Chapter 2: RMS Planimetric Error & Distance
Key Formula: Allowable RMS Error = 0.5 mm × Scale Denominator | Result in metres = Denominator / 2000
| Scale | RMS (mm) | RMS (m) | Exam |
|---|---|---|---|
| 1:2,500 | 1,250 | 1.25 | 2075 Bhadra Q2b |
| 1:5,000 | 2,500 | 2.50 | 2074 Magh / 2071 Bhadra |
| 1:15,000 | 7,500 | 7.50 | 2071 Bhadra Q2c |
| 1:20,000 | 10,000 | 10.00 | 2079 Chaitra Q2a |
| 1:25,000 | 12,500 | 12.50 | 2078 Chaitra / 2076 Bhadra |
| 1:30,000 | 15,000 | 15.00 | 2074 Magh Q2b |
Q1. RMS 1:20,000 VERIFIED [3] 2079 Chaitra Q2a
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// Allowable RMS Planimetric Error
RMS = 0.5 mm × 20,000 = 10,000 mm
RMS = 10,000 / 1000
RMS = 10.0 m
Q2. Distance A(24.5°N, 83.5°E) → B(24.6°N, 84.5°E) VERIFIED [3] 2079 Chaitra Q2b
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// Distance between two geographic points
Given: R = 6378 km
Δlat = 24.6 − 24.5 = 0.1°
Δlon = 84.5 − 83.5 = 1.0°
Mean lat = (24.5 + 24.6) / 2 = 24.55° cos(24.55) = 0.9096
d_NS = 0.1 × π/180 × 6378 = 11.13 km
d_EW = 1.0 × π/180 × 6378 × 0.9096 = 101.25 km
Distance = √(11.13² + 101.25²)
Distance = √(123.88 + 10251.56) = 101.86 km
Q5. Difference: 1° Latitude vs 1° Longitude VERIFIED 2076 Bhadra Q2c
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// 1 degree latitude vs 1 degree longitude
Given: R_eq = 6,378,137 m R_pole = 6,356,752.31 m (WGS84)
1° lat = 2π × R_pole / 360
= 2 × 3.14159 × 6,356,752.31 / 360 = 110,946.26 m
1° lon = 2π × R_eq / 360
= 2 × 3.14159 × 6,378,137 / 360 = 111,319.49 m
Diff = 111,319.49 − 110,946.26 = 373.23 m ≈ 0.373 km
1° longitude at equator is longer because the equatorial radius is larger than the polar radius (Earth is oblate).
Chapter 3: Raster Compression / Encoding
Full Raster: rows × cols cells |
RLE: (value, count) per row |
Quadtree: Recursive 4-quadrant subdivision |
% Decrease: (Full − Encoded) / Full × 100
2074 Bhadra Q1b — 6×6 Grid NEW [6]
| 25 | 25 | 25 | 25 | 25 | 25 |
| 25 | 25 | 25 | 25 | 25 | 25 |
| 25 | 25 | 25 | 25 | 25 | 25 |
| 5 | 5 | 5 | 10 | 10 | 10 |
| 5 | 5 | 5 | 12 | 12 | 12 |
| 5 | 5 | 5 | 15 | 12 | 12 |
Full Raster = 6 × 6 = 36 cells
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// Run-Length Encoding (row by row)
Row 1: (25,6) // 1 pair
Row 2: (25,6) // 1 pair
Row 3: (25,6) // 1 pair
Row 4: (5,3)(10,3) // 2 pairs
Row 5: (5,3)(12,3) // 2 pairs
Row 6: (5,3)(15,1)(12,2) // 3 pairs
Total RLE = 10 pairs = 20 values
% Decrease = (36 − 20) / 36 × 100
% Decrease = 44.44%
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// Block Encoding: (value, r1, c1, r2, c2)
Block 1: (25, 1,1, 3,6) // 3×6 = 18 cells
Block 2: ( 5, 4,1, 6,3) // 3×3 = 9 cells
Block 3: (10, 4,4, 4,6) // 1×3 = 3 cells
Block 4: (12, 5,4, 5,6) // 1×3 = 3 cells
Block 5: (15, 6,4, 6,4) // 1×1 = 1 cell
Block 6: (12, 6,5, 6,6) // 1×2 = 2 cells
Total = 6 entries × 5 values = 30 values
% Decrease = (36 − 30) / 36 × 100 = 16.67%
2081 Chaitra — 8×8 Grid NEW
| 88 | 88 | 75 | 34 | 9 | 9 | 9 | 9 |
| 88 | 88 | 33 | 75 | 9 | 9 | 9 | 9 |
| 88 | 88 | 88 | 34 | 9 | 9 | 9 | 9 |
| 77 | 77 | 77 | 77 | 9 | 9 | 9 | 9 |
| 56 | 56 | 56 | 56 | 56 | 56 | 10 | 10 |
| 56 | 56 | 56 | 56 | 33 | 25 | 10 | 10 |
| 56 | 56 | 56 | 56 | 33 | 33 | 33 | 9 |
| 56 | 56 | 56 | 56 | 33 | 33 | 33 | 33 |
Full Raster = 8 × 8 = 64 cells
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// Run-Length Encoding (row by row)
Row 1: (88,2)(75,1)(34,1)(9,4) // 4 pairs
Row 2: (88,2)(33,1)(75,1)(9,4) // 4 pairs
Row 3: (88,3)(34,1)(9,4) // 3 pairs
Row 4: (77,4)(9,4) // 2 pairs
Row 5: (56,6)(10,2) // 2 pairs
Row 6: (56,4)(33,1)(25,1)(10,2) // 4 pairs
Row 7: (56,4)(33,3)(9,1) // 3 pairs
Row 8: (56,4)(33,4) // 2 pairs
Total RLE = 24 pairs = 48 values
% Decrease = (64 − 48) / 64 × 100 = 25.0%
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// Quadtree Decomposition (8×8)
Level 0 (8×8): Not uniform → split
NW (4×4): Not uniform → split 4 sub-quads
NW-NW(2×2): 88 ✓ NW-NE(2×2): mixed → 4 leaves
NW-SW(2×2): mixed → 4 leaves NW-SE(2×2): mixed → 4 leaves
// NW total: 1 + 1 + 5 + 5 + 5 = 17 nodes
NE (4×4): all 9 = 1 node
SW (4×4): all 56 = 1 node
SE (4×4): Not uniform → split 4
SE-NW: mixed(4+1) SE-NE: 10 ✓(1) SE-SW: 33 ✓(1) SE-SE: mixed(4+1)
// SE total: 1 + 5 + 1 + 1 + 5 = 13 nodes
Already Solved (in Chapter 3 HTML)
The following grids are fully solved with interactive solver in Chapter 3:
• 2080 Chaitra Q7 (8×8 — values 25,31,27,35,3)
• 2076 Bhadra Q3b (8×8 — values 1,2,3,4,5)
• 2075 Bhadra Q3a (8×8 — values 12,10,9,7,2,6,4,1,3,8,5)
• 2073 Magh Q3 (8×8 — F,W,O,S)
• 2073 Bhadra Q3c (8×8 — values 12,5,6,61,62,7,9)
• 2080 Chaitra Q7 (8×8 — values 25,31,27,35,3)
• 2076 Bhadra Q3b (8×8 — values 1,2,3,4,5)
• 2075 Bhadra Q3a (8×8 — values 12,10,9,7,2,6,4,1,3,8,5)
• 2073 Magh Q3 (8×8 — F,W,O,S)
• 2073 Bhadra Q3c (8×8 — values 12,5,6,61,62,7,9)
Chapter 7: IDW Interpolation & Suitability Analysis
IDW Interpolation VERIFIED [4] 2074 Bhadra Q3b
Given: Four known values: 160, 170, 180, 90 at distances: 1000, 400, 600, 120 m
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// IDW Formula: Z = Σ(Zi/di^p) / Σ(1/di^p)
// ── p = 1 (Linear) ──
Point 1: Z=160 d=1000 w=1/1000 = 0.00100 wZ = 0.160
Point 2: Z=170 d= 400 w=1/400 = 0.00250 wZ = 0.425
Point 3: Z=180 d= 600 w=1/600 = 0.00167 wZ = 0.300
Point 4: Z= 90 d= 120 w=1/120 = 0.00833 wZ = 0.750
Σw = 0.01350 ΣwZ = 1.635
Z(p=1) = 1.635 / 0.01350 = 121.11
// ── p = 2 (Quadratic — GIS default) ──
Point 1: w=1/1000² = 0.0000010 wZ = 0.000160
Point 2: w=1/400² = 0.0000063 wZ = 0.001063
Point 3: w=1/600² = 0.0000028 wZ = 0.000500
Point 4: w=1/120² = 0.0000694 wZ = 0.006250
Σw = 0.0000795 ΣwZ = 0.007973
Z(p=2) = 0.007973 / 0.0000795 = 100.32
// p=2 gives more weight to nearest point (Z=90 at d=120)
// That's why p=2 result is closer to 90
Rice Crop Suitability NEW [14] 2074 Magh Q6
Given: Cell size 25 m. Weights: Slope 50%, Soil 30%, Landuse 20%. Threshold > 6.
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// Weighted Suitability = Slope×0.5 + Soil×0.3 + LU×0.2
// Suitable if Score > 6
// Code Slope Soil LU Score Result
Code 1: 10×0.5 + 8×0.3 + 8×0.2 = 9.0 SUITABLE
Code 2: 8×0.5 + 4×0.3 + 1×0.2 = 5.4 NOT suitable
Code 3: 6×0.5 + 10×0.3 + 4×0.2 = 6.8 SUITABLE
Code 4: 4×0.5 + 1×0.3 + 8×0.2 = 3.9 NOT suitable
Code 5: 1×0.5 + 1×0.3 + 10×0.2 = 2.8 NOT suitable
// Area of suitable rice crop land
Suitable cells = 5 (cells with score > 6)
Cell area = 25 m × 25 m = 625 m²
Total suitable area = 5 × 625 = 3,125 m²
Suitable area = 3,125 m² = 0.3125 hectares
Chapter 9: DEM Hydrology — River & Watershed
D8 Codes: E=1, SE=2, S=4, SW=8, W=16, NW=32, N=64, NE=128
Sink: Cell lower than ALL 8 neighbors | Fill: min_neighbor + 0.5
River: FA ≥ threshold cells | Watershed: All cells draining to outlet
Sink: Cell lower than ALL 8 neighbors | Fill: min_neighbor + 0.5
River: FA ≥ threshold cells | Watershed: All cells draining to outlet
Q1. 2080 Chaitra Q10 — 30m DEM VERIFIED [2+2+2+3+3]
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// ── (a) Sink Detection & Filling ──
Sink 1: D3 = 68
Neighbors: C2=97 D2=88 E2=114 C3=84 E3=92 C4=81 D4=78 E4=76
Min neighbor = 76 (E4) 68 < 76 → SINK
Fill: 76 + 0.5 = 76.5
Sink 2: D5 = 73
Neighbors: C4=81 D4=78 E4=76 C5=84 E5=96 C6=103 D6=85 E6=119
Min neighbor = 76 (E4) 73 < 76 → SINK
Fill: 76 + 0.5 = 76.5
// ── (b) Flow Direction (D8) ──
// A B C D E F
Row 1: SE E SE S SW S
Row 2: SE SE SE S SW SW
Row 3: SE E E SE S SW
Row 4: E E E E OUT W
Row 5: NE E E E N NW
Row 6: NE NE N N W NW
// ── (c) Flow Accumulation ──
// A B C D E F
R1: 0 0 1 0 0 0
R2: 0 1 0 4 0 1
R3: 0 1 4 17 2 0
R4: 0 3 4 0 35 0
R5: 0 1 4 7 1 0
R6: 0 0 0 1 0 0
// ── (d) River Network ──
Threshold = 4500 m² / (30×30) = 5 cells
River cells: D3(FA=17) E4(FA=35) D5(FA=7)
// ── (e) Watershed ──
Outlet = E4 (all cells converge here)
Watershed = all 36 cells
Area = 36 × 30 × 30 = 32,400 m²
Q2. 2079 Chaitra Q8 — 100m DEM VERIFIED [10]
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| 1 | 78 | 72 | 69 | 71 | 58 | 49 |
| 2 | 74 | 67 | 56 | 49 | 46 | 50 |
| 3 | 69 | 40 | 44 | 37 | 37 | 48 |
| 4 | 64 | 58 | 55 | 22 | 22 | 24 |
| 5 | 78 | 61 | 47 | 21 | 21 | 19 |
| 6 | 74 | 53 | 34 | 12 | 11 | 12 |
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// ── Sink & Fill ──
Sink: B3 = 40 min neighbor = 44 (C3)
Fill: 44 + 0.5 = 44.5
// ── Flow Accumulation ──
// A B C D E F
R1: 0 0 0 0 0 0
R2: 0 1 1 2 2 0
R3: 0 6 7 5 4 0
R4: 0 0 0 15 5 1
R5: 0 0 0 17 0 8
R6: 0 2 4 23 35 9
// ── River Network ──
Threshold = 40,000 m² / (100×100) = 4 cells
River: B3(6) C3(7) D3(5) E3(4) D4(15) E4(5)
D5(17) F5(8) C6(4) D6(23) E6(35) F6(9)
// ── Watershed ──
All 36 cells → 36 × 10,000 = 360,000 m² = 36 hectares
Q5. 2075 Magh Q3b — 100m, Threshold 6 NEW
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | 510 | 500 | 490 | 480 | 470 |
| 2 | 505 | 495 | 490 | 475 | 480 |
| 3 | 500 | 485 | 455 | 470 | 475 |
| 4 | 495 | 490 | 480 | 465 | 470 |
| 5 | 490 | 485 | 475 | 455 | 450 |
| 6 | 500 | 490 | 480 | 460 | 445 |
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Sink: C3 = 455 min neighbor = 465 Fill = 465.5
// Flow Accumulation
// A B C D E
R1: 0 0 1 0 2
R2: 0 1 0 2 0
R3: 0 3 11 0 0
R4: 0 0 0 14 0
R5: 0 2 4 6 16
R6: 0 0 0 1 26
River (FA≥6): C3(11) D4(14) D5(6) E5(16) E6(26)
// Watershed
Outlet = E6 (445, lowest edge cell)
Catchment = 27 cells × 10,000 = 270,000 m² = 27 ha
Q6. 2075 Bhadra Q9a — Flow Direction NEW
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| 1 | 58 | 52 | 55 | 53 | 56 | 58 |
| 2 | 55 | 40 | 42 | 45 | 51 | 55 |
| 3 | 48 | 34 | 20 | 33 | 48 | 52 |
| 4 | 33 | 23 | 28 | 27 | 25 | 38 |
| 5 | 17 | 20 | 21 | 22 | 23 | 24 |
| 6 | 12 | 10 | 15 | 18 | 16 | 17 |
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Sink: C3 = 20 min neighbor = 23 Fill = 23.5
// Flow Accumulation
// A B C D E F
R1: 0 0 0 0 0 0
R2: 0 3 1 2 1 0
R3: 0 0 12 2 1 0
R4: 0 15 0 0 3 0
R5: 17 0 1 5 0 1
R6: 18 30 7 0 4 2
River (FA≥5): C3(12) B4(15) A5(17) D5(5)
A6(18) B6(30) C6(7)
Largest watershed = 31 cells × 10,000
= 310,000 m² = 31 ha
Q7. 2074 Magh Q5b — Threshold 4 cells NEW [10]
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| 1 | 109 | 106 | 104 | 103 | 114 | 115 |
| 2 | 107 | 105 | 102 | 100 | 112 | 113 |
| 3 | 105 | 103 | 100 | 95 | 110 | 112 |
| 4 | 106 | 105 | 99 | 90 | 108 | 110 |
| 5 | 107 | 110 | 98 | 94 | 110 | 111 |
| 6 | 108 | 111 | 96 | 95 | 112 | 115 |
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Sink: D4 = 90 min neighbor = 94 Fill = 94.5
// Flow Accumulation
// A B C D E F
R1: 0 1 0 1 0 0
R2: 0 0 2 3 1 0
R3: 0 3 5 17 1 0
R4: 0 2 3 27 4 0
R5: 1 0 1 35 0 1
R6: 0 0 1 1 0 0
River (FA≥4): C3(5) D3(17) D4(27) E4(4) D5(35)
Catchment = 36 cells × 30×30
= 32,400 m²
Q8. 2074 Bhadra Q2b — 100m, Threshold 10 NEW [12]
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | 480 | 500 | 490 | 480 | 485 |
| 2 | 505 | 495 | 490 | 475 | 480 |
| 3 | 500 | 485 | 455 | 470 | 475 |
| 4 | 495 | 490 | 480 | 465 | 470 |
| 5 | 490 | 485 | 475 | 455 | 450 |
| 6 | 500 | 490 | 480 | 460 | 445 |
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Sink: C3 = 455 min neighbor = 465 Fill = 465.5
// Flow Accumulation
// A B C D E
R1: 2 0 0 0 0
R2: 0 0 0 3 0
R3: 0 2 10 1 0
R4: 0 0 0 14 0
R5: 0 2 4 6 16
R6: 0 0 0 1 26
River (FA≥10): C3(10) D4(14) E5(16) E6(26)
Catchment at E6 (445) = 27 × 10,000 = 270,000 m² = 27 ha
Q9. 2073 Magh Q7 — 30m, Pour Point (4,4) NEW [10]
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | 118 | 112 | 109 | 111 | 115 |
| 2 | 114 | 107 | 96 | 89 | 92 |
| 3 | 109 | 40 | 84 | 77 | 79 |
| 4 | 104 | 99 | 95 | 62 | 65 |
| 5 | 68 | 68 | 67 | 61 | 64 |
| 6 | 62 | 60 | 58 | 56 | 50 |
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Sink: B3 = 40 min neighbor = 84 Fill = 84.5
// Flow Accumulation
// A B C D E
R1: 0 0 0 0 0
R2: 0 1 1 2 1
R3: 0 4 5 5 2
R4: 0 0 0 17 3
R5: 1 1 0 18 0
R6: 2 5 7 8 29
River (FA≥4): B3(4) C3(5) D3(5) D4(17) D5(18)
B6(5) C6(7) D6(8) E6(29)
Watershed = 30 cells × 900 = 27,000 m² = 2.7 ha
Q11. 2072 Ashwin Q5b NEW [8]
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | 100 | 98 | 97 | 98 | 99 |
| 2 | 105 | 103 | 102 | 99 | 100 |
| 3 | 107 | 105 | 95 | 100 | 101 |
| 4 | 109 | 106 | 104 | 102 | 104 |
| 5 | 115 | 112 | 110 | 109 | 108 |
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Sink: C3 = 95 min neighbor = 99 Fill = 99.5
// Flow Accumulation
// A B C D E
R1: 1 4 24 2 0
R2: 0 1 0 14 0
R3: 0 0 7 3 1
R4: 0 3 1 2 0
R5: 0 0 0 0 0
River (FA≥5): C1(24) D2(14) C3(7)
Watershed = 25 cells × 10,000 = 250,000 m²
Q12. 2071 Magh Q4a — 4×4, Threshold 9 NEW [2+2+2]
| A | B | C | D | |
|---|---|---|---|---|
| 1 | 156 | 144 | 138 | 142 |
| 2 | 148 | 134 | 112 | 98 |
| 3 | 138 | 106 | 88 | 74 |
| 4 | 128 | 116 | 110 | 44 |
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Sinks: None // All water flows to D4=44 (corner)
// Flow Direction
R1: SE SE SE S
R2: SE SE SE S
R3: E E SE S
R4: NE NE E OUT
// Flow Accumulation
R1: 0 0 0 0
R2: 0 1 1 2
R3: 0 3 7 5
R4: 0 0 0 15
River (FA≥9): D4(FA=15) Area = 16 × 10,000 = 160,000 m²
Q13. 2071 Bhadra Q7 — 5×5, Threshold 5 NEW [12]
| A | B | C | D | E | |
|---|---|---|---|---|---|
| 1 | 93 | 90 | 89 | 90 | 89 |
| 2 | 91 | 88 | 82 | 79 | 78 |
| 3 | 89 | 71 | 72 | 73 | 74 |
| 4 | 86 | 83 | 82 | 70 | 69 |
| 5 | 86 | 84 | 82 | 75 | 68 |
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Sink: B3 = 71 min neighbor = 72 Fill = 72.5
// Flow Accumulation
// A B C D E
R1: 0 0 0 0 0
R2: 0 1 1 2 1
R3: 0 7 10 3 2
R4: 0 1 0 18 3
R5: 0 0 1 0 24
River (FA≥5): B3(7) C3(10) D4(18) E5(24)
Watershed = 25 cells × 10,000 = 250,000 m²
Q14. 2081 Chaitra — LiDAR 10m, 1000 m² NEW
| A | B | C | D | E | F | |
|---|---|---|---|---|---|---|
| 1 | 125 | 123 | 121 | 120 | 118 | 117 |
| 2 | 124 | 118 | 119 | 117 | 116 | 118 |
| 3 | 123 | 120 | 118 | 116 | 114 | 115 |
| 4 | 122 | 118 | 116 | 115 | 113 | 114 |
| 5 | 121 | 118 | 114 | 112 | 112 | 114 |
| 6 | 120 | 117 | 115 | 113 | 109 | 112 |
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Sinks: None // Water flows toward E6=109
// Flow Accumulation
// A B C D E F
R1: 0 0 0 0 0 0
R2: 0 4 0 2 2 0
R3: 0 0 0 1 6 1
R4: 0 1 1 1 11 0
R5: 0 1 6 12 13 0
R6: 0 1 0 0 30 0
Threshold = 1000 m² / (10×10) = 10 cells
River: E4(11) D5(12) E5(13) E6(30)
Watershed = 31 cells × 100 = 3,100 m²