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1. What are the salient features of the 73rd Constitutional Amendment Act in India? (4)
2. Explain the Three-Tier Governance of Panchayati Raj System in India and explain the functions of each layer of government. (4)
3. What is meant by fiscal decentralisation? Explain the major roles of State Finance Commission (SFC). (4)
4. What is Grama Sabha? Explain the composition and functions of Grama Sabha. (4)
5. Describe the sources of financing the plans at Grama Panchayat level. (4)
6. What is ‘social audit’? Explain the objectives and significance of social auditing. (5)
7. If X ~ Nₚ(0, I), find the characteristic function of Y = X′AX, where A is any real matrix of order p × p. (5)
8. If state k is persistent null, then for every j, limₙ→∞ pⱼₖ⁽ⁿ⁾ → 0; and if state k is aperiodic, persistent non-null, then prove that limₙ→∞ pⱼₖ⁽ⁿ⁾ → Fⱼₖ/μₖₖ. (5)
9. Let X₁, X₂, …, Xₙ be a random sample of size n taken from a Poisson distribution with parameter λ. Find the UMVUE of e⁻λ. (5)
10. A random sample of size n is taken from a Poisson distribution with parameter λ. Find the most powerful critical region of size α for testing H₀: λ = λ₀ against H₁: λ = λ₁, where λ₁ > λ₀. (5)
11. Show that in simple random sampling without replacement "The probability of selecting a specified unit of the population at any given draw is equal to the probability of it being selected at the first draw" (5)
12. “Planning premises are the anticipated environment in which plans are expected to operate”. Explain this statement by clearly defining the purpose of establishing planning premises. Also list the various external premises in this context. (5)
13. "Planned change requires change agents" discuss. Also describe the roles of change agents in implementing organizational change. (5)
14. "Planning is the basis and action is the essence of Controlling". Discuss. (5)
15. Explain the role of Prof. P. C. Mahalanobis as the architect of Indian Economic Planning (5)
16. Describe the role of NITI Aayog in economic planning. (5)
17. Find a basis of ℝ³ with respect to which the matrix of the linear map T: ℝ³ → ℝ³ defined by T(x, y, z) = (5x − 6y − 6z, −x + 4y + 2z, 3x − 6y − 4z) is diagonal. (5)
18. Examine whether the series ∑ [x²ⁿ⁺¹/(2n + 1) − xⁿ⁺¹/(2n + 2)] is uniformly convergent on (0, 1). Justify your answer. (5)
19. A mass of 20 kg attached to a spring with spring constant 320 N m⁻¹ and brought to rest. Find the position of the mass at time t = π/2 if a force equal to 40 cos 2t is applied to it. (5)
20. Use dynamic programming to minimise z = y₁² + y₂² + y₃² subject to y₁ + y₂ + y₃ ≥ 15 and y₁, y₂, y₃ ≥ 0. (5)
21. Find the spectrum of A : ℓ² → ℓ² defined by A(x₁, x₂, x₃, …) = (x₁, x₂/2, x₃/3, …). (5)
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