Let S = {1, 2, 3, 4} and ∗ be the binary operation defined by a ∗ b = a. Compute the Cayley table for (S, ∗). Is ∗ commutative? Is ∗ associative? Justify your answers.

State Lagrange’s theorem. What are the possible orders of subgroups of a group of order 12?

Define a semigroup. Give an example of an infinite semigroup.

Define a subgroup of a group. Check whether 

is a subgroup of the the group of 2 × 3 matrices over ℂ under addition

Define an abelian group. Give an example of a non-abelian group. (You don’t need to prove that your example is a group. You have to only prove that it is non-abelian.)

Discuss the role of evaluation in RRR by highlighting its various types.

Describe the importance and principles of Disaster Resilient Reconstruction.

What do you understand by Livelihood Creation in Reconstruction, Rehabilitation and Recovery?

'Pre-disaster Planning is integral to Reconstruction, Rehabilitation and Recovery. Comment.

Examine the role of Non-governmental Organisations in Reconstruction, Rehabilitation and Recovery or RRR.

What are the Guiding Principles and Issues in Post-Disaster Recovery?

Write a note on the different types of Rehabilitation.

Define the concept of Reconstruction and bring out its various steps in Reconstruction Process

Explain the types and methods of damage assessment.

Disasters and Development are two sides of the same coin' Elucidate.

Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.

i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.

ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.

iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.

iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.

If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.

Find the orthogonal canonical reduction of the quadratic form-x2 + y2 + z² + 4xy + 4xz. Also, find its principal axes.

Let (x₁, x₂, x₃) and (y₁, y₂, y₃) represent the coordinates with respect to the bases B₁ = {(1,0,0), (1, 1, 0), (0, 0, 1)}, B₂ = {(1,0,0), (0, 1, 1), (0, 0, 1)}. If    find the representation of Q in terms of  (y₁, y₂, y₃).

Consider the linear operator T: C³ → C³, defined by

i) Compute T* and check whether Tis self-adjoint.

ii) Check whether T is unitary.

Let V be the vector space of all real valued functions that are twice differentiable in R and

S = {cos x, sin. x, x cos x, x sin x}.

Check that S is a linearly independent set over R. (Hint: Consider the equation

a ₀cos x + a_₁ sin x + a₂x cos x + a₃x sin x.

(Put x = 0, π, π/2,π/4, etc. and find a₁.)