Class 10th Maths Chapter Wise Formulas

Concepts

Description

Examples/Formula

Natural Numbers

Counting numbers starting from 1.

N = {1, 2, 3, 4, 5, …}

Whole Numbers

Counting numbers including zero.

W = {0, 1, 2, 3, 4, 5, …}

Integers

All positive numbers, zero, and negative numbers.

…, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …

Positive Integers

All positive whole numbers.

Z+ = 1, 2, 3, 4, 5, …

Negative Integers

All negative whole numbers.

Z– = -1, -2, -3, -4, -5, …

Rational Number

Numbers expressed as a fraction where both numerator and denominator are integers and the denominator is not zero.

Examples: 3/7, -5/4

Irrational Number

Numbers that cannot be expressed as a simple fraction.

Examples: π, √5

Real Numbers

All numbers that can be found on the number line, including rational and irrational numbers.

Includes Natural, Whole, Integers, Rational, Irrational

Euclid’s Division Algorithm

A method for finding the HCF of two numbers.

a = bq + r, where 0 ≤ r < b

Fundamental Theorem of Arithmetic

States that every composite number can be expressed as a product of prime numbers.

Composite Numbers = Product of Primes

HCF and LCM by Prime Factorization

Method to find the highest common factor and least common multiple.

HCF = Product of smallest powers of common factors;
LCM = Product of greatest powers of each prime factor; HCF(a,b) × LCM(a,b) = a × b

Category

Description

Formula/Identity

General Polynomial Formula

Standard form of a polynomial

F (x) = anxn + bxn-1 + an-2xn-2 + …….. + rx + s

Special Case: Natural Number n

Difference of powers formula

an – bn = (a – b)(an-1 + an-2b +…+ bn-2a + bn-1)

Special Case: Even n (n = 2a)

Sum of even powers formula

xn + yn = (x + y)(xn-1 – xn-2y +…+ yn-2x – yn-1)

Special Case: Odd Number n

Sum of odd powers formula

xn + yn = (x + y)(xn-1 – xn-2y +…- yn-2x + yn-1)

Division Algorithm for Polynomials

Division of one polynomial by another

p(x) = q(x) × g(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x). Here p(x) is divided, g(x) is divisor, q(x) is quotient, g(x) ≠ 0 and r(x) is remainder.

Types of Polynomials

General Form

Zeroes

Formation of Polynomial

Relationship Between Zeroes and Coefficients

Linear

ax+b

1

f(x)=a (x−α)

α=−b/a​

Quadratic

ax2+bx+c

2

f(x)=a (x−α)(x−β)

Sum of zeroes α+β=−b/a ​; Product of zeroes, αβ= c/a​

Cubic

ax3+bx2+cx+d

3

f(x)=a (x−α)(x−β)(x−γ)

Sum of zeroes, α+β+γ=−b/a​; Sum of product of zeroes taken two at a time, αβ+βγ+γα = c/a​; Product of zeroes, αβγ= −ad​

Quartic

ax4+bx3+cx2+dx+e

4

f(x)=a(x−α)(x−β)(x−γ)(x−δ)

Relationships become more complex; involves sums and products of zeroes in various combinations.

S. No.

Types of Linear Equation

General form 

Description

Solutions

1.

Linear Equation in one Variable

ax + b=0

Where a ≠ 0 and a & b are real numbers

One Solution 

2.

Linear Equation in Two Variables

ax + by + c = 0

Where a ≠ 0 & b ≠ 0 and a, b & c are real numbers

Infinite Solutions possible

3.

Linear Equation in Three Variables

ax + by + cz + d = 0

Where a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c, d are real numbers

Infinite Solutions possible

Concept

Description

Quadratic Equation

A polynomial equation of degree two in one variable, typically written as f(x) = ax² + bx + c, where ‘a,’ ‘b,’ and ‘c’ are real numbers, and ‘a’ is not equal to zero.

Roots of Quadratic Equation

The values of ‘x’ that satisfy the quadratic equation f(x) = 0 are the roots (α, β) of the equation. Quadratic equations always have two roots.

Quadratic Formula

The formula to find the roots (α, β) of a quadratic equation is given by: (α, β) = [-b ± √(b² – 4ac)] / (2a), where ‘a,’ ‘b,’ and ‘c’ are coefficients of the equation.

Discriminant

The discriminant ‘D’ of a quadratic equation is given by D = b² – 4ac. It determines the nature of the roots of the equation.

Nature of Roots

Depending on the value of the discriminant ‘D,’ the nature of the roots can be categorized as follows:

– D > 0: Real and distinct roots (unequal).

– D = 0: Real and equal roots (coincident).

– D < 0: Imaginary roots (unequal, in the form of complex numbers).

Sum and Product of Roots

The sum of the roots (α + β) is equal to -b/a, and the product of the roots (αβ) is equal to c/a.

Quadratic Equation in Root Form

A quadratic equation can be expressed in the form of its roots as x² – (α + β)x + (αβ) = 0.

Common Roots of Quadratic Equations

Two quadratic equations have one common root if (b₁c₂ – b₂c₁) / (c₁a₂ – c₂a₁) = (c₁a₂ – c₂a₁) / (a₁b₂ – a₂b₁).

Both equations have both roots in common if a₁/a₂ = b₁/b₂ = c₁/c₂.

Maximum and Minimum Values

For a quadratic equation ax² + bx + c = 0:

Roots of Cubic Equation

– If ‘a’ is greater than zero (a > 0), it has a minimum value at x = -b/(2a).

– If ‘a’ is less than zero (a < 0), it has a maximum value at x = -b/(2a).

If α, β, γ are roots of the cubic equation ax³ + bx² + cx + d = 0, then:

– α + β + γ = -b/a

– αβ + βγ + λα = c/a

– αβγ = -d/a

Concept

Description

Arithmetic Progressions (AP)

A sequence of terms where the difference between consecutive terms is constant.

Common Difference

The constant difference between any two consecutive terms in an AP. It is denoted as ‘d’. d=a2– a1 = a3 – a2 = …

nth Term of AP

an = a + (n – 1) d,, where ‘a’ is the first term, ‘n’ is the term number, and ‘d’ is the common difference.

Sum of nth Terms of AP

Sn= n/2 [2a + (n – 1)d], where ‘a’ is the first term, ‘n’ is the number of terms, and ‘d’ is the common difference.

Chapter 6: Triangles 

Concept

Description

Similar Triangles

Triangles with equal corresponding angles and proportional corresponding sides.

Equiangular Triangles

Triangles with all corresponding angles equal. The ratio of any two corresponding sides is constant.

Criteria for Triangle Similarity

Angle-Angle-Angle (AAA) Similarity

Two triangles are similar if their corresponding angles are equal.

Side-Angle-Side (SAS) Similarity

Two triangles are similar if two sides are in proportion and the included angles are equal.

Side-Side-Side (SSS) Similarity

Two triangles are similar if all three corresponding sides are in proportion.

Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides proportionally.

Converse of Basic Proportionality Theorem

If in two triangles, corresponding angles are equal, then their corresponding sides are proportional and the triangles are similar.

Formulas Related to Coordinate Geometry

Description

Formula

Distance Formula

Distance between two points A(x1, y1) and B(x2, y2)

AB= √[(x2 − x1)2 + (y2 − y1)2]

Section Formula

Coordinates of a point P dividing line AB in ratio m : n

P={[(mx2 + nx1) / (m + n)] , [(my2 + ny1) / (m + n)]}

Midpoint Formula

Coordinates of the midpoint of line AB

P = {(x1 + x2)/ 2, (y1+y2) / 2}

Area of a Triangle

Area of triangle formed by points A(x1, y1), B(x2, y2) and C(x3, y3)

(∆ABC = ½ |x1(y2 − y3) + x2(y3 – y1) + x3(y1 – y2)|

Category

Formula/Identity

Description/Equivalent

Arc Length in a Circle

l =r × θ

l is arc length, r is radius, θ is angle in radians

Radian and Degree Conversion

Radian Measure = π/180 × Degree Measure

Conversion from degrees to radians

Degree Measure= 180/π × Radian Measure

Conversion from radians to degrees

Trigonometric Ratio

Formula

Description

sin θ

P / H

Perpendicular (P) / Hypotenuse (H)

cos θ

B / H

Base (B) / Hypotenuse (H)

tan θ

P / B

Perpendicular (P) / Base (B)

cosec θ

H / P

Hypotenuse (H) / Perpendicular (P)

sec θ

H / B

Hypotenuse (H) / Base (B)

cot θ

B / P

Base (B) / Perpendicular (P)

Reciprocal Ratio

Formula

Equivalent to

sin θ

1 / (cosec θ)

Reciprocal of cosecant

cosec θ

1 / (sin θ)

Reciprocal of sine

cos θ

1 / (sec θ)

Reciprocal of secant

sec θ

1 / (cos θ)

Reciprocal of cosine

tan θ

1 / (cot θ)

Reciprocal of cotangent

cot θ

1 / (tan θ)

Reciprocal of tangent

Identity

Formula

Pythagorean Identity

sin2 θ + cos2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ ⇒ cos2 θ = 1 – sin2 θ

Cosecant-Cotangent Identity

cosec2 θ – cot2 θ = 1 ⇒ sin2 θ = 1 – cos2 θ ⇒ cos2 θ = 1 – sin2 θ

Secant-Tangent Identity

sec2 θ – tan2 θ = 1 ⇒ sec2 θ = 1 + tan2 θ ⇒ tan2 θ = sec2 θ – 1

Important Concepts in Chapter 9 Trigonometry

Line of Sight

The line formed by our vision as it passes through an item when we look at it.

Horizontal Line

A line representing the distance between the observer and the object, parallel to the horizon.

Angle of Elevation

The angle formed above the horizontal line by the line of sight when an observer looks up at an object.

Angle of Depression

The angle formed below the horizontal line by the line of sight when an observer looks down at an object.

Concept

Description

Circle

A circle is a closed figure consisting of all points in planes that are equidistant from a fixed point called the center.

Radius

The radius of a circle is the distance from the center to any point on the circle’s circumference.

Diameter

The diameter of a circle is a line segment that passes through the center and has endpoints on the circle’s circumference. It is twice the length of the radius.

Chord

A chord is a line segment with both endpoints on the circle’s circumference. A diameter is a special type of chord that passes through the center.

Arc

An arc is a part of the circumference of a circle, typically measured in degrees. A semicircle is an arc that measures 180 degrees.

Sector

A sector is a region enclosed by two radii of a circle and an arc between them. Sectors can be measured in degrees or radians.

Segment

A segment is a region enclosed by a chord and the arc subtended by the chord.

Circumference

The circumference of a circle is the total length around its boundary. It is calculated using the formula: Circumference = 2πr, where ‘r’ is the radius.

Area of a Circle

The area of a circle is the total space enclosed by its boundary. It is calculated using the formula: Area = πr², where ‘r’ is the radius.

Central Angle

A central angle is an angle whose vertex is at the center of the circle, and its sides pass through two points on the circle’s circumference.

Inscribed Angle

An inscribed angle is an angle formed by two chords in a circle with its vertex on the circle’s circumference.

Tangent Line

A tangent line to a circle is a straight line that touches the circle at only one point, known as the point of tangency.

Secant Line

A secant line is a straight line that intersects a circle at two distinct points.

Concentric Circles

Concentric circles are circles that share the same center but have different radii.

Circumcircle and Incircle

The circumcircle is a circle that passes through all the vertices of a polygon, while the incircle is a circle that is inscribed inside the polygon.

Formulas of Areas Related to Circles

Concept

Description

Formula

Area of a Circle

The space enclosed by the circle’s circumference

Area=πr2

Circumference of a Circle

The perimeter or boundary line of a circle

Circumference=2πr or πd

Area of a Sector

The area of a ‘pie-slice’ part of a circle

Area of Sector= (θ/360​) × πr2 (θ in degrees)

Length of an Arc

The length of the curved line forming the sector

Length of Arc= (θ/360​) ​× 2πr (θ in degrees)

Area of a Segment

Area of a sector minus the area of the triangle formed by the sector

Area of Segment = Area of Sector – Area of Triangle

Formulas Related to Surface Areas and Volumes

Geometrical Figure

Total Surface Area (TSA)

Lateral/Curved Surface Area (CSA/LSA)

Volume

Cuboid

2(lb + bh + hl)

2h(l + b)

l × b × h

Cube

6a²

4a²

a³

Right Circular Cylinder

2πr(h + r)

2πrh

πr²h

Right Circular Cone

πr(l + r)

πrl

1/3πr²h

Sphere

4πr²

2πr²

4/3πr³

Right Pyramid

LSA + Area of the base

½ × p × l

1/3 × Area of the base × h

Prism

LSA × 2B

p × h

B × h

Hemisphere

3πr²

2πr²

2/3πr³

Statistical Measure

Method/Description

Formula

Mean

Direct method

X = ∑fi xi / ∑fi​​

Assumed Mean Method

X = a + ∑fi di / ∑fi
,(where di = xi – a)

Step Deviation Method:

X = a + ∑fi ui / ∑fi × h

Median

Middlemost Term

For even number of observations: Middle term
For odd number of observations: (n+1/2) th term

Mode

Frequency Distribution

Mode=1+[f1−f02f1−f0−f2]×hMode=1+[2f1​−f0​−f2​f1​−f0​​]×h

where l = lower limit of the modal class,

f1 =frequency of the modal class,

f0 = frequency of the preceding class of the modal class,

f2 = frequency of the succeeding class of the modal class,

h is the size of the class interval.

Type of Probability

Description

Formula

Empirical Probability

Probability based on actual experiments or observations.

Empirical Probability = Number of Trials with expected outcome / Total Number of Trials

Theoretical Probability

Probability based on theoretical reasoning rather than actual experiments.

Theoretical Probability = Number of favorable outcomes to E / Total Number of possible outcomes