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KENDRIYA VIDYALAYA SANGATHAN
ZONAL INSTITUTE OF EDUCATION AND TRAINING MYSURU
पाठ्यक्रम निदेशक / COURSE DIRECTOR
सुश्र ी मीनाक्षी जैन/Ms. MENAXI JAIN (
Deputy Commissioner & Director
KVS Zonal Institute of Education and Training, Mysore
ASSOCIATE COURSE DIRECTOR
Mr. AGIMON A CHELLAMCOTT
Principal, KV Idukki, Ernakulam Region
RESOURCE PERSONS
1. Mr. R S N ACHARYULU, PGT(Maths),
MALKAPURAM, HYDERABAD REGION
2. Mr. EVLN VAMSI KRISHNA, PGT(Maths),
TIRUPATI No.1, HYDERABAD REGION
Co-ordinated by: Mr. D SREENIVASULU,
Training Associate (Mathematics), ZIET MYSURU
i
DIRECTOR’S MESSAGE……
It is with profound delight and utmost pride that I announce the publication of our study
material of CLASS XII (APPLIED MATHEMATICS) for the session 2023-24. It’s my
firm belief that access to quality education should know no boundaries, transcending
social and economic constraints. Our collective vision is to empower all students with the
tools for success and intellectual growth.
With their steadfast dedication PGT-MATHEMATICS of Bangalore, Chennai,
Ernakulam & Hyderabad regions of Kendriya Vidyalaya Sangathan have invested their
knowledge, expertise, and passion into meticulously crafting these study materials to
complement the classroom learning experience of the students. These materials serve as
invaluable aids for self-study since they are comprehensive, well-structured, and
presented in a manner that is easy to comprehend.
It is with pleasure that I place on record my commendation for the commitment and
dedication of the team of teachers which included Mr. D. SREENIVASULU, Training
Associate (MATHEMATICS) from ZIET Mysore who has been the Coordinator of this
assignment and all the concerned PGT- Mathematics subject experts from the four feeder
regions of ZIET Mysore.
Wishing you all the very best in your academic journey!
MENAXI JAIN
DIRECTOR
ZIET MYSORE
iii
INDEX
S.NO CHAPTER/UNIT Page Numbers
1 CURRICULAM^ 1 - 8
2^1 - NUMBERS, QUANTIFICATION AND NUMERICAL
APPLICATIONS
3^2 - ALGEBRA(MATRICES^ AND DETERMINANTS)^ 32 - 54
4 3A^ -^ DIFFERENTIATION AND ITS APPLICATIONS^ 55 - 92
5 3B^ -^ INTEGRATION AND ITS APPLICATIONS^ 93 - 112
6 3C^ -^ DIFFERENTIAL EQUATIONS AND MODELING^ 113 - 131
7^4 -^ PROBABILITY^ DISTRIBUTIONS^ 132 - 156
8^5 -^ INFERENTIAL^ STATISTICS^ 157 - 185
9^6 -^ INDEX NUMBERS AND^ TIME-BASED^ DATA^ 186 - 207
10^7 -^ FINANCIAL MATHEMATICS^ 208 - 231
11^8 -^ LINEAR PROGRAMMING^ 232 - 272
CURRICULUM APPLIED MATHEMATICS (CODE – 241) CLASS XII SESSION: 2023 - 24
Number of Paper: 1
Total number of Periods: 240 (35 Minutes Each)
Time:3 Hours
Max Marks:
No. Units No. of
Periods
Marks
I Numbers, Quantification and Numerical
Applications
II Algebra 20 10
III Calculus 50 15
IV Probability Distributions 35 10
V Inferential Statistics 10 05
VI Index Numbers and Time-based data 30 06
VII Financial Mathematics 50 15
VIII Linear Programming 15 08
Total 240 80
Internal Assessment 20
2.3 Algebra of Matrices ● Perform operations like addition & subtraction on matrices of same order ● Perform multiplication of two matrices of appropriate order ● Perform multiplication of a scalar with matrix ● Addition and Subtraction of matrices ● Multiplication of matrices (It can be shown to the students that Matrix multiplication is similar to multiplication of two polynomials) ● Multiplication of a matrix with a real number 2.4 Determinants ● Find determinant of a square matrix ● Use elementary properties of determinants ● Singular matrix, Non-singular matrix ● |AB|=|A||B| ● Simple problems to find determinant value 2.5 Inverse of a matrix
- Define the inverse of a square matrix
- Apply properties of inverse of matrices - Inverse of a matrix using: a) cofactors If A and B are invertible square matrices of same size, i) (AB)-^1 =B -^1 A –^1 ii) (A-^1 )-^1 =A iii) (AT)-^1 = (A-^1 )T 2.6 Solving system of simultaneous equations using matrix method, Cramer’s rule and
- Solve the system of simultaneous equations using i) Cramer’s Rule ii) Inverse of coefficient matrix
- Formulate real life problems into a system of simultaneous linear equations and solve it using these methods
- Solution of system of simultaneous equations upto three variables only (non- homogeneous equations) UNIT- 3 CALCULUS
Differentiation and its Applications
3.1 Higher Order Derivatives
- Determine second and higher order derivatives
- Understand differentiation of parametric functions and implicit functions - Simple problems based on higher order derivatives
- Differentiation of parametric functions and implicit functions (upto 2 nd^ order) 3.2 Application of Derivatives
- Determine the rate of change of various quantities
- Understand the gradient of tangent and normal to a curve at a given point
- Write the equation of tangents and normal to a curve at a given point
- To find the rate of change of quantities such as area and volume with respect to time or its dimension
- Gradient / Slope of tangent and normal to the curve
- The equation of the tangent and normal to the curve (simple problems only) 3.3 (^) Marginal Cost and Marginal Revenue using derivatives
- Define marginal cost and marginal revenue
- Find marginal cost and marginal revenue
- Examples related to marginal cost, marginal revenue, etc.
3.4 Increasing /Decreasing Functions
- Determine whether a function is increasing or decreasing
- Determine the conditions for a function to be increasing or decreasing - Simple problems related to increasing and decreasing behaviour of a function in the given interval 3.5 (^) Maxima and Minima
- Determine critical points of the function
- Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
- Find the absolute maximum and absolute minimum value of a function
- Solve applied problems
- A point x= c is called the critical point of f if f is defined at c and f ′(c) = 0 or f is not differentiable at c
- To find local maxima and local minima by: i) First Derivative Test ii) Second Derivative Test
- Contextualized real life problems
Integration and its Applications
3.6 Integration • Understand and determine
indefinite integrals of simple functions as anti-derivative
- Integration as a reverse process of differentiation
- Vocabulary and Notations related to Integration 3.7 Indefinite Integrals as family of curves
- Evaluate indefinite integrals of simple algebraic functions by method of: i) substitution ii) partial fraction iii) by parts
- Simple integrals based on each method (non- trigonometric function) 3.8 Definite Integrals as area under the curve ● Define definite integral as area under the curve ● Understand fundamental theorem of Integral calculus and apply it to evaluate the definite integral ● Apply properties of definite integrals to solve the problems ● Evaluation of definite integrals using properties 3.9 (^) Application of Integration ● Identify the region representing C.S. and P.S. graphically ● Apply the definite integral to find consumer surplus-producer surplus Problems based on finding ● Total cost when Marginal Cost is given ● Total Revenue when Marginal Revenue is given ● Equilibrium price and equilibrium quantity and hence consumer and producer surplus
Differential Equations and Modeling
3.10 Differential Equations ● Recognize a differential equation ● Find the order and degree of a differential equation ● Definition, order, degree and examples
4.6 Normal Distribution ● Understand normal distribution is a Continuous distribution ● Evaluate value of Standard normal variate ● Area relationship between Mean and Standard Deviation
- Characteristics of a normal probability distribution
- Total area under the curve = total probability = 1
- Standard Normal Variate: Z = 𝑥−^ 𝜇^ where 𝜎 x = value of the random variable 𝜇 = mean 𝜎 = S.D. UNIT - 5 INFERENTIAL STATISTICS 5.1 (^) Population and Sample
- Define Population and Sample
- Differentiate between population and sample
- Define a representative sample from a population
- Differentiate between a representative and non- representative sample
- Draw a representative sample using simple random sampling
- Draw a representative sample using and systematic random sampling
- Population data from census, economic surveys and other contexts from practical life
- Examples of drawing more than one sample set from the same population
- Examples of representative and non-representative sample
- Unbiased and biased sampling
- Problems based on random sampling using simple random sampling and systematic random sampling (sample size less than 100) 5.2 (^) Parameter and Statistics and Statistical Interferences
- Define Parameter with reference to Population
- Define Statistics with reference to Sample
- Explain the relation between Parameter and Statistic
- Explain the limitation of Statistic to generalize the estimation for population
- Interpret the concept of Statistical Significance and Statistical Inferences
- State Central Limit Theorem
- Explain the relation between Population-Sampling Distribution-Sample
- Conceptual understanding of Parameter and Statistics
- Examples of Parameter and Statistic limited to Mean and Standard deviation only
- Examples to highlight limitations of generalizing results from sample to population
- Only conceptual understanding of Statistical Significance/Statistical Inferences
- Only conceptual understanding of Sampling Distribution through simulation and graphs
5.3 t-Test (one sample t-test and two independent groups t-test) ● Define a hypothesis ● Differentiate between Null and Alternate hypothesis ● Define and calculate degree of freedom ● Test Null hypothesis and make inferences using t-test statistic for one group / two independent groups ● Examples and non-examples of Null and Alternate hypothesis (only non- directional alternate hypothesis) ● Framing of Null and Alternate hypothesis ● Testing a Null Hypothesis to make Statistical Inferences for small sample size ● (for small sample size: t- test for one group and two independent groups ● Use of t-table UNIT – 6 INDEX NUMBERS AND TIME BASED DATA 6.4 Time Series ● Identify time series as chronological data ● Meaning and Definition 6.5 (^) Components of Time Series ● Distinguish between different components of time series ● Secular trend ● Seasonal variation ● Cyclical variation ● Irregular variation 6.6 Time Series analysis for univariate data ● Solve practical problems based on statistical data and Interpret the result ● Fitting a straight line trend and estimating the value 6.7 Secular Trend ● Understand the long term tendency ● The tendency of the variable to increase or decrease over a long period of time 6.8 (^) Methods of Measuring trend ● Demonstrate the techniques of finding trend by different methods ● Moving Average method ● Method of Least Squares UNIT - 7 FINANCIAL MATHEMATICS 7.1 Perpetuity, Sinking Funds
- Explain the concept of perpetuity and sinking fund
- Calculate perpetuity
- Differentiate between sinking fund and saving account - Meaning of Perpetuity and Sinking Fund - Real life examples of sinking fund - Advantages of Sinking Fund - Sinking Fund vs. Savings account 7.3 Calculation of EMI
- Explain the concept of EMI
- Calculate EMI using various methods
- Methods to calculate EMI: i) Flat-Rate Method ii) Reducing-Balance Method
- Real life examples to calculate EMI of various types of loans, purchase of assets, etc. 7.4 Calculation of Returns, Nominal Rate of Return
- Explain the concept of rate of return and nominal rate of return
- Calculate rate of return and nominal rate of return
- Formula for calculation of Rate of Return, Nominal Rate of Return
UNIT 1: NUMBERS, QUANTIFICATION AND NUMERICAL APPLICATIONS
SOME IMPORTANT RESULTS/CONCEPTS
Modulo Arithmetic:
Euclid ‘s Division Lemma:
For integers a, b(≠ 0), we have 𝒂 = 𝒃𝒒 + 𝒓, where 𝑞, 𝑟 ∈ 𝛧and 0 ≤ 𝑟 < |𝑏|
Modulo Arithmetic is the arithmetic of remainders.𝒂𝒎𝒐𝒅𝒃 = 𝒓
Where mod (modulo) gives the remainder after a is divided by b
Note: 1. 𝑎 𝑚𝑜𝑑 𝑎 = 0
2.If a<b, then 𝑎 𝑚𝑜𝑑 𝑏 = 𝑎
Properties
1. a mod b = ( a + kb ) mod b; where k is any integer
2. (A + B) mod C = (A mod C + B mod C ) mod C
3 .(A - B) mod C = (A mod C - B mod C) mod C
4. (A × B) mod C = (A mod C × B mod C) mod C
Congruence Modulo:
Two positive integers a and b are said to be congruence modulo m if a and b satisfy the
following conditions:
i) ( a-b ) is divisible by m
ii) a mod m = b mod m
Notation used for congruence modulo is:𝑎 ≡ 𝑏(𝑚𝑜𝑑𝑚)
Property 5: If a ≡ b (mod m ) where a, b and m are positive integers then ak ≡ bk ( mod m) for any positive
integer k.
Allegation and Mixture:
Ratio(𝑅)^ = C.P.of Dearer (d)−mean price (m) 𝑚𝑒𝑎𝑛𝑝𝑟𝑖𝑐𝑒(𝑚)−𝐶 .𝑃𝑜𝑓𝑐ℎ𝑒𝑎𝑝𝑒𝑟(𝑐)^ = 𝑑−𝑚 𝑚−𝑐
Quantity of Liquid left after n operations =𝑥( 1 −
𝑦 𝑥
Where, 𝑥− Original amount,𝑦 − taken out and 𝑛 − Number of times
Boats and Streams:
Let the speed of the boat in the still water be 𝑥𝑘𝑚/ℎand sped of the stream be 𝑦𝑘𝑚/ℎ. Then
1. Downstream Speed (𝑢) = 𝑥 + 𝑦𝑘𝑚/ℎ and Upstream 𝑆𝑝𝑒𝑒𝑑(𝑣) = 𝑥 − 𝑦𝑘𝑚/ℎ
2. Speed of Boat =
𝑢+𝑣 2
and Speed of Stream =
𝑢−𝑣 2
Pipes and Cisterns:
A pipe connected to a tank or cistern which fills it is known as inlet pipe and the pipe connected to the
tank which drains or empties it is known as outlet pipe.
When a tank is connected to many pipes (inlets and outlets), then the difference between the sum of the
work done by inlets and the sum of the work
done by outlets gives the filled part of the tank.
- Let a pipe fill a tank in x number of hours, then it can fill (1/x) th portion of the tank in one hour.
- If a pipe can empty a tank in y number of hours, then it can empty out (1/y)th portion of the tank in
one hour.
- The portion of tank they can fill together in one hour ( 1 𝑥
1 𝑦
𝑡ℎart of a tank in 1 hour, then the time taken by the pipe to fill the tank completely is x
hours
- Two pipes can fill a tank in x and y hours respectively. If both the pipes are opened simultaneously,
then time taken by both the pipes to fill the tank is
𝑥𝑦 𝑥+𝑦
hours.
- If two pipes A and B together can fill a tank in x hours and the pipe A alone can fill the tank in y
hours, then time taken by pipe B alone to fill the tank is
𝑥𝑦 𝑦−𝑥
hours.
- If a pipe A can fill a tank in x hours and a pipe B can empty the full tank in y hours (where y > x),
then net part filled in 1 hour is
𝑦−𝑥 𝑥𝑦
- If a pipe A can fill a tank in x hours and a pipe B can empty the full tank in y hours (where x >y ),
then net part emptied in 1 hour
𝑥−𝑦 𝑥𝑦
- Three pipes A, B and C can fill a tank in x y, and z hours respectively. If all the three pipes are
opened simultaneously, then time taken by all the pipes to fill it is
𝑥𝑦𝑧 𝑥𝑦+𝑦𝑧+𝑥𝑧
hours.
- If two pipes are filing a tank at the rate of x hours and y hours respectively and a third pipe is
emptying it at the rate of z hours, then in one hour the part of the tank filled is
1 𝑥
1 𝑦
1 𝑧
hours
Time taken to fill the tank is
𝑥𝑦𝑧 (𝑦𝑧+𝑧𝑥−𝑥𝑦)
2 [( 3 × 7 ) + 5 ]𝑚𝑜𝑑 4 is
a.3 b.2 c.4 d.
Ans: b.
Solution: [( 3 × 7 ) + 5 ]𝑚𝑜𝑑 4 = 26 𝑚𝑜𝑑 4
26 = 6 × 4 + 2 ⇒ 26 ≡ 2 (𝑚𝑜𝑑 4 ) = 2
3 𝑥 ≡ 4 (𝑚𝑜𝑑 7 ) then positive values of 𝑥 are
a. { 4 , 11 , 18 …. } b. { 11 , 18 , 25 …. } c. { 4 , 8 , 12 …. } d. { 1 , 8 , 15 …. }
Ans : a. { 4 , 11 , 18 …. }
Solution: x ≡ 4 (mod 7 )^ ⇒ x = 7 k + 4 , k = 0 , 1 , 2 …
4 Milk and water in two vessels are in the ratio 5:3 and 5:4 respectively .in what ratio liquid in both
vessels be mixed to obtain a new mixture in which ratio of milk and water is 7:5 respectively.
a. 3:2 b.3:5 c. 2:3 d. 2:
Ans : c. 2:
Solution:
7 12
5 9
5 8
7 12
1 36
1 24
5 In what ratio must rice at Rs 69 per kg be mixed with rice at Rs 100 per kg so that the mixture
be worth Rs 80 per kg?
a. 11:20 b. 11: 10 c. 20:11 d. 10:
Ans: c. 20:
Solution: Rice at 69 : rice at 100 = 100 − 80 ∶ 80 − 69 = 20 : 11
6 In a 50m race A can give a start of 5m to B and a start of 14 m to C .In the same race how much
start can B give to C?
a. 9m b.10 m c. 12m .d.11 m
Ans: b.10 m
𝐵 𝐶
𝐵 𝐴
×
𝐴 𝐶
45 36
50 𝑥
B can give a start of C by 50 − 40 = 10 𝑚
7 A runs 1 2
3
times as fast as B. If A gives B a start of 80 m, how far is the running post so that A
and B may reach it at the same time.
a. 100m b.120m c.200 m d.240 m
Ans: c.200 m
Solution: 𝐴: 𝐵 = 1
2 3
Let 𝐴 = 5 𝑥𝐵 = 3 𝑥
A gives B a start of 80 m , 2 𝑥 = 80 𝑥 = 40
Distance = 5 × 40 = 200
Two pipes A and B together can fill a pipe in 4 hours ,pipe B take 6 hours more than A to fill the 8 tank
,if they opened separately .The time taken by A to fill the tank alone is
a. 2 hours b. 4 hours c. 6hours d. 8 hours
Ans: c. 6hours
Solution:
Let Time taken by A = 𝑥ℎ𝑟𝑠 and B = (𝑥 + 6 )ℎ𝑟𝑠
Time taken by A and B together = 4 hours
In 1 hour they can fill
1 4
part of the tank
4 ( 2 𝑥 + 6 ) = 𝑥^2 + 6 𝑥
𝑥^2 − 2 𝑥 − 24 = 0
9 - 41 mod7 is ……
a. - 6 b. 5 .c. 1 d. - 1
Ans: .c. 1
Solution: − 41 = 7 × − 6 + 1
Assertion and Reason Type questions
The following questions consist of two statements, one labelled as ‘Assertion (A)’ and the other
labelled as ‘Reason (R)’. You are to examine these two statements carefully and decide if the Assertion
(A) and Reason (R) are individually true and if so, whether the Reason (R) is the correct explanation
for the given Assertion (A). Select your answer to these items using the codes given below and then
select the correct option.
Codes:
A. Both A and R are individually true and R is the correct explanation of A
B. Both A and R are individually true but R is not the correct explanation of A
C. A is true but R is false
D. A is false but R is true
1 Assertion A :( 486 + 729 )𝑚𝑜𝑑 12 ≡ 3
Reason R : (𝑎 + 𝑏)𝑚𝑜𝑑𝑛 = 𝑎(𝑚𝑜𝑑𝑛) + (𝑏 𝑚𝑜𝑑 𝑛)
( 486 + 729 )𝑚𝑜𝑑 12 = 1215 𝑚𝑜𝑑 12 = 101 × 12 + 3
Ans :A) Both Assertion and Reason are true and R is the correct explanation of A
2 Assertion A : 186 × 93 𝑚𝑜𝑑 7 ≡ 2
Reason R :𝑎. 𝑏(𝑚𝑜𝑑 𝑛) = (𝑎 𝑚𝑜𝑑𝑛). (𝑏 𝑚𝑜𝑑𝑛)
Ans :D) A is false but R is true
3 Assertion A : The ratio of copper and Zinc in brass is 13:7. In 100 kg of Brass there is 35 kg of
Zinc
Reason R : Ratio =
Amount of Zinc in 100 kg Brass= 100 ×
7 20
Ans : B) A and R are true but R is not the correct explanation of A
4 Assertion A : Rohan can row with a speed of 16 km /hr in still water. If the speed of stream is
12km/hr ,then speed of Rohan with stream will be 26 km/hr
Reason R : If the speed of a boat in still is xkm/hr and speed of the stream is ykm/hr, then speed
of down- stream will be x+y km/hr
Ans: 𝑥 = 16 𝑘𝑚/ℎ𝑟 𝑦 = 12 𝑘𝑚/ℎ𝑟
Ans :D) Assertion is False Reason is True
5 Assertion A : If a is any positive real number then 𝑎 + 1
𝑎
Reason R : Let a and b be distinct positive real numbers then
𝑎+𝑏 2
Ans : A ) A and R are true and R is the correct explanation of A
6 Assertion A : Two pipes A and B can fill a tank in 20 hrs and 30 hrs respectively.If both the pipes
open simultaneously, then pipe A should be closed after 8 hrs so that the tank is filled in 18 hrs.
Reason R : Time taken to fill the tank is positive and the time taken to empty a tank is taken
negative.
Ans : B) A and R are true but R is not the correct explanation of A
7 Assertion A : In a 50 m race, A can give a start of 5m to B and a start of 14 m to C ,the B gives a
start of 40 m to C
Reason R : A gives a start of x m means that before the start of race, A is at the starting point and B
is ahead of A by x m, B give a start of 10 m to C.
Ans :D ) Assertion is False Reason is True
8 Assertion A :It is currently 8 am .It will be 4 am in next 500 hrs.
Reason R : We know that time repeats in every 24 hours.
So , 500 𝑚𝑜𝑑 24 = 20
Therefore 500hrs is equivalent to 20 hrs.
Ans :B)Both A and R are true but R is not the correct explanation of A
