Class – 12 Relations and Functions

Function means a correspondence from one value x of the first set to another value y of the second
set.

Domain is defined as the set of all the values that the function can input while it can be defined.

Range are all the values that come out as output of the function involved.

Codomain is the set of values that have the potential of coming out as outputs of a function.

Types of Functions

There are many different types of functions that we encounter in mathematics. We will define them
here:

1. One-One functions

Also called Injective functions, are the functions where each input is connected to a unique output.
If x1 is connected to y1, then there will be no other input x in the domain which will be connected to
y1.

2. Many-One function

It is the opposite of One-One functions. In this type, more than one input x’s can be connected to
the same y.

3. Onto functions

Also called Surjective functions, they are the functions where each of the y that exist in the
codomain is connected to x.

4. Into functions

5. They can contain some values in codomain which are not connected to the input set.

A function that is both one-one and onto is called Bijective functions. If we are told to find the
inverse of a function, the function must be bijective otherwise it is not possible to find the inverse.

Some functions that are commonly encountered are:

1. Constant function

There is only one output value for all the input values of x. Their graph appears as a straight line
parallel to the x-axis.

2. Polynomial functions

These functions look like as shown below

f(x)=anxn+an−1xn−1+⋯ + a0,
The highest power of x is called the degree of the polynomial. Here, an, an−1, ................, a1, a0 are the
coefficients of respective powers of x. The linear function and quadratic functions are special cases
of polynomial functions.

3. Linear functions

These functions are just a linear combination of variables with coefficients. For example, y = 2x + 3 is
a linear function. They appear as straight lines in the x-y axis graph.

4. Trigonometric functions

These functions are composed of various trigonometric ratios.

There are many other types of functions as well that we have not touched upon in this blog. Some
are very complex that you will encounter later.

Types of Relations

Let us define a set A. We can now form a relation R from A to A. Let A = {1,2} and now, let us look at
different types of relations that can be:

1. Universal Relation

If all elements of A are related to every element of A, then it is a universal relation. For example,the
universal relation R on A will be, {(1,1), (2,2), (1,2), (2,1)}.

2. Reflexive Relation

For every element ‘a’ in set A, if (a, a) exists in the relation R, then R is reflexive. For example, if (1,1)
and (2,2) are both part of relation R, hence, R is reflexive.

3. Symmetric Relation

For every ordered pair (a,b) in R, if the pair (b, a) also exists in R, then R is symmetric. If (1,2) and
(2,1) are both present in R, it is symmetric.

4. Transitive Relation

For every (a,b) and (b,c) that exist in R, if (a,c) is present then R is transitive.

5. Equivalence Relation

If a relation R is reflexive, symmetric, and transitive, then R is an equivalence relation.

6. Empty Relation

If none of the elements of set A are related to other elements of set A, then R is empty.
Chapter – 2 Inverse Trigonometric Functions
Chapter – 3 Matrices
1. The elements a ij of a matrix for which i = j are called the diagonal elements of a matrix and the
line along which all these elements lie is called the principal diagonal or the diagonal of the matrix.

2. Properties of transpose of the matrices-

(i) (A+ B)’ = A’ + B’

(ii) (KA)’ = KA’, where K is constant

(iii) (AB)’= B’A’

(iv) (A’)’ = A

3. Properties of Matrix addition-

(i) Matrix Addition is Commutative – If A and B be two m × n matrices, then A+ B= B+A

(ii) Matrix Addition is Associative – If A, Band C be three m × n matrices, then (A+ B) + C= A+ (B+ C)

4. Properties of Multiplication of a Matrix by a Scalar-

(i) If K1 and K2 are scalars and A be a matrix, then (K1 + K2)A = K1A + K2A.

(ii) If K1 and K2 are scalars and Abe a matrix, then K1 (K2A) = (K1 K2)A.

(iii) If A and Bare two matrices of the same order and K, a scalar, then K (A + B) = KA + KB.

(iv) If K1 and K2 are two scalars and A is any matrix then (K1 + K2)A = K1A + K2A.

(v) If A is any matrix and K be a scalar. then (–K)A = – (KA) = K (–A).

5. Properties of Matrix Multiplication-

(i) Associative law for Multiplication – If A, B and C be three matrices of order m × n and n × p and p
× q, respectively, then (AB) C =A(BC).

(ii) Distributive Law – If A, B, C be three matrices of order m × n, n × p and n × q respectively. then A

× (B+ C) = A×B+ A×C

(iii) Matrix Multiplication is not commutative. i.e., A×B ≠ B×A

(iv) The existence of multiplicative Identity: For every square matrix A, there exists an identity matrix
of same order such that IA= AI = A.

6. If A be any n × n square matrix, then A . (AdjA) = (AdjA) . A= |A|. In where In is an n × n unit matrix

7. (i) Only square matrix can have inverse

(ii) The matrix B= A–1, will also be a square matrix of same order A.

(iii) The square matrix A is said to be invertible if A–1

exists.

8. Every invertible matrix possesses a unique inverse.

Chapter – 4 Determinants

The determinant of a matrix is a number defined only for square matrices. It is used in the
analysis of linear equations and their solution.
Properties of Determinants

Property 1 : The value of determinant is not changed when rows are changed into columns
and columns into rows

Property 2 : If any two rows or columns of a determinant are interchanged, the sign of the
determinant changes but its magnitude remains the same

Property - 3 : A determinant having two rows or two columns identical has the value zero

Property - 4 : Multiplying all the elements of a row (or column) by a scalar (a real number) is
equivalent to multiplying the determinant by that scalar.

Property - 5 : A determinant can be split into a sum of two determinants along any row or
column

Some other basic properties of determinants are:

1) Suppose In is the Identity matrix of order n n, then det(I) = 1.

2) If the matrix XT is the transpose of matrix X, then det(XT) = det(X)

3) If X-1 is the inverse matrix of X then det(X-1) = 1 / det(X)

4) If two matrices X and Y are of equal size then det(XY) = det(X) det(Y)

5) The determinant of a matrix is zero if each element of the matrix is zero.

6) In a triangular matrix, the determinant is equal to the product of the diagonal elements.

7) Multiplication of a row (column) of a determinant by a constant: Multiplication of the elements of

any row (or column) by the same number is equivalent to multiplying the determinant by that
number.

8) Laplace’s theorem: An nth-order determinant can be calculated using the Laplace’s formulas.

Expansion of the determinant along the ith row is given by the formula: det A = n∑j = 1aijAij, i = 1, 2,
…, n

Expansion of the determinant along the jth column is expressed in the form: detA = n∑i = 1aijAij, j =
1, 2, …, n

9) Minor: The first minor Mij associated with the element aij of an nth order square matrix A is the
determinant of order (n−1) obtained from the matrix A by deleting the ith row and the jth column.

10) Cofactor: The cofactor Aij is the minor Mij multiplied by (−1) raised to the (i + j) power. Aij =
(−1)i + j Mij

Chapter – 5 Continuity and Differentiability

Chapter – 6 Applications of Derivatives
Chapter – 7 Integrals

The functions that give the derivative, that is, the function f is called antiderivative or integral
of f′.The process of finding the antiderivative is called integration.

Methods of Intergration
Some Identities used
Integration Formulas:
Chapter – 8 Application of Integrals

Definition of integral: An integral is a function, of which a given function is the derivative.

Integration is basically used to find the areas of the two-dimensional region and for
computing volumes of three-dimensional objects.

Indefinite Integrals: These are the integrals that do not have a pre-existing value of limits; thus
making the final value of integral indefinite.

Area under the curve:

The area under the curve can be calculated through three simple steps.

• First, we need to know the equation of the curve(y = f(x)), the limits across which the
area is to be calculated, and the axis enclosing the area.
• Secondly, we have to find the integration (antiderivative) of the curve.
• Finally, we need to apply the upper limit and lower limit to the integral answer and
take the difference to obtain the area under the curve.

Area between Two curves

Chapter – 9 Differential equations

Differential functions: An equation that contains the derivative of a function is called a

differential function.

Order of Differential Equations: The order of a differential equation is the highest order of
the derivative appearing in the equation. Consider the following differential equations,

Degree of Differential Equations: The degree of the differential equation is the power of
the highest ordered derivative present in the equation.

Types of Differential Equations: The different types of differential equations are-

• Ordinary Differential Equations: An equation that contains only one independent

variable and one or more of its derivatives with respect to the variable.

• Homogeneous Differential Equations: A differential equation in which the degree

of all the terms is the same is known as a homogenous differential equation.

• Non-homogenous Differential Equations: A differential equation in which the

degree of all the terms is not the same is known as a homogenous differential
equation.
Solving Differential Equations: A solution that can be obtained from the general solution by
giving particular values to arbitrary constants is called a particular solution. Let us
understand differential equation solver by one example: (dy/dx) = x2y + y

• Step 1: Divide the above differential equation by y.

(1/y)(dy/dx) = (x2 + 1)
We consider y and x both as variables and write this as
(dy/y) = (x2 + 1)dx

• Step 2: Now integrate L.H.S. with respect to y and with respect to x.

∫(1/y)dx = ∫(x2 + 1)dx

• Step 3: After integrating, we get:

log y = (x3/3) + x + c
So, this is how the differential equation is solved.

General Solution – The solution which contains as many as arbitrary constants as the order of the
differential equation is called the general solution of the differential equation.

Particular Solution – Solution obtained by giving particular values to the arbitrary constants in the
general solution of a differential equation is called a particular solution.

Important Notes:

• We can use the following notations for derivatives. (dy/dx) = y', (d2y/dx2) = y'', (d3y/dx3) =
y'''
• The order and degree of a differential equation should be always positive integers.

Chapter – 10 Vector Algebra

Vector: Vectors are geometrical entities that have magnitude and direction. A vector can be
represented by a line with an arrow pointing towards its direction and its length represents
the magnitude of the vector. For a Vector carry a point A to point B.

The length of the line between the two points A and B is called the magnitude of the vector

The direction of the displacement of point A to point B is called the direction of the vector

The initial point of a vector is also called the tail whereas the terminal point is called the
head.

If (x,y,z) are the components of a vector A, then the magnitude of AB is given by,
||A|| = √ (x2+y2+z2)
The magnitude of a vector is a scalar value.
Properties of Vectors: The following properties of vectors help in better understanding of
vectors

Types of Vectors: The vectors are termed as different types based on their magnitude,
direction, and their relationship with other vectors. Let us explore a few types of vectors and
their properties:

• Zero Vectors: Vectors that have 0 magnitude are called zero vectors,
denoted by 0→ = (0,0,0). The zero vector has zero magnitudes and no
direction. It is also called the additive identity of vectors.
• Unit Vectors: Vectors that have magnitude equals to 1 are called unit
vectors, denoted by a^. It is also called the multiplicative identity of vectors.
The magnitude of a unit vectors is 1. It is generally used to denote the
direction of a vector.
• Position Vectors: Position vectors are used to determine the position and
direction of movement of the vectors in a three-dimensional space. The
magnitude and direction of position vectors can be changed relative to other
bodies. It is also called the location vector.
• Equal Vectors: Two or more vectors are said to be equal if their
corresponding components are equal. Equal vectors have the same
magnitude as well as direction. They may have different initial and terminal
points but the magnitude and direction must be equal.
• Negative Vectors: A vector is said to be the negative of another vector if they
have the same magnitudes but opposite directions. If vectors A and B have
equal magnitude but opposite directions, then vector A is said to be the
negative of vector B or vice versa.
• Parallel Vectors: Two or more vectors are said to be parallel vectors if they
have the same direction but not necessarily the same magnitude. The angles
of the direction of parallel vectors differ by zero degrees. The vectors whose
angle of direction differs by 180 degrees are called antiparallel vectors, that
is, antiparallel vectors have opposite directions.
 • Orthogonal Vectors: Two or more vectors in space are said to be orthogonal
if the angle between them is 90 degrees. In other words, the dot product of
orthogonal vectors is always 0. a.b = |a|.|b|Cos90° = 0.
• Co-initial Vectors: Vectors that have the same initial point are called co-
initial vectors.

Vectors Formula: Different mathematical operations can be applied to vectors such as

addition, subtraction, and multiplication

Addition of Vectors: The addition of vectors is commutative and associative. There are two
laws of vector addition:

• Triangle Law of Addition of Vectors: The law states that if two sides of a
triangle represent the two vectors (both in magnitude and direction) acting
simultaneously on a body in the same order, then the third side of the triangle
represents the resultant vector.

• Parallelogram Law of Addition of Vectors: The law states that if two co-initial
vectors acting simultaneously are represented by the two adjacent sides of a
parallelogram, then the diagonal of the parallelogram represents the sum of the
two vectors, that is, the resultant vector starting from the same initial point.
Subtracting Vectors: The subtraction of vectors is similar to the addition of vectors. But here
only the sign of one of the vectors is changed in direction and added to the other vector.

Multiplying Vectors: Vectors can be multiplied but their methods of multiplication are slightly
different from that of real numbers. There are two different ways to multiply vectors:

• Dot Product: The individual components of the two vectors to be multiplied are
multiplied and the result is added to get the dot product of two vectors.
a.b = (a1 i + b1 j + c1 k) . (a2 i + b2 j + c2 k) = (a1, b1, c1).(a2, b2, c2) = (a1.a2) +
(b1.b2) + (c1.c2)

• Another way to determine the dot product of two vectors A and B is to determine the
product of the magnitudes of the two vectors and the cosine of the angle between
them.
A→.B→ = |A||B| cosθ
• Cross Product: The vector components are represented in a matrix and a
determinant of the matrix represents the result of the cross product of the
vectors.

A→×B→ = (b1c2 - c1b2, a1c2 - c1a2, a1b2 - b1a2)

• Another way to determine the cross product of two vectors A and B is to determine
the product of the magnitudes of the two vectors and the sine of the angle between
them.
A→×B→ = |A||B| sinθ / n^
Scalar Multiplication of Vectors: A scalar is a real number that has no direction. When a scalar is
multiplied by a vector, we multiply the scalar by each component of the vector. The operation
of multiplying a vector by a scalar is called scalar multiplication. When a vector a = (a1, a2, a3) =
a1 i + a2 j + a3 k is multiplied by a scalar r, the resultant vector is:

ra = (ra1, ra2, ra3) = (ra1)e1 + (ra2)e2 + (ra3)e3

• If r is negative, then the direction of the resultant vector changes direction by 180
degrees.
• Scalar multiplication is distributive over vector addition, that is, r(a+b) = ra + rb

Scalar Triple Product of Vectors: Scalar triple product of vectors is the dot of one vector with the
cross product of the other two vectors. If any two vectors in a scalar triple product are equal, then
the scalar triple product is zero. If the scalar triple product is equal to zero, then the three vectors a,
b, and c are said to be coplanar.

Also, a.(b × c) = b.(c × a) = c.(a × b)

Angle Between Two Vectors: The angle between two vectors can be calculated using the dot
product formula. Let us consider two vectors a and b and the angle between them to be θ. Then,
the dot product of two vectors is given by a.b = |a||b| cosθ.

θ = cos-1 (a.b/|a||b|)

Chapter – 11 Three Dimensional Geometry

Chapter – 12 Linear Programming

Linear Programming Problems – Problems which concern with finding the minimum or
maximum value of a linear function Z (called objective function) of several variables (say x
and y), subject to certain conditions that the variables are non-negative and satisfy a set of
linear inequalities (called linear constraints) are known as linear programming problems.

Objective function– A linear function z = ax + by, where a and b are constants, which has
to be maximised or minimised according to a set of given conditions, is called a linear
objective function.

Decision Variables – In the objective function z = ax + by, the variables x, y are said to be
decision variables.

Constraints – The restrictions in the form of in e qualities on the variables of a linear

programming problem are called constraints. The condition x 0, y 0 are known as non –
negative restrictions.

Feasible Region–The common region determined by all the constraints including non–
negative constraints x,y >= 0 of linear programming problem is known as feasible region (or
solution region) If we shad c the region according to the given constraints, then the shaded
areas is the feasible region which is the common area of the regions drawn under the given
constraints.

Feasible Solution – Each points within and on the boundary of the feasible region
represents feasible solution of constraints. In the feasible region there are infinitely many
points which satisfy the given condition.

Optimal Solution – Any point in the feasible region that gives the optimal value (maximum
or minimum) of the objective function is called an optimal solution.

Theorem 1 – Let R be the feasible region (convex polygon) for a linear programming
problem and let Z = ax + by be the objective function. When Z has an optimal value
(maximum or minimum), where the variables x and y are subject to constraints described by
linear inequalities, the optimal value must occur at a corner point of the feasible region.

Theorem 2 – Let R be the feasible region for a linear programming problem, and let Z = ax +
by be the objective function. If R is bounded them the objective function Z has both
maximum and minimum value on R and each of these occurs at a corner point of R.

Formulation of LPP – Formulation of LPP means converting verbal description of the given
problem into mathematical form in terms of objective function, constraints and non negative
restriction:

(i) Identification of the decision variables whose value is to be determined.

(ii) Formation of an objective function as a linear function of the decision variables.

(iii) Identification of the set of constraints or restrictions. Express them as linear inequation
with appropriate sign of equality or inequality.

(iv) Mention the non-negative restriction for the decision variables.

2. Solving the LPP–

(i) First of all formulate the given problem in terms of mathematical constraints and an
objective function.

(ii) The constraints would be inequations which shall be plotted and relevant area shall be
shaded.

(iii) The corner points of common shaded area shall be identified and the coordinates
corresponding to these points shall be substituted in the objective function.

(iv) The coordinates of one corner point which maximize or minimize the objective function
shall be optimal solution of the given problem. If feasible region is unbounded, then a
maximum or a minimum value of the objective function may not exist. However, if it exists, it
must occur at a corner point of feasible region
Chapter – 13 Probability

Probability can be defined as the ratio of the number of favourable outcomes to the total number of
outcomes of an event.

Experiment: A trial or an operation conducted to produce an outcome is called an experiment.

Sample Space: All the possible outcomes of an experiment together constitute a sample space. For
example, the sample space of tossing a coin are head and tail.

Favourable Outcome: An event that has produced the desired result or expected event is called a

favourable outcome. For example, when we roll two dice, the possibility of getting number 4 is (1,3),
(2,2), and (3,1).

Trial: A trial denotes doing a random experiment.

Event: The total number of outcomes of a random experiment is called an event.

Probability of an event P(E) = Number of favourable outcomes / Sample Space

Important Notes:

1. Probability is a measure of how likely an event is to happen.

2. Probability is represented as a fraction and always lies between 0 and 1
3. An event can be defined as a subset of sample space.

Conditional probability: It is the probability for one event to occur with some relationship to
one or more other events.

Properties of conditional probability –

(i) If F be an event of a sample space s of an experiment, then P (S/F) = P (F/F) = 1 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

(ii) P (A ∩ B/F) = P(A/F) + P (B/F) – P (A ∩ B/F) IF A and Bare disjoint event then P(A ∩ B/F)
= P(A/F) + P (B/F)

(iii) P (E / F) = 1 – P (E/F) or P (E∩/F) = 1 – P (E/F)

Multiplication Theorem On Probability – Let E and F be two events associated with a
sample space S. P(E ∩ F) denotes the probability of the event that both E and F occur,
which is given by P (E ∩F) = P (E) P (F/E) = P (F) P (E/F), provided P (E) ≠ 0 and P(F) ≠ 0

Independent Event–

(i) Events E and F are independent if P (E ≠ F) = P (E) × P (F)

(ii) Two events E and F are said to be independent if P (E/F) = P (E) and P(F/E) = P(F)
provided P (E) ≠ 0 and P (F) 0

(iii) Three events E, F and G are said to be independent or mutually independent if P(E ∩F
∩G) = P(E) P(F) P(G).

Random Variable – A random variable is a real valued function whose domain is the
sample space of random experiment.

Bayes’ formula: It describes the probability of occurrence of an event in relation to any

condition. It is useful for the case of conditional probability.

Bernoulli Trial –Trials of a random experiment are said to be Bernoulli’s trials, ifthey satisfy
the following conditions:

(i) The trials should be independent.

(ii) Each trial has exactly two outcomes ex- success or failure.

(iii) The probability of success remains the same in each trial.

(iv) Number of trials is finite.

Mean of Random Variable – let X be a random variable whose possible values are x1 , x2,
…… xn if P1, P2, …… Pn are the corresponding probabilities, then mean of X,

Variance of a Random Variable:

Binomial Distribution: The binomial distribution represents the probability for 'x' successes of an
experiment in 'n' trials, given a success probability 'p' for each trial at the experiment.