NWFP PUBLIC SERVICE COMMISSION
Competitive Examination
For The Provincial Management Service (BPS-17)
PURE MATHEMATICS – PAPER-I
- Attempt Any FIVE questions in all, selecting at least TWO questions from each SECTION.
- Extra attempt of question or part will not be considered.
- Candidate must draw two straight lines (=================) at the end of the answer to separate each question attempted in the Answer Book.
Q.1. (a) Define integral domain and prove that every finite integral domain is a field.
(b) If L is a finite extension of K and M is a finite extension of L, then M is a finite extension of K, with [M : K] = [M : L][L : K].
Q.2. (a) The Housing Department of the NWFP Government plans to undertake four housing projects and lists material requirements for each house in each of the project as follows:
| Project 1 | Project 2 | Project 3 | Project 4 | |
|---|---|---|---|---|
| Paint (in 100 gallons) | 1 | 2 | 1 | 1.5 |
| Wood (in 10,000 cubic ft) | 3 | 4 | 2.5 | 2.5 |
| Bricks (in millions) | 1 | 2 | 1.5 | 1 |
| Labour (in 1000 hrs) | 10 | 10 | 9 | 8 |
If the supplier delivers 6,850 gallons paint, 1,420,000 cubic ft of wood, 64 millions bricks and 4,44,000 hours of labour, find the number of houses built for each project.
(b) State Cayley – Hamilton Theorem and verify the Theorem for the matrix M, if
[
| 2 | -1 | 1 |
| -1 | 2 | -1 |
| 1 | -1 | 2 |
]
Also find inverse of the matrix M if exist.
Q.3. (a) Define Cyclic Group.
Let G be a group and let a ∈ G, then H = {an | n ∈ ℤ} is a subgroup of G.
(b) Prove that if φ : G → G’ is an isomorphism of G with G’ and e is the identity of G, then φ(e) is the identity in G’
and also prove that φ(a-1) = [φ(a)]-1 for all a ∈ G.
Q.4. (a) Let W be the subset of ℝ3 defined by
[
| x1 |
| x2 |
| x3 |
]
| x1 = 2x2, x3 = -2x2, x2 any real number }
verify that W is a subspace of ℝ3.
(b) Let V be the vector space of (2 × 2) matrices, and let W be the subspace
[
| 0 | a12 |
| a21 | 0 |
]
, a12 and a21 are real scalars }
Define matrices B1, B2 and B3 in W by
[
| 0 | 2 |
| 1 | 0 |
],
B2 =
[
| 0 | 1 |
| 2 | 0 |
] and
B3 =
[
| 0 | 5 |
| 1 | 0 |
]
Show that the set {B1, B2, B3} is linearly dependent and express B3 as a linear combination of B1 and B2. Also show that {B1, B2} is linearly independent set.
Q.5. (a) Find the vector equation of the plane containing the points A(0, 1, 1), B(2, 1, 0), C(-2, 0, 1):
(i) in parametric form
(ii) in scalar product form
(b) Find the equation of the tangent line to the curve x = 2t + 4, y = 3t2 – 2t + 4 at t = 1 without eliminating the parameter.
Q.6. (a) Find the equation of the surface x2 + y2 + z2 = 2 in:
(i) Cylindrical co-ordinate (ii) Spherical co-ordinate
(b) Find the equation of the plane whose points are equidistant from (2, -1, 1) and (3, 1, 3).
Q.7. (a) Find the total arc length of the cardioid r = 1 + cos θ.
(b) Find an equation of the sphere with centre at (1, 1, -3) that is tangent to the plane x – 2y + 2z = 4.
Q.8. (a) Convert from Rectangular to Cylindrical and Spherical coordinates (-3√3, 3, 0).
(b) Find the distance from the point P(3, 0, -2) to the line L : x = 2 + t, y = -1 – t, z = 3t.