Nbhm m. a. and m.sc. scholarship test 2011

NATIONAL BOARD FOR HIGHER MATHEMATICS
M. A. and M.Sc. Scholarship Test
September 24, 2011
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 6 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 one or more
than one statement may qualify. Write none if none of the statements
qualify, 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 ratio-
nals, 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.
We denote by GLn(R) (respectively, GLn(C)) the group (under matrix
multiplication) of invertible n × n matrices with entries from R (re-
spectively, C) and by SLn(R) (respectively, SLn(C)), the subgroup of
matrices with determinant equal to unity. The trace of a square matrix
A will be denoted tr(A) and the determinant by det(A).
The derivative of a function f will be denoted by f .
All logarithms, unless speciﬁed otherwise, are to the base e.
• Calculators are not allowed.
2

Section 1: Algebra
1.1 Given that the sum of two of its roots is zero, solve the equation:
6x4
− 3x3
+ 8x2
− x + 2 = 0.
1.2 From the following subgroups of GL2(C), pick out those which are
abelian:
a. the subgroup of invertible upper triangular matrices;
b. the subgroup S defined by
S =
a b
−b a
; a, b ∈ R, and |a|2
+ |b|2
= 1 .
c. the subgroup U defined by
U =
a b
−b a
; a, b ∈ C, and |a|2
+ |b|2
= 1 .
1.3 Let
S3
= (x1, x2, x3, x4) ∈ R4
:
4
i=1
x2
i = 1 .
This can be identified with the set U of Question 1.2c above via the identi-
fication
a = x1 + ix2, b = x3 + ix4
and hence automatically acquires a group structure. Compute the inverse of
the element (1
2
, 1
2
, 1
2
, 1
2
) in this group.
1.4 Pick out the pairs which are conjugate to each other in the respective
groups:
a.
1 1
0 1
and
1 0
1 1
in GL2(R);
b.
1 1
0 1
and
1 0
1 1
in SL2(R);
c.
1 3
0 2
and
1 0
0 2
in GL2(R).
1.5 Let R be a (commutative) ring (with identity). Let I and J be ideals
in R. Pick out the true statements:
a. I ∪ J is an ideal in R;
b. I ∩ J is an ideal in R;
c.
I + J = {x + y : x ∈ I, y ∈ J }
is an ideal in R.
3

1.6 Pick out the rings which are integral domains:
a. R[x], the ring of all polynomials in one variable with real coeffcients;
b. C1
[0, 1], the ring of continuously differentiable real-valued functions on
the interval [0, 1] (with respect to pointwise addition and pointwise multipli-
cation);
c. Mn(R), the ring of all n × n matrices with real entries.
1.7 Let W ⊂ R4
be the subspace defined by
W = {x ∈ R4
: Ax = 0}
where
A =
2 1 2 3
1 1 3 0
.
Write down a basis for W.
1.8 let V be the space of all polynomials in one variable with real coefficients
and of degree less than, or equal to, 3. Define the linear transformation
T(α0 + α1x + α2x2
+ α3x3
) = α0 + α1(1 + x) + α2(1 + x)2
+ α3(1 + x)3
.
Write down the matrix of T with respect to the basis
{1, 1 + x, 1 + x2
, 1 + x3
}.
1.9 Let A be a 2 × 2 matrix with real entries which is not a diagonal matrix
and which satisfies A3
= I. Pick out the true statements:
a. tr(A) = −1;
b. A is diagonalizable over R;
c. λ = 1 is an eigenvalue of A.
1.10 Let A be a symmetric n × n matrix with real entries, which is positive
semi-definite, i.e. xT
Ax ≥ 0 for every (column) vector x, where xT
denotes
the (row) vector which is the transpose of x. Pick out the true statements:
a. the eigenvalues of A are all non-negative;
b. A is invertible;
c. the principal minor ∆k of A (i.e. the determinant of the k × k matrix
obtained from the first k rows and first k columns of A) is non-negative for
each 1 ≤ k ≤ n.
4

Section 2: Analysis
2.1 Evaluate:
lim
x→0
(1 + 3x2
)5 cot x+2 cosecx
x .
2.2 Test the following series for convergence:
a. ∞
n=1
(3
√
n3 + 1 − n);
b.
1
1.2.3
+
3
2.3.4
+
5
3.4.5
+
7
4.5.6
+ · · ·
2.3 Find the sum of the inﬁnite series:
1
6
+
5
6.12
+
5.8
6.12.18
+
5.8.11
6.12.18.24
+ · · ·
2.4 Let [x] denote the largest integer less than, or equal to, x ∈ R. Find the
points of discontinuity (if any) of the following functions:
a. f(x) = [x2
] sin πx, x > 0;
b. f(x) = [x] + (x − [x])[x]
, x ≥ 1/2.
2.5 Pick out the uniformly continuous functions:
a. f(x) = cos x cos π
x
, x ∈]0, 1[;
b. f(x) = sin x cos π
x
, x ∈]0, 1[;
c. f(x) = sin2
x, x ∈ [0, ∞[;
2.6 Evaluate: n
k=1
kekx
, x ∈ R{0}.
2.7 Let f : R → R be a function which is diﬀerentiable at x = a. Evaluate
the following:
a.
lim
x→a
an
f(x) − xn
f(a)
x − a
;
b.
lim
n→∞
n
k
j=1
f a +
j
n
− kf(a) .
2.8 Find the cube roots of −i.
2.9 Evaluate:
Γ
z + 2
z
dz
where Γ is the semi-circle z = 2eiθ
, θ varying from 0 to π.
2.10 Find the points z in the complex plane where f (z) exists and evaluate
it at those points:
a. f(z) = x2
+ iy2
;
b. f(z) = zIm(z), where Im(z) denotes the imaginary part of z.
5

Section 3: Geometry
3.1 Let BC be a fixed line segment of length d in the plane. Let A be a
point which moves such that sum of the lengths AB and AC is a constant
k. Find the maximum value of the area of the triangle ∆ABC.
3.2 Let A = (0, 1) and B = (2, 0) in the plane. Let O be the origin and
C = (2, 1). Let P move on the segment OB and let Q move on the segment
AC. Find the coordinates of P and Q for which the length of the path con-
sisting of the segments AP, PQ and QB is least.
3.3 A regular 2N-sided polygon of perimeter L has its vertices lying on a
circle. Find the radius of the circle and the area of the polygon.
3.4 Let BC be a fixed line segment of length d and let S be a fixed point
whose distance from the line BC is 2a. A point A moves such that the alti-
tude of the triangle ∆ABC from the vertex A is equal to the length of the
line segment AS. Find the minimum possible value of the area of the triangle
∆ABC.
3.5 Pick out the bounded sets:
a. S is the set of all points in the plane such that the product of its distances
from a fixed pair of orthogonal straight lines is a constant;
b. S = {(x, y) : 4x2
− 2xy + y2
= 1};
c. S = (x, y) ; x
2
3 + y
2
3 = 1 .
3.6 A circle in the plane R2
centred at C and of unit radius moves without
slipping on the positive x-axis with C moving in the upper half-plane. Write
down the parametric equations of the locus of the point P on the circle which
coincides with the origin at the initial position of the circle and the parameter
θ being the angle through which the radius CP has turned from the initial
vertical position.
3.7 What are the direction cosines of the line joining the point (1, −8, −2)
to the point (3, −5, 4) in R3
?
3.8 Find the equation of the plane passing through the point (1, −2, 1) and
which is perpendicular to the planes 3x+y+z −2 = 0 and x−2y+z +4 = 0.
3.9 Find the equation of the plane containing the line
x − 1
2
=
y + 1
−1
=
z − 3
4
and which is perpendicular to the plane x + 2y + z = 12.
3.10 A moving plane passes through a fixed point (a, b, c) (which is not the
origin) and meets the coordinate axes at the points A, B and C, all away
from the origin O. Find the locus of the centre of the sphere passing through
the points O, A, B and C.
6

Nbhm m. a. and m.sc. scholarship test 2011

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