Monthly Archives: December 2021

In This Post we are  providing Chapter-10 STRAIGHT LINES NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON STRAIGHT LINES

Question-1

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 Question-2

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 Question-3

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 Question-4

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 Question-5

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 Question-6

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 Question-7

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 Question-8

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Question 9

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 Question-10

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Question 11

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 Question-12

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 Question-13

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 Question-14

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Question-15

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In This Post we are  providing Chapter-9 SEQUENCES AND SERIES NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON SEQUENCES AND SERIES

1. A side of an equilateral triangle is 20 cm long. A second equilateral triangle is inscribed in it by joining the mid-points of the sides of the first triangle. This process is continued for third, fourth, fifth, triangles. Find the perimeter of the sixth inscribed equilateral triangle.

Sol: Let the given equilateral triangle be ∆ ABC with each side of 20 cm.
By joining the mid-points of this triangle, we get another equilateral triangle of side equal to half of the length of side of ∆ABC.
Continuing in this way, we get a set of equilateral triangles with side equal to half of the side of the previous triangle.
Now,
Perimeter of first triangle = 20 x 3 = 60 cm;
Perimeter of second triangle = 10 x 3 = 30 cm;
Perimeter of third triangle = 5×3 = 15 cm;

2. In a potato race 20 potatoes are placed in a line at intervals of 4 m with the first potato 24 m from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
Sol: Distance travelled to bring first potato = 24 + 24 = 2 x 24 = 48 m
Distance travelled to bring second potato = 2(24 + 4) = 2 x 28 = 56 m
Distance travelled to bring third potato = 2(24 + 4 + 4) = 2 X 32 = 64 m; and so on…
Clearly, 48, 56, 64,… is an A.P. with first term 48 and common difference 8. Also, number of terms is 20.
Total distance run in bringing back all the potatoes,

3. Find the rth term of an A.P. sum of whose first n terms is 2n +3n²

Sol: Sum of k terms of A.P., S_n = 2n + 3n²

4. 1f the sum of p terms of an AP. is q and the sum of q terms is p, then show that the sum of p + q terms is —(p + q). Also, find the sum of first p — q terms (where, p > q).

Sol: Let first term and common difference of the A.P. be a and d, respectively. Given, S_p = q

5. If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is

Sol:  Let the first term and common difference of given A.P. be a and d, respectively.

6. If x, 2y and 3z are in A.P. where the distinct numbers x, y and z are in G.P., then the common ratio of the G.P.is

Sol: Since x, 2y and 3z are in A.P., we get

7. Let S_n denote the sum of the first n terms of an A.P. If S_2n = 3S_n, then S_3n : S_n  is equal to
Sol Let first term be a and common difference be d.
Then, S_2n = 3S_n

8. If a, b and c are in G.P., then the value of a−bb−c is equal to _________
Sol: Given that, a, b and c are in G.P.

9. Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.
Sol:

 True                                                            –
Consider the progression a, a + d, a + 2d, … and sequence of prime number 2, 3, 5, 7, 11,…
Clearly, progression is a sequence but sequence is not progression because it does not follow a specific pattern.

10.Find the sum to n terms of the series 

33 ​ 34

Ans.a_n = 1² + 2² + 3² + — +n²

35 ​ 36

37 ​ 38

39 ​ 40

41 ​ 42

43 ​ 44

45 ​ 46

47 ​ 48

49 ​ 50

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51

11.Show that the sum of (m+n)^th and (m-n)^th terms of an A. P. is equal to twice the m^th term.

52 ​ 53

Ans. a_m+n = a + (m+n-1) d

54 ​

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6 Marks Questions

12. 150 workers were engaged to finish a job in a certain no. of days 4 workers dropped out on the second day, 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work find the no. of days in which the work was completed  

Ans. a = 150, d = -4 

If total works who would have worked all n days 150(n-8)

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13.  Prove that the sum to n terms of the series 

Ans. Sn = 11 + 103 + 1005 + —- + n terms

Sn = (10+1) + (102 + 3) + (103 + 5) + —– + [10n + (2n-1)]

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14. The ratio of A M and G. M of two positive no. a and b is m : n show that

Ans. 

Sq both side

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15. Between 1 and 31, m number have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m-1)th no. is 5:9 find the value of m.

Ans.

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Textbook Question And Answer:

Q.1 Describe the five major sectors in an economy according to the macroeconomic point of view.[3-4 Marks]
ANSWER: An economy may be’ divided into different sectors depending on the nature of study.

1. Producer sector engaged in the production of goods and services.
2. Household sector engaged in the consumption of goods and services.
   Note: Households are taken as the owners of factors of production.
3. The government sector engaged in activities like taxation and subsidies.
4. Rest of the world sector engaged in exports and imports.
5. Financial sector (or financial system) engaged in the activity of borrowing and lending.

Q.2 What are the four factors of production and remunerations to each of these called? [ 1 Mark]
ANSWER:

Q.3 What are the important features of a capitalist economy? [3-4 Marks]
ANSWER: Features of capitalist economy are:

1. Private ownership of land and capital.
2. Profit is the only motive.
3. Free play of the market forces of demand and supply.
4.  Government looks after growth, stability and social justice in the economy.

Q.4 Describe the Great Depression of 1929. [3-4 Marks]
ANSWER: The Great Depression took place in 1929 which adversely affected the developed economies of Europe and North America. It continued for 10 years. There was extreme fall in aggregate demand due to fall in income, which led to a vicious circle of poverty.

Q.5 Distinguish between stock and flow. Between net investment and capital which is a stock and which is a flow? Compare net investment and capital with flow of water into a tank. [3-4 Marks]
ANSWER:

Net investment is a flow whereas capital is a stock. Amount of water in a tank at a particular point of time is a stock concept, whereas amount of water flowing into it is a flow concept.

Very Short Answer Type:

Q.1 What is meant by circular flow of income?
  ANSWER: It refers to flow of money income or the flow of goods and services across different sectors of the economy in a circular form.

Q.2 What are the three phases of circular flow of income?
ANSWER: Production Phase, Distribution Phase and Disposition Phase.

Question 3. Give the meaning of factor income.
Ans: Income earned by factor of production by rendering their productive services in the production process is known as Factor Income.

Q.4 What is meant by transfer income?
ANSWER: Income received without rendering any productive services is known as Transfer Income.

Q.5 Out of factor income and transfer income which one is a unilateral concept?
ANSWER: Transfer income.

Q.6 Define current transfers.[CBSE 2003]
ANSWER: Transfers made from the current income of the payer and added to the current income of the recipient (who receive) for consumption expenditure are called current transfers

Q.7 Define capital transfers.
ANSWER: Capital transfers are defined as transfers in cash and in kind for the purpose of investment to recipient made out of the wealth or saving of a donor.

Q.8 What is the meaning of final goods?
ANSWER: These are those which are used for:

1. Personal consumption (like bread purchased by consumer household), or
2. Investment or capital formation (like building, machinery purchased by a firm)

Q.9 What is meant by intermediate goods?
ANSWER:  These are those, which are used for:

1.  Further processing (like sugar used for making sweets), or
2. Resale in the same year (If car purchased by a car dealer for resale).

Q.10 What is meant by consumption goods?
ANSWER: Consumption goods are those goods which satisfy the wants of consumers directly.

Q.11Define capital goods.
ANSWER: Capital goods are defined as all goods produced for use in future productive processes.

Q.12 Give an example of a person who is staying abroad for a period more than one year and still he is treated as normal resident of India.
ANSWER: An Indian working in Indian Embassy in the USA will be treated as normal resident of India.

Short Answer Type Question:

Q.1 Explain the basis of classifying goods into intermediate and final goods. Give suitable examples. Or [CBSE 2010]
Distinguish between intermediate products and final products. Give examples. ‘ [CBSE 2009]
ANSWER:

Q.2 Define consumption goods and what are its categories.
ANSWER:  Consumption goods are those which satisfy the wants of the consumers directly.
For example, cars, television sets, bread, furniture, air-conditioners, etc. Consumption goods can further be subdivided into the following categories:

1. Durable goods: These goods have an expected life time of several years and of relatively high value. They are motor cars, refrigerators, television sets, washing machines, air-conditioners, kitchen equipments, computers, communication equipments etc.
2.  Semi-durable goods: These goods have an expected life time of use of one year or slightly more. They are not of relatively great value. Examples are clothing, furniture, electrical appliances like fans, electric irons, hot plates and crockery.
3. Non-durable goods: Goods which cannot be used again and again, i.e., they lose their identity in a single act of consumption are known as non durable goods. These are food grains, milk and milk products, edible oils, beverages, vegetables, tobacco and other food articles.
   goods which satisfy the human wants directly. They cannot be seen or touched, i.e., they are intangible in nature. These are medical care, transport and communications, education, domestic services rendered by hired servants, etc.

Q.3 Define capital goods and its categories.
Or
Define ‘capital goods’.[CBSE Foreign 2011]
ANSWER:

1. Capital goods are defined as all goods produced for use in future productive processes.
2. For example, All the durable goods like cars, trucks, refrigerators, buildings, air crafts, air-fields and submarines used to produce goods and services for sale in the market are a part of capital goods.
3. Stocks of raw materials, semi finished and finished goods lying with the producers at the end of an accounting year are also a part of capital goods.
4. Some more examples of capital goods are machinery, equipment, roads and bridges.
5. These goods require repair or replacement over time as their value depreciate over a period of time.

Q.4 Distinguish between consumption goods and capital goods. Which of these are final goods? [CBSE Delhi 2010]
ANSWER:  

Q.5 Differentiate between Current transfers and Capital Transfers.
ANSWER:

Question And Answer:

Q.1An index number which accounts for the relative importance of the items is known as
(i) weighted index
(ii) simple aggregative index
(iii) simple average of relatives
ANSWER:
(i) An index number becomes a weighted index when the relative importance of items is taken care of weighted index is the weighted average of different goods.

Q.2In most of the weighted index numbers the weight pertains to
(i) base year
(ii) current year
(iii) both base and current year
ANSWER:
(i) In general, the base period weight is preferred in calculating the weighted index number but as per Laspeyre’s method it uses the base year quantity as weight, Paache uses current year quantities as weight and Fisher’s Index Method uses both base and current year quantities.

Q.3 The impact of change in the price of a commodity with little weight in the index will be
(i) small
(ii) large
(iii) uncertain
ANSWER:
(i) An equal rise in the price of an item with little weight will have lower implications for the overall change in the price ;ndex than that of an Item with more weight.

Q.4 A consumer price index measures changes in
(i) retail prices
(ii) wholesale prices
(iii) producers’prices
ANSWER:
(i) Consumer Price Index (CPI), also known as the cost of living index, measures the average change in retail prices which show the most accurate impact of price rise on the cost of living of common people.

Q.5 The item having the highest weight in consumer price index for industrial workers is
(i) food
(ii) housing
(iii) clothing
ANSWER:
(i) As weight and Fisher’s index method uses both base and current year quantities.
Food is given around 57% weight in CPI for industrial workers as it constitutes the major proportion of their total consumption.

Q.6 QIn general, inflation is calculated by using
(i) wholesale price index
(ii) consumer price index
(iii) producer’s price index
ANSWER:
(i) The WPI is widely used to measure the rate of inflation. The weekly inflation rate is given by
XtXt1Xt−1×100
where X, and X_t-1 to the WPI for the (t)th and (t- 1)th weeks.

Q.7 Why do we need an index number?
ANSWER:
Index number enables us to calculate a single measure of change of a large number of items. The index numbers are needed for the general and specific purpose they are

• Measurement of Change in the Price Level or the Value of ‘ Money Index number measures the value of money during different periods of time as well as we can use it to know the Impact of the change in the value of money on different sections of society. It can be worked out to correct the inflationary and deflationary gaps in the system.
• Information of Foreign Trade Index of export and import provides useful information regarding foreign trade which helps in formulating the policies of export and import.
• Calculating Real Wages CPI are used in calculating the purchasing power of money and real wage as follows
  • Purchasing power of money = 1/Cost of living index
  • Real wage = (Money wage/Cost of living index) × 100
• Measuring and Comparing Output Index of Industrial Production (IIP) gives us a quantitative figure about the change in production in the industrial sector and thus helps in comparing industrial output in different periods. Similarly, agricultural production index provides us an estimate of the production index provides us an estimate of the production in agricultural sector.
• Policy Making of Government With the help of index numbers government determines the minatory and fiscal prey and take nassery steps to develop the country.
• Indicating Stock Prices Sensex and NIFT are index numbers of share prices on BSE and NSE respectively. They serve as a useful guide for investors in the stock market. If the sensex and nifty are rising, investors have positive expectations about the future performance of the economy and it is an appropriate time for investment.

Q.8 What are the desirable properties of the base period?
ANSWER:
Base period should have the following properties

• The base year should be a normal period and periods in which extraordinary events have occurred should not be taken as base periods as they are not appropriate for general comparisons.
• Extreme values should not be selected as base period.
• The period should not be too far in the past as comparison with current period cannot be done with such base year as policies, economic and social conditions change with time.
• Base period should be updated periodically.

Q.9 Why is it essential to have different CPI for different categories of consumers?
ANSWER:
The Consumer Price Index (CPI) in India is calculated for different categories as under

• CPI for industrial workers.
• CPI for urban non-manual employees.
• CPI for agricultural labourers.

The reason behind calculation of three different CPIs is that the consumption pattern of the three groups (i.e., industrial workers, urban non-manual workers and agricultural labourers) differs significantly from each other. Therefore, to assess the impact of the price change on the cost of living of the three groups, component items included in the index need to be given different weights for each of the group. This necessitates the calculation of different CPI for different categories of consumers.

Q.10 What does a consumer price index for industrial workers measure?
ANSWER:
Consumer price index for industrial workers measures the average change in retail prices of a basket of commodities which an industrial worker generally consumes. Consumer price index for industrial workers is increasingly being considered the appropriate indicator of general inflation, which shows the most accurate impact of price rise on the cost of living of common people.

The items included in CPI (Consumer Price Index) for industrial workers are food, pan, supari, tobacco, fuel and lighting, housing, colthing, and miscellaneous expenses with food being accorded the highest weight. This implies that the food price changes have a significant impact on the CPI.

Q.11What is the difference between a price index and a quantity index?
ANSWER:
The difference between a price index and a quantity index is as follows

• Price index numbers measure and allow for comparison of the prices of certain goods while quantity index number measure the changes in the physical volume of production, construction or employment.
• Price index numbers are more widely used as compared to quantity index numbers.
• Price index is known as unweighted index number while quantity index number is known was weighted index numbers.

Q.12 Is the change in any price reflected in a price index number?
ANSWER:
No, the change in any price is not reflected in a price index number. Price index numbers measure and permit comparison of the prices of certain goods included in the basket being used to compare prices in the base period with prices in the current period. Moreover, an equal rise in the price of an item with large weight and that of an item with low weight will have different implications for the overall change in the price index.

Q.13 Can the CPI number for urban non-manual emplyees represent the changes in the cost of living of the President of India?
ANSWER:
The CPI for the urban non-manual employees cannot represent the changes in the cost of living of the President of India. This is because the consumption basket of an average non-manual employee does not consist of the items that would be a part of the consumption basket of the President of India.

Q.14The monthly per capita expenditure incurred by workers for an industrial centre during 1980 and 2005 on the following items are given below. The weights of these items are 75, 10, 5, 6 and 4 respectively.
Prepare a weghted index number for cost of living for 2005 with 1980 as the base.

ANSWER:

Q.15 Read the following table carefully and give your comments.
ANSWER:
Index of Industrial Production Base 1993-94

The following conclusions can be made by analysing the above table

• Manufacturing industry has the highest weight of 79.58% in Index of Industrial Production (IIP) while mining and quarrying and electricity industries account for 10.73% and 10.69% respectively.
• Manufacturing Industry has registered the highest growth among all industrial sectors in both the years 1996-97 and 2003-04.
• Mining and quarrying has registered the lowest growth rate in both the years.
• The General Index shows that industrial increased by 30.8% in 1996.-97 as compared to 1993-94 and by 89% in 2003-04.

Q.16 Try to list the important items of consumption in your family.
ANSWER:
(This is a general example. You can use the actual consumption items in your family).
The following items constitute the total consumption needs for a family

• Food
• Clothing
• House-Rent/EMI of Housing loan
• Education
• Electricity
• Entertainment and recreation
• Miscellaneous expenses

Q.17 If the salary of a person in the base year is ? 4,000 per annum and the current year salary is ? 6,000 by how much should his salary rise to maintanin the same standard of living if the CPI is 400?
ANSWER:
Base CPI = ₹ 100
Current CPI = ₹400
Base Year Salary = ₹ 4,000
Current Year Salary = ₹ 6,000
When Base CPI is ₹100, then the salary is = ₹ 4,000
Current salary equivalent to base year salary = (Base year salary/100) × CPI of current year
When Current CPI is ₹ 400, then the salary should be
= 4,000100×400 = ₹ 16,000 100
Thus, his salary should be X 16,000 to maintain his purchasing power. Therefore, in the current year his salary should increase by ₹ 16,000 – ₹ 6,000 = ₹ 10,000 so as to maintain the same level of living in the current year as that of the base year.

Q.18The consumer price index for June, 2005 was 125. The food index was 120 and that of other items
What is the percentage of the total weight given to food?
ANSWER:

Let the total weight = 100
W₁ denotes weight of food
W₂ denotes weight of other items
So,

Multiplying both sides of Eq. (i) by 135 and subtracting Eq. (ii) from (i) we get

So, W₁ = 100015 = 66.67
Substituting the value of in the Eq. (i), we get
W₁ + W₂ = 100
or 6667 + W₂ = 100
W₂ = 33.33
Therefore, percentage of total weight given to food is 66.67% and other items 33.33%.

Q.19 An enquiry into the budgets of the middle class families in a certain city gave the following information

What is the cost of living index of 2004 as compared with 1995?
ANSWER:

Cost of Living Index = 134.50
Thus, the price rose by 34.50% during 1995 and 2004.

Q.20 Record the daily expenditure quantities bought and prices paid per unit of the daily purchases of your family for two weeks. How has the price change affected your family?
ANSWER:
This is a practical exercise. Record the daily expenditure, quantities bought and prices paid per unit of the daily purchases of your family for two weeks and try to analyse if quantities purchased decrease with rise in price of the respective items and also note if the percentage change in quantity brought about by a percentage change in price differ for different types of items.

Q.21Given the following data

Source Economic Survey, Government of India 2004-2005
(i) Calculate the inflation rates using different index numbers.
(ii) Comment on the relative values of the index numbers.
(iii) Are they comparable?
ANSWER:
(i) (a) Inflation using CPI of Industrial Workers

(b) Inflation using CPI of Non-maunal Employees

(c) Inflation using CPI of Agricultural Labourers

(d) Inflation using WPI

(ii) The inflation rate calculated using CPI industrial worker with the base year 1982 is the highest and inflation rate calculated using WPI with the base year 1993-94 is the least.
(iii) No the index number are not comparable because of the following reasons

• Base periods for CPI of industrial workers, urban non-manual workers, agricultural labourers and WPI are different.
• Commodities and their weightage in different index number may be different.

Question And Answer:

Q.1The unit of correlation coefficient between height in feet and weight in kgs is
(a) kg/feet
(b) percentage
(c) non-existent
ANSWER:
(c) Correlation coefficient (r) has no unit. It is a pure number. It meansss units of measurement are not part of r.

Q.2The range of simple correlation coefficient is
(a) 0 to infinity
(b) minus one to plus one
(c) minus infinity to infinity
ANSWER:
(b) The value of the correlation coefficient lies between minus one and plus one, -1 ≤ r ≤ 1. If the value of r is outside this range it indicates error in calculation.

Q.3 If r_XY is positive the relation between X and Y is of the type
(a) when Y increases X increases
(b) when Y decreases X increases
(c) when Y increases X does not change
ANSWER: :
(a) If r is positive the two variables move in the same direction. e.g., when the price of coffee rises, the demand for tea also rises as coffee is a substitute of tea. Therefore, the r between price of coffee and demand for tea will be positive.

Q.4 If r_XY = 0, the variable X and Y are
(a) linearly related
(b) not linearly related
(c) independent
ANSWER:
(b) If r_XY = 0, it means the two variables are uncorrelated and there is no linear relation between them. However, other types of relation may be there and they may not be independent.

Q.5 Of the following three measures which can measure any type of relationship?
(a) Karl Pearson’s coefficient of correlation
(b) Spearman’s rank correlation
(c) Scatter diagram
ANSWER:
(c) The scatter diagram gives a visual presentation of the relationship and is not confined to linear relations. Karl Pearson’s coefficient of correlation and Spearman’s rank correlation are strictly the measures of linear relationship.

Q.6If precisely measured data are available the simple correlation coefficient is
(a) more accurate than rank correlation coefficient
(b) less accurate than rank correlation coefficient
(c) as accurate as the rank correlation coefficient
ANSWER:
(a) Rank correlation should be used only when the variables cannot be measured precisely, generally it is not as accurate as the simple correlation coefficient as all the information concerning the data is not utilised in this.

Q.7 Why is r preferred to covariance as a measure of association?
ANSWER:
Both, correlation coefficient and covariance measure the degree of linear relationship between two variables, but correlation coefficient is generally preferred to covariance due to the following reasons

• The correlation coefficient (r) has no unit.
• The correlation coefficient is independent of origin as well as scal

Q.8 Can r lie outside the -1 and 1 range depending on the type of data?
ANSWER:
No the value of the correlation coefficient lies between minus one and plus one, -1 ≤ r ≤ 1. If the value of r is outside this range in any type of data, it indicates error in calculation.

Q.9 Does correlation imply causation?
ANSWER:
No, correlation measures do not imply causation. Correlation measures co-variation and not causation.
Correlation does not imply cause and effect relation. The knowledge of correlation only gives us an idea of the direction and intensity of change in a variable when the correlated variable changes. The presence of correlation between two variables X and Y simply means that when the value of one variable is found to change in one direction, the value of the other variable is found to change either in the same direction (i.epositive change) or in the opposite direction (i.e., negative change), in a definite way.

Q.10 When is rank correlation more precise than simple correlation coefficient?
ANSWER:
Rank correlation is more precise than simple correlation coefficient in the following situations

• When the Measurements of the Variables are Suspect e.g., in a remote village where measuring rods or weighing scales are not available, height and weight of people cannot be measured precisely but the people can be easily ranked in terms of height and weight.
• When Data is Qualitative It is difficult to quantify qualities such as fairness, honesty etc. Ranking may be a better alternative to quantification of qualities.
• When Data has Extreme Values Sometimes the correlation coefficient between two variables with extreme values may be quite different from the coefficient without the extreme values. Under these circumstances rank correlation provides a better alternative to simple correlation.

Q.11Does zero correlation mean independence?
ANSWER:
No, zero correlation does not mean independence. If there is zero correlation (r_XY = 0), it means the two variables are uncorrelated and there is no linear relation between them. However, other types of relation may be there and they may not be independent.

Q.12 Can simple correlation coefficient measure any type of relationship?
ANSWER:
No, simple correlation coefficient can measure only linear relationship.

Q.13 List some variables where accurate measurement is difficult.
ANSWER:
Accurate measurement is difficult in case of

• Qualitative variables such as beauty, intelligence, honesty, etc.
• It is also difficult to measure subjective variables such as poverty, development, etc which are interpreted differently by different people.

Q.14 Interpret the values of r as 1, -1 and 0.
ANSWER:

• If r = 0 the two variables are uncorrelated. There is no linear relation between them. However, other types of relation may be there and hence the variables may not be independent.
•  If r= 1 the correlation is perfectly positive. The relation between them is exact in the sense that if one increases, the other also increases in the same proportion and if one decreases, the other also decreases in the same proportion.
• If r = -1 the correlation is perfectly negative. The relation between them is exact in the sense that if one increases, the other decreases in the same proportion and if one decreases, the other increases in the same proportion.

Q.15 Why does rank correlation coefficient differ from Pearsonian correlation coefficient?
ANSWER:
Rank correlation coefficient differs from Pearsonian correlation coefficient in the following ways

• Rank correlation coefficient is generally lower or equal to Karl Pearson’s coefficient.
• Rank correlation coefficient is preferred to measure the correlation between qualitative variables as these variables cannot be measured precisely.
• The rank correlation coefficient uses ranks instead of the full set of observations that leads to some loss of information.
• If extreme values are present in the data, then the rank correlation coefficient is more precise and reliable.

Q.16 Calculate the correlation coefficient between the heights of fathers in inches (X) and their sons (Y).
ANSWER:

Note Answer: printed in NCERT is incorrect.

Q.17 Calculate the correlation coefficient between X and Y and comment on their relationship.

ANSWER:

As the value of r is zero, so there is no linear correlation between X and Y.

Q.18Calculate the correlation coefficient between X and Y and comment on their relationship.

ANSWER:

As the correlation coefficient between the two variables is + 1, so the two variables are perfectly positive correlated.

Question And Answer:

Q.1A measure of dispersion is a good supplement to the central value in understanding a frequency distribution. Comment.
ANSWER:
Dispersion is the extent to which values in a distribution differ from the avarage of the distribution. Knowledge of only average is insufficient as it does not reflect the quantum of variation in values.

Measures of dispersion enhance the understanding of a distribution considerably by providing information about how much the actual value of items in a series deviate from the central value, e.g., per capita income gives only the average income but a measure of dispersion can tell you about income inequalities, thereby improving the understanding of the relative living standards of different sections of the society. Through value of dispersion one can better understand the distribution.

Thus a measure of dispersion is a good supplement to the central value in understanding a frequency distribution.

Q.2Which measure of dispersion is the best and how?
ANSWER:
Standard Deviation is considered to be the best measure of dispersion and is therefore the most widely used measure of dispersion.

• It is based on all values and thus provides information about the complete series. Because of this reason, a change in even one value affects the value of standard deviation.
• It is independent of origin but not of scale.
• It is us’eful in advanced statistical calculations like comparison of variability in two data sets.
• It can be used in testing of hypothesis.
• It is capable of further algebraic treatment.

Q.3Some measures of dispersion depend upon the spread of values whereas some calculate the variation of values from a central value. Do you agree?
ANSWER:
Yes, it is true that some measures of dispersion depend upon the spread of values, whereas some calculate the variation of values from the central value. Range and Quartile Deviation measure the dispersion by calculating the spread within which the value lie. Mean Deviation and Standard Deviation calculate the extent to which the values differ from the average or the central value.

Question 4.
Q.4In town, 25% of the persons earned more than ₹ 45,000 whereas 75% earned more than 18,000. Calculate the absolute and relative values of dispersion.
ANSWER:
25% of the persons earned more than ₹ 45,000. This implies that upper quartile Q₃ = 45,000 75% earned more than 18,000. This implies that lower quartile Q₁ =18,000
Absolute Measure of Dispersion = Q₃ – Q₁ = 45,000 – 18,000 = 27,000
Relative Measure of Dispersion
Co-efficient of Quartile Deviation

Q.5 The yield of wheat and rice per acre for 10 districts of a state is as under

Calculate for each crop,
(i) Range
(ii) QD
(iii) Mean’Deviation about Mean
(iv) Mean Deviation about Median
(v) Standard Deviation
(vi) Which crop has greater variation?
(vii) Compare the value of different measures for each crop.
ANSWER:
(i) Range
(a) Wheat Highest value of distribution (H) = 25
Lowest value of distribution (L) = 9
Range = H – L = 25 – 9 = 16
(b) Rice Highest value of distribution (H) = 34
Lowest value of distribution (L)=12
Range = H – L = 34 – 12 = 22
(ii) Quartile Deviation
(a) Wheat Arranging the production of wheat in increasing order 9, 10, 10, 12, 15, 16, 18, 19, 21, 25
Q₁ = N+14th item = 10+14th item = 114th item
= 2.75th item
= Size of 2nd item + 0.75 (size of 3rd item – size of 2nd item)
= 10 + 0.75(10 – 10)
= 10 + 0.75 × 0
= 10
Q₃ = 3(N+1)4th item = 3(10+1)4th item
= 334th item = 8.25th
= Size of 8th item + 0.25 (size of 9th item – size of 8th item)
= 19 + 0.25(21 – 19)
= 19 + 0.25 × 2
= 19 + 0.50 = 19.50
Quartile Deviation = Q3−Q12=19.50−102=9.502 = 4.75
(b) Rice Arranging the data of production of rice
12, 12, 12, 15, 18, 18, 22, 23, 29, 34 item
Q₁ = N+14th item = 10+14th item
= 2.75 th item
= Size of 2nd item + 0.75 (size of 3rd item – size of 2nd item)
= 12 + 0.75(12 – 12) = 12 + 0.75 × 0
= 12

= 8.25th item
= Size of 8th item + 0.25 (size of 9th item – size of 8th item)
= 23 + 0.25(29 – 23)
= 23 + 0.25 × 6
= 23 + 1.5
= 24.5
Quartile Deviation = Q3−Q12=24.5−122=12.502 = 6.25

(iii) Mean Deviation about Mean
(a) Wheat

(b) Rice

(iv) Mean Deviation about Median
(a) Wheat

(b) Rice

(v) Standard Deviation
(a) Wheat

(b) Rice

(vi) Coefficient of Variation
(a) Wheat
CV =σX¯¯¯¯¯×100=5.0415.5×100 = 32.51
(b) Rice
CV =σX×100=7.1619.5×100 = 36.71
Rice crop has greater variation as the coefficient of variation is higher for rice as compared to that of wheat.
(vii) Rice crop has higher Range, Quartile Deviation, Mean Deviation about Mean, Mean Deviation about Median, Standard Deviation and Coefficient of Variation.

Q.6In the previous question, calculate the relative measures of variation and indicate the value which , in your opinion, is more reliable.
ANSWER:
(i) Coefficient of Range
(a) Wheat

The coefficient of variation is more reliable than all other measures.

Q.7A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their scores in five previous scores which are

Which batsman should be selected if we want,
(i) a higher run-getter, or
(ii) a more reliable batsman in the team?
ANSWER:
Batsman X

(i) Average of Batsman X is higher than that of Batsman Y, so he should be selected if we want a high scorer.
(ii) The Batsman Y is more reliable than Batsman X. This is because the coefficient of variation of Batsman X is higher than that of Batsman Y.

Q.8To check the quality of two brands of light bulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.

(i) Which brand gives higher life?
(ii) Which brand is more dependable?
ANSWER:
For Brand A

For Brand B

(i) The average life of bulb of Brand B is comparatively higher than that of Brand A.
(ii) The bulbs of Brand B are more dependable as CV of Brand B is lesser than CV of Brand A.
Q.9 Average daily wage of 50 workers of a factory was ₹ 200 with a Standard Deviation of ₹ 40. Each worker is given a raise of ₹ 20. What is the new average daily wage and standard deviation? Have wages become more or less uniform?
ANSWER:
N = 50
x¯¯¯ = 200
σ = 40
Average wage =  Total wages  Number of workers 
200 =  Total wages 50
So, total wages = 200 × 50 = ₹ 10,000
Now, increase in wage rate = ₹20
Total raise = 50 × 20= ₹ 1,000
Total wage after raise = ₹ 10,000 + 1,000 = ₹ 11,000
New average wage =  New total wages  Number of workers =11,00050 = ₹220
Thus, Mean increases by the amount of increase in wage of each worker as the absolute increase was equal for all.
Standard Deviation will remain the same that is ₹40 as Standard Deviation is independent of origin and hence addition of equal amount in all the values will not cause any change in the Standard Deviation.
Uniformity of wages can be seen by coefficient of variation.
Previously, the coefficient of variation was
CV = σX¯¯¯¯¯×100 = (40/200) × 100 = 20
The new coefficient of variation after wage increase is given by
CV = σx×100 = (40/220) × 100 = 18.18
This shows that wages have become more uniform now as the new CV is lower.

Q.10If in the previous question, each worker is given a hike of 10% in wages, how are the Mean and Standard Deviation values affected?
ANSWER:
Average wage = ₹ 200
Hike in wages = 10%
Since arithmetic mean is not independent of scale, the mean will also increase by 10%.
= 10100×200 = ₹ 20
Hence, the new Mean will be
200 + 20 = ₹ 220
Standard Deviation is also not independent of scale, hence, the Standard Deviation will also increase by 10% Initial Standard Deviation = ₹ 40
So, New Standard Deviation = ₹40 +10% of 40
= ₹ (40 + 4)
= ₹ 44

Q.11Calculate the Mean Deviation using Mean and Standard Deviation for the following distribution.

ANSWER:

Q.12The sum of 10 values is 100 and the sum of their squares is 1090. Find out the coefficient of variation.
ANSWER:

Question And Answer:

Q.1 Which average would be suitable in the following cases?
(i) Average size of readymade garments.
(ii) Average intelligence of students in a class.
(iii) Average production in a factory per shift.
(iv) Average wages in an industrial concern.
(v) When the sum of absolute deviations from average is least.
(vi) When quantities of the variable are in ratios.
(vii) In case of open-ended frequency distribution.
ANSWER:
(i) Mode Average size of any ready made garments should be the size for which demand is the maximum. Hence, the modal value which represents the value with the highest frequency should be taken as the average size to be produced.
(ii) Median It is the value that divides the series into two equal parts. Therefore, Median will be the best measure for calculating the average intelligence of students in a class as it will give the average intelligence such that there are equal number of students above and below this average. It will not be affected by extreme values.
(iii) Arithmetic Mean The average production in a factory per shift is best calculated by Arithmetic Mean as it will capture all types of fluctuations in production during the shifts.
(iv) Arithmetic Mean Arithmetic Mean will be the most suitable measure. It is calculated by dividing the sum of wages of all the workers by the total number of workers in the industrial concern. It gives a fair idea of average wage bill taking into account all the workers.
(v) Arithmetic Mean The algebraic sum of the deviations of values about Arithmetic Mean is zero. Hence, when the sum of absolute deviations from average is the least, then mean could be used to calculate the average.
(vi) Median Median will be the most suitable measure in case the variables are in ratios as it is least affected by the extreme values.
(vii) Median Median is the most suitable measure as it can be easily computed even in case of open ended frequency distribution and will not get affected by extreme values.

Q.2 Indicate the most appropriate alternative from the multiple choices provided against each question.
(i) The most suitable average for qualitative measurement is
(a) Arithmetic mean
(b) Median
(c) Mode
(d) Geometric mean
(e) None of these
ANSWER:
(b) Median is the most suitable average for qualitative measurement because Median divides a series in two equal parts thus representing the average qualitative measure without being affected by extreme values.

(ii) Which average is affected most by the presence of extreme items?
(a) Median
(b) Mode
(c) Arithmetic Mean
(d) Geometric Mean
(e) Harmonic Mean
ANSWER:
(c) It is defined as the sum of the values of all observations divided by the number of observations and therefore it is. affected the most by extreme values

(iii) The algebraic sum of deviation of a set of n values from AM is
(a) n
(b) 0
(c) 1
(d) None of these
ANSWER:
(b) This is one of the mathematical properties of arithmetic mean that the algebraic sum of deviation of a set of n values from AM is zero.

Q.3 Comment whether the following statements are true or false.
(i) The sum of deviation of items from median is zero.
(ii) An average alone is not enough to compare series.
(iii) Arithmetic mean is a positional value.
(iv) Upper quartile is the lowest value of top 25% of items.
(v) Median is unduly affected by extreme observations.
ANSWER:
(i) False
This mathematical property applies to the arithmetic mean and not to median.
(ii) True
Average is not enough to compare the series as it does not explain the extent of deviation of different items from the central tendency and the difference in the frequency of values. These are measured by measures of dispersion and kurtosis.
(iii) False
Median is a positional value.
(iv) True
The upper quartile also called the third quartile, has 75 % of the items below it and 25 % of items above it.
(v) False
Arithmetic mean is unduly affected by extreme observations.

Q.4If the arithmetic mean of the data given below is 28, find (a) the missing frequency and (b) the median of the series

ANSWER:
(a) Let the missing frequency br f₁.
Arithmetic Mean = 28

or 2240 -2100 = 35f₁ = 28f₁
or 140 = 7f₁
f₁ = 20
Hence, the missing frequency is 20.
(b)

So, the Median class = Size of (N2)th item = 50th term.
50th item lies in the 57th cumulative frequency and the corresponding class interval is 20-30.

Q.5The following table gives the daily income of ten workers in a factory. Find the arithmetic mean.

ANSWER:

N = 10
X¯¯¯¯=ΣXN=240010=240
Arithmetic Mean = ₹ 240

Q.6 Following information pertains to the daily income of 150 families. Calculate the arithmetic mean.

ANSWER:

Q.7 The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.

ANSWER:

So, the median class = Size of (N2) th item = 190 item
190th lies in the 129 th cumulative frequency and the corresponding class interval is 200-300.

Median size of land holdings = 241.22 acres

Q.8The following series relates to the daily income of workers employed in a firm. Compute (a) highest income of lowest 50% workers, (b) minimum income earned by the top 25% workers and (c) maximum income earned by lowest 25% workers.

ANSWER:

(a) Highest income of lowest 50% workers will be given by the median. Σf = N = 65
Median class = Size of (N2)th item = Size of (652)th item=325 th item
32.5th item lies in the 50th cumulative frequency and the corresponding class interval is 24.5 – 29.5.

(b) Minimum income earned by top 25% workers will be given by the lower quartile Q₁.
Class interval of Q₁ = (N4)th item
= (654)th item = 1625th item
16.25th item lies in the 30th cumulative frequency and the corresponding class interval is 19.5 – 24.5

(c) Maximum income earned by lowest 25% workers will be given by the upper quartile Q₃.
Class interval of Q₃ = (N4)th item
= 3(654)th item
= 3 × 1625th item
= 48.75th item
48.75th item lines in 50th item and the corresponding class interval is 24.5-29.5

Q.9The following table gives production yield in kg per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode production yield.

ANSWER:
(i) Mean

(ii) Median

(iii) Mode
Grouping Table

Analysis Table

(i) Mean

(ii) Median

(iii) Mode
Grouping Table

Analysis Table

In This Post we are  providing Chapter-8 BINOMINAL THEOREM NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON BINOMINAL THEOREM

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1. Find Hence evaluate 

Ans.

Put 

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2. Show that is divisible by 64, whenever n is positive integer.

Ans. 

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3. Find the general term in the expansion of  

Ans. 

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4. In the expansion of prove that coefficients of and are equal.

Ans. 

Put  and  respectively

Coeff of is 

Coeff of is H.P

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5. Expand 

Ans. 

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6. Find the sixth term of the expansion if the binomial coefficient of the third term from the end is 45.

Ans. The binomial coeff of the third term from end = binomial coeff of the third term from beginning = 

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7. Find a if the 17th and 18th terms of the expansion are equal.

Ans. 

ATQ put  and 17

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8. Find the term independent of in the expansion of 

Ans. 

Put 

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9. If the coeff of and terms in the expansion of are equal find 

Ans.

Coeff are

and 

ATQ 

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10. Show that the coeff of the middle term in the expansion of is equal to the sum of the coeff of two middle terms in the expansion of 

Ans. As is even so the expansion has only one middle term which is  term

Coeff of is 

Similarly being odd the other expansion has two middle term i.e

and term

 i.e and 

The coeff are  and 

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11. Find the value of if the coeff of and terms in the expansion of  are equal.

Ans. 

Put 

And 

ATQ 

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12. Find the 13th term in the expansion of 

Ans. 

Put 

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13.Find , if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of is 

Ans.Fifth term from the beginning in the expansion of is

How fifth term from the end would be equal to in term from the beginning

ATQ 

14. The sum of the coeff. 0f the first three terms in the expansion of being natural no. is 559. Find the term of expansion containing 

Ans.The coeff. Of the first three terms of are and 

Therefore, by the given condition

On solving we get 

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15.Show that the middle term in the expansion of is 

Ans.As is even, the middle term of the expansion term

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In This Post we are  providing Chapter-7 PERMUTATIONS AND COMBINATIONS NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON PERMUTATIONS AND COMBINATIONS

1.In how many ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colours.

Ans.No. of ways of selecting 9 balls

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2.Find if 

Ans.

 Rejected. Because if we put the no. in the factorial is –ve.

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3.If find 

Ans.

4.What is 

Ans.Is multiplication of consecutive natural number

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5.If what is the value of 

Ans. then 

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6.How many words, with or with not meaning each of 2 vowels and 3 consonants can be flamed from the letter of the word DAUGHTER?

Ans.In the word DAUGHTER There are 3 vowels and 5 consonants out of 3 vowels, 2 vowels can be selected in ways and 3 consonants can be selected in ways 5 letters 2 vowel and 3 consonant can be arranged in 5! Ways

Total no. of words 

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7.Convert the following products into factorials 

Ans.

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8.Evaluate 

Ans.

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9.Evaluate 

Ans.

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10.Find if 

Ans.Given 

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11.Evaluate 

Ans.

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12.Convert into factorial 2.4.6.8.10.12

Ans. 

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13.How many words with or without meaning can be formed using all the letters of the word ‘EQUATION’ at a time so that vowels and consonants occur together

Ans. In the word ‘EQUATION’ there are 5 vowels [A.E.I.O.U.] and 3 consonants [Q.T.N]

Total no. of letters = 8

Arrangement of 5 vowels = 

Arrangements of 3 consonants = 

Arrangements of vowels and consonants = 

Total number of words 

14. How many words, with or without meaning can be made from the letters of the word MONDAY. Assuming that no. letter is repeated, it

(i) 4 letters are used at a time

(ii) All letters are used but first letter is a vowel?

Ans. Part-I In the word MONDAY there are 6 letters

4 letters are used at a time

Total number of words 

Part-II  All letters are used at a time but first letter is a vowel then OAMNDY

2 vowels can be arranged in 2! Ways

4 consonants can be arranged in 4! Ways

Total number of words   

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15.Prove that 

Ans.Proof L.H.S.

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In This Post we are  providing Chapter-6 LINEAR INEQUALITIES NCERT MOST IMPORTANT QUESTIONS for Class 11 MATHS which will be beneficial for students. These solutions are updated according to 2021-22 syllabus. These MCQS  can be really helpful in the preparation of Board exams and will provide you with a brief knowledge of the chapter.

NCERT MOST IMPORTANT QUESTIONS ON LINEAR INEQUALITIES

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1. Solve the inequality 

Ans. 

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2. Solve the inequality 

Ans. 

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3. Solution set of the in inequations and is.

Ans. 

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4. Solve. Show the graph of the solution on number line.

Ans.

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5. Solve the inequality. 

Ans. 

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6. Solve when is an integer.

Ans. 

7. Ravi obtained 70 and 75 mark in first unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks.

Ans. Let Ravi secure marks in third test

ATQ 

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8. Find all pairs of consecutive odd natural no. both of which are larger than 10 such that their sum is less than 40.

Ans. Let and be consecutive odd natural no.

ATQ

From (i) and (ii)

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9. A company manufactures cassettes and its cost equation for a week is C=300+1.5and its revenue equation is R=2, where is the no. of cassettes sold in a week. How many cassettes must be sold by the company to get some profit?

Ans. Profit = revenue-cost

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10. The longest side of a is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the is at least 61 cm find the minimum length of the shortest side.

Ans. Let shortest side be cm then the longest side is cm and the third side cm.

ATQ 

Length of shortest side is 9 cm.

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11. In drilling world’s deepest hole it was found that the temperature T in degree Celsius, km below the surface of earth was given by At what depth will the tempt. Be between c and 

Ans. Let km is the depth where the tempt lies between and 

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12. A man wants to cut three lengths from a single piece of board of length 91 cm. The second length is to be 3 cm longer than the shortest and the third length is to be twice as long as the shortest. What are the possible lengths of the shortest board if the third piece is to be at least 5 cm longer than the second.

Ans. Let the shortest length be cm, then second length is (+3) cm and the third length is 2 cm.

ATQ 

Again ATQ

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13. The water acidity in a pool is considered normal when the average pH reading of three daily measurements is between 7.2 and 7.8 If the first pH reading are 7.48 and7.85, find the range of pH value for the third reading that will result in the acidity level being normal.

Ans. Let third reading be  then

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14. Solve graphically 

Ans.  

Put (1,0) in eq. (i)

false

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15. Solve graphically

Ans. 

	0	2
	3	0

Put in eq. ……….

 which is false

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