Prasanna

Students can Download Chapter 13 Probability Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 13 Probability

Introduction
In this chapter we study the concept of conditional probability, multiplication rule of probability and independence of events, Baye’s theorem, Probability distribution and its mean and variance, Bernoulli Trials, Binomial Distribution, and its mean and variance.

A. Basic Concepts
I. Conditional Probability
Let A and B are two events associated with the sample space of a random experiment. The probability of occurrence of the event A that the event B has already happened is called Conditional Probability of A given B, denoted by P(A/B).

Properties:

1. Let A and B be events of a sample space S of an experiment, then P(S/A) = 1, P(A/A) = 1
2. If A and B are two events of a sample space S and E is an event of S such that P(E) ≠ 0, then P((A U B)/E) = P(A/E) + P(B/E) – P((A ∩ B)/E)
3. P(A’/B) = 1 – P(A/B).

II. Multiplication Theorem
If A and B be two events associated with a random experiment, then

1. P(A ∩ B) = P(B) × P(A/B), if P(B) ≠ 0
2. P(A ∩ B) = P(A) × P(B/A), if P(A) ≠ 0

P(A ∩ B ∩ C) = P(A) × P(B / A) × P(C /(A ∩ B)).

III. Independent Events
Two events are said to be independent if the probability of occurrence of any one of the event does not affect the occurrence of the other.

1. P(A/B) = P(A)
2. P(A ∩ B) = P(A) × P(B)
3. P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
4. Two events associated with a random experiment cannot be both mutually exclusive and independent.
5. If P(A ∩ B) ≠ P(A) × P(B), the A and B are dependent events.
6. If A and B are independent events, then
   • A and \(\bar{B}\) are independent events,
   • \(\bar{A}\) and \(\bar{B}\) are independent events.

IV. Theorem of total probability
Let {E₁, E₂,…., E_n}be a partition of the sample space S, and suppose that each of the events E₁, E₂, …., E_n 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(E_n)P(A/E_n)
= \(\sum_{i=1}^{n}\)P(E_i)P(A/E_i).

V. Baye’s Theorem
If E₁, E₂,…., E_n are n non-empty events which constitute a partition of sample space S, then

VI. Probability Distribution
The Probability Distribution of a random variable X is the system of numbers

Where the real numbers x₁, x₂,…….., x_n are the possible values of the random variable X and p₁, p₂,……, P_n is the probability of the each possible values of the random variable X.
P₁ + P₂+…….+P_n = 1 and 0 ≤ p_t ≤ 1.

1. The mean of the above Probability Distribution is denoted by µ, is also called Expectation of X.
   ie; Mean = µ = E(X) = \(\sum_{i=1}^{n}\)x_ip_i
2. The Variance of the above Probability Distribution is denoted by σ²,
   ie; Variance = σ² = E(X²) – [E(X)]²

VII. Bernoulli Trial
Trials of a random experiment are called Bernoulli trial, if it satisfies the following conditions:

1. There should be a finite number of trials.
2. The trials should be independent.
3. Each trial has exactly two outcomes: success or failure.
4. The probability of success remains the same in each trial.

VIII. Binomial Distribution
Binomial Distribution, denote by B(n, p) is given (p + q)ⁿ where p represents probability of success, q represent probability of failure and n represents the number of trials. The probability of x success is
P(X = x) = nC_xq^{n – x}p^x.

1. Mean = np
2. Variance = npq
3. Standard Deviation = \(\sqrt{n p q}\).

Students can Download Chapter 12 Linear Programming Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 12 Linear Programming

Introduction
A special class of optmisation problems such as finding maximum profit, minimum cost, or minimum use of resources, etc, is Linear Programming Problems. In this chapter we study some linear programming problems and their solutions graphically.

A. Basic Concepts
A linear Programming Problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called the objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.
A few important LPP are;

• Diet Problem.
• Manufacturing Problem.
• Transportation Problem.

1. The common region determined by the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a LPP is called the feasible region.

2. Points within and on the boundary of the feasible region represents feasible solution of the constraints. Any point outside the feasible region is an infeasible solution.

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

I. Corner point Method

1. Find the feasible region of the LPP and determine its corner points (vertices).
2. Evaluate the objective function Z = ax + by at each corner points. Let M and m be the maxjmum and minimum values at these points.
3. If the feasible region is bounded, M, and m respectively are the maximum and minimum values of the objective function.
4. If the feasible region is unbounded, then
   • M is the maximum value of the objective function, if the open half-plane determined by ax + by > M has no points in common with the feasible region. Otherwise the objective function no maximum value.
   • m is the minimum value of the objective function, if the open half-plane determined by ax + by < m has no point in common with the feasible region. Otherwise, the objective function has no minimum value.

Students can Download Chapter 11 Three Dimensional Geometry Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 11 Three Dimensional Geometry

Introduction
To refer a point in space we require a third axis (say z-axis) which leads to the concept of three dimensional geometry. In this chapter we study the concept of direction cosines, direction ratios, equation of a line and a plane, angle between two lines and two planes, angle between a line and a plane, shortest distance between two skew lines, distance of a point from a plane.

Basic concepts
I. Direction cosines and direction ratios
Consider a directed line passing through the origin makes angles α, β, and γ with the positive
direction x-axis, y-axis, and z-axis. Then α, β, and γ are called direction angles. The cosine of α, β, and γ are called direction cosines. Generally cos α = l, cos β = m and cos γ = n . Any scalar multiple of direction cosines are called direction ratios.

1. If (a, b, c) is the coordinate of a point P then a,b,c is a direction ratio of the directed line passing along P and origin. Direction cosines will be

2. l² + m² + n² = 1

3. Direction ratios of a line segment passing through two points(x₁, y₁, z₁) and(x₂, y₂, z₂) is
x₂ – x₁, y₂ – y₁, z₁ – z₂

4. The angle between two lines having direction ratios a₁, b₁, c₁ and a₂, b₂, c₂ is

5. If direction ratios are proportional then the lines a, b, c, are parallel.ie; \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

6. If a₁a₂ + b₁b₂ + c₁c₂ = 0 then the two lines are perpendicular.

II. Line in Space
Equation of line when a point and parallel direction ratios are given:
1. Vector equation:
\(\bar{r}=\bar{a}+\lambda \bar{b}\), where \(\bar{a}\) is a point, \(\bar{b}\) is a parallel vector and λ is a parameter, for different values of λ we get parallel lines.

2. Cartesian equation:

where (x₁, y₁, z₁) is a point and a, b, c is a parallel direction ratios.

Equation of line when two points are given:
1. Vector equation:
\(\bar{r}=\bar{a}+\lambda(\bar{b}-\bar{a})\), where \(\bar{a}\) and \(\bar{b}\) are points and λ is a parameter, for different values of λ we get parallel lines.

2. Cartesianequation:

where(x₁, y₁, z₁) and (x₂, y₂, z₂) are two points.

Angle between two lines:
1. Vector form:

be two lines and θ be the angle between them, then cosθ =

• If parallel \(\overline{b_{1}}=k \overline{b_{2}}\), k scalar.
• If perpendicular \(\overline{b_{1}} \overline{b_{2}}\) = 0.

2. Cartesian form:


be two lines and θ be the angle between them, then

• If parallel \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
• If perpendicular a₁a₂ + b₁b₂ + c₁c₂ = 0.

Shortest distance between skew lines:
Lines which are neither interesting nor parallel are known as skew lines. Shortest distance between two skew lines is
1. Vector form:

be two skew lines, then d =

2. Cartesian form:

3. Distance between parallel lines,

III. Plane in space
Normal Form:
1. Vector Equation:
\(\bar{r}\).\(\hat{n}\) = d, where \(\hat{n}\) is a unit vector perpendicular to the Plane, and d is the perpendicular distance of the Plane from the origin. The general vector equation of a Plane is \(\bar{r} \bar{m}=d\), where \(\bar{m}\) is any vector perpendicular to the plane and cfis a constant.

2. Cartesian equation:
lx + my + nz = d, where l, m, n are direction cosines perpendicular to the Plane and dis the perpendicular distance of the Plane from the origin. The general cartesian equation of a Plane is ax + by + cz = d, where a, b, c are direction ratios perpendicular to the plane, and d is a constant.

Equation of plane when a point and a perpendicular vector is given:
1. Vector Equation:
\((\bar{r}-\bar{a}) \bar{m}=d\), where \(\bar{m}\) is a vector perpendicular to the Plane and \(\bar{a}\) is a point on the plane.

2. Cartesianequation:
a(x – x₁) + b(y – y₁) + c(z – z₁) = d, where a, b, c are direction ratios perpendicular to the plane and (x₁, y₁, z₁) is a point on the plane.

Equation of a plane passing through three non-collinear points:
1. Vector Equation:
\((\bar{r}-\bar{a}) \cdot[(\bar{b}-\bar{a}) \times(\bar{c}-\bar{a})]=0\), where \(\bar{a}, \bar{b}, \bar{c}\) are points on the plane.

2. Cartesian equation:

Where, (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) are points on the plane.

3. Intercept form of the equation of a Plane:
Let a, b, c are the x-intercept, y-intercept and z- intercept made by a plane, then the equation of x y z such a Plane is \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\).

4. Angle between two Planes:
Angle between two planes is same as the angle between there perpendicular vectors.

5. Vector Form:

be two Planes and θ be the angle between them, then cos θ

6.
(i) if parallel \(\overline{m_{1}}=k \overline{m_{2}}\), k scalar.
(ii) if perpendicular \(\bar{m}_{1} \overline{m_{2}}\) = 0

7. Cartesian form:
a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂ be two lines and θ be the angle between them, then

• If parallel \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)
• If perpendicular a₁a₂ + b₁b₂ + c₁c₂ = 0

Angle between a line and a Plane:

Plane passing through the intersection of two given planes:
The equation of family of Planes passing through the intersection of the Planes a₁x + b₁y + c₁z = d₁ and a₂x + b₂y + c₂z = d₂ is a₁x + b₁y + c₁z – d₁ + λ(a₂x + b₂y + c₂z – d) = 0.

Distance of a point from a Plane:
The perpendicular distance of the point (x₁, y₁, z₁) from a Plane ax + by + cz = d is

The distance between parallel Planes ax + by + cz = d₁ and ax + by + cz = d₂ is

Students can Download Chapter 10 Vector Algebra Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 10 Vector Algebra

Introduction
Physical quantities we deal are of two types, one that can be specified using a single real number which gives its magnitude and the other which involves the idea of direction as well as magnitude. The first type is called scalar quantity and the second is vector quantity. In this chapter we analyses the basic concepts about vectors, various operations, and their algebraic and geometrical properties.

I. Types of vectors

1. Equal Vectors: Vectors having same magnitude and direction regardless of the positions of their initial points.
2. Collinear Vectors: Vectors which are parallel to the same line, irrespective of their magnitude and direction.
3. Like and Unlike Vectors: Collinear vectors having same direction are like vectors and opposite direction are unlike vectors.
4. Unit Vectors: Vectors with magnitude unity.

II. Component form of a vector
Let i, j, k be the unit vectors along the x-axis, y-axis, z-axis respectively. The point P(x, y, z) be a point in space. Then the position vector of the point P can be expressed in component form as
1. If li + mj + nk is unit vector, then l,m,n are direction cosines along the vector.
2. If P (a, b, c) is a point on space, then a, b, c are direction ratios and

are direction cosines along the vector \(\overline{O P}\).

III. Addition of Vectors

\(\overline{A B}+\overline{B C}+\overline{C A}=\overline{0}\) is known as triangle law of vector addition.

IV. Multiplication of a vector by a scalar
Let \(\bar{a}\) = a₁i + a₂j + a₃k be a vector and λ be a scalar. Then the product of the vector \(\bar{a}\) by a scalar is denoted by λ\(\bar{a}\) and the new vector formed has a magnitude λ|\(\bar{a}\)|.
λ\(\bar{a}\) = λa₁i + λa₂j + λa₃k

V. Vector joining two points
If P(a₁, a₂, a₃) and Q(b₁, b₂, b₃) are two points, then the vector joining P and Q is the vector \(\overline{P Q}\).
ie: \(\overline{P Q}\) = (b₁ – a₁)i + (b₂ – a₂)j + (b₃ – a₃)k

VI. Section Formula
If \(\bar{a}\) and \(\bar{b}\) be the position vectors of the points A and B respectively, then the position vector of the point P which divides AB in the ratio l:m

VII. Dot (Scalar) Product of vectors

VIII. Cross (vector) Product of Vectors

Geometrical meaning of vector product.

• \(\bar{a} \times \bar{b}\) is a vector perpendicular to \(\bar{a}\) and \(\bar{b}\).
• \(|\bar{a} \times \bar{b}|\) gives the area of a parallelogram with adjacent sides \(\bar{a}\) and \(\bar{b}\).

• i × i = j × j = k × k = 0,
• i × j = k, j × k = i, k × i = j
• j × i = -k, k × j = -i, i × k = -j

IX. Box (Scalar Triple) Product of Vectors

Properties:
1. Since \(\bar{b} \times \bar{c}\) is a vector, \([\bar{a} \bar{b} \bar{c}]\) is a scalar quantity.

2. |\([\bar{a} \bar{b} \bar{c}]\)| is the volume of the parallelopiped with a adjacent sides vector \(\bar{a}, \bar{b}, \bar{c}\).

3. If \(\bar{a}\) = a₁i + a₂j + a₃k; \(\bar{b}\) = b₁i + b₂j + b₃k and \(\bar{c}\) = c₁i + c₂j + c₃k, then

4. if \(\bar{a}, \bar{b}, \bar{c}\) be any three vectors, then \([\bar{a} \bar{b} \bar{c}]\) = \([\bar{b} \bar{c} \bar{a}]=[\bar{c} \bar{a} \bar{b}]\) (cyclic permutation of three vectors does not change the value of the scalar triple product).

5. In scalar triple product, the dot and cross can be interchanged.ie,

6. If any two vectors are interchanged the sign of box product is changed but magnitude remains the same.

7. If any two vectors are equal or proportional then the value of box product is zero.

8. Three vectors \(\bar{a}, \bar{b}, \bar{c}\) are coplanar if and only if \([\bar{a} \bar{b} \bar{c}]\) = 0.

Students can Download Chapter 9 Differential Equations Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 9 Differential Equations

Introduction
An equation involving derivatives of a dependent variable with respect to one or more independent variables is called a Differential Equation. In this chapter we study the method formation of a Differential Equation and solving of a Differential Equation.

I. Degree and Order of a DE
Order of a DE is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given DE.

Degree of a DE is defined as the exponent of highest differential coefficient appearing in the equation provided the equation is made into polynomial form in all differential coefficient.

II. Formation of a DE
To form a DE from a given function we differentiate the function successively as many times as the number of arbitrary constants in the equation and eliminate the arbitrary constant.

III. Solution of a DE
1. Variable Separable Type:
A DE of the form mdx = ndy Where m is a function in x alone or a constant and n is a function y alone or a constant.
Solution is ∫mdx = ∫ndy + c.

2. Homogeneous DE:
A DE of the form \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)}\), where f(x, y) and g(x, y) are homogeneous equations in x and y. Solution is put y = vx ⇒ \(\frac{d y}{d x}=v+x \frac{d v}{d x}\) after simplification DE will be converted into variable separable type.

3. Linear DE:
A DE of the form \(\frac{d y}{d x}\) + Py = Q, where P and Q dx are function in x alone or a constant.
Solution is IF = e^∫Pdx
⇒ y(IF) = ∫Q(IF)dx + c.

A DE of the form \(\frac{d x}{d y}\) + px = Q, where P and Q are function in y alone or a constant.
Solution is IF = e^∫Pdy
⇒ x(IF) = ∫Q(IF)dy + c.

Students can Download Chapter 8 Application of Integrals Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 8 Application of Integrals

Introduction
In this chapter we study the specific application of definite integrals to find the area under simple curves, area between lines and arcs of circles, parabolas, and ellipses.

Area under Simple Curves
Area = \(\int_{a}^{b}\)f(x)dx = \(\int_{a}^{b}\)ydx

Area = \(\int_{a}^{b}\)f(y)dy = \(\int_{a}^{b}\)xdy

Area = \(\int_{a}^{c}\)f(x)dx – \(\int_{c}^{b}\)f(x)dx

Area = \(\int_{a}^{b}\)f₂(x)dx – \(\int_{a}^{b}\)f₁(x)dx

Students can Download Chapter 7 Integrals Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 7 Integrals

Introduction
Integration is the reverse process of differentiation. The development of integral calculus is outcome of the efforts to solve the problems to find the function when its derivative is given and to find the area bounded by the graph of a function under certain conditions. In this chapter we study different method of find indefinite integral and definite integrals of certain functions and its properties.

A. Basic Concepts
I. Integration
Let \(\frac{d}{d x}\)F(x) = f(x). then we write ∫f(x)dx = F(x) + C.
These integrals are called indefinite integrals and C is the constant of integration.

1. Indefinite integral is a collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upward or downwards along the y-axis.
2. ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx
3. For any real number k, ∫[kf(x)]dx = k∫f(x)dx

II. Some Standard Results

• ∫xⁿ dx = \(\frac{x^{n+1}}{n+1}\) + C
• ∫\(\frac{1}{x}\)dx = log|x| + C
• ∫e^xdx = e^x + C
• ∫a^xdx = \(\frac{a^{x}}{\log a}\) + C
• ∫sin x dx = -cosx + C
• ∫cos xdx = sin x + C
• ∫sec²xdx = tanx + C
• ∫cosecx cotx dx = -cosecx + C
• ∫secx tanx dx = secx + C
• ∫cosec²x dx = -cotx + C
• ∫tan x dx = log|sec x| + C
• ∫cot xdx = log|sin x| + C
• ∫sec xdx = log|sec x + tan x| + C
• ∫cosec x dx = log|cosec x – cot x| + C

III. Some methods of Integration
1. If \(\frac{d}{d x}\) F(x) = f (x) and ∫f(x)dx = F(x) + C then ∫f(ax + b)dx = \(\frac{1}{a}\) F(ax + b) + C.

2. ∫[f(x)]ⁿ f'(x)dx = \(\frac{[f(x)]^{n+1}}{n+1}\) + C
\(\int \frac{f^{\prime}(x)}{f(x)} d x\) = log[f(x)| + C

3. ∫e^x[f(x) + f'(x)]dx = e^xf(x) + C

4. Substitution Method:
The given integral I = ∫f(x)dx is transformed into another form by changing the independent variable x to t by substituting x = g(t). So that \(\frac{d x}{d t}\) = g'(t) ⇒ dx = g'(t)dt
∴ I = ∫f(x)dx = ∫f(g(t))g'(t)dt.

5.

6.

7. Integration using partial fractions:
Consider Integrals of the form ∫\(\frac{P(x)}{Q(x)}\)dx, where P(x) and Q(x) are polynomials in x and Q(x) ≠ 0. If the degree of P(x) is less than Q(x), then the rational function is proper function otherwise improper function.

If \(\frac{P(x)}{Q(x)}\) is improper function, first it should be converted to proper by long division and now it takes the form \(\frac{P(x)}{Q(x)}\) = T(x) + \(\frac{P_{1}(x)}{Q(x)}\) Where T(x) is polynomial in x and \(\frac{P_{1}(x)}{Q(x)}\) is a proper function.

Now if \(\frac{P(x)}{Q(x)}\) is proper function we factorise the denominator Q(x) into simpler polynomials and decompose into simpler rational function. For this we use the following table.

8.

9. Integration by Parts:
∫f(x)g(x)dx = f(x)∫g(x)dx – ∫(f'(x) ∫g(x)dx)dx
Here the priority of taking first function and second function is more important, for this use order of the letters in words ILATE, where

• • I- Inverse Trigonometric Function.
  • L – Logarithmic Function.

• A – Algebraic Function.
• T -Trigonometric Function.
• E – Exponential Function.

IV. Definite Integral
A definite integral has a unique value. A definite integral is denoted by \(\int_{a}^{b}\)f(x)dx, where a is the upper limit and b is the lower limit of the integral. If \(\frac{d}{d x}\) F(x) = f(x) and ∫f(x)dx = F(x) + C , then
\(\int_{a}^{b}\)f(x)dx = F(b) – F(a).

1. Definite integral as the sum of a limit:
Let f(x) be continuous function defined on a closed interval [a, b]. Then \(\int_{a}^{b}\)f(x)dx is area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis.

Students can Download Chapter 6 Application of Derivatives Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 6 Application of Derivatives

Introduction
In this chapter we analyses the physical and geometrical applications of derivatives in real life such as to determine rate of change, to find tangents and normal to a curve, to find turning points, intervals in which the curve is increasing and decreasing, to find approximate value of certain quantities.

I. Rate of Change
\(\frac{d y}{d x}\), we mean the rate of change of y with respect to x. If s is the displacement function in terms of time t and v the velocity at that time. Then, \(\frac{d s}{d t}\) = velocity, Acceleration = \(\frac{d v}{d t}=\frac{d^{2} s}{d t^{2}}\)

II. Tangents and Normals
If a tangent line to the curve y = f(x) makes an angle θ with the positive direction of the x-axis, then f'(x) = slope of the tangent = tanθ.
Equation of tangent to the curve y = f (x) at the point (x₁, y₁): y – y₁ = f'(x₁)(x – x₁)
Equation of normal to the curve y = f (x) at the point (x₁, y₁): y – y₁ = –\(\frac{1}{f^{\prime}\left(x_{1}\right)}\)(x – x₁)

III. Increasing and decreasing functions
Nature of a function on a given interval;
Strictly increasing on [a, b]: f'(x) > 0, x ∈ (a, b) Increasing on [a, b]: f'(x) ≥ 0, x ∈ (a, b)
Strictly decreasing on [a, b]: f'(x) < 0, x ∈ (a, b) Decreasing on[a, b]: f'(x) ≤ 0, x ∈ (a, b).
1. Between two consecutive points at which f'(x) = 0 the function has only one nature either it is increasing or decreasing, not both.

IV. Approximation
Consider a function y = f(x). Let ∆x denote a small increment in x and ∆y be the corresponding increment in y. Then, ∆y can be approximated by dy, where dy = \(\frac{d y}{d x}\) × ∆x.

V. Maxima and Minima
A function y = f(x) is said to have a local maximum at x = a, if f(a) is the maximum value obtained by the function in the neighbourhood of x = a.

A function y = f(x) is said to have a local minimum at x = a, if f(a) is the minimum value obtained by the function in the neighbourhood of x = a.

Point on the curve at which f'(x) = 0 is called stationary point or turning point. The following are methods to find the local maximum and local minimum at points where f'(x) = 0.

First Derivative Test:

1. If f'(c) = 0 and f'(x)changes its sign from positive to negative from left to right of x = c, then the point is a local maximum point.
2. If f'(c) = 0 and f'(x) changes its sign from negative to positive from left to right of x = c, then the point is a local minimum point.
3. If f'(c) = 0 and if there is no change of sign for f'(x) from left to right of x = c, then the point is a inflexion point.

Second Derivative Test:

1. If f'(c) = 0 and f”(c) < 0 , then x = c is a local maximum point.
2. If f'(c) = 0 and f”(c) > 0, then x = c is a local minimum point.
3. If f'(c) = 0 and f”(c) = 0, then the test fails and go to first derivate test for checking maxima and minima.

Absolute Maxima and Minima:
Let f(x) be a function defined on [a, b] and if , f'(x) = 0 ⇒ x = x₁, x₂, x₃,……etc, then

1. Absolute maximum value of
   = max{f(a), f(x₁), f(x₂),…….f(b)}
2. Absolute minimum value of
   = min{f(a), f.(x₁), f(x₂),…..f(b)}

Students can Download Chapter 5 Continuity and Differentiability Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 5 Continuity and Differentiability

Introduction
As a continuation of limits and derivatives studied in the previous years, now we are entering into a very important concept continuity and its graphical peculiarities. We also learn different methods of differentiation and introduce new class of functions such as exponential and logarithmic functions.

I. Continuity
Continuity of a function at a point
A function f(x) is said to be continuous at a point ‘a’ if the following conditions are satisfied.

1. f(a) should be defined.
2. Left limit should be equal to right limit, ie; \(\lim _{x \rightarrow a^{-}}\)f(x) = \(\lim _{x \rightarrow a^{+}}\) f(x)
3. f(a) should be equal to the limit of the function at ‘a’. \(\lim _{x \rightarrow a^{-}}\)f(x) = \(\lim _{x \rightarrow a^{+}}\) f(x) = f(a).

Continuity of a function
A function f(x) is said to be continuousif the function is continuous at every point on its domain. Some standard continuous functions are mentioned below;

1. Constant function f(x) = c, c-constant.
2. Identity function f(x) = x.
3. Modulus function f(x) = |x|.
4. Exponential function f(x) = e^x.
5. Logarithmic function f(x) = log x.
6. Polynomial function
   f(x) = a₀ + a₁x + a₂x² +…….+ a_nxⁿ
7. Rational function
   f(x) = \(\frac{p(x)}{q(x)}\), P(x) & q(x) are polynomial function and q(x) ≠ 0.
8. Trigonometric and inverse trigonometric function.

Graphical approach:
If there is a break in the graph of a function then it is not continuous.

Algebra of Continuous functions
Suppose f and g be two real functions in their respective domains then the following are true.
1. f + g, f – g, f.g, \(\frac{f}{g}\) [g(x) ≠ 0], fog, gof are all continuous functions.

II. Differentiability
Differentiability at a point:
A function f(x) is said to be differentiable at a point ‘c’ if the following limit exists and the value of the limit is known as the first derivative of f(x) at
x = c denoted by f'(c) or \(\left(\frac{d y}{d x}\right)_{x=c}\)

Derivative of a function:
The function defined by f'(x) = \(\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\)
wherever the limit exists is defined to be the derivative of f. The derivative of f is denoted by
f ‘(x) or \(\frac{d y}{d x}\) or y’ or y₁
1. Every differentiable function is continuous. But the converse need not be.true, eg; f(x) = |x|.

Some Standard Results:

Algebra of derivatives:
Let f(x) and g(x) be two differentiable functions, then
1. \(\frac{d}{d x}\) (f(x) ± g(x)) = \(\frac{d}{d x}\)f(x) ± \(\frac{d}{d x}\) g(x).

2. Product Rule:
\(\frac{d}{d x}\)(f(x).g(x)) = \(\frac{d}{d x}\) f(x) × g(x) + f(x) × \(\frac{d}{d x}\) g(x)

3. Quotient Rule:

4. Chain Rule:
Let y be a real function which is a composite of two functions h(x) and g(x), ie; f(x) = h(g(x)) .then
\(\frac{d}{d x}\) f(x) = h'(g(x)) × g'(x).

5. Implicit Differentiation:
Here differentiate both sides of the function with respect to × and solve for \(\frac{d y}{d x}\).

6. Logarithmic Differentiation:
Function with are complicated Rational functions and of the form f(x) = u(x)^v(x) is differentiated using Logarithmic Differentiation method. Here first take logarithm on both sides of the function and proceed as in implicit differentiation.

7. Parametric Differentiation:
Relation between two variable x and y which are expressed in the formx = f(t), y = g(t) is said to be parametric form with parameter t.
Here

8. Second Order Derivative:
If f'(x) is differentiable we may differentiate once again with respect to x.
Then, \(\frac{d}{d x}\left(\frac{d y}{d x}\right)\) is called the Second Derivate of f with respective to x, denoted by \(\frac{d^{2} y}{d x^{2}}\) or f”(x) or y₂ or y”.

III. Rolle’s theorem
Let f: [a, b] → R be a continuous function on [a, b] and differentiable on (a, b), such that f(a) = f(b) , where a and b are some real numbers. Then there exists some c ∈ (a, b) such that f'(c) = 0.

IV. Mean Value theorem
Let f: [a, b] → R be a continuous function on [a, b]and differentiable on (a, b). Then there
exists some c ∈ (a, b) such that f’(c) = \(\frac{f(b)-f(a)}{b-a}\).

Students can Download Chapter 4 Determinants Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 4 Determinants

Introduction
Determinants have wide applications in engineering, Science, Economics, Social Science, etc. In this chapter we study about the various properties of determinants, minors, cofactors, applications in finding the area of triangle, adjoint, and inverse of a square matrix, and consistency and in consistency of linear equations.

A. Basic Concepts
I. Determinant
Determinant is a real number associated with a square matrix. The determinant of matrix A is denoted by |A|. The value of a determinant is obtained by the sum of products of elements of a row (column) with corresponding cofactors.

• |AB| = |A||B|
• |Aⁿ| = |A|ⁿ

Properties:
(i) The value of a determinant remains the same if its rows and columns are interchanged.
ie; |A^T| = |A|.

(ii) If any two rows (or columns) of a determinant are interchanged, then sign of determinant changes.

(iii) If any two rows (or columns) of a determinant is identical, then value of determinant is zero.

(iv) If each element of a row (or columns) of a determinant is multiplied by a constant k, then its value gets multiplied by k.

(a) If A is a square matrix of order n, then
|KA| = kⁿ|A|

(v) If any two rows (or columns) of a determinant is proportional, then value of determinant is
zero.

(vi) If some or all elements of a row or column of a determinant are expressed as sum of two (or more) terms, then the determinant can be expressed as sum of two (or more) determinants.

(vii) If, to each element of any row or column of a determinant, the equimultiples of corresponding elements of other row (or column) are added, then value of determinant remains the same, ie; the value of determinant remains same if we apply the operation R_i → R_i + kR_j or C_i → C_i + kC_j.

(viii) If the elements of a row or column are multiplied with cofactors of any other row or column, then their sum is zero.
ie; for example; a₁₁C₂₁ + a₁₂C₂₂ + a₁₃C₂₃ = 0.

If the elements of a row or column are multiplied with cofactors of the corresponding row or column, then their sum is |A|.
ie; for example; a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ = |A|.

1. Minor of an element:
The minor of an element a_ij is the determinant obtained by deleting the ith row and jth column, usually denoted by M_ij.

2. Cofactor of an element:
A signed Minor is called cofactor, ie; C_ij = (-1 )^{i + j} M_ij. The matrix obtained by replacing all elements by its cofactor is called cofactor matrix.

3. Adjoint Matrix:
The transpose of a cofactor matrix is. called Adjoint Matrix, usually denoted by adj(A)

Properties:

• A × adj(A) = adj(A) × A = I|A|
• If A is a square matrix of order n, then |adj(A)| = |A|^{n – 1}
• adj(AB) = adj(A)adj(B)

II. Inverse of a Matrix
A square matrix A is invertible if |A| ≠ 0 and A inverse is denoted by A^-1 ie; A^-1 = \(\frac{a d j(A)}{|A|}\)
Properties:

• (A^-1)^-1 = A
• (AB)^-1 = B^-1A^-1
• (A^T)^-1 = (A^-1)^T
• If A is a square matrix of order n, then adj(adj(A)) = |A|^n-2 × A.

III. Application of Determinants
1. Area of a triangle whose vertices are (x₁, y₁), (x₂, y₂), (x₃, y₃) is

(i) If Area = 0 then the points are collinear.

2. Solving of system of linear equations using
matrix method:
Consider the system of linear equations
a₁x + b₁y + C₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Convert the linear equation into matrix form AX = B, where

Then the solution is given by X = A^-1B

• If |A| ≠ 0 the system is consistent and has unique solution.
• If |A| = 0 and adj(A) × B ≠ 0,the system is inconsistent and has no solution.
• If |A| = Oandadj(A) × B = 0 ,the system may be consistent and has infinitely many solutions.

In order to find these infinitely many solutions, replace one of the variable by k (say z = k) and solve any two of the given equations for x and y in terms of k

Students can Download Chapter 3 Matrices Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 3 Matrices

Introduction
The term ‘matrix’ was first used in 1850 by the famous English Mathematician James Joseph Sylvester. In 1858 Arther Cayley began the Systematic development of the theory of matrices. Matrix was first used for the study of linear equations and linear transformations. Now it is largely used in disciplines like statistics, physics, chemistry, psychology, etc.

A. Basic Concepts
I. Matrix
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.
1. Order of a Matrix:
A matrix having m rows and n column is called a matrix of order m × n, generally denoted by

Where 1 ≤ i ≤ m, 1 ≤ j ≤ n  i, j ∈ N.

II. Types of Matrix

• Column Matrix: A matrix having only one column is called Column Matrix.
• Row Matrix: A matrix having only one row is called Row Matrix.
• Square Matrix: A matrix having equal number of row and column is called Square Matrix.
• Diagonal Matrix: A Square matrix having all its non-diagonal entries zero is called Diagonal Matrix.
• Scalar Matrix: A Square matrix having all its non-diagonal entries zero and equal diagonal elements is called Scalar Matrix.
• Identity Matrix: A Square matrix having all its non-diagonal entries zero and diagonal elements unity is called an Identity Matrix.
• Zero Matrix: A matrix having all elements zero is called Zero Matrix.

III. Operations on matrices
1. Equality of Matrices:
Two matrices are equal if they are of same order and corresponding elements are equal.

2. Addition:
Addition is possible only if the two matrices are of same order and the operation is done by adding the corresponding elements in each Matrix. The addition of Matrix A and B is denoted by A + B.
Properties:

• Matrix addition is Commutative.
• Matrix addition is Associative.
• Zero Matrix is the additive identity.
• – A is the additive inverse of matrix A.

3. Scalar Multiplication:
The multiplication of a matrix by a scalar number k is done by multiplying each entries of A by k and matrix thus obtained is kA.

4. Difference:
Difference is possible only if the two matrices are of same order and the operation is done by subtracting the corresponding elements in each Matrix. The difference of Matrix A and B is denoted by A – B.

5. Multiplication:
Multiplication is possible only if the number of column of first matrix is equal to the number of rows of the second. The operation is done by multiplying the element in the first row of the first matrix with the corresponding elements in the first column in the second matrix.

This is continued till the rows in the first matrix finish. The multiplication of Matrix A and B is denoted by A × B or AB.
Properties:

• Matrix multiplication is Non-Commutative.
• Matrix multiplication is Associative, le; A(BC) = (AB)C
• Matrix multiplication is Distributive over addition, ie; A(B + C) = AB + AC
• Identity Matrix is the multiplicative identity, le; AI = IA.

IV. Transpose of a Matrix
The transpose of a matrix A is obtained by interchanging the row and column of A and is denoted by A^T.
Properties:

• [A^T]^T = A
• [kA]^T = kA^T
• [A + B]^T = A^T + B^T
• [AB]^T = B^T A^T

1. Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = A.
Properties:
In a symmetric matrix the corresponding elements on both sides of the main diagonal will be same.

2. Skew Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = -A.
Properties:

• In a Skew Symmetric matrix the corresponding elements on both sides of the main diagonal differ only in sign.
• For any square matrix A with real entries, A + A^T is Symmetric matrix, and A – A^T is Skew Symmetric matrix.
• Any square matrix can be expressed as the sum of a Symmetric and Skew symmetric matrix.
  ie; A = \(\frac{1}{2}\)(A + A^T) + \(\frac{1}{2}\)(A – A^T)
• If A and B are Symmetric matrices of the same order, AB is Symmetric if and only if AB = BA.
• If A and B are Symmetric matrices of the same order, (AB + BA) is Symmetric and (AB – BA) is Skew Symmetric.

V. Elementary Operation on Matrix
There are 6 operations on matrix, 3 for row and 3 for column.

1. The interchange of any two rows or two columns, symbolically denoted as R_i ↔ R_j or c_i ↔ c_j.
2. The multiplication of the elements of any row or column by a non-zero number, symbolically
   denoted as R_i ↔ kR_j or C_i ↔ kC_j.
3. The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number, symbolically denoted as R_i ↔ R_i + kR_j or C_i ↔ C_i + kC_j.

VI. Invertible Matrices
A square matrix B is said to be the inverse of a matrix A if AB = I = BA, then B is generally denoted
as A^-1.

1. Inverse of a square matrix, if it exists, is unique.
2. If A and B are invertible matrices of the same order, then (AB)^-1 = B^-1A^-1

Students can Download Chapter 2 Inverse Trigonometric Functions Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 2 Inverse Trigonometric Functions

Introduction
Trigonometric functions are real functions which are not objective and thus its inverse does not exist. In this chapter we study about the restrictions on domains and ranges of trigonometric functions which ensure the existence of their inverse and observe its graphical peculiarities.

A. Concepts
I. Functions
sin^-1 x : [-1, 1 ] → [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]
cos^-1 x: [-1, 1] → [0, π]
tan^-1 x : R → \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)
cosec^-1 x : R – (-1, 1) → [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)] – {0}
sec^-1 x : R -(-1, 1) → [0, π] – {\(\frac{\pi}{2}\)}
cot^-1 x : R → (0, π)

II. Properties
1. sin (sin^-1 x) = x, x ∈ [-1, 1]
sin^-1(sinx) = x, x ∈ [-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\)]
cos(cos^-1 x) = x, x ∈ [-1, 1]
cos^-1(cosx) = x, x ∈ [o, π]
tan(tan^-1 x) = x, x ∈ R
tan^-1(tan x) = x, x ∈ \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

2. sin^-1(-x) = -sin^-1(x), x ∈ [-1, 1]
tan^-1(-x) = -tan^-1(x), x ∈ R
cosec^-1(-x) = -cosec^-1(x), x ∈ R -(-1, 1)
cos^-1(-x) = π – cos^-1(x), x ∈ [-1, 1]
cot^-1(-x) = π – cot^-1(x), x ∈ R
sec^-1(-x) = π – sec^-1(x), x ∈ R -(-1, 1)
sin^-1(x) + cos^-1(x) = \(\frac{\pi}{2}\), x ∈ [-1, 1].

3. cosec^-1(x) + sec^-1(x) = \(\frac{\pi}{2}\), |x| ≥ 1
tan^-1(x) + cot^-1(x) = \(\frac{\pi}{2}\), x ∈ R

4. sin^-1 x

5. cos^-1 x

6. tan^-1(x) + tan^-1(y) =

7. tan^-1(x) – tan^-1(y) =

8. 2 tan^-1 x

9. sin^-1 x ± sin^-1 y

Students can Download Chapter 1 Relations and Functions Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 1 Relations and Functions

Introduction
A relation from a non-empty set A to a non-empty set B is a subset of A × B. In this chapter we study different types of relations and functions, composition of functions, and binary operations.

Basic Concepts
I. Types of Relations:
Here we study different relations in a set A

Empty Relation:
R : A → A given by R = Φ ⊂ A × A.

Universal Relation:
R : A → A given by R = A × A.

Reflexive Relation:
R : A → A with (a, a) ∈ R, ∀a ∈ A.

Symmetric Relation:
R : A → A with
(a, b) ∈ R ⇒ (b, a) ∈ R, ∀a, b ∈ A.

Transitive Relation:
R : A → A with (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R,
∀a, b, c ∈ A.

Equivalence Relation:
R : A → A which is Reflexive, Symmetric, and T ransitive.

Equivalence Class:
Let R be an Equivalence Relation in a set A. If a ∈ A, then the subset {x ∈ A, (x, a) ∈ R} of A is called the Equivalence Class corresponding to ‘a’ and it is denoted [a].

II. Types of Functions
One-One or Injective function.
A function f : A → B is said to be One-One or Injective, if the image of distinct elements of A under fare distinct.
i.e; f(x₁) = f(x₂) ⇒ x₁ = x₂
Otherwise f is a Many-One function.

1. Graphical approach:
If lines parallel to x-axis meet the curve at two or more points, then the function is not one-one.

Onto or Surjective function:
A function f : A → B is said to be Onto or Surjective, if every element of B is some image of some elements of A under f.
ie; If for every element y ∈ Y then there exists an element x in A such that f(x) = y.

Bijective function:
A function f : A → B is said to be Bijectiveit it is both One-One and Onto.

Composition of Functions.
Let f : A → B and g : B → C be two functions. Then the composition of f and g denoted by is gof defined
gof : A → C and gof (x) = g(f(x)).

1. If f : A → B and g : B → C are One-One, then gof : A → C is One-One.
2. If f : A → B and g : B → C are Onto, then gof : A → C is Onto.
3. If f : A → B and g : B → C are Bijective, ⇔ gof : A → C is Bijective.

Inverse Function:
If f : A → B is defined to be invertible, if there exists a function g : B → A such that gof = I_A and
fog = I_B. The function g is called the inverse of ‘f’ and is denoted by f^-1.

1. If function f : A → B is invertible only if f is bijective.
2. (gof)^-1 = f^-1og^-1.

III. Binary Operations
A binary operation ‘*’ on a set A is a function * : A × A → A, defined by a * b, a, b ∈ A.

1. * : A × A → A is commutative if a * b = b * a, ∀a, b ∈ A.
2. * : A × A → A is associative if a * (b * c) = (a * b) * c, ∀a, b, c ∈ A.
3. e ∈ A is the identity element for the binary operation * : A × A → A if a * e = a = e * a, ∀a ∈ A.
4. An elements a ∈ A is invertible for the binary operation * : A × A → A, if there exists an element b ∈ A such that a * b = e = b * a, where ‘e’ is the identity element for the operation ‘*’. Then ‘b’ is denoted by a^-1.

Students can Download Chapter 16 Chemistry in Everyday Life Notes, Plus Two Chemistry Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Chemistry Notes Chapter 16 Chemistry in Everyday Life

Drugs and their Classifcation
Drugs- chemicals of low molecular masses which interact with macromolecular targets and produce a biological response. When this response is therapeutic and useful, these chemicals are called medicines. Medicines are used in diagnosis, prevention, and treatment of diseases. Chemotherapy – Use of chemicals for therapeutic effect.

1. Classification of Drugs:
(a) On the Basis ofPharmacobgical Effect:
Provides them the whole reange of drugs available forthe treatment of a particular type of problem, useful to doctors, e.g. Analgesics – pain killing effect, Antiseptics – kill or arrest the growth of microorganisms.

(b) On the Basis of Drug Action:
Based on the action of a drug on a particular biochemical process, e.g. Antihistamines – inhibit the action of histamine.

(c) On the Basis of Chemical Structure:
Based on the chemical structure of the drug. These drugs share common structural features and often have similar pharmacological activity, e.g. Sulphonamides

(d) On the Basis of Molecular Targets:
Based on molecular targets, most useful for medicinal chemists. Drugs usually interact with biomolecules called target molecules or drug targets.

Drug-Target Interaction
Enzymes and receptros are important drug targets. Enzymes are proteins which perform the role of biological catalysts in the body. Receptors are proteins which are crucial to communication system in the body.

1. Enzymes as Drug Targets:
(a) Catalytic Action of Enzymes:
In their catalytic activity enzymes perform two major functions:

• To hold the substrate molecule in a suitable position, so that it can be attacked by the reagent effectively.
• To provide functional groups that will attack the substrate and carry out chemical reaction.

(b) Drug-Enzyme Interaction:
Drugs can block the binding site of the enzymes and prevent the binding of substrate, or can inhibit the catalytic activity of the enzyme. Such drugs are called enzyme inhibitors. This can be done in two different ways:

• Competitive Inhibitors: drugs which compete with the natural substrate for their attachment on the active sites of enzymes.
•  Allosteric Site: sites other than the active site of the enzyme. Some drugs bind to the allosteric site of the enzyme which changes the shape of the active site in such a way that substrate cannot recognise it.

Receptors as Drug Targets:
Majority of the receptor proteins are embedded in the cell membrane in such a way that their small part possessing active site projects out of the surface of the membrane and opens on the outside region of the cell membrane.

Chemical messengers – Chemicals involved in the transmission of message between two neurons and that between neurons to muscles. These are received at the binding site of the receptor proteins. To accommodate a messenger, shape of the receptor site changes. This brings about the transfer of message into the cell.

Antagonists – drugs that bind to the receptor site and inhibit its natural function. These are useful when blocking of a message is required.

Agonists – drugs that mimic the natural messenger by switching on the receptor. These are useful when there is lack of natural chemical messenger.

Therapeutic Action of Different Classes of Drugs
1. Antacids:
Drugs which control acidity in stomach. In early times, antacids such as NaHCO₃ or mixture of Al(OH)₃ and Mg(OH)₂ were used.

Histamine, stimulates the secretion of pepsin and hydrochloric acid in the stomach. Its discovery helped in the treatment of hyperacidity. The drug cimetidine (Tegamet) prevents the interaction of histamine with the receptors present in the stomach wall.

This resulted in release of lesser amount of acid. This drug was in use until another drug ranitidine (Zantac) was discovered.

2. Antihistamines:
Drugs which act against histamines, a potent vasodilator. It contracts the smooth muscles in the bronchi and gut and relaxes other muscles in the walls of fine blood vessels. It is also responsible for the nasal congestion associated with common cold and allergic response to pollen, e.g. Brompheniramine (Dimetapp), Terfenadine (Seldane) act as antihistamines.

3. Neurologically Active Drugs:
(a) Tranaquilizers:
Chemical compounds used for the treatment of stress and mild or even severe mental diseases. These relieve anxiety, stress, excitement by inducing a sense of well-being, e.g. Chlorodiazepoxide and meprobamate (mild tranquilizers suitable for relieving tension).

Antidepressant drugs – Drugs that inhibit the enzymes which catalyse the degradation of the important neurotransmitter, noradrenaline. Thus, noradrenaline is slowly metabolised and can activate its receptor for longer periods of time, thus counteracting the effect of depression.
e.g. Iproniazid, Phenelzine. Equanikised in controlling depression and hypertension.

Barbiturates – important class of tranquilizers which are derivatives of barbituric acid. These are hypnotic (sleep producing agents), e.g. veronal, amytal, nembutal, luminal and seconal.

(b) Analgesics:
Reduce or abolish pain without causing impairment of consciousness, mental confusion, incoordination or paralysis or some other disturbances of nervous system. They are two types,

(i) Non-narcotic Analgesics:
Drugs effective in relieving skeletal pain such as due to arthritis, also reduce fever (antipyretic), e.g. Aspirin, paracetamol. Aspirin is also used in the prevention of heart attacks because of its anti blood clotting action.

(ii) Narcotic Analgesic:
Drugs which releive pain and produce sleep in medicinal doses. In poisonous doses they produce stupor, coma, convulsions and ultimately death.They cause addiction, e.g. Morphine, Heroin, Codeine.

4. Antimicrobials:
Destroy/prevent development or inhibit the pathogenic action of microbes such as bacteria, fungi, virus or other parasites selectively. Antibiotics, antieptics and disinfectants are antimicrobial drugs.

(a) Antibiotics:
Chemical substances which are produced by microorganism/partly by chemical synthesis and are capable of destroying or inhibiting other micro organisms.
e.g. Salvarsan – Used in the treatment of syphilis. Antibiotics have either cidal(killing) effect ora static (inhibitory) effect on microbes.

Antibiotics are also classified into broad spectrum and narrow spectrum antibiotics based on their spectrum of action (range of bacteria or other microorganism that are affected by a certain antibiotic).

(1) Broad Spectrum Antibiotics:
Antibiotics which attack a wide range of Gram-positive and Gram-negative bacteria, e.g. Tetracyclin, steptomycin, chloramphenicol, vancomycin, ofloxacin etc. Ampicillin and Amoxycillin are synthetic modifications of pencillins. These have broad spectrum.

(2) Narrow Spectrum Antibiotics:
Effective mainly against Gram-positive or Gram-negative bacteria, e.g. Pencilline.

(3) Limited Spectrum Antibiotics: Antibiotics effective against a single organism or disease.
Dysidazirine – Antibiotic which is toxic towards certain strains of cancer cells.

(b) Antiseptic and Disinfectants:
Chemicals which either kill or prevent the growth of microorganisms. Antiseptics are applied to the living tissues such as wounds, cuts, ulcers and skin deseases. These are not injected like antibiotics, e.g. Furacine, soframidne.

• Dettol- Commonly used antiseptic, it is a mixture of chloroxylenol and terpineol.
• Bithional – added to soaps to impart antiseptic properties.
• Tincture iodine – 2 – 3% solution of iodine in alcohol-water mixture.
• Dilute aq. solution of boric add- weak antiseptic for eyes.
• Iodoform – antiseptic for wounds.

Disinfectants are applied to inanimate objects such as floors, drainage system, instruments, etc.
e.g. 1% of phenol, SO₂ (in very low concentration) and Cl₂ (0.2 to 0.4 ppm) are common disinfectants. Some substance can act as an antiseptic as well as disinfectant by varying concentration.
e.g. 0.2% phenol – antiseptic
1 % phenol – disinfectant.

5. Antifertility Drugs:
Drugs used to control population. Birth control pills essentially contain a mixture of synthetic estrogen and progesterone derivatives. Progesterone suppresses ovulation. Synthetic progesterone derivatives are more potent than progesterone. e.g. Norethindrone.
Ethynylestradbl(novestrol) – eastrogen derivative used in combination with progesterone derivative.

Chemicals in Food
Chemical are added to food for preservation, enhancing their appeal and adding nutritive value.

1. Artificial Sweetening Agents:
Chemical substances which give sweetening effect to food. Commonly used artificial sweetening agents: Aspartame – 100 times sweeter than cane sugar, methyl ester of dipeptide formed from aspartic acid and phenylalanine. Use of aspartame is limited to cold foods and soft drinks because it is unsatble at cooking temperature.

Saccharine – 550 times sweeter than cane sugar. It is the first popular artificial sweetening agent. It is excreted from the body in urine unchanged. It appears to be entirely inert and harmless when taken. Its use is of great value to diabetic persons and people who need to control in take of calories.

Sucrlose – 600 times sweeter than cane sugar. It is the trichloro derivative of sucrose. Its appearance and taste are like sugar. It is stable at cooking temperature. It does not provide calories.

Alitame – 2000 times sweeter than cane sugar. It is high potency sweetner. The control of sweetness of food is difficult while using it.

2. Food Preservatives:
They prevent spoilage of food due to microbial growth, e.g. Sugar, table salt, vegetable oils, sodium benzoate (C₆H₅ COONa), salts of sorbic acid, and propanoic acid.

Cleansing agents
Detergents and soaps are commonly used as cleansing agents.

1. Soaps:
Sodium/pottassium salts of higher fatty acids or long chain fatty acids like palmitic acid, stearic acid, oleic acid etc. Glyceryl ester of fatty acids is treated with aqueous NaOH solution. This reaction is known as saponification.

In this reaction, esters of fatty acids are hydrolysed and the soap obtained remains in colloidal form. Soap is precipitated from the solution by adding NaCI. Pottassium soaps are soft to the skin than sodium soaps. These can be prepared by using KOH solution in place of NaOH.

Types of Soaps
Toilet soap:
Prepared by using better grade of fats and oils with suitable soluble hydroxide. Colour and perfumes are added to make them more attractive.

Medicated soaps:
These contain substances of medicinal value. In some soaps deodorants are added, e.g. Shaving soaps contain glycerol to prevent rapid drying. A gum called rosin is added which forms sodium rosinate and lathers well.

Laundry soaps:
These contain fillers like sodium rosinate, sodium silicate, borax and soium carbonate. Hard water contains calcium and magnesium ions which form insoluble calcium and magnesium soaps respectively when sodium or potassium soaps are dissolved in hand water.

These insoluble soaps separate as scum in water and are useless as cleansing agent. This precipitate adheres onto the fibre of the cloth as gummy mass and hinders good washing.

2. Synthetic Detergents:
Cleaning agents which have all properties of soaps but actually do not contain any soap. These can be used both in soft and hard water. They are mainly classified into 3 categories.

• Anionic detergents: They are sodium salts of sulphonated long chain alcohols or hydrocarbons, e.g. Sodium dodecylbenzene sulphonate.
• Cationic detergents: They are quarternary ammonium salts of amines with acetate, chlorides or bromides as anions. Cationic part possess a long hydrocarbon chain and a positive charge on nitrogen atom. e.g. Cetyltrimethylammonium bromide.
• Non-ionic detergents: They do not contain any ion in their constitution, e.g. detergent formed by the reaction between stearic acid and polyethylene glycol. Liquid dishwashing detergents are non-ionic type.

The hydrocarbon chain of synthetic detergents is highly branched. Bacteria cannot degrade this easily. Slow degradation of detergents leads to their accumulation. The branching of the hydrocarbon chain is now a days controlled and kept to the minimum. Unbranched chains can be biodegraded more easily and hence pollution is prevented.

Supplementary Material
Antioxidants in Food:
These are important and necessary food additives which help in food preservation by retarding the action of oxygen on food. These are more reactive towards oxygen than the food material which they are protecting, e.g. Butylated Hydroxy Toluene (BHT), Butylated Hydroxy Anisole (BHA).

The addition of BHA to butter increases its shelf life from months to years. Sometimes BHT and BHA along with citric acid are added to produce more effect. Sulphur dioxide and sulphite are useful antioxidants for wine and beer, sugarsyrups and cut, peeled or dried fruits and vegetables.

Students can Download Chapter 15 Polymers Notes, Plus Two Chemistry Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Chemistry Notes Chapter 15 Polymers

The word polymer is coined from two Greek words poly means many and mer means unit. They are also called macromolecules.

Monomers – the repeating units of a polymer derived from some simple and reactive molecules. Polymerisation – process of formation of polymers from respective monomers.

e.g. nCH₂ = CH₂ (ethene) → -(CH₂ – CH₂)-_n (polyethene).

Classification of Polymers
1. Based on Source:

• Natural Polymers: naturally occuring polymers, found in plants and animals, e.g. cellulose, rubber,
  wool, starch.
• Semi-synthetic Polymers: ploymers obtained by modifying the naturally occuring polymers.
  e.g. cellulose nitrate, cellulose acetate (rayon)
• Synthetic Polymer: polymers synthesised by chemical processes, e.g. nylon 6,6, Buna – S, etc.

2. Based on the Structure of Polymers:

• Linear Polymers: polymers consisting of long and straight chains, e.g. polythene, polyvinyl chloride (PVC), etc.
• Branched Chain Polymers: polymers containing linear chains having some branches, e.g. low density polythene (LDPE).
• Cross Linked Polymers: polymers formed from bi-functional and tri-functional monomers. They contain strong covalent bonds between various linear chains, e.g. bakelite, melamine, etc.

3. Based on mode of polymerisation:
(1) Addition Polymers:
Polymers formed by the repeated addition of monomers possessing double or tripple bonds, e.g. polyethene, PVC, polystyrene, etc.

Homopolymers:
Addition polymers formed by the polymerisation of a single monomeric species.
e.g. polythene, polystyrene, etc.

Copolymers:
Polymers made by addition polymerisation of two different monomers.
e.g. Buna-S, Buna-N, etc.

(2) Condensation polymers:
Polymers formed by repeated condensation reaction between two different bi-functional ortri-functional monomeric units, e.g. nylon – 6,6.

4. Based on Molecular Forces:
(1) Elastomers:
Rubber-like solids with elastic properties, polymer chains are held together by the weakest intermolecular forces which stretching. The ‘cross links’ introduced help to retract the polymer to its original position after the force is released.
e.g. buna – S, buna – N, neoprene, etc.

(2) Fibres:
Thread forming solids which possess high tensile strength and high modulus due to strong intermolecular forces like H-bonding.
e.g. nylon -6,6, polyesters(terylene), etc.

(3) Thermoplastic Polymers:
Linear or slightly branched long chain molecules capable of repeatedly softening on heating and hardnening on cooling. The inter molecular forces of attraction are intermediate between elastomers and fibres.
e.g. polyethene, polystryne, polyvinyls, etc.

(4) Thermosetting polymers:
Cross linked or heavily branched molecules, which on heating undergo extensive cross linking in moulds and again become infusible. These cannot be reused.
e.g. bakelite, urea-formaldehyde resins, etc.

Types of Polymerisation Reactions
a. Addition Polymerisation or Chain Growth Polymerisation:
Same or different monomers (unsaturated compounds) add together on a large scale through the formation of either free radicals or ionic species.
1. Free radical mechanism-It is characterised by 3 steps.

• Initiation – a free radical is generated in presence of organic peroxide catalyst.
• Propagation – The bigger radicals formed carries the reaction forward.
• Termination – The product radical reacts with another radical to form the polymerised product. e.g. polymerisation of ethene to form polyethene in presence of benzoyl peroxide.

2. Preparation of Some Important Addition Polymers
(a) Polythene:
There are two types of polythene.
(i) Low Density Polythene (LDP):
Obtained by the polymerisation of ethene under high pressure of 1000 to 2000 atm and temperature 350 to 570 K, has a highly branched structure.
Uses: In the insulation of electrical wires; manufacture of squeeze bottles, toys and flexible pipes.

(ii) High Density Polythene (HDP):
Formed when ethene is polymerised in a hydrocarbon solvent in presence of Ziegler-Natta catalyst (Triethylaluminium and TiCl₄ at 333 K – 343 K and 6 – 7 atm. It has high density due to close packing, chemicaly inert, more tougher and harder. Uses: for making buckets, dust bins, bottles, pipes, etc.

(b) Polytetrafuoroethene (Teflon):
Formed by the polymerisation of tetrafluoroethene in presence of free radical or persulphate catalyst at high pressure.
Uses: for the preparation of oil seals, gaskets, nonstick surface coated utensils.

(c) Polyacrylonitrile (PAN):
Formed by the addition polymerisation of acrylonitrile in presence of peroxide catalyst.

Uses: as a substitue for wool in making commercial fibres as orlon or acrilan.

b. Condensation Polymerisation or Step Growth Polymerisation:
It involves repetitive condensation reaction between two bi-functional monomers.
(1) Polyamides:
Polymers possessing amide (-CO-NH-) linkages.
(i) Nylon 6, 6:
Prepared by the condensation of hexamethylenediamine and adipic acid. Uses:in making sheets, bristles for brushes and in textile industry.

(ii) Nylon 6:
Obtained by heating caprolactam with water at a high temperature.

Uses: manufacture of tyre cords, fabrics and ropes.

(2) Polysters:
Polycondensation products of dicarboxylic acids and diols.
(i) Dacron or Teriyne:
Manufactured by heating a mixture of ethylene glycol and terephthalic acid at 420 to 460 K in the presence of zinc acetate-antimony trioxide catalyst.

Uses: in blending with cotton and wool fibres, as glass reinforcing materials in safety helmets.

(3) Phenol-Formaldehyde Polymer:
Phenol and formaldehyde undergo condensation reaction in the presence of either an acid or a base catalyst. The initial linear product formed is called Novolac. It is used in paints.

Novalac on heating with formaldehyde undergoes cross linking to form bakelite. Uses: for making combs, phonograph records, electrical switches and handles of various utensils.

(4) Melamine – Formaldehyde Polymer:
Formed by the condensation polymerisation of melamine and formaldehyde. Use: for making unbreakable crockery.

c. Copolymerisation:
Polymerisation reaction in which a mixture of more than one monomeric species is allowed to polymerise and form copolymer, e.g. Buna – S. It is quite tough and is a good substitute for natural rubber. Uses: for the manufacture of automobile tyres, floortiles, foorwear components, cable insulation, etc.

d. Natural Rubber:
Natural polymer, possess elastic properties, obtained from rubber latex, polymer of isoprene (2 – methyl-1, 3-butadiene), also called cis-1, 4-polyisoprene.

(1) Vulcanisation of Rubber:
Process of heating natural rubber with sulphur at about 373 K to 415 K To improve its physical properties. On vulcanisation, sulphur forms cross links at the reactive sites of double bonds and thus the rubber gets stiffened.

(2) Synthetic Rubber:
Any vulcanisable rubber-like polymer.
(i) Neoprene (polychloropren): Formed by the free radical polymerisation of chloroprene.

It has supreior resistance to vegetable oils and mineral oils. Uses: for manufacturing conveyor belts, gaskets and hoses.

(ii) Buna – N:
Prepared by the copolymerisation of 1,3 – butadiene and acrylonitrile in the presence of a peroxide catalyst.

It is resistanttothe action of petrol, lubricating oil and oiganic solvents. Uses: in making oil seals, tank lining, etc.

Biodegradable Polymers
Polymers which can overcome environmental problems caused by polymeric solid waste materials, can be broken into small fragments by enzyme catalysed reaction, e.g.
(1) Poly β -hydroxybutyrate – co – β- hydroxy valerate (PHBV):
Obtained by the copolymerisation of 3 – hydroxybutanoic acid and 3-hydroxypentanoic acid. Uses: in speciality packaging, orthopaedic devices, in controlled release of drugs.

(2) Nylon -2, Nylon -6:
Alternating polyamide copolymer of glycine and amino caproic acid, biodegradable.
H₂N – CH₂ – COOH + H₂N – (CH₂)₅ – COOH → -(NH – CH₂ – CO – NH – (CH₂)₅ – CO )-_n

Polymers of Commercial Importance