Class – X
Formulae and Basic Concepts
Chapter – 1 (Number System)
1. A number is prime if it has only two factors, 1 and itself
2. Every composite number can be expressed as a product of prime factors.
3. H C F of two numbers = Product of the smaller power of each common factor in the
numbers.
4. L C M of two numbers = Product of the greatest power of each prime factor
involved in the numbers.
5. HCF × LCM = Product of Two Numbers
Chapter – 2 (Polynomials)
1. Standard Form of Linear Polynomial - ax + b
Standard Form of Quadratic Polynomial - ax 2 + bx + c
Standard Form of Cubic Polynomial - ax 3 + bx 2 + cx + d
2. Relationship between the zeroes and coefficient of polynomial
b
(i) Zero of the linear polynomial = − a
b
(ii) Sum of zeroes of quadratic polynomial = S = α + β = −
a
c
Product of zeroes of quadratic polynomial = P = αβ = a
b
(iii) Sum of zeroes of cubic polynomial = α + β + γ = − a
d
Product of zeroes of cubic polynomial = αβγ = − a
c
Sum of product of two zeroes taken at a time = αβ + βγ + αγ = a
3. Quadratic polynomial when sum and product of zeroes are given = x 2 − (α + β)x + αβ
or x 2 − Sx + P
4. Geometrically, the zeros of the polynomial 𝑓(𝑥) are the 𝑥 coordinates of the points where
the graph 𝑦=𝑓 (𝑥 ) intersects the 𝑥 axis.
5. A polynomial of degree ‘n’ can have at most ‘n’ real zeros.
6. If α + β and α β are given / known, then
(i) α2 + β2 = (α + β)2 − 2αβ
(ii) α3 + β3 = (α + β)3 − 3αβ(α + β)
(iii) α4 + β4 = (α2 + β2 )2 − 2(αβ)2
Chapter – 3 (Pair of Linear Equations in two Variables)
1. Each solution (𝑥, 𝑦) of a linear equation 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 which represents a line,
corresponds to a point on the line.
2. A pair of linear equations in two variables 𝑥,𝑦 is 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 +
𝑐2 = 0
3. 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 represents
, 𝑎 𝑏
(i) Intersecting lines if 𝑎1 ≠ 𝑏1
2 2
𝑎1 𝑏1 𝑐
(ii) Parallel lines if 𝑎 = 𝑏 ≠ 𝑐1
2 2 2
𝑎1 𝑏1 𝑐
(iii) Coincident lines if 𝑎 = 𝑏 = 𝑐1
2 2 2
4. A pair of linear equations 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 is said to be
𝑎 𝑏
(i) Consistent and unique solution if 1 ≠ 1
𝑎2 𝑏2
𝑎1 𝑏 𝑐
(ii) Consistent and infinite number of solutions, if = 𝑏1 = 𝑐1
𝑎2 2 2
𝑎1 𝑏1 𝑐1
(iii) Inconsistent and no solution if 𝑎 = 𝑏 ≠ 𝑐
2 2 2
5. Certain basic facts to know:
(i) Equation of 𝑦 − 𝑎𝑥𝑖𝑠 is 𝑥 = 0 and equation of 𝑥 − 𝑎𝑥𝑖𝑠 is 𝑦 = 0
(ii) Equation of line parallel to 𝑥 −axis is 𝑦 = 𝑏
(iii) Equation of line parallel to 𝑦 −axis is 𝑥 = 𝑎
6. If unit’s digit is 𝑥 and ten’s digit is 𝑦, then the number formed is 10𝑦 + 𝑥
7. If the pair of linear equations is such that coefficients of 𝑥 and 𝑦 are interchanged in the
two equations then ‘Add the two equations and Subtract the two equations’. E.g., 254𝑥 +
309𝑦 = −55, 309𝑥 + 254𝑦 = 55
Chapter – 4 (Quadratic Equations)
1. Standard form of Quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0
2. Quadratic Formula: Roots of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are given by 𝑥 =
−𝑏±√𝐷
where 𝐷 = 𝑏 2 − 4𝑎𝑐 is called the discriminant
,
2𝑎
3. Nature of Roots of quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0:
(i) Two distinct real roots if 𝑏 2 − 4𝑎𝑐 > 0
(ii) Two real and equal (coincident) roots if 𝑏 2 − 4𝑎𝑐 = 0
(iii) No real roots if 𝑏 2 − 4𝑎𝑐 < 0
Chapter – 5 (Arithmetic Progressions)
1. General form of an A.P is 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
2. nth term of A.P 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑
3. nth term from the end = 𝑙 − (𝑛 − 1)𝑑, 𝑙 is the last term
𝑛 𝑛 𝑛
4. Sum of 𝑛 terms of A.P = 2 [2𝑎 + (𝑛 − 1)𝑑] or 2 [𝑎 + 𝑙] or 2 [𝑎 + 𝑎𝑛 ]
𝑛(𝑛+1)
5. Sum of first 𝑛 natural numbers = 1 + 2 + 3 + ⋯ + 𝑛 = 2
6. Sum of first 𝑛 even natural numbers = 2 + 4 + 6 + ⋯ + 2𝑛 = 𝑛(𝑛 + 1)
7. Sum of first 𝑛 odd natural numbers 1 + 3 + 5 + ⋯ + (2𝑛 − 1) = 𝑛2
8. 𝑎𝑛 = 𝑆𝑛 − 𝑆𝑛−1 i.e, 𝑎1 = 𝑆1 , 𝑎2 = 𝑆2 − 𝑆1 , 𝑎3 = 𝑆3 − 𝑆2 and so on
9. Three consecutive terms in A.P are taken as 𝒂 − 𝒅, 𝒂, 𝒂 + 𝒅 with common difference ′𝑑′
10. Four consecutive terms in A.P are taken as 𝒂 − 𝟑𝒅, 𝒂 − 𝒅, 𝒂 + 𝒅, 𝒂 + 𝟑𝒅 with common
difference ′2𝑑′
Formulae and Basic Concepts
Chapter – 1 (Number System)
1. A number is prime if it has only two factors, 1 and itself
2. Every composite number can be expressed as a product of prime factors.
3. H C F of two numbers = Product of the smaller power of each common factor in the
numbers.
4. L C M of two numbers = Product of the greatest power of each prime factor
involved in the numbers.
5. HCF × LCM = Product of Two Numbers
Chapter – 2 (Polynomials)
1. Standard Form of Linear Polynomial - ax + b
Standard Form of Quadratic Polynomial - ax 2 + bx + c
Standard Form of Cubic Polynomial - ax 3 + bx 2 + cx + d
2. Relationship between the zeroes and coefficient of polynomial
b
(i) Zero of the linear polynomial = − a
b
(ii) Sum of zeroes of quadratic polynomial = S = α + β = −
a
c
Product of zeroes of quadratic polynomial = P = αβ = a
b
(iii) Sum of zeroes of cubic polynomial = α + β + γ = − a
d
Product of zeroes of cubic polynomial = αβγ = − a
c
Sum of product of two zeroes taken at a time = αβ + βγ + αγ = a
3. Quadratic polynomial when sum and product of zeroes are given = x 2 − (α + β)x + αβ
or x 2 − Sx + P
4. Geometrically, the zeros of the polynomial 𝑓(𝑥) are the 𝑥 coordinates of the points where
the graph 𝑦=𝑓 (𝑥 ) intersects the 𝑥 axis.
5. A polynomial of degree ‘n’ can have at most ‘n’ real zeros.
6. If α + β and α β are given / known, then
(i) α2 + β2 = (α + β)2 − 2αβ
(ii) α3 + β3 = (α + β)3 − 3αβ(α + β)
(iii) α4 + β4 = (α2 + β2 )2 − 2(αβ)2
Chapter – 3 (Pair of Linear Equations in two Variables)
1. Each solution (𝑥, 𝑦) of a linear equation 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 which represents a line,
corresponds to a point on the line.
2. A pair of linear equations in two variables 𝑥,𝑦 is 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 +
𝑐2 = 0
3. 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 represents
, 𝑎 𝑏
(i) Intersecting lines if 𝑎1 ≠ 𝑏1
2 2
𝑎1 𝑏1 𝑐
(ii) Parallel lines if 𝑎 = 𝑏 ≠ 𝑐1
2 2 2
𝑎1 𝑏1 𝑐
(iii) Coincident lines if 𝑎 = 𝑏 = 𝑐1
2 2 2
4. A pair of linear equations 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 ; 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 is said to be
𝑎 𝑏
(i) Consistent and unique solution if 1 ≠ 1
𝑎2 𝑏2
𝑎1 𝑏 𝑐
(ii) Consistent and infinite number of solutions, if = 𝑏1 = 𝑐1
𝑎2 2 2
𝑎1 𝑏1 𝑐1
(iii) Inconsistent and no solution if 𝑎 = 𝑏 ≠ 𝑐
2 2 2
5. Certain basic facts to know:
(i) Equation of 𝑦 − 𝑎𝑥𝑖𝑠 is 𝑥 = 0 and equation of 𝑥 − 𝑎𝑥𝑖𝑠 is 𝑦 = 0
(ii) Equation of line parallel to 𝑥 −axis is 𝑦 = 𝑏
(iii) Equation of line parallel to 𝑦 −axis is 𝑥 = 𝑎
6. If unit’s digit is 𝑥 and ten’s digit is 𝑦, then the number formed is 10𝑦 + 𝑥
7. If the pair of linear equations is such that coefficients of 𝑥 and 𝑦 are interchanged in the
two equations then ‘Add the two equations and Subtract the two equations’. E.g., 254𝑥 +
309𝑦 = −55, 309𝑥 + 254𝑦 = 55
Chapter – 4 (Quadratic Equations)
1. Standard form of Quadratic equation is 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0
2. Quadratic Formula: Roots of the quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 are given by 𝑥 =
−𝑏±√𝐷
where 𝐷 = 𝑏 2 − 4𝑎𝑐 is called the discriminant
,
2𝑎
3. Nature of Roots of quadratic equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0:
(i) Two distinct real roots if 𝑏 2 − 4𝑎𝑐 > 0
(ii) Two real and equal (coincident) roots if 𝑏 2 − 4𝑎𝑐 = 0
(iii) No real roots if 𝑏 2 − 4𝑎𝑐 < 0
Chapter – 5 (Arithmetic Progressions)
1. General form of an A.P is 𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, …
2. nth term of A.P 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑
3. nth term from the end = 𝑙 − (𝑛 − 1)𝑑, 𝑙 is the last term
𝑛 𝑛 𝑛
4. Sum of 𝑛 terms of A.P = 2 [2𝑎 + (𝑛 − 1)𝑑] or 2 [𝑎 + 𝑙] or 2 [𝑎 + 𝑎𝑛 ]
𝑛(𝑛+1)
5. Sum of first 𝑛 natural numbers = 1 + 2 + 3 + ⋯ + 𝑛 = 2
6. Sum of first 𝑛 even natural numbers = 2 + 4 + 6 + ⋯ + 2𝑛 = 𝑛(𝑛 + 1)
7. Sum of first 𝑛 odd natural numbers 1 + 3 + 5 + ⋯ + (2𝑛 − 1) = 𝑛2
8. 𝑎𝑛 = 𝑆𝑛 − 𝑆𝑛−1 i.e, 𝑎1 = 𝑆1 , 𝑎2 = 𝑆2 − 𝑆1 , 𝑎3 = 𝑆3 − 𝑆2 and so on
9. Three consecutive terms in A.P are taken as 𝒂 − 𝒅, 𝒂, 𝒂 + 𝒅 with common difference ′𝑑′
10. Four consecutive terms in A.P are taken as 𝒂 − 𝟑𝒅, 𝒂 − 𝒅, 𝒂 + 𝒅, 𝒂 + 𝟑𝒅 with common
difference ′2𝑑′
