maths study material – 2015-16

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Study Material

Class XII - Mathematics

2015-2016

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KENDRIYA VIDYALAYA SANGATHAN

ZONAL INSTITUTE OF EDUCATION AND

TRAIING, MYSORE

Study Material

Class XII Mathematics

2015-2016

Prepared/Revised by

Mr. S.K.Sivalingam

PGT (Maths), Faculty, KVS ZIET Mysore

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,

Shri. Santosh Kumar Mall, IAS Commissioner, KVS

. . , ()

Shri. G.K. Srivastava, IAS Additional Commissioner (Administration)

()

Shri. U N Khaware Additional Commissioner (Academics)

. ()

Dr. Shachi Kant Joint Commissioner (Training)

. ()

Shri. M Arumugam Joint Commissioner (Finance)

. . ()

Dr. V. Vijayalakshmi Joint Commissioner (Academics)

. . ()

Dr. E. Prabhakar Joint Commissioner (Personnel)

. . ()

Shri. S Vijaya kumar Joint Commissioner (Admn)

Acknowledgement

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FOREWORD The seven PGTs working as members of faculty at KVS,ZIET Mysore - Mr. K

Arumugam (Physics), Mr.S. Kalusivalingam (Maths), Mr. M Reddenna (Geo.), Mr. S.

Murugan (History), Mr. Hari Shankar (Hindi), Mr. Joseph Paul (Econ.) and Mr. U.P

Binoy (English) prepared Study Materials for Class XII for the academic year 2015-

2016 in their respective subjects.

All these study materials focus on some select aspects, namely;

Gist of lessons/chapter

Marking scheme (CBSE)

Important questions

Solved Question papers with value point

Tips for scoring well in the examination

The above mentioned seven members of faculty at ZIET Mysore have put in a lot of efforts and prepared the materials in a period of two months. They deserve commendation for their single-minded pursuit in bringing out these materials. The teachers of these subjects namely, Physics, Mathematics, Geography, History, Hindi, Economics and English may use the materials in the month of January & February 2016 for Pre-Board Examination revision and practice purposes. It is hoped that these materials will help the students perform better in the forthcoming Board Examinations. The teachers are requested to go through the materials thoroughly, and feel free to send their opinions and suggestions for the improvement of these materials to [email protected]. Dr. E.T ARASU Deputy Commissioner & Director KVS, ZIET Mysore

mailto:[email protected]

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PREFACE

It is a matter of great pleasure that after receiving encouragement from our Deputy Commissioner

DR. E.T Arasu, I now present the thoroughly revised latest edition of STUDY MATERIAL OF CLASS XII

MATHEMATICS based on the latest syllabus and revised question paper pattern to be followed from 2015

onwards.

THE SALIENT FEATURES OF THIS STUDY MATERIAL ARE AS FOLLOWS;

It covers the syllabus given by CBSE for the class XII Mathematics

Gist of each chapter

Marking scheme given by CBSE ( 2015)

Two sets of solved question papers with value points

Two sets of model question papers -to be answered

Tips for pre-examination and during examination

The material can be used for the purpose of revision

All the concepts of the subject have been included in the material

I am sure this material will serve the purpose of helping students perform better in the Board Examination.

However, suggestions and comments from the teachers and the students for the improvement of this

material will be highly appreciated.

Place: Mysore S.K.Sivalingam

Date: 17/12/2015 Faculty ZIET Mysore

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Index

S.No. Particulars Page No.

1 Marking Scheme 7

2 Gist of Chapters Chapters 1 to 13 10

3 Important model questions

Chapters 1 to 13 60

4 Solved Question Paper with value points

Sample Paper 1 88

5 Solved Question Paper with value points

Sample Paper 2 105

6 Model Question Paper 1 115

7 Model Question Paper 2 118

8 Tips for scoring well 122

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MARKING SCHEME

Course Structure

Unit Topic Marks

I. Relations and Functions 10

II. Algebra 13

III. Calculus 44

IV. Vectors and 3-D Geometry 17

V. Linear Programming 6

VI. Probability 10

Total 100

Unit I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function. Binary operations.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric functions.

Unit II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operation on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

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2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit functions. Concept of exponential and logarithmic functions.

Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

2. Applications of Derivatives

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

5. Differential Equations

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Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

2. Three - dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Unit V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded and unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability. independent events, total probability, Baye's theorem, Random variable and its probability distribution, mean and variance of random variable. Repeated independent (Bernoulli) trials and Binomial distribution.

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GIST OF CHAPTERS

Chapter1. Relations and Functions

Introduction:-

Any set of ordered pairs (x,y) is called a relation in x and y. Furthermore,

The set of first components in the ordered pairs is called the domain of the relation.

The set of second components in the ordered pairs is called the range of the relation.

Relation: - Let A and B be two sets. Then a relation R from set A to Set B is a subset of .

Types of relations: -

Empty relation: A relation R in a set A is called empty relation, if no element of A is related to any element

of A i

Universal relation :A relation R in a set A is called universal relation, if each element of A is related to every

element of A, i.e., R = A A.

Equivalence relation.: A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric

and transitive

A relation R in a set A is called (i) reflexive, if (a, a) R, for every aA, (ii) symmetric, if (a, b) R implies that (b, a) R, for all a, b A. (iii) transitive, if (a, b) R and (b, c) R implies that (a, c) R, for all a, b, c A. Function:

A function is a relation between a set of inputs and a set of permissible outputs with the property that each

input is related to exactly one output. An example is the function that relates each real number x to its

square x2.

A function is a relation for which each value from the set the first components of the ordered pairs is

associated with exactly one value from the set of second components of the ordered pair.

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Types of functions:

One-one (or injective):

A function f : X Y is defined to be one-one (or injective), if the images

of distinct elements of X under f are distinct, i.e., for every a, b in X, f (a) = f (b)

implies a = b. Otherwise, f is called many-one.

Horizontal line test:

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To check the injectivity of the function, f (x) =2x. Draw a horizontal line such that this line cuts the graph only at one place. Such types of functions are known as one-one functions.

In this case where the line cuts the graph of a function at more than one place, the functions are not one-

one.

Onto (or surjective):

A function f : X Y is said to be onto (or surjective), if every element of Y is the image of some element of X

under f, i.e., for every y in Y, there exists an element x in X such that f (x) = y.

Bijective function:

A function f : X Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

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Composition of Functions and Invertible Function

Let f : A B and g : B C be two functions. Then the composition of f and g, denoted by gof, is defined as the

function gof: A C given by gof(x) = g(f (x)), for all x A.

Inverse of a Bijective Function Let f: A B be a function. If, for an arbitrary x A we have f(x) = y B, then the function, g: B A, given

by g(y) = x, where y B and x A, is called the inverse function of f.

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Binary Operations

A binary operation on a set A is a function : A A A. We denote (a, b) by a b.

Binary Operation: A binary operation * defined on set A is a function from A A A. * (a, b) is

denoted by a * b.

Binary operation * defined on set A is said to be commutative iff a * b = b * a. Binary operation * defined on set A is called associative iff a *(b * c) = (a*b)*c If * is Binary operation on A, then an element e in A is said to be the identity element iff a * e = e * a for every a in A Identity element is unique. If * is Binary operation on set A, then an element b is said to be inverse of a A iff a * b = b * a = e Inverse of an element, if it exists, is unique.

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Chapter 2. Inverse Trigonometric Functions

Inverse trigonometric functions or cyclometric functions - are the so-called inverse functions of the

trigonometric functions, when their domain are restricted to principal value branch to make the

trigonometric functions bijective. The principal inverses are listed in the following table.

Name Usual notation Definition Domain of x for

real result

Range of usual

principal value

(radians)

Range of usual principal

value

(degrees)

arcsine y = sin-1 x x = sin y 1 x 1 /2 y /2 90 y 90

arccosine y = cos-1 x x = cos y 1 x 1 0 y 0 y 180

arctangent y = tan-1 x x = tan y all real numbers /2 < y < /2 90 y 90

arccotangent y = cot-1 x x = cot y all real numbers 0 < y < 0 < y < 180

arcsecant y = sec-1 x x = sec y x 1 or 1 x 0 y < /2 or /2 <

y

0 y < 90 or 90 < y

180

arccosecant y = csc-1 x x = csc y x 1 or 1 x /2 y < 0 or 0 < y

/2

90 y 0 or 0 < y

90

Let us take a function , , . The inverse of this function is defined as (We may

also write , provided the function is both one-one and onto).

http://en.wikipedia.org/wiki/Radianhttp://en.wikipedia.org/wiki/Degree_(mathematics)http://en.wikipedia.org/wiki/Sinehttp://en.wikipedia.org/wiki/Cosinehttp://en.wikipedia.org/wiki/Trigonometric_functionshttp://en.wikipedia.org/wiki/Cotangenthttp://en.wikipedia.org/wiki/Trigonometric_functions#Reciprocal_functionshttp://en.wikipedia.org/wiki/Cosecant

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Very important: does not mean . This is just a notation given to write Inverse Trigonometric Function.

We can easily verify etc. Further any is not an image. Hence, the

function does not exist, as is not one-one and onto.

Now consider the function defined by . It is both one-one and onto which

can be visualized from the above graph. Hence, exists.

In fact, if we restrict the domain of the sin function to any one of the intervals , ,

, .. etc., keeping the range the function becomes one-one and onto.

Then is a function whose domain is and range is any one of the intervals

, , , .. etc., [please recall that if exists, and interchange their

domain and range.

Every branch of the graph of corresponding to branches , , ,

gives a oneone and onto function. The graph in the branch called the principle value

branch.

We can also justify our answer from the chapter Trigonometric Equations and General Values.

has infinitely many solutions, , , , etc.. The value is

called the principle value.

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We may say that . Also value of is called the principle value.

Similarly we restrict the domains of other trigonometric functions to make it invertible the table given below

shows the domain and the range of the six basic inverse trigonometric function.

The values of domain and range indicated for different trigonometric, inverse trigonometric functions can be observed

from the graphs of trigonometric functions and their corresponding inverse trigonometric functions.

We can draw the graph of a inverse function of a function by taking image on line y = x.

Let us apply this to draw graph of y = .

2. he graph of the function y = and y =

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3. The graph of the function y = tanx and y =

Note: Whenever no branch of an inverse function is mentioned, we consider the Principal value branch.

PROPERTIES OF INVERSE CIRCULAR FUNCTIONS

1. (a)

(b)

(c)

2. (a)

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(b)

(c)

3. (a)

(b)

(c)

4. (a)

(b)

(c)

5. (a)

(b)

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Chapter 3. Matrices

Definition: A matrix A is defined as an ordered rectangular array of numbers in m rows and n columns.

1. Row matrix: A matrix can have a single row A [ a11 a12 a13 a1n] 2. Column matrix: - A matrix can have a single column A = 3. Zero or null matrix: A matrix is called the zero or null matrix if all the entries are 0. 4. Square matrix: - A matrix for which horizontal and vertical dimensions are the same (i.e., an matrix). 5. Diagonal matrix: - A square matrix A is called diagonal matrix if aij = 0 for . 6. Scalar matrix: - A diagonal matrix A is called the scalar matrix if all its diagonal elements are equal. 7. Identity matrix : A diagonal matrix A is called the identity matrix if aij = 1 for i = j , it is denoted by In. 8. Upper triangular matrix: - A square matrix A is called upper triangular matrix if aij = 0 for 9. Lower triangular matrix :- A square matrix A is called lower triangular matrix if aij = 0 for

Matrix operations

1. Definition: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of rows and columns.

2. Addition If A and B are matrices of the same type then the sum is a matrixC = obtained by adding the corresponding elements aij+bij i.e. A+B = C if aij+bij =cij

3. Matrix addition is commutative , associative and distributive over multiplication

A + B = B + A

A + (B + C) = (A+ B) + C

A (B + C) = AB + AC

(A+B)C= AC + BC Subtraction If A and B are matrices of the same type then the difference is a matrix D = obtained by subtracting the corresponding elements aij - bij i.e. A - B = C if aij - bij =dij

4. Equal matrices Two matrices are said to be equal if they have the same order and their corresponding elements are also equal i.e. A = B if aij = bij for all I, j .

5. Scalar multiplication- If A and B are matrices of the same order and k, m are scalars then, scalar multiplication is defined as kA=[kaij].

6. K(A+B) = KA + KB

7. (m+n) A = mA+ nA

http://mathworld.wolfram.com/Matrix.html

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8. (mk)A = m(kA) =k(mA)

9. Matrix multiplication

Definition: When the number of columns of the first matrix is the same as the number of rows in the

second matrix then matrix multiplication can be performed.

Let A and B . Then the product of A and B is the matrix C, which has dimensions mxp. The ijth element of

matrix C is found by multiplying the entries of the ith row of A with the corresponding entries in the jth

column of B and summing the n terms. The elements of C are:

Note: AxB is not equal to BxA

10. Transpose of Matrices: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A =

(aij) and the transpose of A is:AT=(aji) where j is the column number and i is the row number of matrix A.

11. Symmetric matrix. For a symmetric matrix A = AT i.e. aij = aji

Skew -symmetric matrix. For a skew-symmetric matrix AT = - A i.e. aji = - aij.

Properties

Note that the diagonal elements of the skew symmetric matrix are 0.

A + AT is a symmetric matrix.

A - AT is a skew - symmetric matrix.

Every square matrix A can be expressed as a sum of a symmetric P and skew symmetric Q matrices

10. Elementary transformation - Following elementary row or column transformations can be applied to

a matrix

Interchange of any two rows or columns.

Multiplication of any row or column by any non-zero number(scalar)

The addition to the elements of any row or column the scalar multiples of any other row or column

Working rule to find A-1 by elementary transformations

a) Write A = InA, apply elementary row transformations to both the matrices A on LHS and In on RHS till you get In= BA. Then B is the required A-1

b) Write A = AIn, apply elementary column transformations to both the matrices A on LHS and In on RHS till you get In= AB. Then B is the required A-1

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NOTE : Apply only one kind of transformations (row or column ) in all the steps in one particular answer.

Chapter 4 Determinants

Every square matrix A is associated with a number, called its determinant and it is denoted by det

(A) or |A| .

Only square matrices have determinants. The matrices which are not square do not have

determinants

(i) First Order Determinant If A = [a], then det (A) = |A| = a

(ii) Second Order Determinant

|A| = a11a22 a21a12

(iii) Third Order Determinant

Evaluation of Determinant of Square Matrix of Order 3 by Sarrus Rule

then determinant can be formed by enlarging the matrix by adjoining the first two

columns on the right and draw lines as show below parallel and perpendicular to the diagonal.

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The value of the determinant, thus will be the sum of the product of element. in line parallel to the

diagonal minus the sum of the product of elements in line perpendicular to the line segment. Thus,

= a11a22a33 + a12a23a31 + a13a21a32 a13a22a31 a11a23a32 a12a21a33.

Properties of Determinants

(i) The value of the determinant remains unchanged, if rows are changed into columns and columns

are changed into rows e.g., |A| = |A|

(ii) In a determinant, if any two rows (columns) are inter changed, then the value of the determinant is

multiplied by - 1.

(iii) If two rows (columns) of a square matrix A are proportional, then |A| = 0.

(iv) |B| = k |A|, where B is the matrix obtained from A, by multiplying one row (column) of A by k.

(v) |kA| = kn

|A|, where A is a matrix of order n x n.

(vi) If each element of a row (or column) of a determinant is the sum of two or more terms, then the

determinant can be expressed as the sum of two or more determinants, e.g.,

(vii) If the same multiple of the elements of any row (or column) of a determinant are added to the

corresponding elements of any other row (or column), then the value of the new determinant remains

unchanged, e.g.,

(viii) If each element of a row (or column) of a determinant is zero, then its value is zero. (ix) If any

two rows (columns) of a determinant are identical, then its value is zero.

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(x) If each element of row (column) of a determinant is expressed as a sum of two or more terms,

then the determinant can be expressed as the sum of two or more determinants.

Important Results on Determinants

(i) |AB| = |A||B| , where A and B are square matrices of the same order.

(ii) (ii) |An

|=|A|n

(iii) If A, B and C are square matrices of the same order such that ith column (or row) of A is the sum of i

th columns (or rows) of B and C and all other columns (or rows) of A, Band C are identical, then |A|

=|B| + |C|

(iv) |In| = 1,where In is identity matrix of order n

(v) |On| = 0, where On is a zero matrix of order n

(vi) If (x) be a 3rd order determinant having polynomials as its elements.

(a) If (a) has 2 rows (or columns) proportional, then (x a) is a factor of (x).

(b) If (a) has 3 rows (or columns) proportional, then (x a)2

is a factor of (x).

(vii) A square matrix A is non-singular, if |A| 0 and singular, if |A| =0.

(viii) Determinant of a skew-symmetric matrix of odd order is zero and of even order is a non- zero

perfect square.

(ix) In general, |B + C| |B| + |C|

(x) Determinant of a diagonal matrix = Product of its diagonal elements

(xi) Determinant of a triangular matrix = Product of its diagonal elements

Minors and Cofactors

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then the minor Mij of the element aij is the determinant obtained by deleting the i

row and jth column.

The cofactor of the element aij is Cij = (- 1)i + j

Mij

Adjoint of a Matrix - Adjoint of a matrix is the transpose of the matrix of cofactors of the give matrix, i.e.,

Properties of Minors and Cofactors

(i) The sum of the products of elements of any row (or column) of a determinant with the

cofactors of the corresponding elements of any other row (or column) is zero

(ii) The sum of the product of elements of any row (or column) of a determinant with the

cofactors of the corresponding elements of the same row (or column) is

Applications of Determinant in Geometry

Let three points in a plane be A(x1, y1), B(x2, y2) and C(x3, y3), then Area =

= 1 / 2 [x1 (y2 y3) + x2 (y3 y1) + x3 (y1 y2)]

SOLVING SYSTEMS OF EQUATIONS USING INVERSE MATRIX METHOD

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DEFINITION: A system of linear equations is a set of equations with n equations and n unknowns, is of the

form of

The unknowns are denoted by x1,x2,...xn and the coefficients

(a's and b's above) are assumed to be given. In matrix form the system of equations above can be written

as:

which can be expressed in matrix equation as AX =B

By pre-multiplying both sides of this equation by A-1 gives:

A-1 (AX) = A-1 B

STEPS

i. Evaluate . Refer the note given below.

ii. Evaluate the cofactors of elements of A.

iii. Form the adjoint of A as the matrix of cofactors

iv. Calculate inverse of A and X.

NOTE - From the above it is clear that the existence of a solution depends on the value of the

determinant of A. There are three cases:

1. If the then the system is consistent with unique solution given by

2. If (A is singular) and adj A .B 0 then the solution does not exist. The system is inconsistent.

3. If (A is singular) and adj A .B = 0 then the system is consistent with infinitely many solutions.to find

these solutions put z = k in two of the equations and solve them by matrix method.

For homogeneous system of linear equations, AX = 0 (B = 0)

1. If the then the system is consistent with trivial solution x = 0, y = 0, z = 0

2. If (A is singular) and adj A .B 0 then the solution does not exist. The system is inconsistent.

3. If (A is singular) and adj A .B = 0 then the system is consistent with infinitely many solutions. to

find these solutions put z = k in two of the equations and solve them by matrix method.

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2

2

d y

dx

Chapter 5 Continuity and Differentiability

Definition

A function f is said to be continuous at x = a if

A function is said to be continuous on the interval [a, b] if it is continuous at each point in the interval.

Definition of derivative : If y = f(x) then y1 =

A function f is differentiable if it is continuous.

When LHD & RHD both exist and are equal then f is said to be derivable or differentiable.

Parametric differentiation: if y =f(t), x= g(t) then,

Logarithmic differentiation: If y = f(x)g(x) then take log on both the sides.

Write logy = g(x) log[f(x)]. Differentiate by applying suitable rule for differentiation.

If y is sum of two different exponential function u and v, i.e. y = u + v. Find by using logarithmic

differentiation separately then evaluate.

Higher Derivatives: The derivative of first derivative is called as the second derivative. It is denoted by y2 or

y or

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FORMULAE

1. 0d

cdx

2. 1n nd

x nxdx

3.1

1n n

d n

dx x x

4. 1d

xdx

5.2

1 1d

dx x x

6. 1

2

dx

dx x

7. logx xd

a a adx

8. x xd

e edx

9. 1

logd

xdx x

10. sin cosd

x xdx

11. cos sind

x xdx

12. 2tan secd

x xdx

13. cos cos cotd

ec x ec x xdx

14. s tand

ec x sec x xdx

15. 2cotd

x cosec xdx

16. 12

1sin

1

dx

dx x

17. 12

1cos

1

dx

dx x

18. 1 21

tan1

dx

dx x

19. 1 21

cot1

dx

dx x

20. 12

1sec

1

dx

dx x x

21. 12

1cos

1

dec x

dx x x

MEAN VALUE THEOREMS

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ROLLES THEOREM

STATEMENT Let f be a real valued function defined on the closed interval [a, b] such that

1. It is continuous on the closed interval [a, b].

2. It is differentiable on the open interval (a, b).

3. f(a) = f(b).

Then, there exist a real number c

GEOMETRICAL INTERPRETATION OF ROLLES THEOREM

Let f(x) be a real valued function defined on [a, b] such that the curve y=f(x) is a continuous curve between

points A (a, f(a)) and B(b, f(b)) and it is possible to draw a unique tangent at every point on the curve y=f(x)

between A and B. Also, there exists at least one ordinates at the end points of the interval [a, b] are equal. Then

there exists at least one point (c, f(c)) lying between A and B on the curve. Let the curve is increasing at A. Now

f(a) = f(b) that means curve has to decrease from some point. Since it is continuous and differentiable so, it must

take smooth turn from somewhere (without forming edge) at that point the tangent parallel to X- Axis i.e. at

that point f(x) = 0. Similarly it can be shown easily if starts decreasing at A then also there exist at least one

point in the given interval s=where f(x) = 0. If it is neither increasing nor decreasing i.e. a parallel line to the X-

axis then, there will be infinite no. of points where f(x) = 0.

Working Rule:

1. We first check whether f(x) satisfies conditions of Rolles Theorem or not.

2. Differentiate the function

3. Equate the derivatives with Zero to get x.

4. If x belongs to given interval then, Rolles Theorem verified.

The following rules may be helpful while solving the problem.

1. A polynomial function is everywhere continuous and differentiable

30

2. The exponential function, sine and cosine function are everywhere continuous and differentiable.

3. Logarithmic function is continuous and differentiable in its domain.

4. Tan x is not continuous

Chapter 6 Application of Derivatives Motion in a straight Line

Suppose a particle is moving in a straight line

Take O as origin in time t the position of the particle be at A where OA= s and in time t+t the position of the particle be

at B where OB=s+s clearly the directed distance of the particle from O is function of time t

s=f (t)

s+s=f (t+t)

Algorithm:

1. Solving any problem first, see what are given data.

2. What quantity is to be found?

3. Find the relation between the point no. (1) & (2).

4. Differentiate and calculate for the final result.

APPROXIMATIONS, DIFFERENTIALS AND ERRORS

Absolute error -

Relative error

Percentage error - Approximation -

1. Take the quantity given in the question as y +k = f(x + h)

2. Take a suitable value of x nearest to the given value.

31

3. Calculate y= f(x) at the assumed value of x.

4. Using differential calculate

5. find the approximate value of the quantity asked in the question as y +k , from the values of y and

evaluated in step 3 and 4.

Tangents and normals

Slope of the tangent to the curve y = f(x) at the point (x0,y0) is given by m.

Equation of the tangent to the curve y = f(x) at the point (x0,y0) is (y - y0) = m (x x0)

Slope of the normal to the curve y = f(x) at the point (x0,y0) is given by -1 / m

Equation of the normal to the curve y = f(x) at the point (x0,y0) is (y - y0) = ( - 1 /m) (x x0)

To curves y = f(x) and y = g(x) are orthogonal means their tangents are perpendicular to each other at the

point of contact

INCREASING AND DECREASING FUNCTION

STRICTLY INCREASING FUNCTION: A function f(x) is said to be a strictly increasing function on (a, b) if

x1 f(x1)

32

x1 f(x1)>f(x2) for all x1, x2(a, b)

Thus, f(x) is strictly increasing on (a, b) if the values of f(x) decrease with increase in the value of x. Graphically, f(x) is

decreasing on (a, b) if the graph y=f(x) moves down as x moves to the right. The graph in fig is the graph of strictly decreasing

on (a, b).

NECESSARY AND SUFFICIENT CONDITION FOR MONOTONICITY OF FUNCTION

In this section we intend to see how we can use derivative of a function to determine where it is increasing and where it is

decreasing.

Necessary condition: if f(x) is an increasing function on(a, b) then the tangent at every point on the curve y=f(x) makes an

acute angle with the positive direction of x- axis.

It is evident from fig that if f(x) is a decreasing function on (a, b), then tangent at every point on the curve y=f(x) makes and

obtuse angle with the positive direction of x- axis.

Thus, f(x) > 0(0( 0 for all , then f(x) increasing on (a, b)

(b) If f(x) < 0 for all , then f(x) decreasing on (a, b)

ALGORITHM FOR FINDING THE INTERVAL IN WHICH A FUNCTION IS INCREASING OR DECREASING.

33

Let f(x) be a real function. To find the intervals of monotonicity of f(x) we proceed as follows:

Step 1 find f(x).

Step II Put f(x)>0 and solve this inequation. For the values of x obtained in step II f(x) is increasing and for the remaining points in its domain it is decreasing.

Or 1. Calculate f(x) =0 for critical points 2. Let c1, c2, c3 are the roots of f(x) = 0. 3. Cut the no line at c1, c2, c3 4. Write the interval in table 5. Consider any point in the interval 6. See the sign of f(x) in that interval and accordingly determine the function is increasing or decreasing in that

interval. Maximum Let f(x) be the real function define on the interval I. then f(x)is said to have the maximum value in I, if there

exists a point a in I such that f(x)f(a) for all x

In such a case, the number f(a) is called the minimum value of f(x) in the interval I and

the point a is called a point of minimum value of f in the interval I.

LOCAL MAXIMA AND LOCAL MINIMA

1.

2. Local maximum A function f(x) is said to attain a local maximum at x=a if there exists a neighborhood (a-,a+-)

of a such that

f(x)

34

f(x)-f(a)>0 for all x in (a-,a+)

The value of the function at x= a i.e. f(a) is called the local minimum value of f(x) at x=a.

The points at which attains either the local maximum or local minimum values are called the extreme values of

f(x).

First Derivatives Test for finding the Local Extremum

Let y=f(x) be a function defined on the interval I and let f be derivable at an interior point c of I. Let f have an extreme

value at x=c,

Case I. Let f have a local maximum value at x=c, then f is an increasing function in the left nbd of x=c .

Thus, f(x) changes continuously from +ve to ve as increases through c.

Case II. Let f have a local minimum value at x=c, then f is an increasing function in the left nbd of x=c

f(x) changes continuously from +ve to +ve as increases through c.

Working rule for first derivative test

Find the sign of f(x) when x is slightly < c and when x is slightly >c.

If f(x) changes sign from +ve to ve as x increases through c, then f has a local maximum value at x=c.

If f(x) changes sign from -ve to +ve as x increases through c, then f has a local minimum value at x=c.

Second Derivative Test for Finding Local Maxima and Local Minima

Suppose a quantity y depends on the another quantity x in a manner shown in fig. it becomes maximum at x1 and

minimum at x2. At these points the tangent to the curve is parallel to the x-axis and hence its slope is 0.

Maxima and minima by using second derivative test

35

Just before the maximum the slope is positive, at the minimum it is zero and just after the maximum it is negative.

Thus decreases at a maximum and hence the rate of change of is negative at a maximum.

Minima

Similarly, at a minimum the slope changes from negative to positive.

Hence with increases of x. the slope is increasing that means the rate of change of slope w.r.t x is positive.

Working Rule:

1. Find f(x).

2. Solve f(x) = 0 within the domain to get critical point let one of the value of x = c.

3. Calculate f(x) at x = c.

4. If f(c) > 0 then, f(x) is minimum at x = c, and if f(c) < 0, then f(x) is maximum at x= c.

Maxima

Similarly, at a maximum the slope changes from positive to negative.

Working Rule:

5. Find f(x).

6. Solve f(x) = 0 within the domain to get critical point let one of the value of x = c.

7. Calculate f(x) at x = c.

8. If f(c) < 0 then, f(x) is minimum at x = c, and if f(c) < 0, then f(x) is maximum at x= c.

Application of Maxima and Minima to problems

Working rule

(i) In order to illustrate the problem, draw a diagram, if possible. Distinguish clearly between the variable and

constants.

(ii) If y is the quantity to be maximized or minimized, express y in terms of a single independent variable with the

help of given data.

(iii) Get dy/dx and d2y/dx2, then equate dy/dx=0 get the value of x.

(iv) Determine the sign of d2y/dx2 at x and proceed.

36

Chapter 7 Integration

Integration is a way of adding slices to find the whole. Integration can be used to find areas,

volumes, central points and many useful things.

The Chain Rule tells us that derivative of g(f(x)) = g'(f(x))f'(x). But what about going the other

way around? What happens if you want to integrate g'(f(x))f'(x)? Well, that's what the

"reverse chain rule" is for. As you can see, a lot of integrals you'll run into can be solved this

way. It is also another way of doing substitution without having to substitute.

Trigonometric substitution

Another substitution technique where we substitute variables with trigonometric functions.

This allows us to leverage some trigonometric identities to simplify the expression into one

that it is easier to take the anti-derivative of.

Division and partial fraction expansion

When you're trying to integrate a rational expression, the techniques of partial fraction

expansion and algebraic long division can be *very* useful.

Integration by parts

When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the

product rule. In this tutorial, we use the product rule to derive a powerful way to take the

anti-derivative of a class of functions--integration by parts.

If u and g are two functions of x then the integral of product of two functions = 1st function -

integral of the product of the derivative of 1st function and the integral of the 2nd function

Write the given integral where you identify the two functions u(x) and v(x) as the 1st

and 2nd function by the order

I inverse trigonometric function

L Logarithmic function

A Algebraic function

T Trigonometric function

E Exponential function

37

Note that if you are given only one function, then set the second one to be the

constant function g(x)=1. integrate the given function by using the formula

DEFINITE INTEGRAL AS LIMIT OF A SUM

Definite integral is closely related to concepts like antiderivative and indefinite integrals. In

this section, we shall discuss this relationship in detail.

Definite integral consists of a function f(x) which is continuous in a closed interval [a, b] and

the meaning of definite integral is assumed to be in context of area covered by the function

f from (say) a to b.

An alternative way of describing is that the definite integral is a

limiting case of the summation of an infinite series, provided f(x) is continuous on [a, b]

The converse is also true i.e., if we have an infinite series of the above form, it can be

expressed as a definite integral.

The Fundamental Theorem of Calculus justifies our procedure of evaluating an

antiderivative at the upper and lower limits of integration and taking the difference.

38

Fundamental Theorem of Calculus

Let f be continuous on [a b]. If F is any antiderivative for f on [a b], then

= F(b)F(a)

Properties

Property 1:

Property 2:

Property 3:

Property4:

Property5:

Property6:

Property 7:

Property 8:

39

1. 1

, 11

nn xx dx n

n

2. 1

1 1

1n ndx

x n x

3. dx x 4.1

logdx xx

5.log

xx aa dx

a 6.

x xe dx e

7. sin cosx dx x 8. cos sinx dx x

9. 2sec tanx dx x 10.2cos cotec x dx x

11. sec tan secx x dx x 12. cos cot cosec x x dx ec x

13. tan log secx dx x 14. cot log sinx dx x

15. sec log sec tanx dx x x 16. cos log cosec cotecx dx x x

17. 12 2

1tan

dx x

a x a a

18. 2 21

log2

dx a x

a x a a x

19.2 2

1log

2

dx x a

x a a x a

20.

1

2 2sin

dx x

aa x

21. 2 22 2

logdx

x x ax a

22.2 2

2 2log

dxx x a

x a

23.2

2 2 2 2 1sin2 2

x a xa x dx a x

a

24.2

2 2 2 2 2 2log2 2

x ax a dx x a x x a

25.2

2 2 2 2 2 2log2 2

x ax a dx x a x x a

40

Chapter 8. Application of Integration

A plane curve is a curve that lies in a single plane. A plane curve may be closed or open.

The area between the curve defined by a positive function and the x axis between

two values of y is called the definite integral of f between these values. The area of a

rectangle is the product of length and breadth the area under a curve will be generated. This

method is generalised to define a variety of integrals that do not describe area.

The standard approach deals with a more general class of functions by imagining that we

divide the interval between a and b into a large number of small strips and estimate the

area of each strip to be the product of its width and some value of f(x) for x within the strip.

The area will then be something like the sum of the areas of the strips. If we then let the

maximum strip length go to zero, we can hope to find the resulting sum of strip areas

approach the true area.

The standard way to do this is to let the i-th strip begin at xi-1 and end at xi; the area of that

strip is estimated as (xi-xi-1)f(x'i) with x'i anywhere in the strip.

A Riemann sum is a sum of the form just indicated: it is a sum over strips of the width of the

strip times a value of the f(x) within the strip. The function is said to be Riemann

integrable if the sum of the area of the strips approaches a constant independent of which

arguments are used within each strip to estimate the area of the strip, as maximum strip

width goes to zero.

The notation used above can be understood from this approach; we are summing the area

of the individual strips, which for a very small interval around x of size dx is estimated to

be f(x)dx, and summing this over all such strips. The integral sign represents the sum which

is not an ordinary sum, but the limit of ordinary sums as the size of the intervals goes to

zero.

http://mathworld.wolfram.com/Curve.htmlhttp://mathworld.wolfram.com/Plane.html

41

Suppose we have a non-negative function f of the variable x, defined in some domain that includes the interval [a, b] with a < b. If f is sufficiently well behaved, there is a well defined area enclosed between the lines x = a, x = b, y = 0 and the curve y = f(x).

That area is called the definite integral of f dx between x = a and x = b (of course only for those functions for which it makes sense).

It is usually written as

If c lies between a and b we obviously have

Area between two curves:-

There are actually two cases that we are going to be looking at.

In the first case we want to determine the area between y = f(x) and y = g(x) on the interval

[a,b]. We are also going to assume that .

The second case is almost identical to the first case. Here we are going to determine the

area between x = f(y) and x =g(y) on the interval [c,d] with .

In this case the formula is,

The area of a triangle is an extension of area of a plane curve.

42

Chapter 9 DIFFERENTIAL EQUATIONS

Definition: An equation containing an independent variable, a dependent variable and

differential coefficients of the dependent variable with respect to the independent variable

is called a differential equation DE.

The ORDER of a differential equation is the highest derivative that appears in the equation.

The DEGREE of a differential equation is the power or exponent of the highest derivative

that appears in the equation.

Unlike algebraic equations, the solutions of differential equations are functions and not just

numbers.

Initial value problem is one in which some initial conditions are given to solve a DE.

To form a DE from a given equation in x and y containing arbitrary constants (parameters)

1. Differentiate the given equation as many times as the number of arbitrary

constants involved in it .

2. Eliminate the arbitrary constant from the equations of y, y, y etc.

A first order linear differential equation has the following form:

The general solution is given by where

called the integrating factor. If an initial condition is given, use it to find the constant C.

Here are some practical steps to follow:

1. If the differential equation is given as , rewrite it in the form

, where

2. Find the integrating factor .

3. Evaluate the integral

4. Write down the general solution .

43

5. If you are given an IVP, use the initial condition to find the constant C.

VARIABLE SEPARABLE FORM The differential equation of the form is called separable,

if f(x,y) = h(x) g(y); that is, - - - (I)

In order to solve it, perform the following STEPS:

1. Solve the equation g(y) = 0, which gives the constant solutions of (I);

2. Rewrite the equation (I) as, and, then,

3. integrate to obtain G(y) = H(x) + C

4. Write down all the solutions; the constant ones obtained from (1) and the ones given

in (3);

5. If you are given an IVP, use the initial condition to find the particular solution. Note

that it may happen that the particular solution is one of the constant solutions given

in (1). This is why Step 4 is important.

Homogeneous Differential Equations

A function which satisfies for a given n is called a homogeneous function of order n.

A differential equation is called a HOMOGENEOUS DIFFERENTIAL EQUATION if it can be

written in the form M(x,y)dx + N(x,y)dy = 0 where M and N are of the same degree.

A first-order ordinary differential equation is said to be homogeneous if it can be written

in the form) Such equations can be solved by the change of variables which transforms

the equation into the separable equation

Steps for Solving a Homogeneous Differential Equation

1. Rewrite the differential in homogeneous form or

2. Make the substitution y = vx or x = vy where v is a variable.

3. Then use the product rule to get

4. Substitute to rewrite the differential equation in terms of v and x or v and y only

5. Follow the steps for solving separable differential equations.

6. Re-substitute v = y/x or v = x / y.

7.

http://mathworld.wolfram.com/First-OrderOrdinaryDifferentialEquation.html

44

Chapter 10 VECTORS

A quantity that has magnitude as well as direction is called a vector. It is denoted by

directed line segment, where A is the initial point and B is the terminal point . The

distance AB is called the magnitude denoted by and the vector is directed from A to

B.

FIXED VECTOR is that vector whose initial point or tail is fixed. It is also known as

localised vector. For example, the initial point of a position vector is fixed at the

origin of the coordinate axes. So, position vector is a fixed or localised vector.

FREE VECTOR is that vector whose initial point or tail is not fixed. It is also known as a

non-localised vector. For example, velocity vector of particle moving along a straight

line is a free vector.

Co-initial vectors: Vectors having the same initial point are called as co-initial vectors.

Negative of a vector: A vector whose magnitude is the same of that of a given vector but in

the opposite direction is called as the negative of the given vector.

POSITION VECTOR gives the position of a point with reference to the origin of the

coordinate system. COLLINEAR VECTORS are those vectors that act either along the

same line or along parallel lines. These vectors may act either in the same direction

or in opposite directions. PARALLEL VECTORS are two collinear

vectors acting along the same direction. EQUAL VECTORS Two

vectors are said to be equal if they have the same magnitude and direction.

ADDITION OF VECTORS:

1. Triangle law of vectors for addition of two vectors. If two vectors can be represented

both in magnitude and direction by the two sides of a triangle taken in the same

order, then the resultant is represented completely, both in magnitude and

direction, by the third side of the triangle taken in the opposite order.

Corollary: 1) If three vectors are represented by the three sides of a triangle taken in order,

then their resultant is zero.

2) If the resultant of three vectors is zero, then these can be represented completely by the

three sides of a triangle taken in order.

2. Parallelogram law of vectors for addition of two vectors.If two vectors

are completely represented by the two sides OA and OB respectively of a

45

parallelogram. Then, according to the law of parallelogram of vectors, the diagonal

OC of the parallelogram will be resultant , such that

MULTIPLICATION OPERATIONS FOR VECTORS:

1. MULTIPLICATION OF A VECTOR BY A SCALAR, When a vector . is multiplied by a real

number, say m, then we get another vector m. The magnitude of mis m times the

magnitude of . If m is positive, then the direction of m is the same as that of . If m is

negative, then the direction of m is opposite to that of .

2. SCALAR MULTIPLICATION OF TWO VECTORS to yield

a b = |a| |b| cos()where:

|a| is the magnitude (length) of vector a

|b| is the magnitude (length) of vector b

is the angle between a and b

It is important to remember that there are two different angles between a pair of

vectors, depending on the direction of rotation. However, only the smaller of the two is

taken in vector multiplication

Some Properties of the Dot Product

i.i = j.j = k.k = 1 and i.j = j.i = i.k = k.i = j.k = k.j = 0.

The dot product is commutative:

The dot product is distributive over vector addition:

The dot product is bilinear:

When multiplied by a scalar value, dot product satisfies:

Two non-zero vectors a and b are perpendicular if and only if a b = 0.

The dot product does not obey the cancellation law: If a b = a c and a 0: then a

(b c) = 0:

If a is perpendicular to (b c), we can have (b c) 0 and therefore b c.

http://en.wikipedia.org/wiki/Commutativehttp://en.wikipedia.org/wiki/Distributivehttp://en.wikipedia.org/wiki/Bilinear_formhttp://en.wikipedia.org/wiki/Perpendicularhttp://en.wikipedia.org/wiki/If_and_only_ifhttp://en.wikipedia.org/wiki/Cancellation_law

46

Vector product:

The Cross Product a b of two vectors is another vector that is at right angles to both.

a b = |a| |b| sin() n

|a| is the magnitude (length) of vector a

|b| is the magnitude (length) of vector b

is the angle between a and b

n is the unit vector at right angles to both a and b

The cross product is anticommutative: a x b = - b x a

| a x b | is the area of the parallelogram formed by a and b

Distributive over addition. This means that a (b + c) = a b + a c

For two parallel vectors a b = 0

i i = j j = k k = 0.

The cross product of a vector with itself is the null vector

If a b = a c and a 0 then: (a b) (a c) = 0 and, by the distributive law above:

a (b c) = 0 Now, if a is parallel to (b c), then even if a 0 it is possible that (b c) 0

and therefore that b c.

SCALAR TRIPLE PRODUCT

It is defined for three vectors in that order as the

scalar . ( )It denotes the volume of the

parallelopiped formed by taking a, b, c as the co-

terminus edges.

i.e. V = magnitude of . = | . |

The value of scalar triple product depends on the cyclic order of the vectors and is

independent of the position of the dot and cross. These may be interchange at pleasure.

However and anti-cyclic permutation of the vectors changes the value of triple product in sign

but not a magnitude.

https://www.mathsisfun.com/algebra/vector-unit.html

47

Properties of Scalar Triple Product:

* If, then their scalar triple product is given by

i.e. position of dot and cross can be interchanged without altering the product.

* , , in that order form a right handed system if . . > 0

i.e. scalar triple product is 0 if any two vectors are equal.

i.e. scalar triple product is 0 if any two vectors are parallel.

Show that

If three vectors are coplanar then .

48

Chapter 11. Three Dimensional Geometry

In space, any point can be expressed as P(x, y, z). The equations of a line passing through a

given point and parallel to a given direction are given by .

The equations of a line which passes through two given points are

.

Two lines in space may be parallel lines or intersecting lines or skew lines. Two parallel lines

or intersecting lines are in the same plane. If two lines do not intersect then there is no

angle between them.

Angle between two lines

Finding the angle between two lines in 2D is easy, just find the angle of each line with the x-

axis from the slope of the line and take the difference. In 3D it is not so obvious, but it can

be shown (using the Cosine Rule) that the angle between two vectors a and b is given by:

Cos =

Unfortunately this gives poor accuracy for angles close to zero; for instance an angle of

1.00E-7 radians evaluates with an error exceeding 1%, and 1.00E-8 radians evaluates as

zero. A similar formula using the sine of the angle Sin = has similar problems with

angles close to 90 degrees.

But combining the two gives Tan = which is accurate for all angles, and since the

(|a||b|) values cancel out the computation time is similar to the other expressions.

49

Shortest distance between two skew lines:

The two key points two remember here are:

1. The shortest line between two lines is perpendicular to both

2. When two vectors are crossed, the result is a vector that is perpendicular to both

Thus the vector representing the shortest distance between AB and CD will be in the same

direction as (AB X CD), which can be written as a constant times (AB X CD). We can then

equate this vector to a general vector between two points on AB and CD, which can be

50

obtained from the vector equations of the two lines. Solving the three equations obtained

simultaneously, we can find the constant and the shortest distance.

Plane:

A plane is a flat (not curved) two dimensional space embedded in a higher number of

dimensions.

In two dimensional space there is only one plane that can be contained within it and that is

the whole 2D space.

Three dimensional space is a special case because there is a duality between points (which

can be represented by 3D vectors) and planes (also represented by 3D vectors). We can

visualise the vector representing a plane as a normal to the plane.

The angle between two planes is defined as the angle between their normals. It is given by

Coplanarity of four points:-

Coplanar points are three or more points which lie in the same plane where a plane is a flat

surface which extends without end in all directions. It's usually shown in math textbooks as

a 4-sided figure. You can see that points A, B, C and D are all coplanar points on a single

plane.

51

The concept of coplanar points may seem simple, but sometimes the questions about it may

become confusing. With a little bit of geometry knowledge and some real-world examples, it

can be mastered even the most challenging questions about coplanar points.

Any three points in 3-dimensional space determine a plane. This means that any group of

three points determines a plane, even if all the points don't look like they're located on the

same flat surface.

Let's look at another real life example. The tissue paper box is covered in leaves. Points have

been placed at the tips of four leaves and labeled W, X, Y and Z. From the first picture, we

can see that points X, Y and Z are coplanar. They form a triangle, and you can visualize that.

If you were to cut through the tissue box and pass through these points, you would have a

piece of the tissue box that would have a plane figure, a triangle, as its base.

52

Chapter 12 Linear Programming

Linear programming is the process of taking various linear inequalities relating to some

situation, and finding the "best" value obtainable under those conditions. A typical example

would be taking the limitations of materials and labor, and then determining the "best"

production levels for maximal profits under those conditions.

In "real life", linear programming is part of a very important area of mathematics called

"optimization techniques". This field of study (or at least the applied results of it) are used

every day in the organization and allocation of resources. These "real life" systems can have

dozens or hundreds of variables, or more. In algebra, though, you'll only work with the

simple (and graphable) two-variable linear case.

The general process for solving linear-programming exercises is to graph the inequalities

(called the "constraints") to form a walled-off area on the x,y-plane (called the "feasibility

region"). Then you figure out the coordinates of the corners of this feasibility region (that is,

you find the intersection points of the various pairs of lines), and test these corner points in

the formula (called the "optimization equation") for which you're trying to find the highest

or lowest value.

The Decision Variables: The variables in a linear program are a set of quantities that need to

be determined in order to solve the problem; i.e., the problem is solved when the best

values of the variables have been identified. The variables are sometimes called decision

variables because the problem is to decide what value each variable should take. Typically,

the variables represent the amount of a resource to use or the level of some activity.

The Objective Function: The objective of a linear programming problem will be to maximize

or to minimize some numerical value. This value may be the expected net present value of a

project or a forest property; or it may be the cost of a project;

The Constraints: Constraints define the possible values that the variables of a linear

programming problem may take. They typically represent resource constraints, or the

minimum or maximum level of some activity or condition.

Linear Programming Problem Formulation: We are not going to be concerned with the

question of how LP problems are solved. Instead, we will focus on problem formulation

translating real-world problems into the mathematical equations of a linear program and

interpreting the solutions to linear programs. We will let the computer solve the problems

for us. This section introduces you to the process of formulating linear programs. The basic

53

steps in formulation are: 1. Identify the decision variables; 2. Formulate the objective

function; and 3. Identify and formulate the constraints. 4. A trivial step, but one you should

not forget, is writing out the non-negativity constraints. The only way to learn how to

formulate linear programming problems is to do it.

The corner points are the vertices of the feasible region. Once you have the graph of the

system oflinear inequalities, then you can look at the graph and easily tell where the corner

points are. You may need to solve a system of linear equations to find some of the

coordinates of the points in the middle.

54

Chapter 13 Probability

Definition: Two events are dependent if the outcome or occurrence of the first affects

the outcome or occurrence of the second so that the probability is changed.

The conditional probability of an event B in relationship to an event A is the probability that

event B occurs given that event A has already occurred. The notation for conditional

probability is P(B|A) [pronounced as probability of event B given A].

The notation used above does not mean that B is divided by A. It means the probability of

event B given that event A has already occurred. The notation used above does not mean

that B is divided by A. It means the probability of event B given that event A has already

occurred.

When two events, A and B, are dependent, the probability of both occurring is:

P(A and B) = P(A) P(B|A)

Definition: Two events, A and B, are independent if the fact that A occurs does not

affect the probability of B occurring.

Some other examples of independent events are:

Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.

Choosing a marble from a jar AND landing on heads after tossing a coin.

Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the

second card.

Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.

Multiplication Rule 1: When two events, A and B, are independent, the probability of

both occurring is:

P(A and B) = P(A) P(B)

Summary: The probability of two or more independent events occurring in

sequence can be found by computing the probability of each

event separately, and then multiplying the results together.

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Summary: The probability of two or more independent events occurring in

sequence can be found by computing the probability of each

event separately, and then multiplying the results together.

Summary: The probability of two or more independent events occurring in sequence

can be found by computing the probability of each event separately, and then multiplying

the results together.

Bayes theorem

Bayes' theorem (also known as Bayes' rule) is a useful tool for calculating conditional

probabilities. Bayes' theorem can be stated as follows:

Bayes' theorem. Let A1, A2, ... , An be a set of mutually exclusive events that together form

the sample space S. Let B be any event from the same sample space, such that P(B) > 0.

Then,

P( Ak | B ) =

P( Ak B )

P( A1 B ) + P( A2 B ) + . . . + P(

An B )

Bayes' theorem (as expressed above) can be intimidating. However, it really is easy to use.

The remainder of this lesson covers material that can help you understand when and how to

apply Bayes' theorem effectively.

When to Apply Bayes' Theorem

Part of the challenge in applying Bayes' theorem involves recognizing the types of problems

that warrant its use. You should consider Bayes' theorem when the following conditions

exist.

The sample space is partitioned into a set of mutually exclusive events { A1, A2, . . . ,

An }.

Within the sample space, there exists an event B, for which P(B) > 0.

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The analytical goal is to compute a conditional probability of the form: P( Ak | B ).

You know at least one of the two sets of probabilities described below.

P( Ak B ) for each Ak

P( Ak ) and P( B | Ak ) for each Ak

Random Variable

When the value of a variable is determined by a chance event, that variable is called

a random variable.

Discrete vs. Continuous Random Variables

Random variables can be discrete or continuous.

Discrete. Within a range of numbers, discrete variables can take on only certain

values. Suppose, for example, that we flip a coin and count the number of heads.

The number of heads will be a value between zero and plus infinity. Within that

range, though, the number of heads can be only certain values. For example, the

number of heads can only be a whole number, not a fraction. Therefore, the number

of heads is a discrete variable. And because the number of heads results from a

random process - flipping a coin - it is a discrete random variable.

Continuous. Continuous variables, in contrast, can take on any value within a range

of values. For example, suppose we randomly select an individual from a population.

Then, we measure the age of that person. In theory, his/her age can take

on any value between zero and plus infinity, so age is a continuous variable. In this

example, the age of the person selected is determined by a chance event; so, in this

example, age is a continuous random variable.

Probability Distribution

A probability distribution is a table or an equation that links each possible value that

a random variable can assume with its probability of occurrence.

Discrete Probability Distributions

The probability distribution of a discrete random variable can always be represented by a

table. For example, suppose you flip a coin two times. This simple exercise can have four

possible outcomes: HH, HT, TH, and TT. Now, let the variable X represent the number of

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heads that result from the coin flips. The variable X can take on the values 0, 1, or 2; and X is

a discrete random variable.

The table below shows the probabilities associated with each possible value of X. The

probability of getting 0 heads is 0.25; 1 head, 0.50; and 2 heads, 0.25. Thus, the table is an

example of a probability distribution for a discrete random variable.

Number of

heads, x

Probability,

P(x)

0 0.25

1 0.50

2 0.25

Note: Given a probability distribution, you can find cumulative probabilities. For example,

the probability of getting 1 or fewer heads [ P(X < 1) ] is P(X = 0) + P(X = 1), which is equal to

0.25 + 0.50 or 0.75.

Continuous Probability Distributions

The probability distribution of a continuous random variable is represented by an equation,

called the probability density function (pdf). All probability density functions satisfy the

following conditions:

The random variable Y is a function of X; that is, y = f(x).

The value of y is greater than or equal to zero for all values of x.

The total area under the curve of the function is equal to one.

Mean and Variance

Just like variables from a data set, random variables are described by measures of central

tendency (like the mean) and measures of variability (like variance). This lesson shows how

to compute these measures for discrete random variables.

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Mean of a Discrete Random Variable

The mean of the discrete random variable X is also called the expected value of X.

Notationally, the expected value of X is denoted by E(X). Use the following formula to

compute the mean of a discrete random variable.

E(X) = x = [ xi * P(xi) ]

where xi is the value of the random variable for outcome i, x is the mean of random

variable X, and P(xi) is the probability that the random variable will be outcome i.

Variability of a Discrete Random Variable

The equation for computing the variance of a discrete random variable is shown below.

2 = { [ xi - E(x) ]2 * P(xi) }

where xi is the value of the random variable for outcome i, P(xi) is the probability that the

random variable will be outcome i, E(x) is the expected value of the discrete random

variable x.

Binomial Distribution

To understand binomial distributions and binomial probability, it helps to understand

binomial experiments and some associated notation; so we cover those topics first.

Binomial Experiment

A binomial experiment is a experiment that has the following properties:

The experiment consists of n repeated trials.

Each trial can result in just two possible outcomes. We call one of these outcomes a

success and the other, a failure.

The probability of success, denoted by P, is the same on every trial.

The trials are independent that is, the outcome on one trial does not affect the

outcome on other trials.

Consider the following statistical experiment. You flip a coin 2 times and count the number

of times the coin lands on heads. This is a binomial experiment because:

The experiment consists of repeated trials. We flip a coin 2 times.

Each trial can result in just two possible outcomes - heads or tails.

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The probability of success is constant - 0.5 on every trial.

The trials are independent; that is, getting heads on one trial does not affect

whether we get heads on other trials. Binomial Distribution

A binomial random variable is the number of successes x in n repeated trials of a binomial

experiment. The probability distribution of a binomial random variable is called a binomial

distribution.

The binomial distribution describes the behaviour of a count variable X if the following

conditions apply:

1: The number of observations n is fixed.

2: Each observation is independent.

3: Each observation represents one of two outcomes ("success" or "failure").

4: The probability of "success" p is the same for each outcome.

Mean and Variance of the Binomial Distribution

The binomial distribution for a random variable X with parameters n and p represents the

sum of n independent variables Z which may assume the values 0 or 1. If the probability that

each Z variable assumes the value 1 is equal to p, then the mean of each variable is equal

to 1*p + 0*(1-p) = p, and the variance is equal to p(1-p). By the addition properties for

independent random variables, the mean and variance of the binomial distribution are

equal to the sum of the means and variances of the n independent Z variables,

so

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1. If f(x) = x + 7 and g(x) = x 7 then find (f g) (7).

2. If the function f is an invertible function find the inverse of f(x) = .

3. If the function f is an invertible function defined as f(x) = then write f -1(x).

4. Show that the relation R in the set of real numbers defined as R = {(a, b): a } is neither

reflexive, symmetric nor transitive.

5. Let T be the set of all triangles in a plane with R as relation in T given by R = {(T1, T2): T1 T2}.

Show that R is an equivalence relation.

6. Let Z be the set of all integers and R be the relation on Z given by R = {(a, b): a,b Z, a b is

divisible by 5}. Show that R is an equivalence relation.

7. If f: R R defined by f(x) = (3 x3) then find (f f) (x).

8. Show that the relation R in the set A = { 1, 2, 3, 4, 5} given by R = {(a, b): is even} is an

equivalence relation.

9. Show that the relation S in the set A = { x : x Z, } given by S = {(a,b): a,b Z,

is divisible by 4} is an equivalence relation.

10. Is the binary operation defined on N given by for all a,b N, commutative?

ii) Is associative?

11. If the binary operation on Z is defined by then find .

12. Let be the binary operation on N given by . Find .

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13. Let f:N N defined by f(n) = for all n N. Find whether the

function is bijective.

14. Let be the binary operation on Q given by =2a + b - ab. Find 3 4.

15. What is the range of f(x) = ?

16. Consider f: R defined by f(x) = 9x2 + 6x 5. Show that f is invertible with

.

17. Let A = and be a binary operation on A defined by (a, b) (c, d) = (a+c, b+d). Show

that is commutative and associative. Also find the identity element for on A if any.

18. If f: R R and g: R R are given by f(x) = sin x and g(x) = 5 x2 find ( )(x).

19. If f(x) = 27x3 and g(x) = find ( ) (x).

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1. Evaluate

2. Evaluate

3. Evaluate

4. Evaluate

5. Evaluate

6. Evaluate

7. Evaluate

8. Evaluate

9. Evaluate

10. Evaluate

11. What is the domain of sin-1 x?

12. Prove that tan-11 + tan-12 + tan-13 = .

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13. Prove that tan-1 + tan-1 + tan-1 +tan-1 = .

14. Prove that tan-1 + tan-1 = .

15. Prove that .

16. Prove that .

17. Prove that tan-1x+tan-1 = .

18. Prove that tan-1 .

19. Prove that .

20. Prove that .

21. Solve tan-12x + tan-13x = .

22. Solve tan-1(x+1) + tan-1(x-1) = tan-1 .

23. Solve .

24. Solve .

25. Solve 2tan-1(Cos x) = tan-1(2 Cosecx)

26. Solve

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27. Prove that .

1. Find x and y if

2. Find x and y if

3. Find x if

4. Find y if

5. Find y if

6. Find y if

7. Using elementary transformations find the inverse of the matrix .



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13. Solve 2x y + z = 3, - x + 2y z = 1 and x y + 2z = 1.

14. Solve 3x 2y +3z = 8, 2x + y z =1 and 4x 3y+2z = 4.

15. Solve x + y + z = 4, 2x + y 3z = -9 and 2x y + z = -1.

16. Solve 2x 3y + 5z = 11, 3x + 2y4z = -5 and x+y-2z=-3.

17. Solve x + y + z = 6, x + 2z = 7 and 3x + y + z = 12.

18. Solve x+2y -3z=-4, 2x+ 3y +2z = 2 and 3x 3y -4z = 11.

19. If A = find A-1. Using A-1 solve 2x 3y + 5z = 16, 3x + 2y 4z = -4 and

x + y - 2z = -3.

20. If A = find A-1. Using A-1 solve 2x + y + 3z = 9, x + 3y z = 2and -2x + y

+ z = 7.

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Differentiation of composite functions: Differentiation by chain rule:

Find the derivative of the following functions wrt x:

1. (x2 3)5. 2.

4123

xx

3.10

1xx

4.

63x x

5. 21 x 6. 23 4x

7. 1

1

x

x

8.

2 1

2 1

x

x

9. 2 3 5x x 10. (x2+ 1)3 (x3 1)4.

11. 34

21

x

x

12. 1 1

1 1

x x

x x

Differentiation of trigonometric functions Find the derivative of the following functions wrt x:

1. Sin 5x 2. Cos (2x 3)

3. Sin3 (5x + 4) 4. Tan104x

5.1 sin2

1 2

x

sin x

6.

1 s

1

co x

cosx

7. 1 tan3

1 tan3

x

x

8. If y = x tan x then prove that x sin2x dy

dx= y2 + y sin2x.

9. If y = a sin x + b cos x then prove that y2 + 2

dy

dx

= a2 + b2.

10. If y = 1 sin2

1 sin2

x

x

then prove that

dy

dx+ sec2 (

4

-x) = 0.

Differentiation of inverse trigonometric functions Find the derivative of the following functions wrt x:

1. Sin12x 2. Tan-15x

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3. Sin-1 21 x 4. 21sin

21

x

x

5. 1 3sin 3 4x x 6. 11sec 22 1x

7.

1

1s1

xxco

xx

8. 211s

2

xco

9. 1tan sec tanx x 10. cos1tan

1 sin

x

x

11. 1 cos1tan1 cos

x

x

12.

1 sin1tan1 sin

x

x

13. 1tan1

a x

ax

14. 211tan21

x

x

15. 11t1

xco

x

16.

2 21 11tana x

ax

17. 1tan21 1

x

x

18. 1 2tan 1x x

19. 1 11tan1 1

x x

x x

20. 1 sin 1 sin1t1 sin 1 sin

x xco

x x

21. 1 2sin 1 1x x x x

22. 1 3 1 3sin 3 4 cos 4 3x x x x

23. If y = 1sin

21

x

x

then prove that 21 1dyx xydx .

Differentiation of log and exponential functions Find the derivative of the following wrt x:

1. 2x

e 2. 2 3xe

3.21 xe 4. 2xe

5. sinxe 6. sin xe

7. tanx xe 8. x xe e

x xe e

9. cosxe x 10. log sec x

11. 2log 3x 12. 1log xx

13. log logx 14. 21 1

log21 1

x

x

15. 1 cos2

log1 cos2

x

x

16.

1 sinlog

1 sin

x

x

17. log x a x b 18. log cos cotecx x

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19. log s tanecx x 20. log s tan2 2

x xec

21.2log 1x x

22.

2log 1x x

23. log sinaxe bx

Differentiation of Implicit functions

Find dy

dxfrom the following:

1. 2 2 25x y 2. 2 2

12 2

x y

a b

3. 3 3 3x y xy 4. 3 3 2 23x y x y xy 5. 4 4 4x y xy 6. 3 3x y y x

7. x y a

8. If 2 21 log 1y x x x

then prove that 21 1 0.dyx xydx

9. If 1 1 0x y y x then prove that

1

21

dy

dx x

.

10. If 2 21 1 1x y y x then prove that 21

0.21

dy y

dx x

11. If 2 21 1x y a x y then prove that 21

0.21

dy y

dx x

12. If x + y = sin(x + y) then prove that dy

dx= - 1.

13. If y log x = x y then prove that

log.

21 log

dy x

dx x

14. ......x x x x

15. sin sin sin sin ......x x x x

16. log log log log ......x x x x

17. 1

1

1

.....

x

x

xx

Logarithmic differentiation Differentiate the following wrt x:

1. xx 2. logxx

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3. sinxx 4. sin xx

5. cossin xx 6. sinn xta x

7. log xx 8. sin xx

9. 22 3cot

2 2

xxxx x

10. 2 1sin cos2 1

xx xxx

11. 2 2x xx x 12. sin sinx xx x

13. s sinsin sco x xx co x 14. tan sinsin tanx xx x

15. tan sinsin tanx xx x

16. If y xx y c then finddy

dx.

17. If 0y xx y then finddy

dx.

18. If y x yx e then prove that

log

21 log

dy x

dx x

.

19. If

x yy xx e

then finddy

dx.

20. If yx y x then finddy

dx.

21.

...xxx

22.

...sinsinsinxxx

Differentiation of parametric functions

Find dy

dxfrom the following:

1. x= at2, y = 2at

2. 21

21

tx

t

, 2

21

ty

t

.

3. x = a cos , y= b sin

4. x = a (1 cos ), y = a ( + sin )

5. x = a (cos + sin ), y = a (sin cos )

6. x = 2cos cos 2, y = 2 sin sin 2.

7.

Higher Derivatives The derivative of the derivative of a function is called as the second derivative.

2''

2 2

d y d dyy or y or

dx dxdx

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3 2'''

3 3 2

d y d d yy or y or

dxdx dx

3 2

3 2

d y d d y

dxdx dx

1. Find the second derivative of x2 + log x wrt x.

2. If y = ex (sin x + cos x) then prove that y2 2y1 + 2y = 0.

3. If y = sec x + tan x then prove that

2 cos

2 21 sin

d y x

dx x

4. If y = cosec x + cot x then prove that

2 sin

2 21 s

d y x

dx co x

5. If y = a sin 2x + b cos 2x then prove that 4 02y y

6. If y = 1nta x then prove that 21 2 012x y xy

7. If y = 2

1nta x then prove that 2

2 21 2 1 2 012

x y x x y

8. If y = 2

1nsi x then prove that 21 2 012x y xy

9. If y = 1sin xe

then prove that 21 012x y xy y

10. If y = 1nta xe

then prove that 2

2 21 2 1 012

x y x x y y

11. If y = sin (log x) + 3 cos (log x) then prove that 2 012

x y xy y

12. If x = a ( sin ), y = a (1 cos ) then find 2y at =

2

.

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1. Show that the rectangle of maximum area that can be inscribed in a circle is a square.

2. Verify LMVT for the function f(x) = x2 + 2x + 3 for [4, 6].

3. Prove that the curves x = y2 and xy = k intersect at right angles if 8k2=1.

4. Find the equation of the tangent to the curve y = which is parallel to the line 4x

2y+5=0

5. Find the intervals in which the function f(x) = is increasing and decreasing.

6. The length x of a rectangle is decreasing at the rate of 5 cm / min and the width y is

increasing at the rate of 4 cm / min. when x = 8 cm y = 6 cm find the rate of change of

a)perimeter b) area of the rectangle.

7. Find the intervals in which the function f given by f(x) = sin x + cos x, is strictly

increasing or strictly decreasing.

8. Find the intervals in which the function f given by f(x) = sin x - cos x, is strictly

increasing or strictly decreasing.

9. If the sum of the hypotenuse and a side of a right triangle is given, then show that the area

of the triangle is maximum when the angle between them is .

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10. A manufacturer sell x items at a price of Rs each. The cost price of x items is Rs

. Find the number of items he should sell to earn maximum profit.

11. Find the equation of the tangent to the curve y = which is parallel to the line 4x

2y+5=0

12. Find the approximate value of f(2.01) where f(x) = 4x3+5x2+2.

13. Find the points on the curve y = x3 at which the slope of the tangent is equal to the y co-

ordinate of the point.

14. Find the equations of the normal to the curve y = x3+2x+6 which are

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