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1
Study Material
Class XII - Mathematics
2015-2016
2
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
3
,
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
4
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]5
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
6
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
7
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).
8
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
9
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.
10
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.
11
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:
12
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.
13
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.
14
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.
15
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/Cosecant16
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.
17
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 =
18
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)
19
(b)
(c)
3. (a)
(b)
(c)
4. (a)
(b)
(c)
5. (a)
(b)
20
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.html21
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
22
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.
23
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.
24
(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
25
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
26
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.
27
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
28
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
29
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.html41
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.html44
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_law46
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.html47
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.
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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 .
8. Using elementary transformations find the inverse of the matrix .
9. Using elementary transformations find the inverse of the matrix .
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10. Using elementary transformations find the inverse of the matrix .
11. Using elementary transformations find the inverse of the matrix .
12. Using elementary transformations find the inverse of the matrix .
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 parallel to the line
x+14y+4= 0.
15. Find the equation o