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IILM Institute for Higher Education
Worksheets: Importance of Mathematics
Academic Year: PGP/2011-12
SESSION 1: AVERAGES
OVERVIEW
Mean (Arithmetic Mean)
The most common measure of central tendency
Affected by extreme values (outliers)
Direct method
For Individual Observation:
Mean = ∑ X / n
For Grouped Data
Mean = ∑ f X / n
Sample Size
1 1 2
N
ii N
XX X X
N N
Population Size
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
Short Cut Method
For Individual Observation:
Mean = A + ∑ d / n where d is deviation from assumed mean A .
For Grouped Data
Mean = A + ∑f d / n
Step Deviation Method
Mean = A + ( ∑ f d / n ) * I where I is length of class Interval.
Median
Robust measure of central tendency
Not affected by extreme values
In an ordered array, the median is the “middle” number
If n or N is odd, the median is the middle number
If n or N is even, the median is the average of the two middle numbers
Individual Observation
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Median = 5 Median = 5
Position of Median = (N + 1 ) / 2 th item
Grouped Data
Position of median = N / 2 th item
Median = L1 + ( N /2 – C.f -1 / F med ) * I
Mode
measure of central tendency
Value that occurs most often
Not affected by extreme values
Used for either numerical or categorical data
There may may be no mode
There may be several modes
Mode = L 1 +( f m – f m-1 / 2 f m – f m-1 + f m+ 1 ) * I
WORKSHEET -1
1. The average income of a person on working for the first five days of the week is Rs. 35 per day
and if he works for the first six days of the week, his average income per day is Rs. 40. Then,
his income for the sixth day is ____________.
2. The median of the series 3, 18, 7, 20, 11, 12, 9, 17, 22 is ____________.
3. The mode of the distribution of values 5,7,9,9,8,5,6,8,7,7,5,7,9,2,7 is ____________.
4. The mean of a set of 10 observations is 4. Another set of 20 observations is added to it which
makes the mean of the combined set equal to 6. The mean of second set is ____________.
5. The following table gives the monthly income of 10 employees in an office:
Income (Rs) 1780 1760 1690 1750 1840 1920
1100 1810 1050 1950
Calculate the arithmetic mean of incomes by direct method and short-cut method.
6. From the following data of the marks obtained by 60 students of a class calculate the arithmetic
mean by direct method and short-cut method:
Marks No. of Students
20 08
30 12
40 20
50 10
60 06
70 04
7. From the following data compute arithmetic mean by direct method and short-cut method:
Marks No. of Students
0-10 05
10-20 10
20-30 25
30-40 30
40-50 20
50.60 10
8. From the following data find the value of median:
Income (in Rs.)
1000 1500 800 2000 2500 1800
Number of persons
24 26 16 20 6 30
9. Calculate the value of mode for the following data:
Marks:
10 15 20 25 30 35 40
Numbers:
8 12 36 35 28 18 9
10. Calculate mode from the following data:
Class Frequency
0-6 12
6-12 24
12-18 36
18-24 38
24-30 37
30-36 06
SESSION 2: MEASURES OF DISPERSIONOVERVIEW
The various measures of central tendencies (averages) alone cannot yield or describe the data, unless all
the observations are same .It is necessary to describe the variability in the data. So Averages need to be
supported by other tools called as Measures of Dispersion.
Dispersion is used to indicate the facts that within a given group , the items differ from one another
in size, or in other words there is a lack of uniformity in their sizes.
The measures of Dispersion (variation) are classified as:
Absolute Measure of Dispersion-are also called an expressed in the same statistical in which
the original data is given as rupees, kg, tones etc.
Relative Measures of Dispersion-are pure numbers which are independent of any statistical
unit and can be used to compare the variability in the two data expressed in different statistical
unit. They are also called as coefficient of variation.
Various Measures Of Dispersion
1. RANGE: It is the difference between the largest and smallest value in the distribution.
R=L-S (Absolute Measure)
Coeff. Of Range = (L-S)/(L+S) (Relative measure)
2. Quartile Deviation: Average amount by which two quartiles differ from the median.
Q.D= (Q3-Q1)/2 (Absolute Measure)
Coeff.of Q.D= (Q3-Q1)/ (Q3+Q1) (Relative Measure)
3. Mean Deviation: Average amount of scatter in data from the average , ignoring the sign of
deviations
M.D = ∑׀x-mean׀/ N (absolute measure)
Coeff.of M.D= M.D./Mean (Relative measure)
3. Standard Deviation: IT is the most popular measure of Dispersion defined as the square root of the
arithmetic mean of the squares of the deviations of the observations from their arithmetic mean. It
is denoted by sigma.
S.D=√ ∑(x-mean)2 /N (absolute measure)
Coeff of S.D= (σ/mean)*100 (Relative measure or coeff .of variation)
Variance= square of S.D.
WORKSHEET-2
1. a) The Jaeger family drove through 6 mid western states on their summer vacation. Gasoline prices
varied from state to state. What is the range of gasoline prices?
$1.79, $1.61, $1.96, $2.09, $1.84, $1.75
b) Ms. Kaiser listed the price of share company from Monday to Saturday. Calculate Range and Coeff.of
Range.
Days : Mon Tues Wed Thurs Fri Sat
Rs : 55 54 52 53 56 58
2. The following table shows the monthly income of 10 families in a town:
Family : 1 2 3 4 5 6 7 8 9 10
Monthly
Income (Rs.) : 7800 7600 6900 7500 8400 9200 11000 8100 10,500 9500
Calculate the Range and coeff. of range.
3.Calculate the Range, Quartile deviation and coeff. Of Quartile deviation from the following data:
C.I. : 8-12 12-16 16-20 20-24 24-28
Freq: 5 12 20 10 3
4.Calculate the mean deviation from the mean for the following data:
Marks: 0-10 10-20 20-30 30-40 40-50 50-60 60-70
No. of students: 6 5 8 15 7 6 3
5. A purchasing agent obtained samples of lamps from two suppliers. He had the samples tested in his
own laboratory for the length of life with the following results:
Length of life(in hrs) Samples from
Company A
Samples from
Company B
700-900
900-1100
1100-1300
10 3
16 42
26 12
1300-1500 8 3
i) Which Company’s lamps have greater average life?
Ii) Which Company’s lamps are more uniform?
6. “After settlement the average weekly wages in a factory increased fromRs.8000 to Rs.12000 and
standard deviation had increased from Rs.100 to Rs.150.After settlement the wage has become higher
and more uniform.” Do you agree?
7. For two firms A and B, the following data are available:
A B
Number of Employees 100 200
Average Salary 1600 1800
Standard deviation of salary 16 18
Compute the following:
i) Which firms pays larger package of salary?
ii) Which firm shows greater variability in the distribution of salary?
8. Verify the correctness of the following statement:
“A Batsmen scored at an average of 60 runs an innings against Pakistan. The standard
Deviation of the runs scored by him was 12.A year later against Australia , his average came down to
50 runs an innings and the standard deviation of the runs scored fell down to 9.Therefore ,it is correct
to say that his performance was worse against Australia and that there was lesser consistency in his
batting against Australia.”
9.The at tendance at 4 Cinema hal ls on a given day was 200,500,300 and 1000
people .
a) Calculate the dispersion of the number of a t tendees
b) Calculate the coefficient of variation
c) . If there were 50 at tendees more in each room on the same day, what
effect would i t have on the dispersion?
10. A marathon race was completed by 5 participants. What is the range of times given in
hours below?
2.7 hr, 8.3 hr, 3.5 hr, 5.1 hr, 4.9 hr
SESSION 3: DIAGRAMATIC AND GRAPHICAL REPRESENTATION OF DATA
OVERVIEW
◦ Sort raw data in ascending order:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
◦ Find range: 58 - 12 = 46
◦ Select number of classes: 5 (usually between 5 and 15)
◦ Compute class interval (width): 10 (46/5 then round up)
◦ Determine class boundaries (limits): 10, 20, 30, 40, 50, 60
◦ Compute class midpoints: 15, 25, 35, 45, 55
◦ Count observations & assign to classes
Frequency Distributions, Relative Frequency Distributions and Percentage Distributions
Data in ordered array:12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class Frequency Relative Frequency Percentage
10 but under 20 3 0.15 15
20 but under 30 6 0.3 30
30 but under 40 5 0.25 25
40 but under 50 4 0.2 20
50 but under 60 2 0.1 10
total 20 1 100
Graphing Numerical Data:
Histogram
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
The Frequency PolygonData in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Tabulating Numerical Data:
Cumulative Frequency
Data in ordered array:
12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58
Class Frequency Cumulative Frequency
10 but under 20 3 3
20 but under 30 6 3 + 6 = 9
30 but under 40 5 9 + 5 = 14
40 but under 50 4 14 + 4 = 18
50 but under 60 2 18 + 2 = 20
total 20 20
Graphing Numerical Data: The Ogive (Cumulative % Polygon)
Tabulating and Graphing Categorical Data:Univariate Data
Investment Category Amount Percentage
(in thousands $)
Stocks 46.5 42.27
Bonds 32 29.09
CD 15.5 14.09
Savings 16 14.55
Total 110 100
Bar Chart(for an Investor’s Portfolio)
Pie Chart (for an Investor’s Portfolio)
Tabulating and Graphing Bivariate Categorical Data
Investment Investor A Investor B Investor C Total Category
Stocks 46.5 55 27.5 129
Bonds 32 44 19 95
CD 15.5 20 13.5 49
Savings 16 28 7 51
Total 110 147 67 324
WORKSHEET 3
Describe briefly the construction of histogram and frequency polygon of a frequency distribution and state their uses:
Prepared a Histogram and a Frequency Polygon from the following data:
Class : 0–6 6–12 12–18 18–24 24–30 30–36f : 4 8 15 20 12 6
Marks obtained by 50 students in a History paper of full marks 100 are as follows:78 25 25 40 30 29 35 42 43 4344 20 48 44 43 48 36 46 48 4736 60 31 47 33 65 68 73 39 1260 20 47 49 51 38 49 35 52 6134 76 79 20 16 70 65 39 60 45
Arrange the data in a frequency distribution table in class intervals of length 5 units and draw a histogram to present the above data.
(a) Represent the following data by Histogram:Weight (Kg.) 35–40 40–45 45–50 50–55 55–60 60–65 No. of Persons 12 30 22 30 18 10
(b) Represent the following frequency distribution by means of a Histogram and superimpose thereon the corresponding frequency polygon and frequency curve:
Salary (’00 Rs.) No. of Employees Salary (’00 Rs.) No. of Employees300–400 20 700–800 115400–500 30 800–900 100500–600 60 900–1000 60600–700 75 1000–1200 40
(cHow many families can be expected to have monthly income between 3500 and 4250 rupees. 4250 – 3500
Hint: ------------------ x 309 = 115.88 = 116. 2000
The following table gives the scholastic aptitude scores of the 50 departmental students of a certain department in a certain university:
345 530 556 354 590 395 515 479 494 420563 444 629 440 485 505 604 490 445 605402 406 730 506 516 472 475 610 586 523691 520 465 468 545 624 582 570 578 505523 575 420 605 527 461 440 585 420 384
Construct a frequency distribution table with appropriate class limits and class boundaries. (Take the length of the class equal to 30 units).
Draw histogram to represent the above frequency distribution.
(a) Draw the ‘less than’ and ‘more than’ ogive curves from the data given below:Weekly wages (‘00 Rs.) 0–20 20–40 40–60 60–80 80–100No. of wages 10 20 40 20 10
(b) Below is given the frequency distribution of marks in Mathematics obtained by 100 students in a class:
Marks 20–29 30–39 40–49 50–59No. of Students 7 11 24 32Marks 60–69 70–79 80–89 90–99No. of Students 9 14 2 1
Draw the ogive (less than or more than type) for this distribution and use it to determine the median.
Age distribution of 200 employees of a firm is given below. Construct a less than ogive curve and hence of otherwise calculate semi-inter-quartile range (Q3–Q1)/2 of the distribution.
Age in years (less than): 25 30 35 40 45 50 55No. of employees : 10 25 75 130 170 189 200
The following table gives the distribution of the wages of 65 employees in a factory:Wages in Rs.(Equal to or more than)
50 60 70 80 90 100 110 120
Number of Employees 65 57 47 31 17 7 2 0Draw a ‘less than’ ogive curve from the above data, and estimate the number of employees earning at least Rs. 63 but less than Rs. 75.
Construct a frequency table for the following data regarding annual profits, in thousands of rupees in 50 firms, taking 25–34, 35–44, etc., as class intervals.
28 35 61 29 36 48 57 67 69 5048 40 47 42 41 37 51 62 63 3331 32 35 40 38 37 60 51 54 5637 46 42 38 61 59 58 44 39 5738 44 45 45 47 38 44 47 47 64
Construct a less than ogive and find:Number of firms having profit between Rs. 37,000 and Rs. 58,000.Profit above which 10% of the firms will have their profits.Middle 50% profit group.
Represent the following data by means of a time series graph.Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999Export (Rs. ‘000) 267 269 263 275 270 280 282 272 265 266Import (Rs. ‘000) 307 310 280 260 275 271 280 280 260 265
Present the following hypothetical data graphically.AREA AND PRODUCTION OF RICE IN INDIA
Year 1987 1988 1989 1990 1991 1992Area (Million Acres) 174.1 177.3 176.1 177.9 179.3 179.1Production (Million Tonnes) 72.5 77.8 74.8 77.2 78.0 74.8
Present the following data about India by a suitable graph:PRODUCTION IN MILLION TONS
Months Highest Price
Lowest Price Months Highest Price
Lowest Price
(Rs.) (Rs.) (Rs.) (Rs.)January 160.0 152.0 July 175.0 163.2February 162.2 156.0 August 175.8 160.0March 165.0 160.3 September 172.2 165.0April 166.5 162.4 October 178.0 168.0May 168.2 160.5 November 171.0 165.0June 170.0 161.9 December 175.5 167.0
(a) Draw a bar chart to represent the following information:Year 1952 1957 1962 1967 1972 1977No. of women M.P.’s 22 27 34 31 22 19
(b) Represent the following data with the help of a bar diagram:Year 1970-
711971-
721972-
731973-
741974-
751975-
761976-77
Notes in Circulation (Rs. Crores)
4,221 4,655 5,272 6,159 6,231 6,572 7,778
(c) In a recent study on causes of strikes in mills, an experimenter collected the following data.
Causes Economic Personal Political Rivalry OthersOccurrences(in Percentage)
58 16 10 6 10
Represent the data by bar chart.
(d) Below are data on the number of films made in different regional and / or other languages in India in different years.
Year 1947 1951 1961 1970 1971 1972 1973No. of films 281 229 303 396 433 414 448
Draw a bar chart to represent the above data.
(a) Represent the following data by a suitable diagram:Item of Expenditure Family A Family BFood 200 250Clothing 100 200House Rent 80 100Fuel and Light 30 40Education 90 210
Total 500 800
(b) Represent the following data by a percentage sub-divided bar-diagram: (cover if possible)Item of Expenditure Family A
(Income Rs. 500)Family B
(Income Rs. 500)Food 150 150Clothing 125 60Education 25 50Miscellaneous 190 70Saving or Deficit +10 -30
(c) Draw a suitable diagram to represent the following data on livelihood patterns in India, U.S.A. and U.K.
Occupation India U.S.A. U.K.Agriculture and ForestryManufacture and CommerceOther Industries and Services
71%15%14%
13%46%41%
5%55%40%
Total 100% 100% 100%
(d) Represent the following data on production of Tea, Cocoa and Coffee by means of a pie diagram.
Tea Cocoa Coffee Total3,260 tons 1,850 tons 900 tons 6,010 tons
(a) Point out the usefulness of diagrammatic representation of facts and explain the construction of volume and pie diagrams.
(b) A Rupee spent on ‘Khadi’ is distributed as follows:Paise
FarmerCarder and Spinner
1935
WeaverWasher-man, Dyer and PrinterAdministrative Agency
288
10Total 100
Present the data in the form of a pie diagram.
(c) Draw a pie diagram for the following data of Sixth Five-Year Plan Public Sector Outlays:Agriculture and Rural DevelopmentIrrigation, etc.EnergyIndustry and MineralsTransport, Communication, etc.Social Services and Others
12.9%12.5%27.2%15.4%15.9%16.1%
(a Raw MaterialsTaxesManufacturing expensesEmployeesOther ExpensesDepreciationDividendsRetained Income
1,689582543470286947551
Total 3,790
SESSION 4 :RATIO AND PROPORTION
OVERVIEW
Ratio
If A and B are two similar quantities , then A/B or A : B is called their ratio. A and B are called as the
terms of the ratio. A is called the antecedent and B is called the consequent. The ratio remains
unchanged by multiplying or dividing the antecedent and consequent by the same number.
Example: If A 's Income is Rs 1000 per month and B's income is Rs 1500 per month , then the ratio
between the incomes of A and B is
1000 / 1500 = 2/ 3
Proportion
A proportion is an equality of two ratios I.e a :b = c : d is a proportion and term a , b , c and d are said
to be in proportion and proportionally expressed as a: b : : c : d a and d are called extremes and b and
c are called as means.
WORKSHEET - 4
1. .The ratio of prices of two cows was 23 : 16 . Two years later when the price of the first had
risen by Rs 477 and that of second by 10% , the ratio of their prices became 20 : 11. Find the
original prices.
2. In mixing tea , 1 kg in every 100 kg is wasted . In what proportion must a dealer mix teas
which cost him Rs 24 and Rs 18 per kg respectively so that cost comes to 20 per kg.
3. A consists of three substances whose volumes are in the ratio of 4 : 5 : 7 . The weights of
equal volume of substances are in the ratio of 3 : 6 : 7 . Find the ratio of the weights of the
substances in the mixture.
4. Divide Rs 6270 among A , B , C so that A receives 3/7 as much as B and C together receive
and B receive 2 / 9 of what A and C together receive.
5. 18 liters are withdrawn from vessel full of wine . It is then filled with water . Then 18 liters of
mixture are drawn and quantity of wine to that of water in it is 16 : 9 , how much does the
vessel hold?
6. Monthly incomes of two persons are in the ratio of 5 : 7 and their monthly expenditure are in
the ratio of 7 : 11 . If each saves Rs 150 a month find their monthly incomes.
7. Two casks A and B are filled with two kind of liquids , mixed in cask A in the ratio of 2 : 7
and in the cask B in the ratio of 1 : 5 . What quantity must be taken from each cask to form a
mixture which shall consist of 2 liters of one kind and 9 liters of other?
8. A cubic centimeter of two metals A and B weighs 0.57 gm and 0.82 gm respectively . An
alloy of the two metal is to be made , in which ratio of A to B is to be in stated proportion. B
y mistake the proportion reversed and weight of the cubic centimeter of alloy thus made fall
short of the required weight by 0.05 gm . Find the actual proportion of two metals in the alloy
both by volume and weight.
9. The proportion of milk and water in three samples is 2 : 1 , 3 : 2 , 5 : 3. A mixture comprising
of equal quantities of all 3 sample is made. The proportion of milk and water in the mixture
is?
10. A ,B and C enter into a partnership by investing Rs 3600 , Rs 4400 and Rs 2800. A is working
partner and gets fourth of the profit for his services and remaining profit is divided amongst
the three in rate of their investments. What is the amount of profit B gets if A gets a total of
Rs 8000.
SESSION 5: MATRIX
OVERVIEW
A matrix is an array of numbers arranged in certain numbers of rows and columns .
If there are m x n numbers aij ( i = 1 to m , j= 1to n), we can write a matrix with m rows and n columns
A1 A2 .... A3
a11 a12 a1n I
B= a21 a22 a2n II
am1 am2 a3n III
1. A matrix having m rows and n columns is called matrix of order m x n. The individual enetries
of the array aij are termed as elements of matrix A.
2. Matrix can b indicated by enclosing an array of numbers by parentheses [ ] or ( )
3. Matrices are usually denoted by capital letters A, B,C while small letters a,b ,c ...etc used to
denote elements of matrix.
Representation of data in matrix form
Matrices can be used to present a given st of data in compact form. For e.g the following matrix gives
transportation cost per unit for each of the three warehouses to each of the four distribution points.
Distribution point
I II IIII IV
I 15 20 25 19
Warehouse II 14 12 32 10
III 12 10 20 21
Types of Matrices
I) Rectangular matrix
A matrix consisting of m rows and n columns where m = n is called as rectangular matrix.
For e.g A = a11 a12 a13
a21 a22 a23
is a recangular matrix of order 2 x 3
II) Square matrix
If the number of rows of a matrix is equal to a column of a matrix , the matrix is said to be a square
matrix. For eg
A = a11 a12 a13
a21 a22 a23
a31 a32 a33
A is a square matrix as number of rows (3) = number of columns (3)
III) Row matrix
A matrix having only one row is called a row matrix or ( row vector).
For eg [ 4 1 2] is a 1 x 3 matrix ( one row three columns)
IV)Column matrix
A matrix having only one column is called as a column matrix or ( column vector)
4
1
2 is a 3 x 1 matrix ( three rows one column)
V) Identity Matrix
A square matrix with each of its diagonal elements equal to unity (1) and no – diagonal elements euql
to zero is called as identity matrix .
The matrix I = 1 0 0
0 1 0 Diagonal elements
0 0 1 is a 3 x 3 identity matrix
VI)Null Matrix
A matrix having all its elements equal to zero is called as a null matrix .
Transpose of a Matrix
The transpose of a matrix is a matrix denoted by A' is obtained by interchange of its rows and columns.
Symbolically if A = [ aij ]m xn then A' = [ aji ] n x m
If A = 1 2 9 then A ' = 1 2
2 4 8 2 4
9 8 ( Interchange of rows and columns)
WORKSHEET - 5
1.Three firms A,B & C supplied 40,35 & 25 truck load of stones and 10,5,8 truck loads of sand
respectively to a contractor. If the cost of stone and sand are Rs 1200 and Rs 500 per truck load
respectively, find the total amount paid by the contractor to each of those firms, by using matrix
method.
2.The annual sale volumes of three products X,Y,Z whose sale prices per unit are Rs 3.50,Rs 2.75,Rs
1.50 respectively, in two different markets 1 and 2 are shown below
Market Product
X Y Z
1 6000 9000 13000
2 12000 6000 17000
3. A1 A2 A3
2 4 6 I
A= 8 10 12 II
14 16 18 III
A1 A2 A3
4 6 8 I
B= 10 12 14 II
16 18 20 III
A1 A2 A3
2 10 8 I
C= 8 12 16 II
10 24 38 III
Matrix A shows the stock of 3 types of items I,II,III in three shops A1,A2,A3. Matrix B shows the
number of items delivered to the three shops at the beginning of a week. Matrix C shows the number of
items sold during that week. Using matrix algebra, find
The number of items imediately after the delivery.
The number of items at the end of the week.
4.the cost vector for three materials cement, wood and steel is given by [700 100 1000] and the
amount of materials needed to construct a house is given by the vector [100 50 20].Using
Proper vector notations,find the vector representing the total cost of material.
5.find the adjoint of the matrix
1 4 0
A = -1 2 2
0 0 2
6.Find Adj.A and verify the Theorem A(adj.A)=(Adj.A)A=IAI I3 for the matrix
1 2 3
A= 1 3 4
1 4 3
7. A company is manufacturing two types of auto cycles for gents and ledies separately,which are
assembled and finished in two workshops W1 and W2.Each type takes 15 hours and 10 hours for
assembly and 5 hrs and 2 hrs for finishing in the respective shops.If total no of hours available are 400
and 120 in work shops W1 and W2 respectively, calculate the number of units of autocycles produced
using matrix method.
8. A manufacturer is manufacturing two types of products A and B.L1 and L2 are two machines which
are used for manufacturing these two types of products.The time taken both by A and B on machines is
given below
Machine L1 Machine L2
Product A 20 hrs 10 Hrs
Product B 10 hrs 20 hrs
If 600 hrs is the time available on each machine, calculate the number of units of each type
manufactured using matrix method.
9.The prices of three commodities P,Q and R are Rs.X,Y and Z per unit respectively. A purchases 4
units of R and sells 3 units of P and 5 units of Q;B purchases 3 units of Q and sells 2 units if P and 1
unit of R;C purchases I unit of P and sells 4 units of Q and 6 units of R.In the process A,B and C earn
Rs.6000,Rs.5000,Rs.13000 respectively. find the prices per unit of P,Q and R.
10.The daily cost of operating a hospital is C,a linear finction of the number of in-patients I,and out-
patients P,plus a cost a,i.e,C=a+bP+dI.
Given the following data from three days,find the values of a,b and d by setting up a linear system of
equations and using the matrix inverse.
Days Cost in Rs No of In-Patients I No of Out-patients P
1 6956 40 10
2 6725 35 9
3 7100 40 12
SESSION 6: PERMUTATION AND COMBINATION
OVERVIEW
Permutation: means arrangement of things. The word arrangement is used, if order of things is considered.
Let n be the positive integer and r be the positive integer less than equal to n . The number of different arrangements of r things taken out of n dissimilar things is denoted by nP r . Each such arrangement is called as permutation of n things taken r at a time.
For e.g. all the arrangements of two letters chosen out of [ a, b , c ] are given by ab , ba , ac , ca , bc , cb
Thus given by 3P2 = 6
Combination: means selection of things. The word selection is used, when the order of things has no importance. Thus if a, b, c are given set of objects and two objections are to be chosen. The different combinations are given by {a, b}, {b, c} and {c, a}. In combination, the order in which the elements are selected does not matter.
Combination is denoted by C. For e.g. 10C5
Concept of Factorial
In order to solve permutation and combination problems we have to use concept of factorial. The product of all consecutive integers starting from 1 to t is denoted by t! and read t – factorial.
Thus 4! = 4 x 3 x 2 x 1
For e.g find value of 6P4 = 6! / 6 – 4! = 6! / 2! = 6 x 5 x 4 x 3 x 2 x1 / 2 x 1 = 360
WORKSHEET- 6
1. How many different words can we make using the letters A, B, E and L ?
2. How many 2 digit numbers can you make using the digits 1, 2, 3 and 4 without repeating the
digits?
3. How many 3 letter words can we make with the letters in the word LOVE?
4. How many lines can you draw using 3 non collinear (not in a single line) points A, B and C on a
plane?
5. We need to form a 5 a side team in a class of 12 students. How many different teams can be
formed?
6. A committee including 3 boys and 4 girls is to be formed from a group of 10 boys and 12 girls. How many different committee can be formed from the group?
7. In a certain country, the car number plate is formed by 4 digits from the digits 1, 2, 3, 4, 5, 6, 7,
8 and 9 followed by 3 letters from the alphabet. How many number plates can be formed?
8. If a university student has to choose 2 science classes from 5 available science classes and 3
other classes from a total of 7 other classes available, how many different groups of classes
there?
9. In how many ways 12 persons may be divided into two groups of 6 persons each?
10. Find the number of arrangement that can be made out of letters of the word ASSASINATION?
SESSION 7 :LINEAR EQUATIONS
OVERVIEW
Equation.
An equation is a statement of equality between two expressions which is not true for all values
of the variables involved in it.
Identity:
An identity is a statement of equality between two expressions which is true for all values of the
variables involved.
Roots or solution of an equation:
The root is a value of the variable for which the equation is satisfied or it’s both sides are equal
to each other.
Degree of an equation:
The highest power of unknown variable with a non zero coefficient is called the degree of the
equation eg. In 2x2 +3x = 0 the highest power of x is 2 so the degree of equation is 2.
Linear equations:
The linear equations are simplest of all the equations and their standard form is ax +b =
0.expression of the form ax = b or ax +b = 0 are linear equations in variable x or equations of
first degree in x.
Simultaneous linear equations:
When an equation is having more than one variable it’s called simultaneous linear equation in
provided number of variables like equations ax +by +c and ax +by +cz are linear equations in
two and three variables respectively provided (a≠0,b≠0) for two variable case and
(a≠0,b≠0,c≠0 ) for three variable case. As the number of variables increases in an equation the
no. of equations will also increase in to the system .The linear simultaneous equations can be
solved by various methods like substitution ,elimination, cross multiplication etc.
Quadratic equation.
Quadratic equation: An equation of the form ax2+bx+c =0 is called a quadratic equation.A
quadratic equation can also be solved by various methods loke factirzattion, formula(shree
dharacharya) etc.
WORKSHEET - 7
Q.1 Solve the problemx/6 –x/5 =x/15-x/3 +7
Q.2 x+4-[2+{x-(2+x)}] = 1/2
Q.3 2x +3y =8 ; 3x + y =5
Q.4 x-2y =3 ; 7y-2x =3
Q.5 3x-7y =20 4x-2y=3
Q.5 y = 3(x+1) ; 4x = y+1
Q 6 x/6 +y/15 =4 ; x/3 –y/12 =19/4
Q.7. 3/x +2/y =13 ; 5/x -3/y = 9
Q.8 2/x +y = -3 ; 1/2x -2y/3 = 1/6.
Q.9 (3x/5 +x/2 ) = (5x/4 -3).
Q.10 (3/(x-6)) + (7/(x-2)) = (10/(x-4)).
Q.11 Monthly income of two persons is in the ratio of 4:5 and their monthly expenditure are in the ratio of 7:9.If each saves Rs. 50 per month find the monthly income of both the persons.
Q.12 For a certain commodity the demand is (D) in kg for a price p in Rs. Is given by D= 100(10-p).The supply equation giving the supply (S) in kg. for a price (p) in Rs. Is S = 75(p-3).the market is such that the demand equals supply.find the market price and the quantity that will be bought and sold.
Q.13 Let the speed of a boat in still water be 10 km per hour.If it can travel 24 km. down stream and 14 km upstream,indicate the speed of the flow of stream.
SESSION 8 : DERIVATIVES
OVERVIEW
In calculus , a branch of mathematics , the derivative is a measure of how a function changes as its
input changes. The process of finding a derivative is called as differentiation. Differentiation is a
method to compute the rate at which a dependent output y changes with respect to change in input x.
The functional relationship is denoted by y = f(x)
where,
y – dependent variable ( The variable which is predicted on the basis of another variable is
called as dependent variable)
x - independent variable ( The variable which is used to predict another variable is called
independent variable)
For eg. When sales are predicted on the basis of advertising expenditure , sales is dependent variable
(y) and advertising expenditure is independent variable (x)
Differentiation helps us to find out the average rate of change in the dependent variable with respect to
change in the independent variable. For e.g two variables are sales (y) and advertising expenditure (x)
such that y is a function of x . Therefore differential coefficient dy / dx represents rate of change in y
with respect to x.
Basic formula of differentiation
Function f(x) Derivative f ' (x)
xn nxn-1 n , is a real number
c, a constant 0
For e.g f(x) = x2 + 4x + 6
f '(x) = 2x2-1 + 4 x 1-1 + 0 ( as 6 is a constant)
= 2x + 4 ( as x0 = 1)
Applications of Derivatives in Business
Cost Function
Total cost has two parts Variable cost and fixed cost. If C (x) denotes cost of producing x units of a
product then C(x) = V(x) + F(x)
If C(x) = - x2 + 10 x + 40
Then F(x) = 40
V(x) = - x2 + 10 x
Marginal cost = If C (x) is the total cost of producing x units then increase in cost in producing one
more unit is called marginal cost at an output level of x units and is given as dC/ dx.
For the above example dC/dx = -2x + 10.
Revenue Function
Revenue R(x), gives the total money obtained by selling x units of a product . If x units are sold at Rs p
per unit , then R(x) = p.x
Marginal Revenue : It is the rate of change in revenue per unit change in output. If R is the revenue
and X is the output then MR = dR/ dx
Profit Function
Profit P(x) is the difference between total revenue R (x) and total cost C(x)
P(x) = R(x) – C(x)
Marginal Profit : Marginal Profit is the rate of change in profit per unit change in output I.e dP/dx
Elasticity Function
Price elasticity of demand : If price of the commodity increases by 1 percent by what percentage
amount of demand has changed. This can be answered by using the concept of elasticity of demand.
The price elasticity of demand is defined as the rate of proportional change in quantity demanded x to
the change in price per unit p.
Ed = -dx.p / dp .x
WORKSHEET - 8
1. A company estimates that the total cost of producing x units of a certain product is given by
C(x) = 400 + 0.02x + 0.0001x2
Find I) Average cost ii) the marginal average cost.
2. If C(x) = 0.01 x + 5 + (500 /x) is a manufacturer's average cost equation , find the marginal
cost function. What is the marginal cost when 100 units are produced ? Interpret the result.
3. If Demand function is x = 20 / p + 1 where p is the price per unit for x units , find the marginal
revenue function.
4. When the price p = 25 , find the elasticity of demand if the demand function q = 100 - 2p.
5. If p = (100 / q + 2 ) - 2 represents the demand function for a product , where p is the price per
unit for q units .Find the marginal revenue.
6. A company has x items produced the total cost C and total revenue R given by the equations
a. C = 100 + 0.015x2 and R = 3x . Find the equation for profit ,marginal cost , marginal
revenue and marginal profit.
7. The total cost of a firm is given by 0.01x3 – 2x2 + 400 x, find
i. MC function
ii. AC function
iii. value of x
8. Find the elasticity of demand of the function x = 100 – 5p at p = 10
9. Find elasticity of a demand function p = -2x2 + 3x + 150 at x = 8
10. If the consumption function is given by C = 8 + 9I3/2 , where I is income of the consumer. Find
the marginal consumption function.
SESSION 9 : Maxima and Minima
OverviewSteps to calculate maxima and minima for Function f (x)
Step 1: Find f ' (x)
Step 2: Solve f ' (x) = 0 to get value of x , Let x = c be one of the values of x
Step 3: Find f '' (x) and then put x = c to get f '' (c)
Conditions If f'' ( c ) < 0 x = c is the point of maxima
If f '' ( c ) > 0 x = c is the point of minima
If f '' ( c) = 0 its neither a point of maxima or minima - Inflection point
Sample Solution
f(x) = x3 - 12 x
Step 1 : f ' (x) = 3x2 – 12
Step 2 : putting f ' (x ) = 0 3x2 – 12 = 0 3x2 = 12 x = + - 2
Step 3 : f '' (x) = Differentiating f ' (x) I.e 3x2 – 12 = 6x
f '' (+ 2) = 6 x 2 = 12 which is positivef '' ( - 2) = 6 x – 2 = -12 which is negative
f '' ( 2) > 0 , function f(x) is minimum at x = 2f '' ( -2 ) < 0 , function f(x) is maximum at x = -2
Maximum value = f ( - 2) = (-2)3 – 12( -2) = 16
Minimum value = f (2) = (2)3 – 12( 2 ) = - 16
WORKSHEET - 9
1.The total profit y in rupees of a company from manufacturing from the manufacturer and sale of x
bottles is given by y = -x2 / 400 + 2x – 80
a) How many bottles must the company sell to achieve maximum profit.
b) What is the profit per bottle when maximum profit is achieved.
2.A firm has revenue function given by R = 8 D where R is gross revenue and D is quantity sold .
The production cost is given by
C = 1 , 50 ,000 + 60 ( D / 900)2
Find the total profit function and the number of units to be sold to get maximum profit.
3.A television company charges Rs 6000 per unit for an order of 50 sets or less sets. The charge is
reduced by Rs 75 per set for each order in excess of 50 . Find the largest size order the company
should allow us as to receive maximum revenue.
4. A manufacturer can sell x items per day at a price p rupees each where p = 125 - 5x / 3. The cost
of production for x items is 500 + 13 x + 0.2 x2 . Find how much he should produce to have a
maximum profit , assuming that all items produced can be sold? What is maximum profit.
5.A manufacturer determines his total cost function is c = q3 / 3 + 2q + 300 , where q is the
number of units produced. At what level of output will average cost per unit be minimum.
6.The cost function of a good produced by a firm is given by the relation C = 65 + 0.025 q 2 and it
can sell goods at Rs 5 per unit . Find the maximum profit and the number of units purchased at
maximum profit.
7.A tour operator charges Rs 136 per passenger for 100 passengers with discount of Rs 4 for each
10 passengers in excess of Rs 100. Determine the number of passengers that will maximize the
amount received from each passenger.
8.Total cost of daily output of q tons of coal is Rs ( 1/10 q3 - 3q2 + 50q ) , what is the value of q
when average cost is minimum ? Verify that this level average cost = marginal cost .
9.The total cost c of output q is given by c = 300q – 10q2 - 1/3q3. . Find the output level at which the
marginal cost and the average cost receive their minima.
10.The market demand law of a firm is given by 4p + q -16 = 0. Find the output level when the
revenue is maximum. Also find the maximum revenue.
