UPSC Combined Geo-Scientist Main 2025 Geophysics Paper I PDF

Section A

Q1. Formulate a linear inverse problem to determine \rho_c (average density of core) and \rho_m (average density of mantle) from the two data points d₁ and d₂ as per the details given below:

(a) d₁: Mass of the Earth
d₂: Moment of inertia

(Assume the Earth as a spherical body)
Also write the expression for least square solution of the inverse problem.

(b) u) (ii) Differentiate between compressional waves and transverse waves. What is natural period of seismometer? At critical damping, the response of

(i) Explain divergence theorem.

(d) (ii) Solve the following using divergence theorem : If \vecF = xy\hati + y^2z\hatj + z²\hatk, evaluate ∬_S (\vecF \cdot \hatn) dS, where S is the unit cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1. 6 2

Section A

Q2. Explain, briefly, seismic body waves using suitable sketches.

(b) (i) Explain, briefly, seismic body waves using suitable sketches.

Section A

Q3. Differentiate between compressional waves and transverse waves.

(b) (ii) Differentiate between compressional waves and transverse waves.

Section A

Q4. What is natural period of seismometer? At critical damping, the response of seismometer to a periodic disturbing signal with frequency ω is given by
u = (Aω^2)/((ω_0² + ω)) cos(ω t - \Delta)Using this equation, define how long-period seismometer becomes displacement meter and short-period seismometer becomes accelerometer. Also give the frequency range of seismic signals recorded by displacement seismometer and accelerometer.

(c) What is natural period of seismometer? At critical damping, the response of seismometer to a periodic disturbing signal with frequency ω is given by
u = (Aω^2)/((ω_0² + ω)) cos(ω t - \Delta)
Using this equation, define how long-period seismometer becomes displacement meter and short-period seismometer becomes accelerometer. Also give the frequency range of seismic signals recorded by displacement seismometer and accelerometer.

Section A

Q5. Explain divergence theorem.

(d) (i) Explain divergence theorem.

Section A

Q6. Solve the following using divergence theorem : If \vecF = xy\hati + y^2z\hatj + z²\hatk, evaluate ∬_S (\vecF \cdot \hatn) dS, where S is the unit cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.

(d) (ii) Solve the following using divergence theorem :
If \vecF = xy\hati + y^2z\hatj + z²\hatk, evaluate ∬_S (\vecF \cdot \hatn) dS, where S is the unit cube defined by 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.