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SET 2


  Question 1

Consider the statement:

“Not all that glitters is gold”

Predicate glitters (x) is true if x glitters and predicate gold (x) is true if x is gold. Which one of the following logical formulae represents the above statement?


A : ∀x glitter (x) => ¬gold(x)
B : ∀xgold(x) => glitter(x)
C : ∃xgold(x) ∧ ¬ glitter(x)
D : ∃ glitter(x) ∧ ¬ gold(x)


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     Correct answer is :D

     Solution :
      It means “It is false that every glitter is gold” or “some glitters are not gold”. Then we can say “atleast one glitter object is not gold”.

  •   Question 2

    Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .



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     Correct answer is :0.25

     Solution :
      The smaller sticks, therefore, will range in length from almost 0 meters up to a maximum of 0.5 meters, with each length equally possible.
    Thus, the average length will be about 0.25 meters, or about a quarter of the stick.

  •   Question 3

    Consider the following system of equations:
    3x + 2y= 1
    4x +7z=1
    x + y + z = 3
    x - 2y + 7z = 0
    The number of solutions for this system is __________________




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     Correct answer is :1


  •   Question 4

    The value of the dot product of the eigenvectors corresponding to any pair of different eigen values of a 4-by-4 symmetric positive definite matrix is ______________.



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     Correct answer is :0

     Solution :
      ( The eigen vectors corresponding to distinct eigen values of real symmetric matrix are orthogonal)

  •   Question 5

    Let the function given below where θ = ∈ [ π/6 , π/2] and f'(θ) denote the derivative of f with respect to θ . Which of the following statement is / are TRUE?
    (I) There exists θ ∈ (π/6 ,&pi/;3) such that f'(θ) = 0
    (I) There exists θ ∈ (π/6 ,π/3) such that f'(θ) ≠ 0


    A : I only
    B : II only
    C : Both I and II
    D : Neither I and II


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     Correct answer is :C

     Solution :
      By mean value theorem

  •   Question 6

    There are 5 bags labelled 1 to 5. All the coins in a given bag have the same weight. Some bags have coins of weight 10 gm, others have coins of weight 11 gm. I pick 1, 2, 4, 8, 16 coins respectively from bags 1 to 5. Their total weight comes out to 323 gm. Then the product of the labels of the bags having 11 gm coins is ___.



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     Correct answer is :12

     Solution :
      Let the weight of coins in the respective bags (1 through 5) be a,b,c,d and e-each of which can take one of two values namely 10 or 11 (gm).
    Now, the given information on total weight can be expressed as the following equation:
    1.a+2.b+4.c+8.d+16.e = 323
    a must be odd  => a = 11
    The equation then becomes: 11+2.b+4.c+8.d+16.e = 323
    =>2.b+4.c+8.d+16.e = 312
    =>b+2.c+4.d+8.e = 156
    b must be even  b = 10
    The equation then becomes: 10+2.c+4.d+8.e = 156
    =>2.c+4.d+8.e = 146 
    => c+2.d+4.e = 73
    c must be odd  c = 11
    The equation now becomes: 11+2.d+4.e = 73 
    =>2.d+4.e = 62
    =>d+2.e = 31
     e = 11 and e = 10
    Therefore, bags labelled 1, 3 and 4 contain 11 gm coins => Required Product = 1*3*4* = 12.

  •   Question 7

    The function f(x) = x sin x satisfies the following equation. f"(x) + f(x) +tcosx = 0. The value of t is______.



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     Correct answer is :-2

     Solution :
      Given f "(x) +f (x)+ t cos x =0
    and f(x)= xsin x
    f '(x)= x cos x + sin x
    f "(x)= x(- sin x) + cos x + cos x
    = 2cos x - xsin x
    = 2cos x - f(x)
    2cos x - f(x)+ f(x) +t cos x = 0
    => 2cos x= -t cos x=> t= -2

  •   Question 8

    A function f(x) is continuous the interval [0,2]. It is known that f(0) = f(2) = -1 and f(1) = 1. Which one of the following statements must be true?

    A : There exists a y in the interval (0,1) such that f(y) = f(y + 1)
    B : For every y in the interval (0,1), f(y) = f(2 - y)
    C : The maximum value of the function in the interval (0.2) is 1
    D : There exists a y in the interval (0,1) such that f(y) =f(2 – y)


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     Correct answer is :A

     Solution :
      Define g(x) =f(x)-f(x+1) in [0,1]. g(0) is negative and g(1) is positive. By intermediate value
    theorem there is y€(0,1) such that g(y)=0
    That is f(y) =f(y+1).
    Thus answer is (a)

  •   Question 9

    Four fair six-sided dice are rolled. The probability that the sum of the results being 22is X/1296.The value of X is ______________.



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     Correct answer is :10

     Solution :
      22 occurred in following ways
    6 6 6 4 -> 4 ways
    6 6 5 5 -> 6 ways
    Required probability = 6+4/2296 = 10/2296 => x=10

  •   Question 10

    A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1- pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)}and the set of all 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10- pennants is ______________.



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     Correct answer is :89

     Solution :
      No twos: 11111111111=> pennant
    Single two: 211111111 => 9!/8!1! = 9 pennants
    Two twos: 22111111 => 8!/6!.2! = 28
    Three twos: 2221111 => 7!/3!.4! = 35
    Four twos: 222211 => 6!/4!.2! = 15
    Five twos: 22222 =>1
    Total = 89 pennants.

  •   Question 11

    Let S denote the set of all functions f:{0,1}4 -> {01} . Denote by N the number of functions from S to the set {0,1}. The value of log2 log2 N is______.



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     Correct answer is :16

     Solution :
      The number of functions from A to B where size of A = |A| and size of B = |B| is |B||A|
    {0,1}4 = {0,1} X {0,1} X {0,1}X {0.1} = 16
    |S| = 216
    N=2|S|
    loglogN= loglog 2|S| = log |S| = log 216 =16

  •   Question 12

    An ordered n-tuple (d1,d2,…,dn) with d1 >= d2 >= .... >= dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1,d2,.......dn respectively. Which of the following 6-tuples is NOT graphic?

    A : (1, 1, 1, 1, 1, 1)
    B : (2, 2, 2, 2, 2, 2)
    C : (3, 3, 3, 1, 0, 0)
    D : ( 3, 2, 1, 1, 1, 0)


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     Correct answer is :C

     Solution :
      According to havel-hakimi theorem
    (1,1,1,1,1,1) is graphic iff<1,1,1,1,0> is graphic
    (0,1,1,1,1) is graphic iff (0,1,1,0) is graphic
    (0,0,1,1) is graphic iff (0,0,0) is graphic
    Since (0,0,0) is graphic (1,1,1,1,1,1) is also graphic.
    (The process is always finding maximum degree and removing it from degree sequence, subtract 1 from each degree for d times from right to left where d is maximum degree)
    (2,2,2,2,2,2) is graphic iff (2,2,22-1,2-1) = (2,2,2,1,1) is graphic
    (1,1,2,2,2

  •   Question 13

    Which one of the following propositional logic formulas is TRUE when exactly two of p, q, and r are TRUE?

    A : ((p <-> q) ∧ r) V (p ∧ q ∧ ~ r)
    B : (~ (p <-> q) ∧ r) V (p ∧ q ∧ ~ r)
    C : ((p -> q) ∧ r) V (p ∧ q ∧ ~ r)
    D : (~ (p <-> q) ∧ r) ∧ (p ∧ q ∧ ~ r)


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     Correct answer is :B

     Solution :
      P = T q=F and r=T
    Option A will become false
    Option C will become false.
    Option D is always false.

  •   Question 14

    The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by p Then 100p= _____________.



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     Correct answer is :11.85 - 11.95

     Solution :
      p= P [at least three computers are working]
    =P (3 or 4 computers working)
    = (4C3) * (6C1) / 10C4 + 4C4 / 10C4 = 5/42
    100p=11.9

  •   Question 15

    Each of the nine words in the sentence ”The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is _____________. (The answer should be rounded to one decimal place.



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     Correct answer is :3.8889


  •   Question 16

    The maximum number of edges in a bipartite graph on 12 vertices is __________________________.



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     Correct answer is :36

     Solution :
      The number of edges in a bipartite graph on n-vertices is atmost n2/4
    The maximum number of edges in a bipartite graph on 12 –vertices is n2 /4 = 12*12 /4 = 36

  •   Question 17

    If the matrix A is such that
    Then the determinant of A is equal to ________.






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     Correct answer is :0

     Solution :
      |A| = 0

  •   Question 18

    A non-zero polynomial f(x) of degree 3 has roots at x = 1,x = 2 and x = 3. Which one of the following must be TRUE?

    A : f(0) f(4) < 0
    B : f(0) f(4) > 0
    C : f(0) + f(4) > 0
    D : f(0) + f(4) < 0


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     Correct answer is :A

     Solution :
      Since, the roots of f(x) = 0 i.e., x = 1, 2, 3 lies between 0 and 4 and f(x) is of degree 3
    f(0) and f(4) are of opposite signs
     f(0).f(4)<0.

  •   Question 19

    Suppose n and p are unsigned int variables in a C program. We wish to set p to nC3 . If n is large, which one of the following statements is most likely to set p correctly?

    A : p = n * (n – 1) * (n-2) / 6;
    B : p = n * (n – 1) / 2 * (n-2) / 3;
    C : p = n * (n – 1) / 3 * (n-2) / 2;
    D : p = n * (n – 1) / 2 * (n-2) / 6.0;


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     Correct answer is :B

     Solution :
      n*(n-1) is an even number so we divide it by 2 and the rest by 3. The output will be same but overflow can be avoided.

  •   Question 20

    In the Newton-Raphson method, an initial guess of x0 = 2 is made and the sequencex0,x1,x2 ... is obtained for the function
    0.75x3 - 2x2 - 2x +4 = 0
    Consider the statements
    (I) x3 = 0 .
    (II) The method converges to a solution in a finite number of iterations.
    Which of the following is TRUE?


    A : Only I
    B : Only II
    C : Both I and II
    D : Neither I and II


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     Correct answer is :A


  •   Question 21

    The product of the non-zero eigenvalues of the matrix is





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     Correct answer is :6


  •   Question 22

    The probability that a given positive integer lying between 1 and 100 (both inclusive) is NOT divisible by 2, 3 or 5 is ______ .



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     Correct answer is :0.259 - 0.261

     Solution :
      Let A = divisible by 2, B = divisible by 3 and C = divisible by 5, then
    n(A) = 50, n(B) = 33, n(C) = 20
    n(A ∩ B) = 16 (100/6)
    n(A ∩ C) = 10 (100/10)
    n(B ∩ C) = 6 (100/15)
    n(A ∩ B ∩ C) = 3 (100/(2*3*5))
    Now find n(A U B U C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ B) -n(B ∩ C) + n(A ∩ B ∩ C)
    (100 - n(A U B U C))/100
    Substituting the values we get answer as 0.26

  •   Question 23

    The number of distinct positive integral factors of 2014 is _________________________



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     Correct answer is :8

     Solution :
      2014 = 2×19×53 i.e., product of prime factors
    Number of distinct positive integral factors of 2014 is (2)×(2)×(2) = 8.

  •   Question 24

    Consider the following relation on subsets of the set S of integers between 1 and 2014. For two distinct subsets U and V of S we say U < V if the minimum element in the symmetric difference of the two sets is in U.
    Consider the following two statements:
    S1: There is a subset of S that is larger than every other subset.
    S2: There is a subset of S that is smaller than every other subset.
    Which one of the following is CORRECT?


    A : Both S1 and S2 are true
    B : S1 is true and S2 is false
    C : S2 is true and S1 is false
    D : Neither S1 nor S2 is true


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     Correct answer is :A

     Solution :
      From given data S1 is true ,since null set is larger than every other set ,and S2 is true since the universal set {1,2,...,2014} is smaller than every other set.
    Both s1 and s2 are true.

  •   Question 25

    A cycle on n vertices is isomorphic to its complement. The value of n is _____.



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     Correct answer is :5

     Solution :
      Consider a cycle on five vertices C5
    C5 and C5` are isomorphic


  •   Question 26

    Which one of the following Boolean expressions is NOT a tautology?

    A : ((a -> b) ∧ (b -> c)) -> (a -> c)
    B : (a <-> c) -> (~ b -> (a ∧ c))
    C : (a ∧ b ∧ c) -> (c ∧ a )
    D : a -> (b -> a)


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     Correct answer is :B


  •   Question 27

    Let X and Y be finite sets and f : X -> Y be a function. Which one of the following statements is TRUE?

    A : For any subsets A and B of X, f (A U B) = |f (A)| + |f (B)|
    B : For any subsets A and B of X, f (A ∧ B) =f (A) ∧ f (B)
    C : For any subsets A and B of X, f (A ∧ B) =min{ f (A) , f (B) }
    D : For any subsets S and T of Y, f-1 (S ∧ T) = f-1 (S) ∧ f-1(T)


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     Correct answer is :D


  •   Question 28

    Let G be a group with 15 elements. Let L be a subgroup of G. It is known that L ≠ G and that the size of L is at least 4. The size of L is _______.



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     Correct answer is :5

     Solution :
      Order of subgroup divides order of group (Lagrange’s theorem).
    3, 5 and 15 can be the order of subgroup. As subgroup has atleast 4 elements and it is not equal to the given group, order of subgroup can’t be 3 and 15. Hence it is 5.

  •   Question 29

    Which one of the following statements is TRUE about every n × n matrix with only real eigenvalues?

    A : If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigenvalues is negative.
    B : If the trace of the matrix is positive, all its eigenvalues are positive.
    C : If the determinant of the matrix is positive, all its eigenvalues are positive.
    D : If the product of the trace and determinant of the matrix is positive, all its eigenvalues are positive.


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     Correct answer is :A

     Solution :
      If the trace of the matrix is positive and the determinant of the matrix is negative then atleast one of its eigen values is negative.
    Since determinant = product of eigen values.

  •   Question 30

    If V1 and V2 are 4-dimensional subspaces of a 6-dimensional vector space V, then the smallest possible dimension of V1 ∧ v2 is _______.



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     Correct answer is :2

     Solution :
      Let the basis of 6-dimensional vector space be {e1, e2, e3,e4, e5, e6}. In order for V1  ∧ V2 to have smallest possible dimension V1 and V2 could be, say, {e1, e2, e3,e4} and {e3, e4, e5, e6} respectively. The basis of V1 ∧ V2 would then be {e3, e4}. => Smallest possible dimension = 2.

  •   Question 31

    Find the value of k





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     Correct answer is :4


  •   Question 32

    Let A be a square matrix size n × n. Consider the following pseudocode. What is the expected output?
    C = 100;
    for i = 1 to n do
    for j = 1 to n do
    {
    Temp = A[ i ] [ j ] + C ;
    A [ i ] [ j ] = A [ j ] [ i ] ;
    A [ j ] [ i ] = Temp – C ;
    }
    for i = 1 to n do
    for j = 1 to n do
    output (A[ i ] [ j ]);


    A : The matrix A itself
    B : Transpose of the matrix A
    C : Adding 100 to the upper diagonal elements and subtracting 100 from lower diagonal elements of A
    D : None of these


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     Correct answer is :A

     Solution :
      In the computation of given pseudo code for each row and column of Matrix A, each upper triangular element will be interchanged by its mirror image in the lower triangular and after that the same lower triangular element will be again re-interchanged by its mirror image in the upper triangular, resulting the final computed Matrix A same as input Matrix A.

  •   Question 33

    With respect to the numerical evaluation of the definite integral, K= ∫ba  x2 dx, where a and b are given, which of the following statements is/are TRUE?
    (I) The value of K obtained using the trapezoidal rule is always greater than or equal to the exact value of the definite integral.
    (II) The value of K obtained using the Simpson’s rule is always equal to the exact value of the definite integral.


    A : I only
    B : II only
    C : Both I and II
    D : Neither I and II


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     Correct answer is :C


  •   Question 34

    The value of the integral given below is



    A : -2π
    B : π
    C :
    D :


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     Correct answer is :A


  •   Question 35

    Let S be a sample space and two mutually exclusive events A and B be such that A ∪ B = S. If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is ______ A



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     Correct answer is :0.25


  •   Question 36

    Consider the set of all functions f :{0,1,...,2014}->{0,1...,2014} such that f (f (i)) = i, for 0 <= i<= 2014 .
    Consider the following statements.
    P. For each such function it must be the case that for every i, f(i) = i,
    Q. For each such function it must be the case that for some i,f(i) = i,
    R. Each such function must be onto.
    Which one of the following is CORRECT?


    A : P, Q and R are true
    B : Only Q and R are true
    C : Only P and Q are true
    D : Only R is true


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     Correct answer is :B

     Solution :
      Let us consider a function (counter example) as
    f(0)=1,f(1) =0,f(2)=3,f(3)=2,.........,f(2012)=2013,f(2013)=2012 andf(2014)=2014
    Clearly f (f (i))= i for 0<= i <=2014
    Here f(i) ≠ i for every i and f(i) = i for some i
    Also f is onto
    Hence , only Q and R are true

  •   Question 37

    There are two elements x,y in a group (G,*) such that every element in the group can be written as a product of some number of x’s and y’s in some order. It is known that
    x * x = y * y = x * y *x * y = y* x * y *x = e
    where e is the identity element. The maximum number of elements in such a group is _________________.




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     Correct answer is :4


  •   Question 38

    If G is a forest with n vertices and k connected components, how many edges does G have?

    A : [n / k]
    B : [n / k]
    C : n-k
    D : n-k+1


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     Correct answer is :C

     Solution :
      Let n1 ,n2 ,.....nk be the number of vertices respectively in K connected components of a forest G, then n1 ?1,n2 ?1,.....,nk ?1 be the number of edges respectively in K connected components and n1 + n2 + ..... + nk = n (number of vertices in G)
    Hence, number of edges in G = number of edges in K connected components
    (n1-1)+(n2-1)+..........+(nk-1) = n-k

  •   Question 39

    Let d denote the minimum degree of a vertex in a graph. For all planar graphs on n vertices with d>=3, which one of the following is TRUE?

    A : In any planar embedding, the number of faces is at least n/2 +2
    B : In any planar embedding, the number of faces is less than n/2 +2
    C : There is a planar embedding in which the number of faces is less than n /2 +2
    D : There is a planar embedding in which the number of faces is at most n /d +1


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     Correct answer is :A

     Solution :
      We know that v + r = e + 2e = n + r ? 2 ...(1)
    Where V= n(number of vertices); r =number of faces and e =number of edges
    Given d>=3 then 3n <=2e
    e>=3n/2
    n+r-2>=3n/2 (using(1))
    r>=3n/2 -n +2 => r>= n/2 +2
    No.of faces is atleast n/2 + 2

  •   Question 40

    The CORECT formula for the sentence, “not all rainy days are cold” is

    A : ∀d (Rainy(d) ∧ ~Cold(d))
    B : ∀d ( ~Rainy(d)->Cold(d))
    C : ∃d(~Rainy(d) -> Cold(d))
    D : ∃d (Rainy(d) ∧ Cold(d))


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     Correct answer is :D


  • MY REPORT
    TOTAL = 40
    ANSWERED =
    CORRECT / TOTAL = /40
    POSITIVE SCORE =
    NEGATIVE SCORE =
    FINAL SCORE =