Square Identities:
\( (a + b)^2 = a^2 + 2ab + b^2 \)
\( (a - b)^2 = a^2 - 2ab + b^2 \)
\( a^2 - b^2 = (a + b)(a - b) \)

Cube Identities:
\( (a + b)^3 = a^3 + b^3 + 3ab(a + b) \)
\( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \)
\( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
\( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)

Three Variable Identity:
\( (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \)

nth Term of AP:
\[ a_n = a + (n-1)d \]

Common Difference:
\[ d = a_2 - a_1 = a_3 - a_2 = ... \]

Sum of First n Terms:
\[ S_n = \frac{n}{2}[2a + (n-1)d] \]
OR
\[ S_n = \frac{n}{2}[a_1 + a_n] \]

Arithmetic Mean:
If a, b, c are in AP, then: \[ b = \frac{a + c}{2} \]

Standard Form:
\[ ax^2 + bx + c = 0 \] where \( a \neq 0 \)

Quadratic Formula (Roots):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Discriminant:
\[ \Delta = b^2 - 4ac \]

• If Δ > 0: Two distinct real roots
• If Δ = 0: Two equal real roots
• If Δ < 0: No real roots

Sum and Product of Roots:
Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
Product of roots: \( \alpha \cdot \beta = \frac{c}{a} \)

One Variable:
\[ ax + b = 0 \] where \( a \neq 0 \)

Two Variables:
\[ ax + by + c = 0 \] where \( a \neq 0 \) or \( b \neq 0 \)

Pair of Linear Equations:
\[ a_1x + b_1y + c_1 = 0 \]
\[ a_2x + b_2y + c_2 = 0 \]

Rectangle:
Area = \( l \times b \)
Perimeter = \( 2(l + b) \)

Square:
Area = \( a^2 \)
Perimeter = \( 4a \)

Triangle:
Area = \( \frac{1}{2} \times base \times height \)
Perimeter = \( a + b + c \)

Circle:
Area = \( \pi r^2 \)
Circumference = \( 2\pi r \)

Reciprocal Relations:
\( \sin \theta = \frac{1}{\cosec \theta} \)
\( \cos \theta = \frac{1}{\sec \theta} \)
\( \tan \theta = \frac{1}{\cot \theta} \)

Pythagorean Identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

Other Identities:
\( \sec^2 \theta - \tan^2 \theta = 1 \)
\( \cosec^2 \theta - \cot^2 \theta = 1 \)

Mean (Arithmetic Average):
\[ \bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i} \]

Median:
For grouped data: \[ M = l + \frac{\frac{n}{2} - F}{f} \times h \]
Where l = lower boundary, F = cumulative frequency before median class, f = frequency of median class, h = class width

Mode:
\[ Mode = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h \]
Where \( f_1 \) = frequency of modal class, \( f_0 \) = frequency before modal class, \( f_2 \) = frequency after modal class