PURE MATHEMATICS, PAPER-I CSS Combined Competitive Exam Paper 2009

PURE MATHEMATICS, PAPER-I

FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION FOR

RECRUITMENT TO POSTS IN BPS-17 UNDER

THE FEDERAL GOVERNMENT, 2009

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TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

NOTE:

(i) Attempt FIVE questions in all by selecting at least THREE questions from

SECTION–A and TWO question from SECTION–B. All questions carry EQUAL

marks.

(ii) Use of Scientific Calculator is allowed.

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SECTION – A

Q.1. (a) Prove that the set Sn of all permutations on a set X of n elements is a group under the operation

‘o’ of composition of permutations. Will (Sn, o) be an abelian group? How do we call this

group? (10)

(b) If G is a group, N a normal subgroup of G, then show that the set G/N of right cosets of N in G

is also a group. How we call this group? Also, if G is finite then show that

( ) ( )

( ).

o N

o G

N

o G = (10)

Q.2. (a) Let cp be a homomorphism of a group G onto another group H with kernel K. Prove that G/K is

isomorphic to H, that is G/K ≈ H. (10)

(b) Let Zn be the set of the congruence classes modulo n, that is,

Zn = {[0], [1], [2], ……….. [n–1]}

Define the two binary operations on Zn under which it is a ring. Prove that the ring Zn is an

integral domain ⇔n is a prime number. (10)

Q.3. (a) Let T : R3 → R3 be the linear mapping defined by:

T(x,y,z) = (x+2y – z, y + z, x+2y – z)

Verify that

Rank (T) + Nullity (T) = dim D(T)

Also find a basis for each Rank (T) and Nullity (T) (10)

(b) If U and W are finite – dimensional subspaces of a vector space V over a field F then prove that

dim(U+W) + dim(U∩W) = dim U + dim W (10)

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Q.4. (a) Let H and K be two subgroups of a group G. Prove that HK is a subgroup of G ⇔ HK = KH.

(10)

(b) Let v1, v2, …….., vn be non-zero eigen vectors of an operator T:V→V belonging to distinct

eigen values λ1, λ2, …….., λn. Show that the vectors v1, v2, ………, vn are linearly independent.

(10)

S.No.

R.No.

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Q.5. (a) Let V be the vector space of n-square matrices over the field IR. Let U and W be the subspaces

of symmetric and antisymmetric matrices, respectively. Show that V = U ⊕ W. (10)

(b) Diagonalize the following matrix:

⎥ ⎥ ⎥

⎦

⎤

⎢ ⎢ ⎢

⎣

⎡− − −

=

6 4 10

4 6 4

4 4 8

M

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SECTION – B

Q.6. (a) Find the lengths of the following curves: (10)

(i) 9y2 = 4x3 from x = 3 to x = 8

(ii) = θ = θ = Δ

θ

to

2

r Sin2 from o

(b) Find the radius of curvature of the given curve at the designated point. (10)

a x (x y)

a x a

y a ln a x a ; ,

2

2 2

2 2

2 2

− +

+ −

+ +

=

Q.7. (a) Show that the two lines

L1 : x = 4 – t, y = –2 +2t, z = 7 – 3t

L2 : x = x = 3 + 2s, y = – 7 – 3s, z = 6 + 4s

are skew. Also find the points on the lines such that the segment joining these points is

perpendicular to both lines and hence find the shortest distance between the given lines. http://www.allvupastpapers.blogspot.com/  

 (10)

(b) Find the equation of the sphere through the circle x2 + y2 + z2 = 1, 2x + 4y + 5z = 6 and touching

the plane z = 0. (10)

Q.8. (a) At a point on a curve r = r(t) at which k ≠ o, show that

[ ]

2 r r

r r r

′× ′′

′ ′′ ′′′

τ =

dt

where r′ = dr (10)

(b) Find the First Fundamental Form and fundamental magnitudes of first order for the sphere

r = (a cos u. cos v, a cos u. Sinv, a Sinu)

Also prove that parametric curves are orthogonal. (10)

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