UNIT–I : NUMBER SYSTEMS
CHAPTER-1
REAL NUMBERS
Fundamental Theorem of Arithmetic
Concepts Covered Fundamental Theorem of Arithmetic:
Topic-1 For any two positive integers a and b,
We have HCF (a, b) × LCM (a, b) = a × b
a× b a× b
or HCF (a, b) = or LCM (a, b) =
LCM ( a , b) HCF ( a , b)
Revision Notes
The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes and this factorisation is unique, apart from the
order in which the prime factors occur. Fundamental theorem of arithmetic is also called a Unique Factorization
Theorem.
Composite number = Product of prime numbers
Or
Any integer greater than 1 can either be a prime numbers or can be written as a unique product of prime
numbers. e.g.,
(i) 2 × 11 = 22 is the same as 11 × 2 = 22.
(ii) 6 can be written as 2 × 3 or 3 × 2, where 2 and 3 are prime numbers.
(iii) 15 can be written as 3 × 5 or 5 × 3, where 3 and 5 are prime numbers.
The prime factorization of a natural number is unique, except to the order of its factors.
e.g., 12 detained by multiplying the prime numbers 2, 2 and 3 together,
12 = 2 × 2 × 3
We would probably write it as
12 = 22 × 3
By using Fundamental Theorem of Arithmetic, we shall find the HCF and LCM of given numbers (two or more).
This method is also called Prime Factorization Method.
Prime Factorization Method to find HCF and LCM:
(i) Find all the prime factors of given numbers.
(ii) HCF of two or more numbers = Product of the smallest power of each common prime factor, involved in the
numbers.
(iii) LCM of two or more numbers = Product of the greatest power of each prime factor, involved in the numbers.
Key Words
Highest Common Factor (HCF): The HCF of two or more numbers is the highest number among all the
common factors of the given numbers.
Prime Numbers: A number that can be divided exactly only by itself and 1.
Fundamental Facts
(1) The Euclidean algorithm is useful for reducing a common fraction to lowest terms.
714 51 × 14 14
For example: = = .
765 51 × 15 15
(2) The concept of LCM is important to solve problem related to racetracks, traffic light etc.
(3) In Mathematics problem, where we pair two objects against each other, the LCM value is useful in optimizing
the quantities of the given objects.
,2 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X
Example
Find the LCM of 40, 36 and 126 by applying the prime factorization method.
Step 1. Factorise each of the given positive integers such as:
40 = 2 × 2 × 2 × 5
36 = 2 × 2 × 3 × 3
and 126 = 2 × 3 × 3 × 7
Step 2. Express them as a product of powers of primes in ascending order of magnitudes of primes:
40 = 23 × 5, 36 = 22 × 32 and 126 = 2 × 32 × 7
Step 3. To find LCM, list all prime factors of 40, 36 and 126 with their greatest exponents as:
\ LCM = 23 × 32 × 5 × 7
= 8 × 9 × 5 × 7
= 2520.
Mnemonics
Concept: Euclid's Division Lemma (a = bq + r)
Mnemonics: Alibaba's best product quotation is assent reward.
Interpretation:
Alibaba's A = a
best B = b
quotation Q = q
assets A = addition of bq and r
reward R = r
Then a = b × q + r.
Irrational Numbers
Topic-2
Concepts Covered Rational & Irrational Numbers.
Revision Notes
p
Rational Numbers: A number in the form , where p and q are co-prime numbers and q ≠ 0, is known as
q
rational number.
3 2
For example: 2, – 3, , − , etc. are rational numbers.
7 5 p
Irrational Numbers: A number is called irrational if it cannot be written in the form q , where p and q are integers
and q ≠ 0. For example, 2 , 3 , 5 , π are irrational numbers.
Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
The sum or difference of a rational and an irrational number is irrational.
The product and quotient of a non-zero rational and an irrational number is irrational.
Key Words
Co-Prime Numbers: Co-prime numbers are those numbers that have only one common
factor. For example: 3 and 5, 11 and 13 etc.
Fundamental Facts
(1) The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to
the Pythagorean Hippasus of metapontum who produced a proof of the irrationality of 2 .
(2) Irrational number are numbers that cannot be expressed as the ratio of two whole numbers.
, Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X 3
Example
Show that 2 3 + 7 is an irrational number.
a
Step 1. Let 2 3 + 7 be a rational number. Since, a rational number can be expressed as , where
b
b ≠ 0 and a & b are integers.
a
Step 2. Then 2 3 +7 =
b
a
or 2 3 = −7
b
1a
or 3 = −7
2 b
Here, L.H.S. = 3 is an irrational.
1a
But, R.H.S. = − 7 is a rational.
2 b
So, it is not possible.
Step 3. Hence, our assumption that 2 3 + 7 is a rational is incorrect.
Hence, 2 3 + 7 is an irrational number.
UNIT–II : ALGEBRA
CHAPTER-2
POLYNOMIALS
Revision Notes
Polynomial: An algebraic expression in the form of anxn + an–1xn–1 +.........+ a2x2+a1x + a0, (where n is a whole
number and a0, a1, a2, ........., an are real numbers) is called a polynomial in one variable x of degree n.
Value of a Polynomial at a given point: If p(x) is a polynomial in x and ‘α’ is any real number, then the value
obtained by putting x = α in p(x), is called the value of p(x) at x = α.
Zero of a Polynomial: A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.
Geometrically, the zeroes of a polynomial p(x) are precisely the X-coordinates of the points, where the graph of y
= p(x) intersects the X-axis.
(i) A linear polynomial has one and only one zero.
(ii) A quadratic polynomial has at most two zeroes.
(iii) A cubic polynomial has at most three zeroes.
(iv) In general, a polynomial of degree n has at most n zeroes.
Graphs of Different types of Polynomials:
l Linear Polynomial: The graph of a linear polynomial p(x) = ax + b is a straight line that intersects X-axis at
one point only.
l Quadratic Polynomial: (i) Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola which opens
upwards, if a > 0 and intersects X-axis at a maximum of two distinct points.
(ii) Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola which opens downwards, if a < 0 and
intersects X-axis at a maximum of two distinct points.
l Cubic polynomial: Graph of cubic polynomial p(x) = ax3 + bx2 + cx + d intersects X-axis at a maximum of three
distinct points.
, 4 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X
Relationship between the Zeroes and the Coefficients of a Polynomial:
(i) Zero of a linear polynomial
(-1)1 Constant term
=
Coefficient of x
−b
If ax + b is a given linear polynomial, then zero of linear polynomial is
a
(ii) In a quadratic polynomial,
Sum of zeroes of a quadratic polynomial
(-1)1 Coefficient of x
=
Coefficient of x 2
Product of zeroes of a quadratic polynomial
(-1)2 Constant term
=
Coefficient of x 2
∴If α and b are the zeroes of a quadratic polynomial ax2 + bx + c, then
b c
α+β = − and αβ =
a a
(iii) If α, β and γ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then
b b c c d d
α + β + γ = (–1)1
= – , αβ + βγ + γα = (–1)2 = and αβγ = (–1)3 =–
a a a a a a
Discriminant of a Quadratic Polynomial: For f (x) = ax2 + bx + c, where a ≠ 0, b2 – 4ac is called its discriminant D.
The discriminant D determines the nature of roots/zeroes of a quadratic polynomial.
Case I: If D > 0, graph of f (x) = ax2 + bx + c will intersect the X-axis at two distinct points, x-coordinates of points
of intersection with X-axis is known as ‘zeroes’ of f (x).
∴ f (x) will have two zeroes and we can say that roots/zeroes of the two given polynomials are real and unequal.
Case II: If D = 0, graph of f (x) = ax2 + bx + c will touch the X-axis at one point only.
∴ f (x) will have only one ‘zero’ and we can say that roots/zeroes of the given polynomial are real and equal.
Case III: If D < 0, graph of f (x) = ax2 + bx + c will neither touch nor intersect the X-axis.
CHAPTER-1
REAL NUMBERS
Fundamental Theorem of Arithmetic
Concepts Covered Fundamental Theorem of Arithmetic:
Topic-1 For any two positive integers a and b,
We have HCF (a, b) × LCM (a, b) = a × b
a× b a× b
or HCF (a, b) = or LCM (a, b) =
LCM ( a , b) HCF ( a , b)
Revision Notes
The Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes and this factorisation is unique, apart from the
order in which the prime factors occur. Fundamental theorem of arithmetic is also called a Unique Factorization
Theorem.
Composite number = Product of prime numbers
Or
Any integer greater than 1 can either be a prime numbers or can be written as a unique product of prime
numbers. e.g.,
(i) 2 × 11 = 22 is the same as 11 × 2 = 22.
(ii) 6 can be written as 2 × 3 or 3 × 2, where 2 and 3 are prime numbers.
(iii) 15 can be written as 3 × 5 or 5 × 3, where 3 and 5 are prime numbers.
The prime factorization of a natural number is unique, except to the order of its factors.
e.g., 12 detained by multiplying the prime numbers 2, 2 and 3 together,
12 = 2 × 2 × 3
We would probably write it as
12 = 22 × 3
By using Fundamental Theorem of Arithmetic, we shall find the HCF and LCM of given numbers (two or more).
This method is also called Prime Factorization Method.
Prime Factorization Method to find HCF and LCM:
(i) Find all the prime factors of given numbers.
(ii) HCF of two or more numbers = Product of the smallest power of each common prime factor, involved in the
numbers.
(iii) LCM of two or more numbers = Product of the greatest power of each prime factor, involved in the numbers.
Key Words
Highest Common Factor (HCF): The HCF of two or more numbers is the highest number among all the
common factors of the given numbers.
Prime Numbers: A number that can be divided exactly only by itself and 1.
Fundamental Facts
(1) The Euclidean algorithm is useful for reducing a common fraction to lowest terms.
714 51 × 14 14
For example: = = .
765 51 × 15 15
(2) The concept of LCM is important to solve problem related to racetracks, traffic light etc.
(3) In Mathematics problem, where we pair two objects against each other, the LCM value is useful in optimizing
the quantities of the given objects.
,2 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X
Example
Find the LCM of 40, 36 and 126 by applying the prime factorization method.
Step 1. Factorise each of the given positive integers such as:
40 = 2 × 2 × 2 × 5
36 = 2 × 2 × 3 × 3
and 126 = 2 × 3 × 3 × 7
Step 2. Express them as a product of powers of primes in ascending order of magnitudes of primes:
40 = 23 × 5, 36 = 22 × 32 and 126 = 2 × 32 × 7
Step 3. To find LCM, list all prime factors of 40, 36 and 126 with their greatest exponents as:
\ LCM = 23 × 32 × 5 × 7
= 8 × 9 × 5 × 7
= 2520.
Mnemonics
Concept: Euclid's Division Lemma (a = bq + r)
Mnemonics: Alibaba's best product quotation is assent reward.
Interpretation:
Alibaba's A = a
best B = b
quotation Q = q
assets A = addition of bq and r
reward R = r
Then a = b × q + r.
Irrational Numbers
Topic-2
Concepts Covered Rational & Irrational Numbers.
Revision Notes
p
Rational Numbers: A number in the form , where p and q are co-prime numbers and q ≠ 0, is known as
q
rational number.
3 2
For example: 2, – 3, , − , etc. are rational numbers.
7 5 p
Irrational Numbers: A number is called irrational if it cannot be written in the form q , where p and q are integers
and q ≠ 0. For example, 2 , 3 , 5 , π are irrational numbers.
Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.
The sum or difference of a rational and an irrational number is irrational.
The product and quotient of a non-zero rational and an irrational number is irrational.
Key Words
Co-Prime Numbers: Co-prime numbers are those numbers that have only one common
factor. For example: 3 and 5, 11 and 13 etc.
Fundamental Facts
(1) The discovery of irrational numbers is usually attributed to Pythagoras, more specifically to
the Pythagorean Hippasus of metapontum who produced a proof of the irrationality of 2 .
(2) Irrational number are numbers that cannot be expressed as the ratio of two whole numbers.
, Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X 3
Example
Show that 2 3 + 7 is an irrational number.
a
Step 1. Let 2 3 + 7 be a rational number. Since, a rational number can be expressed as , where
b
b ≠ 0 and a & b are integers.
a
Step 2. Then 2 3 +7 =
b
a
or 2 3 = −7
b
1a
or 3 = −7
2 b
Here, L.H.S. = 3 is an irrational.
1a
But, R.H.S. = − 7 is a rational.
2 b
So, it is not possible.
Step 3. Hence, our assumption that 2 3 + 7 is a rational is incorrect.
Hence, 2 3 + 7 is an irrational number.
UNIT–II : ALGEBRA
CHAPTER-2
POLYNOMIALS
Revision Notes
Polynomial: An algebraic expression in the form of anxn + an–1xn–1 +.........+ a2x2+a1x + a0, (where n is a whole
number and a0, a1, a2, ........., an are real numbers) is called a polynomial in one variable x of degree n.
Value of a Polynomial at a given point: If p(x) is a polynomial in x and ‘α’ is any real number, then the value
obtained by putting x = α in p(x), is called the value of p(x) at x = α.
Zero of a Polynomial: A real number k is said to be a zero of a polynomial p(x), if p(k) = 0.
Geometrically, the zeroes of a polynomial p(x) are precisely the X-coordinates of the points, where the graph of y
= p(x) intersects the X-axis.
(i) A linear polynomial has one and only one zero.
(ii) A quadratic polynomial has at most two zeroes.
(iii) A cubic polynomial has at most three zeroes.
(iv) In general, a polynomial of degree n has at most n zeroes.
Graphs of Different types of Polynomials:
l Linear Polynomial: The graph of a linear polynomial p(x) = ax + b is a straight line that intersects X-axis at
one point only.
l Quadratic Polynomial: (i) Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola which opens
upwards, if a > 0 and intersects X-axis at a maximum of two distinct points.
(ii) Graph of a quadratic polynomial p(x) = ax2 + bx + c is a parabola which opens downwards, if a < 0 and
intersects X-axis at a maximum of two distinct points.
l Cubic polynomial: Graph of cubic polynomial p(x) = ax3 + bx2 + cx + d intersects X-axis at a maximum of three
distinct points.
, 4 Oswaal CBSE Revision Notes Chapterwise & Topicwise, MATHEMATICS (STANDARD), Class-X
Relationship between the Zeroes and the Coefficients of a Polynomial:
(i) Zero of a linear polynomial
(-1)1 Constant term
=
Coefficient of x
−b
If ax + b is a given linear polynomial, then zero of linear polynomial is
a
(ii) In a quadratic polynomial,
Sum of zeroes of a quadratic polynomial
(-1)1 Coefficient of x
=
Coefficient of x 2
Product of zeroes of a quadratic polynomial
(-1)2 Constant term
=
Coefficient of x 2
∴If α and b are the zeroes of a quadratic polynomial ax2 + bx + c, then
b c
α+β = − and αβ =
a a
(iii) If α, β and γ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then
b b c c d d
α + β + γ = (–1)1
= – , αβ + βγ + γα = (–1)2 = and αβγ = (–1)3 =–
a a a a a a
Discriminant of a Quadratic Polynomial: For f (x) = ax2 + bx + c, where a ≠ 0, b2 – 4ac is called its discriminant D.
The discriminant D determines the nature of roots/zeroes of a quadratic polynomial.
Case I: If D > 0, graph of f (x) = ax2 + bx + c will intersect the X-axis at two distinct points, x-coordinates of points
of intersection with X-axis is known as ‘zeroes’ of f (x).
∴ f (x) will have two zeroes and we can say that roots/zeroes of the two given polynomials are real and unequal.
Case II: If D = 0, graph of f (x) = ax2 + bx + c will touch the X-axis at one point only.
∴ f (x) will have only one ‘zero’ and we can say that roots/zeroes of the given polynomial are real and equal.
Case III: If D < 0, graph of f (x) = ax2 + bx + c will neither touch nor intersect the X-axis.
