Nbhm m. a. and m.sc. scholarship test 2008

NATIONAL BOARD FOR HIGHER MATHEMATICS
M. A. and M.Sc. Scholarship Test
September 20, 2008
Time Allowed: 150 Minutes
Maximum Marks: 30
Please read, carefully, the instructions on the following page
1

INSTRUCTIONS TO CANDIDATES
• Please ensure that this question paper booklet contains 8 numbered
(and printed) pages. The reverse of each printed page is blank and can
be used for rough work.
• There are three parts to this test: Algebra, Analysis and Geometry.
Each part consists of 10 questions adding up to 30 questions in all.
• Answer each question, as directed, in the space provided for it in the
answer booklet, which is being supplied separately. This question
paper is meant to be retained by you and so do not answer questions
on it.
• In certain questions you are required to pick out the qualifying state-
ment(s) from multiple choices. None of the statements, or more than
one statement may qualify. Write none if none of the statements qual-
ify, or list the labels of all the qualifying statements (amongst (a),(b)
and (c)).
• Points will be awarded in the above questions only if all the correct
choices are made. There will be no partial credit.
• N denotes the set of natural numbers, Z - the integers, Q - the rationals,
R - the reals and C - the ﬁeld of complex numbers. Rn
denotes the n-
dimensional Euclidean space. The symbol ]a, b[ will stand for the open
interval {x ∈ R | a < x < b} while [a, b] will stand for the corresponding
closed interval; [a, b[ and ]a, b] will stand for the corresponding left-
closed-right-open and left-open-right-closed intervals respectively. The
symbol I will denote the identity matrix of appropriate order. All
logarithms, unless speciﬁed otherwise, are to the base e.
• Calculators are not allowed.
2

Section 1: Algebra
1.1 Let α, β and γ be the roots of the polynomial
x3
+ 2x2
− 3x − 1.
Compute:
1
αβ
+
1
βγ
+
1
γα
.
1.2 Let G be a cyclic group of order 8. How many of the elements of G are
generators of this group?
1.3 Which of the following statements are true?
(a) Any group of order 15 is abelian.
(b) Any group of order 25 is abelian.
(c) Any group of order 55 is abelian.
1.4 A real number is said to be algebraic if it is the root of a non-zero poly-
nomial with integer coefficients. Which of the following real numbers are
algebraic?
(a) cos 2π
5
(b) e
1
2
log 2
(c) 5
1
7 + 7
1
5
1.5 Let Z +
√
3Z denote the ring of numbers of the form a + b
√
3, where a
and b ∈ Z. Find the condition that a + b
√
3 is a unit in this ring.
1.6 Let Fp denote the field Z/pZ, where p is a prime. Let Fp[x] be the asso-
ciated polynomial ring. Which of the following quotient rings are fields?
(a) F5[x]/{x2
+ x + 1}
(b) F2[x]/{x3
+ x + 1}
(c) F3[x]/{x3
+ x + 1}
1.7 Let G denote the group of invertible 2 × 2 matrices with entries from F2
(the group operation being matrix multiplication). What is the order of G?
1.8 Let A be a 3 × 3 upper triangular matrix with real entries. If a11 =
1, a22 = 2 and a33 = 3, determine α, β and γ such that
A−1
= αA2
+ βA + γI.
3

1.9 Let V be a vector space such that dim(V ) = 5. Let W and Z be sub-
spaces of V such that dim(W) = 3 and dim(Z) = 4. Write down all possible
values of dim(W ∩ Z).
1.10 Let T : R2
→ R2
be the linear map which maps each point in R2
to its
reﬂection on the x-axis. What is the determinant of T? What is its trace?
4

Section 2: Analysis
2.1 Evaluate:
lim
x→0
(1 − sin x cos x)cosec2x
.
2.2 Evaluate:
lim
n→∞
1
n6
n
k=1
k5
.
2.3 Determine if each of the following series is absolutely convergent, condi-
tionally convergent or divergent:
(a)
∞
n=1
(−1)n cos nx
n2
, x ∈ R.
(b)
∞
n=1
(−1)n n
n + 2
.
(c)
∞
n=1
(−1)n−1 1
2n + 3
.
2.4 Let f : R2
→ R be a mapping such that f(0, 0) = 0. Determine which
of the following are jointly continuous at (0, 0):
(a)
f(x, y) =
x2
y2
x2 + y2
, (x, y) = (0, 0).
(b)
f(x, y) =
xy
x2 + y2
, (x, y) = (0, 0).
(c)
f(x, y) =
x sin 1
y
+ y sin 1
x
if xy = 0
0 otherwise.
2.5 Which of the following functions are uniformly continuous?
(a) f(x) = sin2
x, x ∈ R.
(b) f(x) = x sin 1
x
, x ∈]0, 1[.
(c) f(x) = x2
, x ∈ R.
5

2.6 Which of the following maps are differentiable everywhere?
(a) f(x) = |x|3
x, x ∈ R.
(b) f : R → R such that |f(x) − f(y)| ≤ |x − y|
√
2
for all x and y ∈ R.
(c) f(x) = x3
sin 1√
|x|
when x = 0 and f(0) = 0.
2.7 Pick out the true statements:
(a) If the series n an and n bn are convergent, then n anbn is also con-
vergent.
(b) If the series n an is convergent and if n bn is absolutely convergent,
then n anbn is absolutely convergent.
(c) If the series n an is convergent, an ≥ 0 for all n, and if the sequence
{bn} is bounded, then n anbn is absolutely convergent.
2.8 Write down an equation of degree four satisfied by all the complex fifth
roots of unity.
2.9 Evaluate:
2 sin
π
2
+ i .
2.10 Let Γ be a simple closed contour in the complex plane described in the
positive sense. Evaluate
Γ
z3
+ 2z
(z − z0)3
dz
when
(a) z0 lies inside Γ, and
(b) z0 lies outside Γ.
6

Section 3: Geometry
3.1 What is the locus of a point which moves in the plane such that the
product of the squares of its distances from the coordinate axes is a positive
constant?
3.2 Let
x(t) =
1 − t2
1 + t2
and y(t) =
2t
1 + t2
.
What curve does this represent as t varies over [−1, 1]?
3.3 Consider the line 2x − 3y + 1 = 0 and the point P = (1, 2). Pick out the
points that lie on the same side of this line as P.
(a) (−1, 0)
(b) (−2, 1)
(c) (0, 0)
3.4 Consider the points A = (0, 1) and B = (2, 2) in the plane. Find the
coordinates of the point P on the x-axis such that the segments AP and BP
make the same angle with the normal to the x-axis at P.
3.5 Let K = {(x, y) | |x| + |y| ≤ 1}. Let P = (−2, 2). Find the point in K
which is closest to P.
3.6 Let S be the sphere in R3
with centre at the origin and of radius R.
Write down the unit outward normal vector to S at a point (x1, x2, x3) on S.
3.7 Pick out the sets which are bounded:
(a) {(x, y) | x
2
3 + y
2
3 = 1}.
(b) {(x, y) | (x + y)(x − y) = 2}.
(c) {(x, y) |x + 2y ≥ 2, 2x + 5y ≤ 10, x ≥ 0, y ≥ 0}.
3.8 Find the length of the radius of the circle obtained by the intersection of
the sphere
x2
+ y2
+ z2
− 2x − 4y − 6z − 2 = 0
and the plane x + 2y + 2z − 20 = 0.
7

3.9 Let λ1 and λ2 be the eigenvalues of the matrix
a h
h b
.
Assume that λ1 > λ2 > 0. Write down the lengths of the semi-axes of the
ellipse
ax2
+ 2hxy + by2
= 1
as functions of λ1 and λ2.
3.10 Let V be the number of vertices, E, the number of edges and F, the
number of faces of a polyhedron in R3
. Write down the values of V, E, F and
V − E + F for the following polyhedra:
(a) a tetrahedron.
(b) a pyramid on a square base.
(c) a prism with a triangular cross section.
8

Nbhm m. a. and m.sc. scholarship test 2008

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