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.