Important Mathematical Formulae for Class - 10

For class 10th, the curriculum usually covers foundational topics across different areas of mathematics. Here are some important mathematical formulae for class 10th, organized by topic:

Algebra

1. Quadratic Equations: The solutions of the quadratic equation \( ax^2 + bx + c = 0 \) are given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
2. Arithmetic Progression (AP):
• nth Term: The nth term of an AP is given by
\[ a_n = a + (n - 1)d \] where \( a \) is the first term and \( d \) is the common difference.







• Sum of First n Terms: The sum of the first \( n \) terms is
\[ S_n = \frac{n}{2} [2a + (n - 1)d] \]
3. Polynomials:
• Relationship between Coefficients and Roots:
For a quadratic polynomial \( ax^2 + bx + c = 0 \), if the roots are \( \alpha \) and \( \beta \), \[ \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a} \]

Geometry

1. Triangles:
• Pythagorean Theorem**: In a right triangle,
\[ c^2 = a^2 + b^2 \] where \( c \) is the hypotenuse.
• Similarity of Triangles: If two triangles are similar, then the corresponding sides are proportional, and the corresponding angles are equal.
2. Circles:
• Area of a Circle:
\[ \text{Area} = \pi r^2 \]
• Circumference of a Circle:
\[ \text{Circumference} = 2\pi r \]
• Length of an Arc:
\[ \text{Arc length} = \frac{\theta}{360^\circ} \times 2\pi r \] where \( \theta \) is the angle in degrees.
• Area of a Sector:
\[ \text{Area} = \frac{\theta}{360^\circ} \times \pi r^2 \]

Trigonometry

Trigonometric Ratios**: For a right triangle with angle \( \theta \): \[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]

Trigonometric Identities: - \[ \sin^2\theta + \cos^2\theta = 1 \] - \[ 1 + \tan^2\theta = \sec^2\theta \] - \[ 1 + \cot^2\theta = \csc^2\theta \]

Mensuration

Surface Area and Volume of Solids:

• Cuboid:
• Surface Area: \( 2(lb + bh + hl) \)
• Volume: \( l \times b \times h \)
• Cube:
• Surface Area: \( 6a^2 \)
• Volume: \( a^3 \)
• Cylinder:
- Curved Surface Area: \( 2\pi rh \) - Total Surface Area: \( 2\pi r (r + h) \) - Volume: \( \pi r^2 h \)
• Cone:
- Curved Surface Area: \( \pi rl \) - Total Surface Area: \( \pi r (r + l) \) - Volume: \( \frac{1}{3}\pi r^2 h \)
• Sphere:
- Surface Area: \( 4\pi r^2 \) - Volume: \( \frac{4}{3}\pi r^3 \)
• Hemisphere:
- Curved Surface Area: \( 2\pi r^2 \) - Total Surface Area: \( 3\pi r^2 \) - Volume: \( \frac{2}{3}\pi r^3 \)
### Statistics 1. **Mean** (Average): For a set of numbers \( x_1, x_2, \ldots, x_n \), \[ \text{Mean} = \frac{x_1 + x_2 + \cdots + x_n}{n} \] 2. **Median**: - For an odd number of observations, the median is the middle value. - For an even number of observations, the median is the average of the two middle values. 3. **Mode**: The value that occurs most frequently in a data set. These formulae cover key concepts from algebra, geometry, trigonometry, mensuration, and statistics, which are essential for class 10th mathematics.