UPSC Previous Year Question Paper 2026-Combined Geo-Scientist (Main) Examination,-GEO-PHYSICS

UPSC Previous year question paper Geo scientist exam

1. (a) Calculate the present- day relative motion at location 22∘S22∘S and 25∘E25∘E in the Botswana using India- Eurasia rotation pole at 24∘N24∘N and 18∘E18∘E . Assume the radius of the Earth to be 6371km6371km and angular velocity is 7.2×10−77.2×10−7 degree/year. 10

2. (b)(i) An earthquake is generated by a rupture along a fault having a length of 100km100km width of 20km20km and an average displacement of 5m5m . Assuming the average modulus of rigidity of the earth's crust to be 30GPa30GPa , calculate the magnetic moment (Mw)(Mw​) of the earthquake. 5

3. (b)(ii) A continental crust is modeled as a homogeneous, isotropic elastic medium with Young's modulus E=75GPaE=75GPa , Poisson's ratio σ=0.26σ=0.26 and density ρ=2.8g/ccρ=2.8g/cc . Determine the shear and bulk moduli and calculate the P−P− and S−S− wave velocities in the medium. 5

4. (c) For a given vector field

F⃗=(x2yz)i^+(xy2z)j^+(xyz2)k^,F=(x2yz)i^+(xy2z)j^​+(xyz2)k^,

(i) Find the divergence of F⃗F

(ii) Evaluate ∇⋅F⃗∇⋅F at point (1, 1, 1)

(iii) Find the curl of F⃗F and comment whether the field is irrotational.

4+2+4=104+2+4=10

1. (d) Differentiate between Forward and Inverse problems. Explain direct and model based inversion techniques with suitable geophysical examples. 10

2. (a) Derive expression for the Geoid height anomaly over a mountain range for Airy compensation model.

Assuming a constant continental crustal thickness of 35km35km , calculate the geoid height anomaly due to a mountain of height 4km4km for Airy- type compensation.

Given : Gravitational constant, G=6.67×10−11m3kg−1⋅s−2G=6.67×10−11m3kg−1⋅s−2

Acceleration due to gravity, g=9.8ms−2g=9.8ms−2

Radius of the Earth, R=6371kmR=6371km

Crustal density =2.8g/cc=2.8g/cc

Mantle density =3.3g/cc=3.3g/cc

1. (b) Explain the concept of seafloor spreading and critically discuss the geophysical evidences supporting it. 10

2. (c) A tectonic region follows the Gutenberg- Richter frequency- magnitude relationship with a b- value of 0-9.

(i) Estimate the ratio of annual occurrence rates of earthquakes with magnitudes M≥5.5M≥5.5 and M≥7.0M≥7.0

(ii) Calculate the total annual seismic energy released by earthquakes with magnitudes between 5-5 and 7-0. 5+5=10

1. (a) Assume that the Earth is spherically symmetric and composed of concentric homogeneous shells with no chemical and phase changes within shells. Pressure and density are assumed to increase purely hydrostatistically. Derive the Adams- Williamson equation for the radial variation of the density in the Earth and discuss it with suitable figure. 10

2. (b) A seismic refraction survey is conducted over a homogeneous horizontally layered Earth. In one model, the crust is 35km35km thick, with P- wave velocity of 5.8km/s5.8km/s , overlying an upper mantle with P- wave velocity of 8.0km/s8.0km/s . In a second model, the crust consists of an upper layer 25km25km thick with a P- wave velocity of 5.8km/s5.8km/s , underlain by a 10km10km thick low- velocity layer with a P- wave velocity of 4.5km/s4.5km/s ; the mantle velocity remains 8.0km/s8.0km/s .

Plot the first- arrival travel- time curves for both models on a single distance - time diagram and discuss the effect of the low- velocity layer on the estimation of the depth to the top of the mantle. 10

1. (c) The data obtained from a velocity reflection survey are t0,1=6.0st0,1​=6.0s , t0,2=15.0st0,2​=15.0s , α~1=6.0km/sα~1​=6.0km/s and α~2=7.0km/sα~2​=7.0km/s where α~1α~1​ and α~2α~2​ represent the time- weighted root- mean- square velocity at first and second interfaces, respectively and t0,1t0,1​ and t0,2t0,2​ are the normal incidence two- way travel time, respectively. Determine the velocity- depth model for the survey, assuming that each subsurface layer has a constant seismic velocity. 10

2. (a) During a geophysical survey, charge accumulation is observed on two underground mineral bodies due to electrostatic induction caused by tectonic stress. The two bodies can be approximated as point charges.

Charge on Body AA q1=+5×10−6Cq1​=+5×10−6C Body BB q2=−8×10−6Cq2​=−8×10−6C

Distance (r) between AA and B=0.40mB=0.40m Relative permittivity of surrounding rock medium ϵr=4ϵr​=4

(i) Calculate the electrostatic force between the two bodies.

(ii) State whether the force is attractive or repulsive.

(iii) Briefly comment on the geophysical significance of such electrostatic forces in subsurface exploration.

Take:ϵ0=8.854×10−12F/m.(5+2+3=10)Take:ϵ0​=8.854×10−12F/m.(5+2+3=10)

1. (b) (A) Describe the geomagnetic field of earth and its components. Establish the relationship between the various components.

(B) At a place on Earth's surface, the angle of dip is 30∘30∘ and total magnetic field is B=5×10−5TB=5×10−5T

Calculate:

(i) the horizontal component H

(ii) the vertical component V

(C) Also, describe how it protects the Earth from solar and cosmic radiation. 4+4+2=104+4+2=10

1. (c) Define Model Resolution matrix. Explain its importance and compute the model Resolution matrix for the following Kernel matrix:

G=[1011](10)G=[11​01​](10)

1. (a) For a given charge and current distribution, the scalar potential VV and the vector potential AA related to the electric field vector EE and magnetic field vector BB by the equations :

B=∇×AB=∇×A

Rewrite Maxwell's equations in terms of these potentials for linear, homogenous media. 10

1. (b) Let A=[3113]A=[31​13​]

Find singular value decomposition (SVD) of matrix AA i.e. A=UΣVTA=UΣVT where UU and VV are orthogonal matrix, ΣΣ is a diagonal matrix of singular values. Also using SVD, determine the rank and 2- norm of AA 10

1. (c) In geophysics, inversion problem often involve finding a model mm (e.g. subsurface velocity, density) that minimizes the misfit function

ϕ(m)=∥dobs−G(m)∥2ϕ(m)=∥dobs​−G(m)∥2

where dobsdobs​ is observed data

G(m)G(m) is forward operator

(A) Explain why Genetic Algorithms (GA) are particularly suitable for geophysical inverse problems highlighting advantages over gradient based methods.

(B) Discuss the theoretical limitations of GA or Simulated Annealing (SA) when applied to geophysical inverse problems, particularly regarding ill-posedness and non-uniqueness. 5+5=10

UPSC Previous year question paper Geo scientist exam

SECTION 'B'

1. (a) Prove that for a laminar flow of fluid or field, the equation of continuity can be written as divV⃗=0divV=0 . 10

2. (b) Explain Ampere's Circuital law in magnetostatics. Elaborate the steps adopted by Maxwell in modifying this equation for time dependent fields. 10

3. (c) With schematic diagrams please explain how transmission lines are used for

(i) Load matching (ii) Impedance measurements

1. (d) (i) Explain different system segments of Global Positioning System (GPS). 5 (ii) Calculate range error in GPS L1L1​ signal due to presence of ionosphere having total electron contents (TEC) of 15 TECU. (Here frequency of L1L1​ signal, f=1575.42MHzf=1575.42MHz ) (1 TECU = 1×10161×1016 ele/m 33 ) 5

2. (a) Assuming that there is no singularity in finite plane of differential equation

y′′−2xy′+2λy=0whereλ=constanty′′−2xy′+2λy=0whereλ=constant

prove that its solution is Hermite's polynomial of order nn . 10

1. (b) What is Green's function? Describe its importance alongwith its properties. 10

2. (c)(i) Explain how a definite Eulerian integral is another form of Beta function. 4

3. (c)(ii) Prove that Beta and Gamma functions are related to each other as below:

β(m,n)=Γ(n)Γ(m)Γ(m+n)(6)β(m,n)=Γ(m+n)Γ(n)Γ(m)​(6)

1. (a) Using Gauss's law, obtain Poisson's equation for a scalar field. Under what conditions does Poisson's equation become identical to Laplace equation? 10

2. (b) Given that D⃗=zρcos⁡2ϕa^z c/m2D=zρcos2ϕa^z​ c/m2 , calculate the charge density at (1,π/4,3)(1,π/4,3) and the total charge enclosed by the cylinder of radius 1 m1 m with −2≤z≤2 m−2≤z≤2 m . 10

3. (c) (i) Explain briefly the different types of magnetohydrodynamic (MHD) waves. 5

(ii) Show that the velocity of magnetosonic wave, VMVM​ is related with the Alfven velocity, VAVA​ and sound velocity, VSVS​ by the relation: VM2=VS2+VA2VM2​=VS2​+VA2​ . 5

1. (a) Using the plane wave propagation in conducting medium prove that the multi-strand wires are better conductor of AC current than a single core wire of same area of cross-section. 10

2. (b) A rectangular waveguide of dimentions a=2cm,b=1cma=2cm,b=1cm operates below 14 GHz. How many TE and TM modes can the waveguide use if it is filled with a medium of σ=0,ϵ=4ϵ0,μr=1σ=0,ϵ=4ϵ0​,μr​=1 . Also calculate the cutoff frequencies of modes. 10

3. (c) Using Maxwell's equations for reflection of plane wave at an oblique angle prove that there are two types of polarizations taking place (i) Parallel polarization (ii) Perpendicular polarization and the equations for them is Fresnel equation. 10

4. (a) Derive the Hydrostatic equation for the atmosphere in a hydrostatic balance:

∂p∂z=−gρ∂z∂p​=−gρ

Where ρρ is the density of air, pp is pressure of the air and gg is the gravitational acceleration. 10

1. (b) Describe basic chemical processes for formation of ionosphere. Explain with proper reactions, why NO+NO+ is a dominant ion in E- region and O+O+ is in F- region? 10

2. (c) Explain different types of ionospheric irregularities and differentiate between Sporadic- E and Spread- F irregularities. 10

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UPSC Previous year question paper Geo scientist exam

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