12th Class

Here is a comprehensive list of important mathematical formulas typically covered in 12th grade mathematics. These formulas are from various topics such as Calculus, Algebra, Geometry, Trigonometry, Probability, and Linear Algebra, which are part of standard 12th-grade curricula.

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1. Algebra

Quadratic Equation

• General form: ( ax^2 + bx + c = 0 )
• Roots using the quadratic formula:
  [
  x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
  ]
  where ( a ), ( b ), and ( c ) are the coefficients of the quadratic equation.

Sum and Product of Roots

For the quadratic equation ( ax^2 + bx + c = 0 ):

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

Cubic Equation

• For the equation ( ax^3 + bx^2 + cx + d = 0 ), the formulas for the sum and product of roots can be derived using Vieta’s relations.

Binomial Theorem (Expansion of ( (a + b)^n ))

[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
]

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2. Calculus

Differentiation

1. Basic Derivatives:

• ( \frac{d}{dx}(x^n) = n x^{n-1} )
• ( \frac{d}{dx}(\sin x) = \cos x )
• ( \frac{d}{dx}(\cos x) = -\sin x )
• ( \frac{d}{dx}(\tan x) = \sec^2 x )
• ( \frac{d}{dx}(e^x) = e^x )
• ( \frac{d}{dx}(\ln x) = \frac{1}{x} )

1. Product Rule:
   [
   \frac{d}{dx}(u \cdot v) = u’v + uv’
   ]
2. Quotient Rule:
   [
   \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u’ – u v’}{v^2}
   ]
3. Chain Rule:
   [
   \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
   ]

Integration

1. Basic Integrals:

• ( \int x^n dx = \frac{x^{n+1}}{n+1} + C ) (for ( n \neq -1 ))
• ( \int \sin x \, dx = -\cos x + C )
• ( \int \cos x \, dx = \sin x + C )
• ( \int \sec^2 x \, dx = \tan x + C )
• ( \int e^x \, dx = e^x + C )
• ( \int \frac{1}{x} \, dx = \ln |x| + C )

1. Definite Integral:
   [
   \int_a^b f(x) \, dx = F(b) – F(a)
   ]
   where ( F(x) ) is an antiderivative of ( f(x) ).
2. Integration by Parts:
   [
   \int u \, dv = uv – \int v \, du
   ]
3. Integration by Substitution:
   If ( x = g(t) ), then:
   [
   \int f(x) \, dx = \int f(g(t)) g'(t) \, dt
   ]

Limits

1. Limit of a function:
   [
   \lim_{x \to a} f(x) = L
   ]
2. Standard Limits:
   [
   \lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to 0} \frac{1 – \cos x}{x} = 0
   ]

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3. Trigonometry

1. Trigonometric Ratios:

• ( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} )
• ( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} )
• ( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} )

1. Pythagorean Identities:
   [
   \sin^2 \theta + \cos^2 \theta = 1
   ]
   [
   1 + \tan^2 \theta = \sec^2 \theta
   ]
   [
   1 + \cot^2 \theta = \csc^2 \theta
   ]
2. 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} )

1. Double Angle Formulas:

• ( \sin 2A = 2 \sin A \cos A )
• ( \cos 2A = \cos^2 A – \sin^2 A )
• ( \tan 2A = \frac{2 \tan A}{1 – \tan^2 A} )

1. Triple Angle Formulas:

• ( \sin 3A = 3 \sin A – 4 \sin^3 A )
• ( \cos 3A = 4 \cos^3 A – 3 \cos A )

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4. Vectors

1. Vector Addition:
   [
   \vec{A} + \vec{B} = \langle A_x + B_x, A_y + B_y \rangle
   ]
2. Dot Product:
   [
   \vec{A} \cdot \vec{B} = A_x B_x + A_y B_y
   ]
3. Cross Product:
   [
   \vec{A} \times \vec{B} = \langle (A_y B_z – A_z B_y), (A_z B_x – A_x B_z), (A_x B_y – A_y B_x) \rangle
   ]

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5. Probability

1. Probability of an Event:
   [
   P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}
   ]
2. Addition Law of Probability:
   [
   P(A \cup B) = P(A) + P(B) – P(A \cap B)
   ]
3. Multiplication Law of Probability:
   [
   P(A \cap B) = P(A) \cdot P(B|A)
   ]
4. Bayes’ Theorem:
   [
   P(A|B) = \frac{P(B|A) P(A)}{P(B)}
   ]
5. Mean of Random Variable:
   [
   \mu = \sum x_i P(x_i)
   ]
6. Variance of Random Variable:
   [
   \sigma^2 = \sum (x_i – \mu)^2 P(x_i)
   ]

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6. Matrices and Determinants

1. Matrix Multiplication:
   [
   C = AB
   ]
   (Dot product of rows of ( A ) with columns of ( B )).
2. Determinant of a 2×2 Matrix:
   [
   \text{det}(A) = \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad – bc
   ]
3. Cofactor Expansion:
   For a 3×3 matrix:
   [
   \text{det}(A) = a(ei – fh) – b(di – fg) + c(dh – eg)
   ]
4. Inverse of a 2×2 Matrix:
   [
   A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}
   ]

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This is a broad summary of the key formulas you’ll likely encounter in 12th-grade mathematics.