Numerical Bank

---
name: numerical-bank
description: Most repeated numerical problem types in Math and Physics. Students who can solve these 15 types can handle ANY numerical on the board exam.
---

# Numerical Bank — 15 Types That Cover ALL Board Numericals

## Mathematics Numericals

### Type 1: Quadratic Word Problems
```
Pattern: "The product of two consecutive numbers is 272. Find them."
Setup: x(x+1) = 272 → x² + x - 272 = 0 → solve using formula
Variations: age problems, speed-distance, area problems
```

### Type 2: AP Sum Problems
```
Pattern: "Find sum of first 20 terms where a=7, d=3"
Formula: Sₙ = n/2[2a + (n-1)d] = 20/2[14 + 57] = 710
Variations: find n given sum, find d given two terms and sum
```

### Type 3: Trigonometric Heights & Distances
```
Pattern: "A tower 30m high casts shadow of 30√3m. Find sun's angle."
Setup: tan θ = 30/(30√3) = 1/√3 → θ = 30°
Variations: two angles, moving towards/away from tower
```

### Type 4: Mensuration Combination
```
Pattern: "A toy is a cone on a hemisphere. Cone height 4cm, radius 3cm. Find TSA."
Steps: TSA = CSA of cone + CSA of hemisphere = πrl + 2πr²
  l = √(r²+h²) = √(9+16) = 5
  TSA = π(3)(5) + 2π(9) = 15π + 18π = 33π = 103.62 cm²
```

### Type 5: Statistics Mean/Median
```
Pattern: Given grouped frequency table, find mean using step-deviation
Steps: 1) Choose assumed mean (a) 2) Calculate uᵢ = (xᵢ-a)/h 3) Apply formula
```

## Physics Numericals

### Type 6: Ohm's Law + Circuits
```
Pattern: "R₁=3Ω, R₂=6Ω in parallel. Battery 3V. Find total R, I, power."
Steps: 1/R = 1/3 + 1/6 = 3/6 → R = 2Ω
  I = V/R = 3/2 = 1.5A
  P = VI = 3 × 1.5 = 4.5W
```

### Type 7: Resistivity Calculation
```
Pattern: "Wire length 1m, area 1mm², resistance 2Ω. Find resistivity."
Formula: R = ρl/A → ρ = RA/l
Convert: A = 1mm² = 1×10⁻⁶ m²
  ρ = 2 × 10⁻⁶ / 1 = 2×10⁻⁶ Ωm
```

### Type 8: Mirror/Lens Formula
```
Pattern: "Object 30cm from concave mirror, f=15cm. Find image position."
Sign convention: u = -30, f = -15 (concave)
  1/v + 1/u = 1/f → 1/v = 1/(-15) - 1/(-30) = -1/30
  v = -30cm (real, inverted, same size)
```

### Type 9: Power of Lens
```
Pattern: "Convex lens f=20cm. Find power."
Convert: f = 20cm = 0.2m
  P = 1/f = 1/0.2 = +5D (positive for convex)
```

### Type 10: Electrical Energy & Cost
```
Pattern: "100W bulb used 10 hours daily for 30 days. Cost at ₹5/kWh?"
Energy = P × t = 100 × 10 × 30 = 30000 Wh = 30 kWh
Cost = 30 × 5 = ₹150
```

## Quick Checks After Every Numerical
```
✅ Did I write units with the answer?
✅ Did I show Given → Formula → Substitution → Answer?
✅ Did I use correct sign convention (for mirror/lens)?
✅ Is my answer reasonable? (negative resistance = wrong somewhere)
✅ Did I convert units? (cm→m for Power, mm²→m² for resistivity)
```