Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key

NATIONAL BOARD FOR HIGHER MATHEMATICS
M. A. and M.Sc. Scholarship Test
September 20, 2014
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 7 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 Miscellaneous.
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.
• Notations
– N denotes the set of natural numbers {1, 2, 3, · · ·}, Z - the inte-
gers, 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.
– We denote by Mn(R) (respectively, Mn(C)), the set of all n × n
matrices with entries from R (respectively, C).
– 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 and the second
derivative by f .
– All logarithms, unless speciﬁed otherwise, are to the base e.
• Calculators are not allowed.
2

Section 1: Algebra
1.1 Find the sign of the permutation σ defined below:
σ =
1 2 3 4
3 1 4 2
.
1.2 Let G be an arbitrary group and let a and b be any two distinct elements
of G. Which of the following statements are true?
(a) If m is the order of a and if n is the order of b, then the order of ab is the
l.c.m. of m and n.
(b) The order of ab equals the order of ba.
(c) The elements ab and ba are conjugate to each other.
1.3 Let G be the group of invertible upper triangular matrices in M2(R). If
we write A ∈ G as
A =
a11 a12
0 a22
,
which of the following define a normal subgroup of G?
(a) H = {A ∈ G | a11 = 1}.
(b) H = {A ∈ G | a11 = a22}.
(c) H = {A ∈ G | a11 = a22 = 1}.
1.4 Give an example of an ideal in the ring C[0, 1] of all continuous real
valued functions on the interval [0, 1] with pointwise addition and pointwise
multiplication as the ring operations.
1.5 Which of the following sets of vectors form a basis for R3
?
(a) {(−1, 0, 0), (1, 1, 1), (1, 2, 3)}.
(b) {(0, 1, 2), (1, 1, 1), (1, 2, 3)}.
(c) {(−1, 1, 0), (2, 0, 0), (0, 1, 1)}.
1.6 Write down a basis for the following subspace of R4
:
V = {(x, y, z, t) ∈ R4
| z = x + y, x + y + t = 0}.
1.7 Let A ∈ M2(R). Which of the following statements are true?
(a) If (tr(A))2
> 4det(A), then A is diagonalizable over R.
(b) If (tr(A))2
= 4det(A), then A is diagonalizable over R.
(c) If (tr(A))2
< 4det(A), then A is diagonalizable over R.
1.8 Let V be the vector space of all polynomials in a single variable with real
coefficients and of degree less than, or equal to, 3. Equip this space with the
standard basis consisting of the elements 1, x, x2
and x3
. Consider the linear
transformation T : V → V defined by
T(p)(x) = xp (x) + 3xp (x) + 2p(x), for all p ∈ V.
Write down the corresponding matrix of T with respect to the standard basis.
3

1.9 With the notations and deﬁnitions of Problem 1.8 above, ﬁnd p ∈ V such
that
xp (x) + 3xp (x) + 2p(x) = 11x3
+ 14x2
+ 7x + 2.
1.10 If α, β and γ are the roots of the equation
x3
− 3x2
+ 4x − 4 = 0,
write down an equation of degree 3 whose roots are α2
, β2
and γ2
.
4

Section 2: Analysis
2.1 Which of the following series are convergent?
(a)
∞
n=1
2n2 + 3
5n3 + 1
.
(b)
∞
n=1
(n + 1)n
nn+3
2
.
(c)
∞
n=1
n2
x(1 − x2
)n
, where 0 < x < 1.
2.2 Evaluate:
lim
n→∞
1 +
1
n
1 +
2
n
· · · 1 +
n
n
1
n
.
2.3 In each of the following, evaluate the limit if the limit exists, or state
that the limit does not exist if that is the case.
(a)
lim
x→0
[x]
x
.
(b)
lim
x→0
x
1
x
.
(c)
lim
x→0
cos(π
2
cos x)
sin(sin x)
.
(Note: The symbol [x] denotes the greatest integer less than, or equal to, x.)
2.4 In each of the following ﬁnd the points of continuity of the function f.
(a)
f(x) =
0, if x ∈ Q,
1, if x ∈ Q.
(b)
f(x) =
0, if x ∈ Q.
x2
− 1, if x ∈ Q.
2.5 Let f : R → R be diﬀerentiable at x = a and let f(a) > 0. Evaluate:
lim
x→a
f(x)
f(a)
1
log x−log a
.
5

2.6 Which of the following functions are diﬀerentiable at x = 0?
(a)
f(x) =
tan−1 1
|x|
, if x = 0,
π
2
, if x = 0.
(b)
f(x) = |x|
1
2 x.
(c)
f(x) =
x2
cos π
x
, if x = 0,
0, if x = 0.
2.7 Let x, y ∈]0, ∞[. Which of the following statements are true?
(a) | log(1 + x2
) − log(1 + y2
)| ≤ |x − y|.
(b) | sin2
x − sin2
y| ≤ |x − y|.
(c) | tan−1
x − tan−1
y| ≤ |x − y|.
2.8 For what values of x ∈ R is the following function decreasing?
f(x) = 2x3
− 9x2
+ 12x + 4.
2.9 Which of the following statements are true?
(a) If n ∈ N, n > 2, then
π
2
0
sinn
x dx is rational when n is even.
(b) If n ∈ N, n > 2, then
π
2
0
sinn
x dx is rational when n is odd.
(c) If n ∈ N, n > 2, then
π
2
0
cosn
x dx is irrational when n is even.
2.10 A right circular cylinder is inscribed in a sphere of radius a > 0. What
is the height of the cylinder when its volume is maximal?
6

Section 3: Miscellaneous
3.1 Given a function u : R → R, define u+
(x) = max{u(x), 0} and u−
(x) =
− min{u(x), 0}. If u1 and u2 are real valued functions defined on R, which
of the following statements are true?
(a) |u1 − u2| = (u1 − u2)+
+ (u1 − u2)−
.
(b) max{u1, u2} = (u1 − u2)+
+ u2.
(c) max{u1, u2} = (u1 − u2)−
+ u1.
3.2 Let a1, · · · , an be positive real numbers. What is the minimum value of
a1
a2
+
a2
a3
+ · · · +
an−1
an
+
an
a1
?
3.3 Let n be a fixed positive integer. For 0 ≤ r ≤ n, let Cr denote the usual
binomial coefficient
n
r
, viz. the number of ways of choosing r objects
from n objects. Evaluate:
C0
1
−
C1
2
+
C2
3
− · · · + (−1)n Cn
n + 1
.
3.4 Let a, b, c ∈ R. Evaluate:
b + c a a
b c + a b
c c a + b
.
3.5 Which of the following statements are true?
(a) For every n ∈ N, n3
− n is divisible by 6.
(b) For every n ∈ N, n7
− n is divisible by 42.
(c) Every perfect square is of the form 3m or 3m + 1 for some m ∈ N.
3.6 Find the sum of the following infinite series:
1 +
2
6
+
2.5
6.12
+
2.5.8
6.12.18
+ · · ·
1
1.2
−
1
2.3
+
1
3.4
− · · ·
1
3!
+
4
4!
+
9
5!
+ · · ·
3.9 Find the area of the triangle in the complex plane whose vertices are the
points representing the numbers 1, ω and ω2
, the cube roots of unity.
3.10 Find the equation of the plane in R3
which passes through the point
(−10, 5, 4) and which is perpendicular to the line joining the points (4, −1, 2)
and (−3, 2, 3).
7

KEY
Section 1: Algebra
1.1 −1
1.2 b,c
1.3 a,b,c
1.4 I = {f ∈ C[0, 1] |f(x) = 0 for all x ∈ S}
for any subset S of [0, 1]
1.5 a,c
1.6 Any two linearly independent vectors
belonging to V (Example: (1, 0, 1, −1) and
(0, 1, 1, −1))
1.7 a
1.8 



2 0 0 0
0 5 2 0
0 0 8 6
0 0 0 11




1.9 1 + x + x2
+ x3
1.10 x3
− x2
− 8x − 16 = 0
Section 2: Analysis
2.1 b,c
2.2 4
e
2.3 a. Limit does not exist; b. 1; c. 0
2.4 a. Emptyset; b. {−1, +1}
2.5 ea
f (a)
f(a)
2.6 b,c
2.7 a,b,c
2.8 1 < x < 2
2.9 b,c
2.10 2a√
3
Section 3: Miscellaneous
3.1 a,b,c
3.2 n
3.3 1
n+1
3.4 4abc
3.5 a,b,c
3.6 4
1
3
3.7 2 log 2 − 1
3.8 2e − 5
3.9 3
√
3
4
3.10 7x − 3y − z + 89 = 0
Note: Please accept any answer which
is correct, but expressed in an equivalent,
though diﬀerent, form, where applicable.
1

Nbhm m. a. and m.sc. scholarship test september 20, 2014 with answer key

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