DU MSC Maths Sample Question Paper

Can you give me sample question paper for MSC mathematics entrance examination organized by Delhi university ?

[Quick Sam]

Here I am giving you sample question paper for MSC mathematics entrance examination organized by Delhi university .


ct A = (0, 1]. Write two real numbers which arc limit points of A. Also, give a real number which is
mit point of A. Justify your answer for one of the three points .

/(.r) =I, for every x e R.
-• Sho\f, that the c ' . ' • I • .. \ .
has at least one solutiOn in the interval [1, 4]. • •
(Hint: One may usc f(x) = (x -4) log x )

5. Check the unifomt convergence of
.., XJ
I z 2
n=tl+n X
on {i) [0, 1] (ii) [ ~ , 1].
For x,y e R, define
d(.r,y) = lx-yj +jxz-Y21•

Check if dis a metric on R? Justify your answer •
Find the inten•al of the convergence of ... ...
(i) LXII (ii) L 2n X
11!.
I n=1 11=1
Section 2 - Algebra
~ Find nil the cigcnva
Also, find the ~igenvcctor corresponding to one of the eigenvalues
9. (n) Show that the linear congruence 12x = 8 (mod 15) has no solution.
(b) Usc Euler's theorem to show that 27~ = 1 (mo~ 8).

Show that/ is a homomorphism. Determine kerf nnd prorc thnt Z/ker/ is isomorphic to {I, -1). f• Determine the dimensions of the following ' 'ector spaces: _
(n) {(x,y, z) e R: x + 2y- 3z = 0}; (b) {(x,y, z) c: R: x-y + z = 0, 2r + ,:,z = 0}.
i:;, I;ind the matrix corresponding to the map rp: RJ--} R2 defined ns
rp(x1 ,x2,x3 ) ~ (2x1 -3.r2 + x3,x1 - x2 - 2x3 )
•
relative to the basis {(1,0,0),(0,1,0),(0,0,1)) of R3 nnd {(1,0),(0, 1)) of R1
•
.
~ Let R be n ring with unity. If
(a/J)2 =a2b2 " Va,b e R
then show that R is a commutath•e ring.
~• Let R be n ring with identity clement e and let S be n subring of R. Cnn S ha\'C an identity clement
different from e? Justify your answer.

DU msc mathematics paper




in a for more detail in atteched in pdf file....

here I am looking for DU Department of Mathematics Entrance Test Papers for this exam preparation so will you plz let me know from where I can collect its paper ??

[sumit]

As you want here I am giving bellow DU Department of Mathematics Entrance Test Papers , so on your demand I am providing same here :
(1) How many elements are there in Z[i]/h3 + ii?
A) infinite B) 3
C) 10 D) finite but not 3 or 10.
(2) Let P be the set of all n n complex Hermitian matrices. Then P is a vector
space over the filed of
A) C B) R but not C
C) both R and C D) C but not R.
(3) Which one of the following is true?
A) There are infinitely many one-one linear transformations from R
4
to R
3
B) The dimension of the vector space of all 3 3 skew-symmetric matrices over
the field of real numbers is 6
C) Let F be a field and A a fixed n n matri over F. If T : Mn(F) → Mn(F) is
a linear transformation such that T(B) = AB for every B ∈ Mn(F), then the
characteristic polynomial for A is the same as the characteristic polynomial
for T
D) A two-dimensional vector space over a field with 2 elements has exactly 3
different basis.
(4) Let V and W be vector spaces over a filed F. Let S : V → W and T : W → V be
linear transformations. Then which one of the following is true?
A) If ST is one-to-one, then S is one-to-one
B) If V = W and V is finite-dimensional such that T S = I, then T is invertible
C) If dim V = 2 and dim W = 3, then ST is invertible
D) If T S is onto, then S is onto.
(5) The order of the automorphism group of Kleins group is
A) 3 B) 4 C) 6 D) 24.
(6) Which one of the following group is cyclic?
A) The group of positive rational numbers under multiplication
M.A./M.Sc. Admission Entrance Test 2014 3
B) The dihedral group of order 30
C) Z3 ⊕ Z15
D) Automorphism group of Z10.
(7) Which one of the following is a field?
A) An infinite integral domain B) R[x]/hx
2 − 2i
C) Z3 ⊕ Z15 D) Q[x]/hx
2 − 2i.
(8) Which one of the following is true for the transformation T : P2(R) → P2(R)
defined by T(f) = f + f
0 + f
00?
A) T is one-to-one but not onto
B) T is onto but not one-to-one
C) T is invertible
D) the matrix of T with respect to the basis {1, 2} is upper triangular.
(9) In Z[x], the ideal of hxi is
A) maximal but not prime B) prime but not maximal
C) both prime and maximal D) neither prime not maximal.
(10) Which one of the following is true for the transformation T : Mn → C defined by
T(A) = tr A =
Pn
i=1 Aii?
A) Nullity of T is n
2 − 1 B) Rank of T is n
C) T is one-to-one D) T(AB) = T(A)T(B) for all A, B ∈ Mnn.
(11) Let W1 = {A ∈ Mn(C) : Aij = 0 ∀ i ≤ j} and W2 is the set of symmetric matrices
of order n. Then the dimension of W1 + W2 is
A) n B) 2n C) n
2 D) n
2 n.
(12) The logarithmic map from the multiplicative group of positive real numbers to the
additive group of real number is
A) a one-to-one but not an onto homomorphism
B) an onto but not a one-to-one homomorphism
C) not a homomorphism
D) an isomorphism.

DU Department of Mathematics Entrance Test Papers