12th Grade Mathematics Formulae

1. Quadratic Formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

2. Derivative of a Function:

\[ \frac{d}{dx}\left[f(x)\right] = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \]

3. Sum of an Arithmetic Progression (AP):

\[ S_n = \frac{n}{2} \left[2a + (n-1)d\right] \]

4. Sum of a Geometric Progression (GP):

\[ S_n = \frac{a(1 - r^n)}{1 - r} \quad (r \neq 1) \]

5. Area of a Triangle (Using Heron’s Formula):

\[ A = \sqrt{s(s-a)(s-b)(s-c)} \quad \text{where} \ s = \frac{a+b+c}{2} \]

6. Standard Equation of a Circle:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

7. Distance Between Two Points:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

8. Slope of a Line:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

9. Definite Integral (Fundamental Theorem of Calculus):

\[ \int_a^b f(x) \, dx = F(b) - F(a) \quad \text{where} \ F'(x) = f(x) \]

10. Logarithmic Property:

\[ \log_a(xy) = \log_a x + \log_a y \]