Numerical Bank

---
target: 495+/500
author: Senior Sibling
name: numerical-bank
description: >-
  High-Yield 12th Grade Numerical Types. 
  Focus on Electrostatics, Optics, Nernst Equation, and Rate Kinetics.
---

# Numerical Bank — High-Yield 12th Grade Types

## 🔭 Physics Numericals

### Type 1: Gauss's Law & Flux
- **Pattern:** Calculating flux through a surface or $E$ field due to symmetric charge distributions.
- **Key Step:** $\Phi = \oint \vec{E} \cdot d\vec{A} = q_{enclosed} / \epsilon_0$.
- **Variation:** Charge inside a cube, infinite line charge ($E = \lambda / 2\pi\epsilon_0r$).

### Type 2: Kirchhoff's Laws (Circuit Analysis)
- **Pattern:** Finding currents in multi-loop circuits.
- **Key Step:** Define loop directions and apply $\Sigma V = 0$.
- **Trap:** Sign convention for $E$ and $IR$ when moving against current/battery.

### Type 3: Mirror/Lens & Prism
- **Pattern:** Finding image position ($v$) or Refractive Index ($n$).
- **Lens Maker Formula:** $1/f = (n-1)(1/R1 - 1/R2)$.
- **Prism:** $n = \sin[(A+Dm)/2] / \sin[A/2]$.

### Type 4: LCR Circuit (AC)
- **Pattern:** Finding Impedance ($Z$), Resonant Frequency, or Phase Angle.
- **Formula:** $Z = \sqrt{R^2 + (X_L - X_C)^2}$ where $X_L = \omega L, X_C = 1/\omega C$.

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## 🧪 Chemistry Numericals

### Type 5: Solutions (Colligative Properties)
- **Pattern:** Calculating Molar Mass or Freezing Point depression.
- **Formula:** $\Delta T_f = i \cdot K_f \cdot m$.
- **Trap:** Forgetting the van't Hoff factor ($i$) for electrolytes.

### Type 6: Electrochemistry (Nernst Equation)
- **Pattern:** Calculating Cell Potential ($E_{cell}$).
- **Formula:** $E_{cell} = E^0_{cell} - (0.0591/n) \log([Prod]/[React])$.
- **Variation:** Calculating Gibbs Free Energy ($\Delta G = -nFE_{cell}$).

### Type 7: Chemical Kinetics (Order of Reaction)
- **Pattern:** Finding Rate Constant ($k$) or Half-life for 1st order reactions.
- **Formula:** $k = (2.303/t) \log([R]_0 / [R])$.
- **Half-life:** $t_{1/2} = 0.693/k$.

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## 📐 Mathematics Numericals

### Type 8: Differential Equations
- **Pattern:** Solving Linear Differential Equations.
- **Steps:** 1) Find $P$ and $Q$. 2) Find I.F. $= e^{\int P dx}$. 3) Apply $y(I.F.) = \int Q(I.F.) dx + C$.

### Type 9: Vectors & 3D Geometry
- **Pattern:** Shortest distance between two skew lines.
- **Formula:** $d = |(b1 \times b2) \cdot (a2 - a1)| / |b1 \times b2|$.

### Type 10: Bayes' Theorem
- **Pattern:** Inverse probability questions (e.g., "Given the outcome, find the probability it came from Bag A").
- **Key:** $P(E1|A) = [P(E1)P(A|E1)] / \Sigma [P(Ei)P(A|Ei)]$.

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## The "Topper Checklist" for Numericals
1. ✅ **GIVEN:** List all values with SI units ($cm \to m$, $\mu C \to C$).
2. ✅ **FORMULA:** State the formula clearly BEFORE substitution.
3. ✅ **SUBSTITUTION:** Show the raw numbers in the formula.
4. ✅ **RESULT:** Final answer with proper units and **Boxed**.
5. ✅ **APPROXIMATION:** Use log tables/calculations carefully; don't round off too early.
6. ✅ **SIGN:** For Physics, double-check lens/mirror signs.