: Efficiency in consumption. 

Explanation : Efficiency in consumption means distribution of produced goods and services to the people for consumption for having maximum satisfaction.

Public distribution system is developed specially to help economically weaker section of society to satisfy their consumption. Rationing shops provide subsidized food and non food items to below poverty line people.

Importance of Macro Economics :

• Functioning of an Economy : It gives an idea of functioning of an economic system and help us to understand the behavioural pattern of aggregate variables.
• Economic fluctuations : It helps to analyse the causes of fluctuation in income, output and employment.
• National Income : It helps to study about National Income and makes possible to formulate correct economic policies.
• Economic Development : It helps us to understand the problems of the developing countries such as poverty, difference in the standards of living, etc., and suggest important steps to achieve economic development.
• Performance of an Economy : It helps us to analyse the performance of an economy where National Income estimates are used to measure the same.
• Study of Macro-economic Variables : Study of macro economic variables are important to understand the working of the economy.
• Level of Employment : Macro economics helps to analyse the general level of employment and output in an economy.  

Given e_D =- 0.2. Since e_D is less than one, it is a case of inelastic demand. If there is 10% increase in the price of the good, there will be less than 10% fall in quantity demanded. Result will be rise in the expenditure.

Concept : Lumping method / Macro economics.

Explanation : Macro economics use lumping method to study the whole economy rather than its part.

Govern ment collects data through census which gives them details about population such as sex ratio, literacy rate, unemployment rate, age composition, etc.

It is true that measure of price elasticity of demand for a normal good carrier minus sign because there is an inverse relationship between P and Q. Whereas there is a positive relationship between price and quantity supplied. Therefore, price elasticity of supply carries plus sign.

Concept : Micro economics / Slicing method. 

Explanation : Micro economics refers to the study of small unit from whole economy. Micro economics uses slicing method to split the whole economy into small individual units. Gauri has used slicing method from micro economics to collect information about the income of a particular firm from various firms.