10th Class

Here’s a comprehensive list of important mathematical formulas for 10th-grade mathematics. These formulas cover topics such as Algebra, Geometry, Trigonometry, Mensuration, and Statistics that are typically studied in the 10th grade.

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1. Algebra

1.1. Polynomials

• General form of a polynomial:
  [
  P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
  ]
  where ( a_n, a_{n-1}, \dots, a_0 ) are constants and ( x ) is the variable.
• Factorization of a quadratic polynomial:
  [
  ax^2 + bx + c = a(x – p)(x – q)
  ]
  where ( p ) and ( q ) are the roots of the polynomial.

1.2. Quadratic Equations

• Standard form of a quadratic equation:
  [
  ax^2 + bx + c = 0
  ]
• Quadratic formula to find the roots:
  [
  x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
  ]
• Sum and Product of Roots:
• Sum of roots: ( \alpha + \beta = -\frac{b}{a} )
• Product of roots: ( \alpha \beta = \frac{c}{a} )

1.3. Arithmetic Progression (AP)

• nth term of an AP:
  [
  a_n = a + (n – 1)d
  ]
  where ( a ) is the first term and ( d ) is the common difference.
• Sum of the first n terms of an AP:
  [
  S_n = \frac{n}{2} [2a + (n – 1) d]
  ]

1.4. Geometric Progression (GP)

• nth term of a GP:
  [
  a_n = a r^{n-1}
  ]
  where ( a ) is the first term, and ( r ) is the common ratio.
• Sum of the first n terms of a GP (when ( r \neq 1 )):
  [
  S_n = \frac{a(1 – r^n)}{1 – r}
  ]

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2. Geometry

2.1. Triangles

• Area of a triangle:
  [
  \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
  ]
• Pythagoras Theorem (for right-angled triangles):
  [
  a^2 + b^2 = c^2
  ]
  where ( a ) and ( b ) are the perpendicular sides and ( c ) is the hypotenuse.

2.2. Circle

• Circumference of a circle:
  [
  C = 2\pi r
  ]
  where ( r ) is the radius.
• Area of a circle:
  [
  A = \pi r^2
  ]
• Length of an arc:
  [
  L = \frac{\theta}{360^\circ} \times 2\pi r
  ]
  where ( \theta ) is the central angle in degrees.
• Area of a sector:
  [
  A = \frac{\theta}{360^\circ} \times \pi r^2
  ]
  where ( \theta ) is the central angle in degrees.

2.3. Quadrilaterals

• Area of a rectangle:
  [
  A = \text{length} \times \text{breadth}
  ]
• Area of a square:
  [
  A = \text{side}^2
  ]
• Area of a parallelogram:
  [
  A = \text{base} \times \text{height}
  ]
• Area of a rhombus:
  [
  A = \frac{1}{2} \times \text{product of diagonals}
  ]
• Area of a trapezium:
  [
  A = \frac{1}{2} \times (a + b) \times h
  ]
  where ( a ) and ( b ) are the lengths of the parallel sides and ( h ) is the height.

2.4. Coordinate Geometry

• Distance formula between two points ( (x_1, y_1) ) and ( (x_2, y_2) ):
  [
  d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}
  ]
• Midpoint formula:
  [
  M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  ]
• Equation of a straight line in slope-intercept form:
  [
  y = mx + c
  ]
  where ( m ) is the slope and ( c ) is the y-intercept.

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3. Trigonometry

3.1. Basic Trigonometric Ratios

For a right-angled triangle with angle ( \theta ):

• Sine:
  [
  \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
  ]
• Cosine:
  [
  \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
  ]
• Tangent:
  [
  \tan \theta = \frac{\text{opposite}}{\text{adjacent}}
  ]

3.2. Trigonometric Identities

• Pythagorean Identity:
  [
  \sin^2 \theta + \cos^2 \theta = 1
  ]
• Other useful identities:
  [
  1 + \tan^2 \theta = \sec^2 \theta, \quad 1 + \cot^2 \theta = \csc^2 \theta
  ]
• Angle Sum and Difference Formulas:
  [
  \sin(A + B) = \sin A \cos B + \cos A \sin B
  ]
  [
  \cos(A + B) = \cos A \cos B – \sin A \sin B
  ]
  [
  \tan(A + B) = \frac{\tan A + \tan B}{1 – \tan A \tan B}
  ]

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4. Mensuration (Surface Area and Volume)

4.1. Surface Area and Volume of Solids

• Surface Area of a Cube:
  [
  A = 6 \times \text{side}^2
  ]
• Surface Area of a Cuboid:
  [
  A = 2(lb + bh + hl)
  ]
  where ( l ), ( b ), and ( h ) are the length, breadth, and height respectively.
• Surface Area of a Sphere:
  [
  A = 4\pi r^2
  ]
• Volume of a Cube:
  [
  V = \text{side}^3
  ]
• Volume of a Cuboid:
  [
  V = l \times b \times h
  ]
• Volume of a Sphere:
  [
  V = \frac{4}{3} \pi r^3
  ]
• Volume of a Cone:
  [
  V = \frac{1}{3} \pi r^2 h
  ]
• Volume of a Cylinder:
  [
  V = \pi r^2 h
  ]

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5. Statistics

5.1. Mean, Median, and Mode

• Mean of a data set:
  [
  \text{Mean} = \frac{\sum \text{(all observations)}}{\text{number of observations}}
  ]
• Median:
• Arrange the data in ascending or descending order, and find the middle value.
• Mode:
• The value that appears most frequently in the data set.

5.2. Probability

• Basic Probability Formula:
  [
  P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
  ]

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