University of Hyderabad (UoH) 2008 M.Sc Mathematics Entrance - Question Paper
Entrance Examination: M.Sc. Mathematics 2. 0 0
Halt Ticket. Number:
Time: 2 hours Part, A: 25 marks
Max. Marks. 75 Part B: 50 marks
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Department of Mathematics and Statistics University of Hyderabad
Note: Throughout this question paper, N stands for the set of all natural numbers, Z stands for the set of all integers, Q stands for the set of all rational numbers , R stands for the set of all real numbers and C stands for the set of all complex numbers.
Part A - 1 mark for each question
1. Let A = e33e'i. Then the smallest positive integer l such that Al = 1 is
(a) 6
(b) 9
(c) 12
(d) 5
2. Consider the vector (1,1,1) in R3. Two linearly independent vectors orthogonal to it are
(a) (1,-1,1) and (1,1,-2)
(b) (-2,1,1) and (1,1-2)
(c) (1,-1,0) and (2,-2,0)
(d) (0,1,-1) and (0,-2,2)
3. The graph of the polynomial (X2 2)(X2 + X +1) will cross the X-axis
(a) 0 times
(b) once
(c) twice
(d) 3 times
4. There exists an integer which is not divisible by the square of a prime number. The negation of this statement is
(a) There exists an integer which is divisible by the square of a prime
(b) Every integer is not divisible by the square of a prime number
(c) Every integer is divisible by the square of a prime number
(d) There exists many integers divisible by the square of a prime number
5. Let f,g : R R be continuous functions whose graphs do not intersect. Then for which function below the graph lies entirely on one side of the X-axis
(a) f
(b) g + f
(c) g - f
(d) gf
6. An example of a function from R R2 with bounded range is
(a) f(t) = (t,t2)
(b) f (t) = (t, sin t)
(c) f (t) = (t, sinh t)
(d) f (t) = (sin t, cos t)
7. The real root of X3 + X +1 = 0 lies between
(a) -2 and -1
(b) -1 and 0
(c) 1 and 2
(d) 2 and 3
8. Which of the following maps is a linear transformation from R3 to R?
(a) T(a, b, c) = a(b + c)
(b) T(a, b, c) = 2(a + b + c)
(c) T(a, b, c) = ab + c
(d) T(a, b, c) = abc
9. The events A\ and A2 occur with probabilities 0.6 and 0.8 respectively. At least one of them occurs with a probablity of 0.9. The probability that both Ai and A2 will occur is
(a) 0
(b) 0.5
(c) 1
(d) cannot be determined from the data given
10. Two students each are randomly placed in n rooms in a hostel. If n of the 2n students are Mathematics students and n are in Statistics, the prob-ablity that each room has Mathematics student and a statistics student is
(a) 2n
(b) 2n!
(c) 4
(d) n*
11. How many positive integers a less than 24 satisfy a8 = 1(mod24)?
(a) 2
(b) 4
(c) 6
(d) 8
12. Suppose f : [0, to) R is continuous
(a) Then f is uniformly continuous on [k, to) for some k > 0
(b) Then | f | is uniformly continuous on [0, to)
(c) If f is uniformly continuous on [k, to), for some k > 0, then f is uniformly continuous on [0, to)
(d) If f is decreasing then f is uniform continuous
13. Let V be a vector space of all polynomials of degree less than or equal 4
4
over Q and W = 4=0 aiX% G Q[X] | a0 is an even integer}. Then
A=0 1
(a) W is not a subspace of V
(b) W is a subspace and dimW < dimV
(c) W is a subspace and dimW = 4
(d) W is a subspace and dimW = 5
14. x2 = 2y2 log y is a solution of
d -2+y2
dy _ 2y
d 2+y2
dy _ 2y
d 22 + y2
dy _ 2y
d 2+2y2
(b) dx = -2
15. Let Ci, C2 be two circles in R2 with centres at points a, b respectively and suppose that Cl n C2 is a singleton set {c}. Let la bl denote the distance between a and b. Then
(a) la bl > la cl
(b) | a b| = | a c| + | c b|
(c) la bl2 = la cl2 + lc bl2
(d) None of these
16. The distance between the straight lines 3x4y+10 = 0 and 3x4y 5 = 0
(a) 0
(b) 3
(c) 5
17. f (x) = ex e x, g(x) = ex + e x Then
(a) Both f and g are even functions
(b) Both f and g are odd functions
(c) f is odd , g is even
(d) f is even, g is odd
18. xn+1 = pxn, x0 = 1. The sequence {xn}
(a) diverges
(b) xn is monotonically increasing and converges to 0
(c) xn is monotonically decreasing and converges to 0
(d) None of the above
19. f (x) is an odd function, g(x) an even function then
(a) f o g is odd
(b) f o g is even
(c) f o f is odd
(d) g o g is odd
20. Let X be a set and f,g : X X be functions. We can say that f o g is bijective if
(a) at least one of f, g is bijective
(b) both f and g are bijective
(c) f is 1 1 and g is onto
(d) f is onto and g is 1 1
21. Let f : [1,1] R be a continuous function such that f1 f (x)dx = 0.
Then
(a) f = 0
(b) f is an odd function
, 1/2
(c) / f (x)dx = 0
J-1/2
(d) None of these
22. Let f,g : R R be functions. We can conclude that h(x) < f(x)Vx G R if we define h : R > R as
(a) min{g(x), f (x) + g(x)}
(b) min{f(x),f(x) + g(x)}
(c) max{g(x),f(x) + g(x)}
(d) max{f(x),f(x) + g(x)}
23. Let X be a non-empty set, f : X X be a function and let A, B C X. Then the identity f (A n B) = f (A) n f (B) is true
(a) always
(b) if f is 1-1
(c) if f is onto
(d) if A U B = X
24. If n > 1000 is a natural number, the remainder when n2 + n + 1 is divided by 4 is
(a) always 1
(b) always 3
(c) 1 or 3
(d) 0 or 2
25. Let X be a finite set with 5 elements. Then the number of 1-1 functions from X x X to X x X is
(a) 5!
(b) (5!)2
(c) 25!
(d) f
Part B - 2 marks for each question
1. The number of 2 x 2 matrices with integer entries that satisfy the polynomial X2 + X + 1 is
(a) atmost 2
(b) exactly 2
(c) infinite
(d) none
2. Let (Q, +) be the group of all rationals under addition and (Q+,.) be the group of positive nonzero rationals under multiplication. Suppose f : Q Q+ is a homomorphism. Then f (17) =
(a) 172
(b) 17
(c) T7
3. Let f be a function from [-1,1] to R
(a) If f is differentiable at 0 with f '(0) = 0 then f (0) = 0
(b) If f (0) = 0 then f is differentiable at 0
(c) If f (0) = 0 then the X-axis is tangent to the graph of f at 0
i. All three statements are false
ii. (a) and (c) are false but (b) is true
iii. (a) and (b) are false but (c) is true
iv. (b) and (c) are false but (a) is true
4. Let fn(x) = (x + n)2 and f (x) = fn(x). Then
(a) limnx J0 fn(x) dx does not exist
(b) limn,x, J0 fn(x) dx exists but fQ f (x) dx does not exist
(c) limn/q1 fn(x) dx = /q1 f (x) dx
(d) limn/q1 fn(x) dx = /q1 f (x) dx
5. Let f (x) = | cos x\ and g(x) = cos |x|. then
(a) both f and g are differentiable at 0
(b) f is differentiable at 0 but g is not
(c) g is differentiable at 0 but f is not
(d) neither f nor g are differentiable at 0
6. Let V1 and V2 be subspaces of R3 given by V1 = {(a, b,c) G R3 \a + b = 2c} and V2 = {(a,b,c) G R3 \a + b c = 0}. Then dim (Vi n V2) is
(a) 0
(b) 1
(c) 2
(d) 3
7. In a bag there are 12 marbles, 11 of which are white and one is red. A child takes out 6 of them, the probability that one of these 6 is red is
(a) Strictly greater then 2
(b) equal to 2
(c) Strictly less then 3
(d) equal to 3
8. Two families of 3 members each have to be seated in a row, in how many ways can it be done so that all members of a family do not sit together?
(a) 648
(b) 504
(c) 120
(d) 324
9. Let Ti,T2 be two linear transformations from a finite dimensional vector space V to another space W. Suppose that Ti, T2 are onto. Then
(a) dim Ker Ti=dim Ker T2
(b) Ker Ti= Ker T2
(c) Ker Ti strictly contained Ker T2
(d) Ti = T2
10. The orthogonal trajectories of the family of curves x + 2y2 = c where c is a constant is
(a) y = 4x
(b) y = 4x
(c) y = eix
(d) y = e-4x
11. A non zero vector common to the space spanned by (1,2,3), (3,2,1) and the space spanned by (1,0,1) and (3,4,3) is
(a) (1,2,3)
(b) (0,-2,-2)
(c) (3,2,0)
(d) (1,1,1)
12. A subset A of C is said to be balanced if whenever a G A and t G R, it is true that aeit G A. Which one of these four subsets is balanced?
(a) The elliptic region {x + iy | X4 + \ < 1}
(b) The upper half plane {x + iy ly > 0}
(c) The Y-axis {x + iy ly = 0}
(d) The annular region {x + iy |1 < x2 + y2 < 2}
13. Let A6 be the set of all positive integers for which 6 is not a factor. Then
(a) A6 is closed under addition
(b) A6 is closed under multiplication
(c) A6 U 6N = N
(d) A6 U 6A6 = N
14. Which is not a group homomorphism?
(a) f : (R, +) (R {0},.) given by f (x) = xex
(b) f : (Q {0},.) (Q {0},.) given by f (x) = 2x
(c) f : (N, +) (R, +) given by f (x) = x + \x\
(d) f : (C, +) (C, +) given by f (x) = 2x
15. For a real number x, let \x\ denote the greatest integer less than or equal to x. Then
(a) \xy\ > \x\\y\ for all x,y G R
(b) \xy\ < \x\\y\ for all x,y G R
(c) \xy\ > \x\ + \y\ for all x,y G R
(d) \xy\ < \x\ + \y\ for all x,y G R
16. Let (xn) be a sequence of positive real numbers. A sufficient condition for (xn) to have no convergent subsequence is
(a) \xn+2 xn+i| > \xn+i xn G N
(b) Vi,j G N, the set {n G N : \xi xn\ < 1} is finite
(c) 'TIT=1 xnk = ro for every increasing sequence (nk) of natural numbers.
(d) none of the above
17. Let P be a real polynomial such that for x G R, P(x) = 0 iff x = 2 or 4. Then
(a) degree of P is 2
(b) P(3) < 0
(c) P'(x) = 0 for some x < 4
(d) P(x) is of the form c(x 2)n(x 4)m where c is a constant
18. Let f : R R be a continuous function with f (0) = 0 and let (xn) be a sequence in R with lim f (xn) = 0. Then
n
(a) lim xn = 0 n
(b) lim xnk = 0, for some subsequence (xnk) k
(c) (xn) is bounded
(d) none of the above
19. Number of generators of the group (Z36, +) is
(a) 1
(b) 6
(c) 12
(d) 35
20. Let xn = o1, and yn = r1. then
n n2 + 1 &n n log n
(a) Sxn is convergent, Syn is divergent
(b) Sxn is convergent, Syn is convergent
(c) Sxn is divergent, Syn is convergent
(d) Sxn is divergent, Syn is divergent
21. Let AAB denote the symmetric difference of A and B. Then A ABAC is the same as
(a) {x | x belongs to all or none of the sets A, B, C}.
(b) {x I x belongs to all or exactly one of the sets A, B, C}.
(c) {x I x belongs to exactly one of the sets A, B, C}.
(d) {x I x belongs to the complement of the union of A, B and C}.
22. Consider the following three conditions on a set A C N :
Condition (1) : A = {ma+nb | m,n G N} for some a,b G N with (a, b) = 1. Condition (2) : N A is finite.
Condition (3) : there exists no G N such that A = {n G N/n > no}. Then (a) (2) (3).
(b) (3) (1) (2).
(c) (1) (2) (3).
(d) (1) (2).
23. The function f (x) = xx on (0, x) has
(a) a local maximum at e-1 but no local minimum.
(b) a local maximum at e-1 and a local minimum at 1.
(c) two local maxima at 1 and e-1 but no local minimum.
(d) neither a local maximum nor a local minimum.
24. Let f (x) = 1 x2/3 for x G [1,1]. Then
(a) f '(c) = 0 for some c G ( 1,0).
(b) f '(c) = 0 for some c G (0,1).
(c) f '(x) is never zero in ( 1,0).
(d) f '(x) is zero in (0,1) at two points.
25. A vector of length 1 in R3 which is orthogonal to the vectors i + 2j + 3A; and 4i + 5j + 6k is
(c) i/s + k
10
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