Q1.
(a) Define cyclic group? Find all the subgroups of a cyclic group of order 12. (10)
(b) Find all the subgroup of S₃? (10)

Q2.
(a) Determine whether the following vectors linearly independent or not.
v₁(1,1,2) , v₂(1,2,5) and v₃(5,3,4). (10)
(b) Solve the system of equations. x + 2y + z = 2 , 3x + y – 2z = 1 , 4x – 3y – z = 3. (10)

Q3. (a) Find the rank of the matrix

 2   2  -1   0   1   0
-1  -1   2  -3   1   0
 1   1  -2   0  -1   0
 0   0   1   1   1   0

(10)

(b) Determine a basis for the null space of matrix

1   2  -3   2  -3
0  -1   1  -2   0
1   1  -1   1  -1
1   2   5  -6  -3

(10)

Q4. (a) Find the Eigen values and Eigen vectors of the matrix

1   0  -1
1   2   1
2   2   3

(10)

(b) Use Cayley-Hamilton Theorem to find A³ if A =

2   1   1
0   1   0
1   1   2

(10)

Q5.
(a) Find the equation of the plane passing through the points (2, -3,1) and containing the line x – 3 = 2y = 3z – 1 (10)
(b) Find the equation of the sphere if the centre is on the line x = y = z and it passes through the points (5,3,0) and (-1,4,1). (10)

Q6.
(a) Find the equation of the tangent plane to the surface z = x² + y² at the point (2,1,5). Find also the equation of normal line at that point. (10)
(b) Transform the equation of the curve ρ = 3 cos θ sin φ into the spherical coordinates and rectangular coordinates. (10)

Q7.
(a) Determine the length of the cycloid x = a(θ + sin θ) , y = a(1 – cos θ) between the cusps. (10)
(b) Determine the curvature at the point (³⁄₂, ³⁄₂) on the folium x³ + y³ = 3xy. (10)