Maths Formulas For Class 10: Revise All Concepts (PDF)

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Maths Formulas For Class 10: The provided statement introduces a comprehensive list of essential mathematical formulas for CBSE Class 10 students, covering topics like Real Numbers, Polynomials, Trigonometry, and more.

This compilation serves as a valuable revision checklist for the upcoming 2024 board exam.

The list includes not only formulas but also important terms and properties, enhancing students’ understanding.

Using this resource for revision aids in reinforcing knowledge and preparing students thoroughly for the exam, emphasizing the significance of regular practice and practical application of these mathematical concepts.

Maths Formulas For Class 10 Revise For Board 2024

PDF covers Complete Concepts and Formulae for following:

Unit I: Number Systems

  • Chapter 1: Real Numbers
  • Chapter 2: Polynomials

Unit II: Algebra

  • Chapter 3: Pair of Linear Equations in Two Variables
  • Chapter 4: Quadratic Equations
  • Chapter 5: Arithmetic Progressions

Unit III: Coordinate Geometry

  • Chapter 6: Coordinate Geometry

Unit IV: Geometry

  • Chapter 7: Triangles

Unit V: Trigonometry

  • Chapter 8: Introduction to Trigonometry
  • Chapter 9: Some Applications of Trigonometry

Unit VI: Mensuration

  • Chapter 10: Circles
  • Chapter 11: Areas Related to Circles
  • Chapter 12: Surface Areas and Volumes

Unit VII: Statistics and Probability

  • Chapter 13: Statistics
  • Chapter 14: Probability

Ch1: Real Numbers

Natural Numbers (N): ${1, 2, 3, 4, 5,…}

Whole Numbers (W): ${0, 1, 2, 3, 4, 5,…}

Integer Number (Z)= {…,−3,−2,−1,0,1,2,3,…}

Rational Numbers: p/q form where p and q are integer

LCM(a,b) x HCF(a,b) = ab

LCM (P, Q, R):

$\frac{{P \cdot Q \cdot R}}{{\text{{HCF}}(P, Q) \cdot \text{{HCF}}(Q, R) \cdot \text{{HCF}}(P, R)}}$

HCF (P, Q, R):

$\frac{{P \cdot Q \cdot R}}{{\text{{LCM}}(P, Q) \cdot \text{{LCM}}(Q, R) \cdot \text{{LCM}}(P, R)}}$

Chapter 2: Polynomials

$(a+b)^2 = a^2 + 2ab + b^2$

$(a-b)^2 = a^2 – 2ab + b^2$

$(x+a)(x+b) = x^2 + (a+b)x + ab$

$a^2 – b^2 = (a+b)(a-b)$

$a^3 – b^3 = (a-b)(a^2 + ab + b^2)$

$a^3 + b^3 = (a+b)(a^2 – ab + b^2)$

$(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

$(a-b)^3 = a^3 – 3a^2b + 3ab^2 – b^3$

Chapter 3: Pair of Linear Equations in Two Variables

  • Linear equation in one variable: (ax + b = 0)
  • Linear equation in two variables: (ax + by + c = 0)

$a_1x + b_1y$ + $c_1 = 0$, $a_2x + b_2y + c_2 = 0$

ScenarioConditionGraphical Example
Unique Solution (One Solution)a1/a2 ≠ b1/b2Lines intersect at a single point2x + 3y = 7 and x – y = 1
Infinitely Many Solutionsa1/a2 = b1/b2 = c1/c2Lines completely overlap3x – 6y = 9 and x – 2y = 3
No Solutiona1/a2 = b1/b2 ≠ c1/c2Lines are parallel4x + 2y = 8 and 2x + y = 5

Ch4: Quadratic Equations

  • $x = \frac{{-b \pm \sqrt{{b^2 – 4ac}}}}{{2a}}$
  • Discriminant: $b^2 – 4ac$

Ch5: Arithmetic Progressions

  • (n)th term of AP: a + (n-1)d
  • Sum of (n) terms in AP: $S_n = \frac{n}{2}[2a + (n-1)d]$
  • Sum of all terms in AP with the last term (l): $\frac{n}{2}(a + l)$

Ch6: Triangles

  • Area of Triangle: A = $\frac{1}{2}bh$
  • Area of Isosceles Triangle: $\frac{1}{4}b\sqrt{4a^2 – b^2}$
  • Area of Right Triangle: A = $\frac{1}{2}\times \text{Base} \times \text{Height}$
  • Area of Equilateral Triangle: A = $\frac{\sqrt{3}}{4} \times \text{side}^2$

Ch7: Coordinate Geometry

  1. Distance Formula: $\sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
  2. Section Formula: $\left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)$
  3. Mid-point Formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Ch8: Introduction to Trigonometry

Basic Trigonometric Ratios:

Sine (sin) of an angle A:
$\sin A = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$

Cosine (cos) of an angle A:
$\cos A = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$

Tangent (tan) of an angle A:
$\tan A = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$

Reciprocal Trigonometric Ratios:

Cosecant (cosec) of an angle A:
$\csc A = \frac{1}{\sin A}$

Secant (sec) of an angle A:
$\sec A = \frac{1}{\cos A}$

Cotangent (cot) of an angle A:
$\cot A = \frac{1}{\tan A}$

Trigonometric Identities:

Pythagorean Identity:
$\sin^2 A + \cos^2 A = 1$
$\csc ^2 A – \cot^2 A = 1$
$\sec^2 A – \tan^2 A = 1$

Reciprocal Identity:
$\csc A = \frac{1}{\sin A}, \quad \sec A = \frac{1}{\cos A}, \quad \cot A = \frac{1}{\tan A}$

Quotient Identity:
$\tan A = \frac{\sin A}{\cos A}$

Complementary Angles:

Complementary Angle Relationship:
$\sin (90^\circ – A) = \cos A, \quad \cos (90^\circ – A) = \sin A, \quad \tan (90^\circ – A) = \cot A$

Trigonometric Ratios for Standard Angles:

Trigonometric Ratios for 0°, 30°, 45°, 60°, 90°:

AngleSine (sin)Cosine (cos)Tangent (tan)Cotangent (cot)
010undefined
30°1/2√3/21/√3√3
45°√2/2√2/211
60°√3/21/2√31/√3
90°10undefined0

Ch9: Some Applications of Trigonometry

Angle of Elevation and Depression:

  • If you are looking up at an object, it’s the angle of elevation.
  • If you are looking down at an object, it’s the angle of depression.

Ch10: Circles

TheoremStatement
Tangent is Perpendicular to RadiusA tangent to a circle is perpendicular to the radius through the point of contact.
Lengths of Tangents from an External PointThe lengths of tangents drawn from an external point to a circle are equal.
Angles in the Same SegmentThe angles in the same segment of a circle are equal.
Angle at the Centre is Twice Angle at the CircumferenceThe angle subtended by an arc at the center of a circle is twice the angle it subtends at any point on the circumference.
Cyclic QuadrilateralThe sum of the opposite angles of a cyclic quadrilateral is 180°.
Angles in a SemicircleAn angle in a semicircle is a right angle.
Equal Chords Subtend Equal AnglesEqual chords of a circle subtend equal angles at the center.
Perpendicular from the Center to a Chord Bisects ItThe perpendicular from the center of a circle to a chord bisects the chord.
Equal Chords are Equidistant from the CenterEqual chords of a circle are equidistant from the center.

Circumference of a Circle:

$C = 2\pi r$

Where (C) is the circumference, and (r) is the radius of the circle.

Area of a Circle:

$A = \pi r^2$

Where (A) is the area, and (r) is the radius of the circle.

Area of a Sector:

$A_{\text{sector}} = \frac{n}{360} \pi r^2$

Where (n) is the central angle of the sector in degrees.

Length of an Arc:

$L_{\text{arc}} = \frac{n}{360} \times 2\pi r$

Where (n) is the central angle of the arc in degrees.

Area of a Segment:

$A_{\text{segment}} = A_{\text{sector}} – A_{\text{triangle}}$

Where $A_{\text{sector}}$ is the area of the sector, and $A_{\text{triangle}}$ is the area of the triangle formed by the radii and the chord of the segment.

Ch13: Statistics

Mean for Grouped Data:

  • Direct Method: $\bar{x} = \frac{\sum f_i \cdot x_i}{N}$
  • Assumed Mean Method: $\bar{x} = A + \frac{\sum f_i \cdot u_i}{N}$
  • Step Deviation Method: $\bar{x} = A + \frac{\sum f_i \cdot u_i}{N \cdot h}$
  • Where:
    • $\bar{x}$ is the mean.
    • $f_i$ is the frequency of the $i^{th}$ class.
    • $x_i$ is the midpoint of the $i^{th}$ class.
    • ( N ) is the total number of observations.
    • ( A ) is the assumed mean.
    • $u_i$ is the deviation of $x_i$ from A.
    • h is the class width.

Mode: The value that occurs most frequently in a data set.

Median: The middle value of a data set when it is ordered.

Mode of Grouped Data:

$\text{Mode} (l) = l + \frac{(f_1 – f_0)}{(2f_1 – f_0 – f_2)} \times h$

where:

l is the lower limit of the modal class

h is the class size

$f_1$ is the frequency of the modal class

$f_0$ is the frequency of the class preceding the modal class

$f_2$ is the frequency of the class succeeding the modal class.

Median for Grouped Data:

$\text{Median} (l) = l + \frac{\frac{n}{2} – cf}{f} \times h$

where:

l is the lower limit of the median class

n is the total number of observations

cf is the cumulative frequency of the class preceding the median class

f is the frequency of the median class

h is the class size.

Ch14: Probability

1. Basic Probability Formula:

P(E) = n(E) / n(S)

  • P(E) represents the probability of an event E occurring.
  • n(E) represents the number of favorable outcomes for event E.
  • n(S) represents the total number of possible outcomes in the sample space.

Example: If you toss a fair coin, the probability of getting heads is 1/2 because there’s 1 favorable outcome (heads) out of 2 possible outcomes (heads or tails).

2. Probability of Not an Event:

P(E’) = 1 – P(E)

  • P(E’) represents the probability of the event E not occurring (also called the complementary event).

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