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Entrance Paper, Study notes of Mathematics

University of HyderabadMathematics

<div><br /></div><div>Closed under addition, closed under multiplication, group homomorphism, positive real numbers, continuous function, number of generators, local maximum, local minimum, uniformly continuous, distance between the straight lines</div><div><br /></div><div><br /></div>

Typology: Study notes

2010/2011

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Department of Mathematics and Statistics
University of Hyderabad
Note: Throughout this question paper, Nstands for the set of all natural
numbers, Zstands for the set of all integers, Qstands for the set of all rational
numbers , Rstands for the set of all real numbers and Cstands for the set of
all complex numbers.
Part A - 1 mark for each question
1. Let λ=e30πi
36 . Then the smallest positive integer lsuch that λl= 1 is
(a) 6
(b) 9
(c) 12
(d) 5
2. Consider the vector (1,1,1) in R3. Two linearly independent vectors or-
thogonal 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→Rbe 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
1
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Entrance Paper and more Study notes Mathematics in PDF only on Docsity!

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 λ = e 3036 πi^. Then the smallest positive integer l such that λl^ = 1 is (a) 6 (b) 9 (c) 12 (d) 5
  2. Consider the vector (1, 1 , 1) in R^3. Two linearly independent vectors or- thogonal 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 (X^2 − 2)(X^2 + 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

  1. An example of a function from R → R^2 with bounded range is (a) f (t) = (t, t^2 ) (b) f (t) = (t, sin t) (c) f (t) = (t, sinh t) (d) f (t) = (sin t, cos t)
  2. The real root of X^3 + X + 1 = 0 lies between (a) -2 and - (b) -1 and 0 (c) 1 and 2 (d) 2 and 3
  3. Which of the following maps is a linear transformation from R^3 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
  4. The events A 1 and A 2 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 A 1 and A 2 will occur is (a) 0 (b) 0. (c) 1 (d) cannot be determined from the data given
  5. 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) (^21) n! (b) (^21) n! (c) (^2) n^2 Cnn (d) (^) (n^2 n!) 2
  6. How many positive integers a less than 24 satisfy a^8 ≡ 1(mod24)?

(d) 15

  1. 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

  1. xn+1 = − 43 xn, x 0 = 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

  1. f (x) is an odd function, g(x) an even function then

(a) f ◦ g is odd (b) f ◦ g is even (c) f ◦ f is odd (d) g ◦ g is odd

  1. Let X be a set and f, g : X → X be functions. We can say that f ◦ 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
  2. Let f : [− 1 , 1] → R be a continuous function such that

− 1 f^ (x)dx^ = 0. Then (a) f ≡ 0 (b) f is an odd function (c)

− 1 / 2

f (x)dx = 0 (d) None of these

  1. Let f, g : R → R be functions. We can conclude that h(x) ≤ f (x) ∀x ∈ 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)}

  1. Let X be a non-empty set, f : X → X be a function and let A, B ⊂ X. Then the identity f (A ∩ B) = f (A) ∩ f (B) is true (a) always (b) if f is 1- (c) if f is onto (d) if A ∪ B = X
  2. If n ≥ 1000 is a natural number, the remainder when n^2 + n + 1 is divided by 4 is (a) always 1 (b) always 3 (c) 1 or 3 (d) 0 or 2
  3. Let X be a finite set with 5 elements. Then the number of 1-1 functions from X × X to X × X is (a) 5! (b) (5!)^2 (c) 25! (d) 25!5! Part B - 2 marks for each question
  4. The number of 2 × 2 matrices with integer entries that satisfy the polyno- mial X^2 + X + 1 is (a) atmost 2 (b) exactly 2 (c) infinite (d) none
  5. 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) 17^2 (b) 17 (c) 171

(a) 648 (b) 504 (c) 120 (d) 324

  1. Let T 1 , T 2 be two linear transformations from a finite dimensional vector space V to another space W. Suppose that T 1 , T 2 are onto. Then (a) dim Ker T 1 =dim Ker T 2 (b) Ker T 1 = Ker T 2 (c) Ker T 1 strictly contained Ker T 2 (d) T 1 = T 2
  2. The orthogonal trajectories of the family of curves x + 2y^2 = c where c is a constant is (a) y = 4x (b) y = − 4 x (c) y = e^4 x (d) y = e−^4 x
  3. 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)
  4. A subset A of C is said to be balanced if whenever a ∈ A and t ∈ R, it is true that aeit^ ∈ A. Which one of these four subsets is balanced? (a) The elliptic region {x + iy | x 42 + y 92 ≤ 1 } (b) The upper half plane {x + iy |y > 0 } (c) The Y -axis {x + iy |y = 0} (d) The annular region {x + iy | 1 ≤ x^2 + y^2 ≤ 2 }
  5. Let A 6 be the set of all positive integers for which 6 is not a factor. Then

(a) A 6 is closed under addition (b) A 6 is closed under multiplication (c) A 6 ∪ 6 N = N (d) A 6 ∪ 6 A 6 = N

  1. 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

  1. For a real number x, let bxc denote the greatest integer less than or equal to x. Then (a) bxyc ≥ bxcbyc for all x, y ∈ R (b) bxyc ≤ bxcbyc for all x, y ∈ R (c) bxyc ≥ bxc + byc for all x, y ∈ R (d) bxyc ≤ bxc + byc for all x, y ∈ R
  2. Let (xn) be a sequence of positive real numbers. A sufficient condition for (xn) to have no convergent subsequence is (a) |xn+2 − xn+1| > |xn+1 − xn| ∀n ∈ N (b) ∀ i, j ∈ N, the set {n ∈ N : |xi − xn| < (^1) j } is finite (c)

k=1 xnk^ =^ ∞^ for every increasing sequence (nk) of natural num- bers. (d) none of the above

  1. Let P be a real polynomial such that for x ∈ 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
  2. Let f : R → R be a continuous function with f (0) = 0 and let (xn) be a sequence in R with (^) nlim→∞f (xn) = 0. Then (a) (^) nlim→∞xn = 0 (b) (^) klim→∞xnk = 0, for some subsequence (xnk ) (c) (xn) is bounded (d) none of the above
  3. Number of generators of the group (Z 36 , +) is

(a) 1 (b) 6

(a) − √ˆi 6 +

√2ˆj √ 3 + √kˆ 6

(b) √ˆi 6 −

√2ˆj √ 3 + √ˆk 6 (c) − √ˆi 6 +

√2ˆj √ 3 − √kˆ 6

(d) √ˆi 6 +

√2ˆj √ 3 − √ˆk 6