RELATIONS & FUNCTIONS
CONCEPTS:1. RELATION 2. TYPES OF RELATIONS a) Empty Relation b) Universal Relation c) Reflexive Relation d) Symmetric Relation e) Transitive Relation 3. EQUIVALENCE RELATION A Relation which is Reflexive, Symmetric and Transitive 4. FUNCTION To recall - identity , constant, linear, polynomial, rati onal, Signum, modulus, greatest integer functions 5. TYPES OF FUNCTIONS a) One – One – One One ( Injective mapping) b) Onto ( Surjective mapping) c) One – One – One One and Onto ( Bijective mapping) 6. COMPOSITION OF FUNCTIONS To recall algebra of functions and to introduce composition of functions. 7. INVERSE OF A FUNCTION Problems based on linear, quadratic and rational functions 8. BINARY OPERATIONS Operation Table 9. PROPERTIES OF BINARY OPERATIONS a) Commutative b) Associative c) Existence of Identity element d) Existence of Inverse element e) To check properties of binary operation using operation table
INVERSE TRIGONOMETRIC FUNCTIONS 1. RECALL ALL THE TRIGONOMETRIC IDENTITIES
2. DOMAIN AND PRINCIPAL VALUE BRANCH OF ALL INVERSE TRIG. FUNCTION The following table gives the inverse trigonometric function (principal branches) along with their domains and r anges sin-1
:
[-1, 1]
2 , 2
cos-1
:
[-1, 1]
[0, ]
cosec-1 :
R – (-1, 1)
2 , 2 - {0}
sec
:
R – (-1, 1)
[0, ] – {
tan-1
:
R
, 2 2
cot-1
:
R
(0, )
-1
2
}
3. PROPERTIES OF INVERSE TRIG. FUNCTIONS 4. APPLICATION PROBLEMS a) Evaluate the Inverse Trig. Functions b) Evaluating the Problems by using Trig. Identities i) a2 + x2 Put x = a tan or x = a cot 2 2 ii) a - x Put x = a sin or x = a cos 2 2 iii) x – a Put x = a sec or x = a cosec c) To reduce into simplest form d) To solve :- for the unknown variable x or e) Problems involving cos -sin >0 if sin -cos >0 if
& 4
0,
, & 4 2
MATRICES 1.Definition 2.Order of a Matrix 3.Construction of a Matrix of given order &identifying entries(example :Find a11, a 22, a23 etc) (Construct a matrix of order 2 3Whoseentries are defined as aij
i2j 2
)
4.Types of matrices i)column matrix, ii)row matrix,iii)square matrix, iv)diagonal matrix, v) scalar matrix vi)identity matrix, vii)zero ornull matrix 5.Equality of matrices-Finding the unknown values x,y if given two matrices are equal 6.Matrix operations, i) addition , ii)subtraction , iii)scalar multiplication-(multiply all the entries of the matrix with the scalar k.) , iv)multiplication of matrices-a)Matrix multiplication need not be commutative b)Matrix multiplication is associative c)Product of two non zero matrices can be a zero matrix. 7.Transpose ofa matrix&its properties i) A / A /
ii) AB B T AT iii)(kA)T=kAT T
iv) A B AT B T 8.Symmetric&skew –symmetric matrices I)AT=A (Symmetric) ii)AT=-A( skew –symmetric) 9.Writing a given square matrix as a sum of a symmetric & a skew symmetric matrices. A A / is symmetric T
symmetric A A is skew 1 1 10. A A A A A /
/
/
2 2 11.Elementary row and column transformations. a).If row transformation is used i)Use A=IA Ii)Use only the following three transformations Ri R j
Ri kRi
Ri Ri kR j , where k is any nonzero real number b).If column transformation is used i)Use A=AI Ii)Use only the following three transformations C i C j
C i kC i
C i C i kC j , where k is any nonzero real number 12.Finding the inverse of a square matrix using elementary row or column transformations 1 AB B 1 A 1 13.
AA 1 A 1 A I 14.Invertible matrix( Let A and B be two square matrices, if AB=BA =Ithen B is called the inverse of the matrix Aand A is called an invertible matrix. Inverse of A is denoted by A -1)
CHAPTER – DETERMINANTS. 1. Explain the difference between a matrix and a determinant. Determinant is a number or a function we associate with a matrix. 2. Determinant of a matrix of order 1 and order 2. 3. Determinant of a matrix of order 3. It can be done by expanding about any row or any column. 4. Properties of determinants. i) The value of the determinant remains unchanged if its rows and columns are interchanged.
ii) iii) iv) v)
vi)
If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes. If any tworows(or columns) of a determinant are identical then the value of the determinant is zero. If each element of a row(or a column of a determinant is multiplied by constant k, then its value gets multiplied by k. If some or all elements of a row or column of a determinant are expressed as sum oftwo(or more) terms, then the determinant can be expressed a s sum of two (or more) determinants. If to each element of a ny row or column of a determinant, the equimultiples of corresponding elements of other row(or column) are adds, the value of determinant remains the same. Ie.RiRi+kR j orCiCi + kC j then value of determinant remains the same. 5. Solving problems using properties of determinants ie.problems involving proving LHS = RHS using properties. 6. Finding the area of a triangle using determinants. 7. Finding minors and co-factors. i+j
8. Aij=(-1) Mij 9. = a11A11+ a12A12 + a13A13.whereAij are cofactors of aij . 10. If elements of a row (or column) are multiplied with cofactors of any other row (or column). Then their sum is zero.
| |
11. Finding the adjoint and inverse of a matrix. 12. A is singular matrix implies = 0. 13. . But if IAI =IkBIwhere k is a constant then order of Matrix A. n-1 14. = . where n is the order of the square matrix A 15. )T = (AT)-1 16. adj(AT) = (adjA)T (n-1)2 17. =
| | | | | |−| | | | | | |
18. If
then
|| ||
adj A =
=kn
where n is the
(Note: to find adjA interchange main
leading diagonal and change the sign of other diagonal. ) n 19. If A is a square matrix of order n then IkAI= k IAI 20. A-1 = -1 T
||
adj A.
21. (A ) =(AT)-1
| |
22.Working rule for solving of linear equations in 3 variables. Step 1: Express the system of linear equation as AX = B find Step 2: If 0, then the system of equation is consistent and has a unique solution given by X = A-1 B. Step 3: If = 0 and (adjA)B 0, then the system is inconsistent. Step 4: If = 0 and (adjA)B = 0, then the system may or may not be consistent. In case it is consistent the system has infinite number of solutions.
| | | | ||
Step 5: In order to find the solutions when system has infinitely many solution, take any one variable equate it to t or k. Find the values of other variables in terms of t or k.
Topic: Continuity and Differentiability Limits 1. lim 0
2. lim 0
3. lim 0
4. lim x a
5. lim x 0
6. lim x a
1 1 1 − − − − 1 + 1 ≠0 ℎ ≠0 ℎ lim
x a
7. Algebra of Limits: lim x a
lim
x a
(i) (ii)
If lim x a
=0 and lim
limit does not exist
=0 and lim
limit of the
x a
If lim x a
x a
function is zero (iii) If lim =0 and lim x a
x a
=0 then f(x) , g(x) can be factorized or
rationalized or simplified using trigonometric identities Continuity
Definition -1 A real function f(x) on a subset of real numbers and let a be a point in the domain of f then f is continuous at a if lim f ( x) f (a)
5
x a
Example: Find if lim x 0
Deinition-2 A real function f(x) is said to be continuous at a if lim f ( x) lim f ( x) f (a) x a
{23, 23, ≤2 >2
x a
Example: Verify that the following function f(x) is continuous at x = 2
Theorem
Suppose f and g are real valued function such that fog is defined at c. If g is con tinuous at c and if f is continuous at g(c) then fog is continuous at c Differentiability Y=f(x)
C (constant)
′ −
Sinx Cosx Tanx Secx Cosecx Cotx ex Logx
0 n Cosx -sinx 2 Sec x secxtanx -cosecxcotx -cosec2x ex 1/x
(u+v) u-v Uv
u’+v’ u’-v’ uv’+vu’
1 √1 1 √11 11 11 √1 1 √ 1 ′ 12√ 1
sin− cos− tan− cot− sec− cosec− √ 1
loga Result: Every differentiable function is continuous but the converse need not be true Chain Rule Let y=vou,
t = u(x) and if both
0
,
exist then
. =
Derivatives of Implicit and explicit functions Example: + 2x (Explicit function) (implicit function) Differentiation of one function with respect to another function 2 Differentiate sin x with respect to cosx
Take u = sin 2x and v=cosx then
×
lloogg / lologglog log log log log × Rules of logarithmic function = = =n Change of base rule
=
loge = 1, log1 = 0, Example: If
, find
Solution: y= u+v then =
where
and
Parametric Form
If x=f(t), y = g(t) then
Second Order Derivatives Let y=f(x) ,
If x=f(t), y = g(t) then
= y1 then
= f’’(x) = y 2
=
Rolles Theorem If a real-valued function ƒ is continuous on a closed interval [a, b], differentiable on the open interval (a, b), andƒ(a) = ƒ(b), then there exists a c in the open interval
(a, b) such that
Mean Value Theorem Let be differentiable on the open interval interval . Then there is at least one point
in
and continuous on the closed such that
CONCEPT MAPPING – INTEGRATION 1.Integration- as an inverse process of differentiation
(sinx) = cosx,
( )=
,
(
)=
We observe that the function cosx is the derivative function of sinx.we say that sinx is
an antiderivative ( or an integral)of cosx.similarly and integrals) of
and
.we note that for any real number c,treated as constant
function,its derivative is0and hence we can writ e , (
are the antiderivatives (or
)=
(sinx +c) = cosx,
(
)=
Thus antiderivative or integrals of the above functions are not unique.Actually there exists infinitely many antiderivatives of each of these functions which can be obtained by choosing C arbitrarily from the set of real numbers( also called constant of integration).for this reason C is customarily referred to as arbitrary constant. More generally
∫ 2.Symbol
(F(x) +c)= f(x),x belongs to I
-- Meaning 3.Comparison between differentiation and integration a. Both are operations on function b. All functions are not differentiable c.All functions are not integrable d .Derivative of a function when it exists it is unique function but the integral of of a function is not so. However they are unique upto an additive constant ie.any two integrals of a function differ by a constant. e. When a polynomial function P(x) is differentiated the result is a polynomial whose degree is one less than the degree of p(x) eg.
∫ (
) = 2x
When a polynomial function is integrated the result is a polynomial whose Degree is one more than that of p(x)
Eg.
dx = + c
f. The derivative of a function has a geometrical meaning- the slope of the tangent to the corresponding curve at a point g.Integration- family of curves placed parallel to each other having parallel tangents at the point of intersection of the curves of the family ,with the lines perpendicular to the axis representing the variable of integration. 4.Methods of integration
↓
Integration by Substitution ↓ Application of
Methods of integration 1. Integration by direct method ↓ ↓
Function
Trigonometric function
Integration using Partial fraction ↓ Partial fraction
+ − − −− -
+
Integration by parts ↓
∫ ∫ =uv-
+ − − − ++ −−− −++ − − − − − − − ++ + − ++ − ++ ∫ ± ∫ ± ∫∫ − ∫ ++∫ + ∫ + ++ ++ ++ ∫∫ √ ± √ ∫∫√ ∫ ∫ ∫ ∫ In integrals
-
+
-
↓
+
+
Integration of some
-
Particular function
+
+
-
,
,
,
,
SOME SPECIAL INTEGRALS 1.
dx
2.
dx
3.
dx
4.
dx
Definite Integrals
- Area of the region bounded by the curve y=f(x) and the ordinates
x=a,x=b and x-axis Calculating
Find the indefinite integral
.Let this be F(x),
= F(b) –F(a)
Definite integrals
↓↓↓↓
Definite integrals based on Definite integrals Properties of integration types of indefinite integrals as a limit of sum definite integral of modulus fn
Some important results.
∫∫ + ∫∫ ∫∫ ∫∫ ∫ − −− −− ∫∫ + − − ∫ −
1. 2. 3. 4. 5. 6. 7. 8.
=
+c
= x+c = -cosx +c = sinx +c x dx = tanx + c x dx = -cotx +c = secx + c = - cosecx + c
9.
dx = -
10.
dx =
+c OR -
+c
11.
=
+c OR -
+c
12.
dx =
+ c
+c OR
+c
∫∫ ∫ ∫∫ ∫ ∫ ∫∫ ∫ ∫ ∫ ∫ ∫ ∫∫ ∫ ∫ ∫ :∫ = ∫ , ∫− 13. 14.
dx =log x +c dx =
+ c
( f(x) +f ’(x) ) dx = f(x) +c 15. Some Properties of Definite Integrals
: : :
=
: :
+
:
,If f(2a-x)=f(x) =0 if f(2a-x)=-f(x)
If f is an even function (ii)
Limit as a sum
∫∑⋯ → ∑∑ ⋯ [ /] ⋯ ⋯ / 3.
dx =
( f(a) + f(a+h)+f (a+2h)+…………+f(a+(n-1)h)
= n (n+1)/2 = n (n+1) (2n+1)/6 = = a(
r≠ 1
INTEGRATION(TIPS TO REMEMBER THE POINTS) 1.Read thoroughly the formulae 2.Prepare a pocket formula dictionary (Writing all the formulae in a note book). 3.Insist the children daily to bring in the class and whenever they get time they should memories the formulae. 4.They have to put those formulae in practice. 5.Daily formula test to be conducted. 6.Students should be practiced so that they can identify the method of integration.
APPLICATIONS OF INTEGRATION
Review 1. 2. 3. 4.
Standard equations of straight lines Equation of circles with Centre at the origin, andcenter at (h,k) Equation of parabolas Equation of ellipse
5. First fundamental theorem of integral calculus 6. Second fundamental theorem of integral calculus 1.Area under the curve y=f(x)and the x-axis and the ordinates at x=a and at x=b is
∫
.
2.Area under the curve x=f(y)and the x-axis and the ordinates at y=c and at y= d is
∫
3.The area bounded by the curve y=f(x) and x-axis and the ordinates x=a and x=b is given by A 1 +A2=A
≥
4.Area under two curves If y=f(x) , y= g(x) where f(x) g(x) in the [a,b] such that the point of intersection of these two curves are given by x=a and x=b obtained by taking common values of y from given equation of two
] ≥
curves then the area between the curves is given by A=
∫[
≤
5.If f(x) g(x) [a,c] and f(x) g(x) in [c,b] where a < c
6.Area of a triangle when the coordinates of the vertices or equations of sides are given.
DIFFERENTIAL EQUATION CONCEPT MAPPING: Definition of differential equation. Order of a differential equation Without radical sign: Order is the order of the highest order derivative appearing in the equation
2 50 √ ,√ ,.
Example:
.
With radical sign : like remove the radical sign by squaring, cubing etc. then Order is the order of the highest order derivative appearing in the equation.
1
Example:
Degree of a differential equation The degree of a differential equation is the degree of the highest order derivative occurring in the equation, when the differential coefficients are made free from radicals if each term involving derivatives of differential equation is polynomial(can be expressed as a polynomial).
Example:
0
Verifying the given function as a solution of the given differential equation. General solution and particular solution(Finding values of the constants by using the given condition) Formation of differential equation Steps involved Write down the given equation of family of curves Differentiate the given equation as many times as the number of arbitrary constants. Eliminate the arbitrary constants from the given equation and the equation obtained by differentiation. Solving a differential equation: Variable separable
, . . . , . , , .
Type-1:
Type-2: integrate.
Type-3:
Homogeneous : If the question is asked to prove homogeneous find then “f” is homogeneous of degree “n”.
Step 1: put y= vx and
Step 2: The equation reduce to variable separable and solve. Some homogeneous equation can be solved by substituting
Linear Differential equation of type :
“x” alone or constants. Step 1: Identify P and Q.
then separate the variables and integrate. , then divide by
and
where “v” is a function of “x” alone.
.
, where P and Q are functions of
∫ ∫ ∫ ∫ . .∫ ∫.∫ . ∫ ∫ ∫ ∫ . .∫ ∫ .∫ . Step 2: Find
and find
(Note: If
, then
Step 3: Solution
Linear Differential equation of type :
, where P and Q are functions of
“y” alone or constants. Step 1: Identify P and Q.
Step 2: Find
Step 3: Solution
and find
(Note: If
, then
+ "" + "" + "" 4 .
The equation representing the parabola having vertex at the origin and axis along the positive direction of X-axis. The family of circles touching y-axis at the orgin. The family of circles touching the x-axis:
VECTOR ALGEBRA
CONCEPT MAPPING DEFINITION OF Vectors Definition of Scalars Position Vector Dc’s and Dr’s Types of vectors Zero Vector Unit Vector
⃗ ⃗ ⃗ ⃗ ⃗
Unit vector in the direction of + or Collinear Vectors Equal Vectors Negative of a Vector Addition of Vectors Multiplication of a Vector by a Scalar Unit Vectors along the coordinate axes. Vector joining two points Section formula Dot product of vectors Projection of vectors on a line Perpendicular vectors Finding the angle between the two vectors
⃗ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗
Finding I + I, I + + I, I - I Squaring of a vector Angle between two vectors. Expressing dot product in rectangular coordinates. Cross product of vectors Cross product of unit vectors. Expressing cross product in rectangular coordinates. Unit vector perpendicular to two given vectors. Angle between two vectors. Area of parallelogram when adjacent sides are given. Area of parallelogram when diagonals are given. Area of a triangle Area of a rectangle when position vectors of A,B,C,D are given. Scalar Triple product of vectors.
Expressing the STP in rectangular coordinates. Geometrical meaning of STP. Vector Triple Product. Properties of dot, cross and STP Dot product Cross product
[⃗,,⃗] ⃗. ⃗ |⃗|. ⃗ ⃗ ⃗ |⃗|. ⃗ ⃗ [ ] ⃗ , , ⃗ ⊥ ⃗. ⃗ ⃗. ⃗ |⃗| ⃗ ⃗⃗ ⃗ [ ] ⃗ ⃗ , , (⃗. ⃗) ⃗(⃗⃗.)⃗ ⃗ ⃗ ̂ ̂ ̂ ̂ ⃗ ⃗ ̂ ̂̂ ̂ ⃗⃗ ̂ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ ̂ ⃗ ⃗ ⃗.⃗ [⃗,,⃗] ⃗ ⃗ ⃗, ⃗ ⃗ ⃗ ⃗⃗.⃗ ⃗ ⃗ , ⃗ , where is perpendicular to both , .
If
Scalar triple product Notation is If coplanar
If parallel
if any two vectors are equal.
If
If
If
Then
Then
Geometrical meaning
Geometrical meaning Volume of a parallelepiped
Then
Geometrical meaning projection of
with as adjacent sides.
Adjacent sides
Concept Mapping Topic: 3-Dimensional Geometry.
1. Direction Cosines and Direction Ratios of a Line.( =
=k )
2. Relation between the direction cosines of a line. ( . ) 3. Direction Cosines of a line passing through two points P(x1,y1,z1) and Q(x2,y2,z2) 4. Equation of a line in a space
⃗
5. Equation of a line through a given point and parallel to a given Vector 6. Derivation of Cartesian form from vector form. 7. Equation of a line passing through two given points. 8. Derivation of Cartesian form from Vector form. 9. Whenever the Equation of a line or a plane is given it is to see that whether the Equation is in standard form. 10. Angle between Two Lines. 11. Shortest Distance between Two Lines 12. Distance between two skew Lines. 13. Distance between two parallel lines 14. Equation of a plane in normal form& Cartesian form. 15. Equation of a plane perpendicular to a given vector and passing through agiven point 16. Equation of a plane passing t hrough three non collinear points. 17. Intercept form of the equation of a plane.
18.Plane passing through the intersection of two given planes. 19.Coplanarity of Two Lines. 20.Angle between Two Planes. 21. Distance of a Point from a Plane. 22.Angle between a Line and a Plane 23. Image of a point w.r.t line and plane. 24. Geometrical interpretation to be given wherever possible. 25. condition for perpendicularity an d parallality of two lines and two planes. Summary of the chapter:
.
Probability
Introduction
Concepts in probability
CONDITIONAL PROBABILITY
PROPERTIES OF CONDITIONAL PROBABILITY
MULLTIPLICATION THEOREM ON PROBABILITY
Enumerating outcomes
Probability
INDEPENDENT EVENTS
PARTITION OF SAMPLE SPACE
THEOREM OF TOTAL PROBABILITY BAYE’S THEOREM
and ITS APPLICATIONS MEAN OF A RANDOM VARIABLE BINOMIAL DISTRIBUTION (BERNOULLI TRIALS)
RANDOM VARIABLE AND ITS PROBABILITY DISTRIBUTION FORMULA TO FIND THE VARIANCE OF A RANDOM VARIABLE PROBLEM SOLVING BASED ON BAYE’S THEOREM& BERNOULLI TRIALS
Formatted: Font:
12 pt
CONCEPT MAPPING:
→
12 pt, Complex Script Fo
→
→
→
→
PROBABILITY Experiments Sample Space(S) Events(A,B,C,E,F,G,….) Conditional Probability If E and F are two events associated with the same sample space of a random experiment, the conditional probability of the event E given that F has occurred, i.e. P (E|F) is given by P(E|F) =P(E∩ F)/P(F) provided P(F) ≠ 0 Properties of conditional probability: Let E and F be events of a sample space S of an experiment, then we have Property 1:P(S|F) = P(F|F) = 1 Property 2: If A and B are any two events of a sample space S and F is an event of S such that P(F) ≠ 0, then P((A B)|F) = P(A|F) + P(B|F) – P((A ∩ B)|F) Property 3:P(E′|F) = 1 − P(E|F) Multiplication Theorem on Probability P(E ∩ F) = P(E) P(F|E) = P(F) P(E|F), provided P(E) ≠ 0 and P(F) ≠ 0 Multiplication rule of probability for more than two events: If E, F and G are three events of sample space, we have P(E ∩ F ∩ G) = P(E) P(F|E) P(G|(E ∩ F)) = P(E) P(F|E) P(G|EF)
∪
→
→
→
→ →
Independent Events: Two events E and F are said to be independent, ifP(F|E) = P (F) provided P (E) ≠ 0 and P (E|F) = P (E) provided P (F) ≠ 0 Let E and F be two events associated with the same random experiment, then E and F are said to be independent ifP(E ∩ F) = P(E) . P (F) Partition of a sample space Let {E1, E2,...,En} be a partition of the sample space S, and suppose that each of the events E1, E2,..., En has nonzero probability of occurrence. Let A be any event associated with S, then P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... + P(En) P(A|En)
→
→
Theorem of total probability P(A) = P(E1) P(A|E1) + P(E2) P(A|E2) + ... + P(En) P(A|En)
→
∪∪∪
Bayes’ Theorem If E1, E2 ,..., En are n non empty events which constitute a partition of sample space S, i.e. E1, E2 ,..., En are pairwise disjoint and E1 E2 ... = S andA is any event of nonzero probability, then
/
P(
)
∑ | |
for any j =1,2,3,…..,n
En
Random Variables and its Probability Distributions o
Probability distribution of a random variable
The probability distribution of a random variable X is the system of numbers x X :x 1 2 .. . x n P(X) :p p . .. pn 1 2 xn where,x1,x2, …. are the possible values of the random variable X and p i (i = 1,2,..., n ) is the probability of the random variable X taking the value xi ∑pi = 1, i = 1, 2,..., n i.e. P(X = xi ) = pi A random variable X has the following probability distribution:
X P(X)
0 0
1 K
2 2K
3 2K
→ → ∑∑==
5
6
7 7k2 +k
Mean of a random variable E (x)= µ= pi 2 Variance of a random variable E (x )= o Bernoulli Trials and Binomial Distribution Bernoulli trials o Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions : (i) There should be a finite number of trials. (ii) The trials should be independent. (iii) Each trial has exactly two outcomes : success or failure. (iv) The probability of success remains the same in each trial. Binomial distribution o The probability of r successes P(X = r) is denoted by P(x) and is given by P(X =r) =nCrqn –rpr,r = 0, 1,..., n. (q = 1 – p) This P(x) is called the probability function of the binomial distribution. A binomial distribution with n-Bernoulli trials and probability of success in each trial as p, is denoted by B(n, p). P(at least r successes), P(at most r successes), P(exactly r successes) P(r>3), P(r<3), P(r=3) cases. o
4 3K
→
→
