Class 10 Maths Notes are the most effective study material for students to better understand the concepts covered in the PCTB textbook. Most of the concepts in the textbook are taken from higher classes that build the student’s basic foundation. In addition to analyzing preparation levels, these mathematics solutions also provide knowledge of new concepts. Students can also check the exercise questions in 10th class textbooks to clear their doubts.
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10th Class Chapter Wise Mathematics Notes PDF Download
Chapter 1 – Quadratic Equation
The 1st chapter start with the definition of quadric definition “An equation which contains the squares of the unknown variable quantity, but no higher power”. To find the quadric equations factorization and completing square methods are used.
Topics covered in chapter 1 Quadric Equation
Quadratic equation in one variable with the help of factorization and by using completing square metho.
Solution by Factorization:
In this method, write the quadratic equation in the standard form as
ax2 + bx + c = 0 (i)
If two numbers r and s can be found for equation (i) such that r + s = b and rs = ac then ax2 + bx + c can be factorized into two linear factors.
Solution by Completing Squares
To sovle the quadratic equation with the help of completing square is illustrated with the following example.
Remember that, For our convenience, we make the co-efficient of x2 is equal to 1 in the method of completing squares.
Summary
An equation which contains the square of the unknown (variable) quantity, but no higher power, is called a quadratic equation or an equation of the second degree.
A second degree equation in one variable x, ax2 + bx + c = 0 where a is not equal to 0 and a, b, c are real numbers, is called the general or standard form of a quadratic equation.
An equation is said to be a reciprocal equation, if it remains unchanged, when x is replaced by 1/x
In exponential equations, variables occur in exponents.
An equation involving expression under the radical sign is called a radical equation.
Any quadratic equation is solved by the following three methods..
- Factorization
- Completing square
- Quadratic formula
Chapter 2 – Theory of Quadratic Equation
In chapter 2, we will learn the discriminant (b2 – 4ac) of quadratic expressions ax2 + bx + c. By solving the quadratic equation we get different kind of roots. We also discuss the nature of the roots of quadratic equation through discriminant.
Roots and Co-efficient of Quadratic Equation
Symmertric function of roots of quadratic equation
A symmertric function is one in which the value of expressions involving them remains unchanged when they are changed.
Summary
Discriminant of the quadratic expression ax2 + bx + c is “b2 -4ac prime prime .
Complex cube roots of unity are ω and ω2
Properties of cube roots of unity.
- The product of three cube roots of unity is one. i.e., (1) (ω) (ω2) = ω3 = 1
- Each of the complex cube roots of unity is reciprocal of the other.
- Each of the complex cube roots of unity is the square of the other.
- The sum of all the cube roots of unity is zero, i.e.,(1 + ω + ω2 = 0
The roots of the quadratic equation ax2 + bx + c = 0, a ≠0
Symmetric functions of the roots of a quadratic equation are those functions in which the roots involved are such that the values of the expressions remain unaltered, when roots are interchanged.
Formation of a quadratic equation if its roots are given; x² – (sum of the roots)x + product of the roots = 0
Synthetic division is the process of finding the quotient and remainder, when a polynomial is divided by a linear polynomial.
A system of equations having a common solution is called a system of simultaneous equations.
Chapter 3 – Variations
Students can learn direct and inverse variation in chapter 3. In direct variation when one quantity increase the other also increase or one decrease the other quantity also decrease. In other words, if a quantity y varies directly with regard to a quantity x. We say that y is directly proportional to x and is written as y ∝ x or y = kx i.e., y/x = k, k ≠ 0
The sign ∝ read as “varies as” is called the sign of proportionality or variation, while k ≠ 0 known as constant of variation.
While in inverse variation, the variables work opposite. When one variable decrease the other increase. In other words, if a quantity y varies inversely with regard to quantity x. We say that y is inversely proportional to x or y varies inversely as x and is written as y ∝ 1/x. i.e., x = k where k ≠ 0 is the constant of variation.
Ratio
A relation between two quantities of the same kind (measured in same unit) is called ratio. If a and b are two quantities of the same kind and b is not zero, then the ratio of a and b is written as a : b or in fraction a
e.g., if a hockey team wins 4 games and loses 5, then the ratio of the games won to games lost is 4: 5 or in fraction 4/5
Remember that:
- The order of the elements in a ratio is important.
- In ratio a: b, the first term a is called antecedent and the second term b is called consequent.
- A ratio has no units
Proportion
A proportion is a statement, which is expressed as an equivalence of two ratios. If two ratios a: b and c: d are equal, then we can write a : b = c:d Where quantities a, d are called extremes, while b, c are called means. Symbolically the proportion of a, b, c and d is written as a:b::c:d or a:b=c:d
Theorems on Proportions
If four quantities a, b, c and d form a proportion, then many other useful properties may be deduced by the properties of fractions.
If a / b = c / d then b: a = d : c
Summary
A relation between two quantities of the same kind is called ratio.
A proportion is a statement, which is expressed as equivalence of two ratios.
If two ratios a: b and c: d are equal, then we can write a / b = c / d
If two quantities are related in such a way that increase (decrease) in one quantity causes increase (decrease) in the other quantity is called direct variation.
If two quantities are related in such a way that when one quantity increases, the other decreases is called inverse variation.
Theorem of Invertendo
If a / b = c / d then b / a = d / c
Theorem of Alternando
If a / b = c / d then a / c = b / d
Unit 4 – Partial Fractions
The 4th chapter start with domain fraction. Thats define as the quotient of two numbers or algebraic expressions is called a fraction. The quotient is indicated by a bar (__). We write, the dividend above the bar and the divisor below the bar. A fraction is an indicated quotient of two numbers or algebraic expressions.
For example, x2 + 2/ x -2 is a fraction with x -2 ≠ 0. If x – 2 = 0 then the fraction is not defined because x – 2 =0 ⟹ x=2 which makes the denominator of the fraction zero.
The rational fraction also subdomain of raction. An expression of the form (N(x))/(D(x)) where N(x) and D(x) are polynomials in x with real coefficients and D(x) ≠ 0 is called a rational fraction. In this unit the proper and improper fraction questions also given thats formulas have been provided for the help of students.
Chapter 5 – Set & Functions
A set is a well-defined collection of objects and it is denoted by capital letters A, B, C etc.
Some Important Sets:
In set theory, we usually deal with the following sets of numbers denoted by standard symbols:
Natural Number:
The Natural Number set start from 1 and no have any end number.
N = The set of natural numbers { 1, 2, 3, 4 ,….}
Whole Number:
The whole number is different from natural. The whole number is start from zero 0 and have no limit.
W = The set of whole numbers = {0, 1, 2, 3, 4 ,… }
Z = The set of all integers = {0, 1, ±2, ±3,…}
E = The set of all even integers = {0, ±2, ±4, ….}
O = The set of all odd integers = {±1, ±3, ±5, …}
P = The set of prime numbers = { 2, 3, 5, 7, 11, 13, 17, … }
R = The set of all real numbers =Q U Q’