(a) (i) Observations were made with a theodolite from the points A and B of baseline AB having horizontal length of 200 m to the two inaccessible points P and Q at the same side of the line AB.

The following observations are done :

∠PAB = 80°, ∠QBA

Question

(a) (i) Observations were made with a theodolite from the points A and B of baseline AB having horizontal length of 200 m to the two inaccessible points P and Q at the same side of the line AB.

The following observations are done :

∠PAB = 80°, ∠QBA = 90°, ∠QAB = 50°, ∠PBA = 45°

The angle of elevation of P from A = 20°

The angle of elevation of Q from A = 12°

Calculate the horizontal distance PQ and difference in elevation between P and Q.

10

(ii) A pair of overlapping vertical photographs was taken with an aerial camera of focal length 150 mm from an altitude of 3000 m above datum. The mean principal base measured is equal to 80 mm. In the common overlap, a flagpole with its base 100 m above datum is observed. What will be the height of the flagpole if the parallax difference between the top and bottom point of the flagpole is 4·6 mm? Also, find the scale of photograph at datum and the distance covered by the aircraft between two successive exposures.

10

(b) (i) Explain the required properties of a good ballast material for railway track. Explain with reasons which ballast material is best suitable for high-speed railway track.

10

(ii) Calculate the minimum depth of ballast required for a broad gauge railway track having sleeper density of M+6, length of rail of 13·0 m and width of sleeper of 25 cm.

5

(c) Briefly discuss the effect of the following on the properties of mortar used for construction work :

(i) Alkali water and seawater

(ii) Low temperature

(iii) Sand and water

15

UPSC Answer Check · Accepted Answer

This multi-part question requires solving numerical problems in (a)(i)-(ii), explaining concepts with calculations in (b)(i)-(ii), and discussing material properties in (c). Allocate approximately 35% time to part (a) for precise trigonometric calculations involving inaccessible point location and aerial photogrammetry; 30% to part (b) for ballast properties and depth calculation; and 35% to part (c) for comprehensive discussion on mortar deterioration factors. Begin each numerical part with clear diagram sketching, show all formulae with proper units, and conclude with practical implications for Indian railway/construction contexts.
- Part (a)(i): Apply sine rule to triangles PAB and QAB to find horizontal distances AP, BP, AQ, BQ; use these to determine PQ via coordinate geometry or cosine rule; calculate elevations using tangent of vertical angles and find elevation difference
- Part (a)(ii): Apply parallax-height relationship h = (H-h₁)×dp/(b+dp) for flagpole height; calculate photographic scale as f/H; determine air base and ground coverage using B = b×H/f
- Part (b)(i): List ballast properties (hardness, angularity, durability, drainage, elasticity); justify crushed stone as best for high-speed tracks citing Indian Railways specifications and reduced maintenance
- Part (b)(ii): Calculate sleeper density M+6 = 13+6 = 19 sleepers per rail length; apply ballast depth formula considering sleeper spacing and bearing area requirements for BG track
- Part (c): Discuss alkali/seawater effects (efflorescence, sulphate attack, corrosion); low temperature effects (retarded hydration, frost damage); sand/water effects (gradation, workability, strength, w/c ratio) with IS code references

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies and applies: sine rule for inaccessible point location in (a)(i), parallax equations for aerial photogrammetry in (a)(ii), ballast functions and high-speed requirements per IRS specifications in (b), and chemical/physical mechanisms of mortar deterioration in (c); cites relevant IS codes (IS 2911, IS 2386, IS 2250)	Identifies most concepts correctly but confuses formulae (e.g., parallax vs. relief displacement) or omits key mechanisms (e.g., sulphate attack in seawater); partial code references	Major conceptual errors: uses wrong triangle for sine rule, confuses air base with photo base, describes ballast as merely 'stones' without functional properties, or lists generic effects without mortar-specific mechanisms
Numerical accuracy	20%	10	All calculations precise to 2-3 decimal places: PQ ≈ 248.6 m, elevation difference ≈ 30.5 m in (a)(i); flagpole height ≈ 23.0 m, scale 1:20,000, ground coverage ≈ 1600 m in (a)(ii); ballast depth ≈ 0.30 m or 30 cm in (b)(ii); proper unit handling throughout	Correct method but arithmetic errors (±5-10% deviation); correct formulae but substitution errors; inconsistent decimal precision; partial credit for correct intermediate steps	Order-of-magnitude errors; wrong formulae entirely; missing units or dimensional inconsistency; no calculation shown for (b)(ii) despite being numerical
Diagram quality	15%	7.5	Clear labeled sketches for (a)(i): baseline AB with triangles PAB, QAB showing all given angles; vertical section showing elevation angles; for (a)(ii): stereo-pair geometry with nadir points, air base, and parallax measurement; neat freehand or constructed diagrams with all variables marked	Diagrams present but incomplete labeling; missing vertical section in (a)(i) or confused stereo geometry in (a)(ii); diagrams described in text without visual representation	No diagrams despite geometric complexity; or diagrams with wrong configuration (e.g., P and Q on opposite sides of AB); illegible sketches without labels
Step-by-step derivation	25%	12.5	Systematic derivation: states given data, draws inference, applies relevant formula with justification, substitutes values with units, simplifies stepwise, and boxes final answers; for (a)(i) shows coordinate calculation or cosine rule for PQ; for photogrammetry derives height from first principles	Steps present but skips key justifications; jumps from formula to answer without intermediate values; correct final answers but unclear derivation path; some parts solved mentally without record	No derivation shown—only final answers; or completely wrong sequence of steps; missing essential steps like calculating individual distances before finding PQ
Practical interpretation	20%	10	Connects calculations to field practice: discusses error propagation in inaccessible point surveying, flight planning implications in aerial survey, ballast selection for Dedicated Freight Corridors/High Speed Rail (Mumbai-Ahmedabad), and mortar durability in coastal/tropical Indian conditions; suggests quality control measures	Brief mention of practical relevance without elaboration; generic statements about 'accuracy' or 'durability' without specific Indian railway/construction context	Purely theoretical treatment; no interpretation of results; misses opportunity to discuss real-world implications despite question inviting applied engineering knowledge

Q3

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