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SET 1


  Question 1

In the following LU decomposition of the matrix , if the diagonal elements of U are both 1, then the lower diagonal entry 122 of L is





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     Correct answer is :5


  •   Question 2

    limx→ ∞ x 1/x

    A :
    B : 0
    C : 1
    D : Not defined


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     Correct answer is :C


  •   Question 3

    For a set A, the power set of A is denoted by 2A. If A = {5, {6}, {7}}, which of the following options are True?
    1. ∅ ∈ 2A
    2. ∅ ⊆ 2A
    3. {5,(6)} ∈ 2 A
    4. {5,(6)} ⊆ 2A


    A : 1 and 3 only
    B : 2 and 3 only
    C : 1,2 and 3 only
    D : 1,2 and 4 only


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     Correct answer is :C


  •   Question 4

    Which one of the following is Not equivalent to p ↔ q?

    A : (¬p ∨ q) ∧(p ∨ ¬q)
    B : (¬p ∨ q) ∧ (q -> p)
    C : (¬p ∧ q) ∨ (p ∧ ¬q)
    D : (¬p ∧ ¬q) ∨ (p ∧ q)


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     Correct answer is :C


  •   Question 5





    A : h(x)/g(x)
    B : -1/x
    C : g(x)/h(x)
    D : x/(1-x)2


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     Correct answer is :A


  •   Question 6

    Suppose L = {p, q, r, s, t} is a lattice represented by the following Hasse diagram.For any xy?L, not necessarily distinct, x ? y and x ? y are join and meet of x, y respectively. Let L3= {(x,y,z): x, y, z ? L} be the set of all ordered triplets of the elements of L. Let pr be the probability that an element (x,y,z) | L3 chosen equiprobably satisfies x (y z) = (x y) (x z). Then



    A : pr = 0
    B : pr = 1
    C : 0 < pr <= 1/5
    D : 1/5 < pr <1


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     Correct answer is :D


  •   Question 7

    Consider the following 2 × 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are –1 and 7. What are the values of a and b?



    A : a = 6,b = 4
    B : a = 4,b = 6
    C : a = 3,b = 5
    D : a = 5,b = 3


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     Correct answer is :D

     Solution :
      Given λ1 = -1 and λ2 = 7 are eigen values of A
    By properties, λ1 + λ2 = sum of diagonal and λ1* λ2= determinant of A
    => 6=1+a and -7 = a-4b
    =>a=5 => -7 = 5-4b
    =>b=3

  •   Question 8







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     Correct answer is :-1


  •   Question 9

    Let G be a connected planar graph with 10 vertices. If the number of edges on each face is three, then the number of edges in G is _______________.



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     Correct answer is :24

     Solution :
      By euler's formula :
    |V|+|R| = |E|+2 _________(1) where |V|, |E|, |R| are respectively number of vertices, edges and faces (regions)
    Given |V| = 10 _______(2) and number of edges on each face is three
    3|R| = 2|E| => |R| = (2/3) |E| __________ (3)
    substituting 2 , 3 in 1 ,we get
    10 + (2/3)|E| = |E| +2 => (|E| /3) = 8 => |E| =24

  •   Question 10







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     Correct answer is :0.99


  •   Question 11

    Let R be the relation on the set of positive integers such that a aRb if and only if a and b are distinct and have a common divisor other than 1. Which one of the following statements about R is true?

    A : R is symmetric and reflexive but not transitive
    B : R is reflexive but not symmetric and not transitive
    C : R is transitive but not reflexive and not symmetric
    D : R is symmetric but not reflexive and not transitive


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     Correct answer is :D

     Solution :
      R is not reflexive as each element can?t be related to itself.
    R is symmetric
    Let a = 3, b = 6 and c = 10 then 3 and 6 have a common division other than 1
    6 and 10 have a common division other than 1
    but 3 &10 have no common division other than 1
    3R6 and 6R10 but 3not related10
    => R is not transitive

  •   Question 12

    Consider the following two statements.
    S1 : if a candidate is known to be corrupt, then he will not be elected
    S2 : if a candidate is kind, he will be elected
    Which one of the following statements follows from S1 and S2 per sound interference rules of logic?


    A : If a person is known to corrupt, he is kind
    B : If a person is not known to be corrupt, he is not kind
    C : If a person is kind, he is not known to be corrupt
    D : If a person is not kind, he is not known to be corrupt


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     Correct answer is :C

     Solution :
      Let P: candidate known to be corrupt
    q: candidate will be elected
    r: candidate is kind
    then S1 = p -> ~q
    =q -> ~p (contrapoitive rule)
    and s2 : r -> q => r -> ~p (transitive rule )
    i.e., If a person is kind, he is not known to be corrupt
    Option is C

  •   Question 13

    The larger of the two eigenvalues of the matrix is ____________.





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     Correct answer is :6

     Solution :
      Characteristic equation is :
    λ2 - 5 λ -6 = 0 => (λ-6) (λ+1) = 0
    => λ = 6,-1
    Larger eigen value is 6

  •   Question 14

    The cardinality of the power set of {0, 1, 2,.......10} is _________



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     Correct answer is :2048

     Solution :
      cardinality of the power set of {0, 1, 2, … , 10} is 112 i.e.., 2048

  •   Question 15

    The number of divisors of 2100 is _______.



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     Correct answer is :36

     Solution :
      Let N =2100
    22 * 3*52 * 7 (i.e.,product of primes)
    Then the number of division of 2100 is
    (2+1).(1+1)e. 36.(2+1).(1+1) i.e. (3)(2)(3)(2) i.e. 36

  •   Question 16

    In a connected graph, a bridge is an edge whose removal disconnects a graph. Which one of the following statements is true?

    A : A tree has no bridges
    B : A bridge cannot be part of a simple cycle
    C : Every edge of a clique with size >= 3 is a bridge (A clique is any compete sub graph of a graph)
    D : A graph with bridges cannot have a cycle


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     Correct answer is :B

     Solution :
      Since, every edge in a tree is bridge (A) is false
    Since, every edge in a complete graph k n(n>= 3) is not a bridge =>c is false
    Since, in a cycle every edge is not a bridge => (B) is true

  •   Question 17

    Let f(X) = x-(1/3) and A denote the area of the region bounded by f(x) and the X-axis, when x varies from – 1 to 1. Which of the following statements is/are TRUE?
    (I) f is continuous in [-1,1]
    (II) f is not bounded in [-1,1]
    (III) A is nonzero and finite


    A : II only
    B : III only
    C : II and III only
    D : I, II and III


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     Correct answer is :C


  •   Question 18

    The number of onto function (surjective function) from set X = {1,2,3,4}to set Y ={a,b,c} is ______.



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     Correct answer is :36


  •   Question 19

    Consider two relations R1(A,B) with the tuples (1.5), (3,7) and R2 (A,C) = (1,7), (4,9). Assume that R(A,B,C) is the full natural outer join of R1 and R2 . Consider the following tuples of the form (A,B,C): a =(1.5,null), b=(1,null,7) c=(3,null,9), d=(4,7,null), e =(1,5,7), f=(3,7,null), g =(4,null,9).Which one of the following statements is correct?

    A : R contains a,b,e,f,g but not c, d.
    B : R contains all of a,b,c,d,e,f,g
    C : R contains e,f,g but not a,b
    D : R contains e but not f,g.


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     Correct answer is :C


  •   Question 20

    A young tableau is a 2D array of integers increasing from left to right and from top to bottom. Any unfilled entries are marked with ∞ , and hence there cannot be any entry to the right of , or below a . The following Young tableau consists of unique entries.
    1 2 5 14
    3 4 6 23
    10 12 18 25
    31

    When an element is removed from a Young tableau, other elements should be moved into its place so that the resulting table is still a Young tableau (unfilled entries may be filled in with a ∞ ). The minimum number of entries (other than 1) to be shifted, to remove 1 from the given Young tableau is _________




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     Correct answer is :5



  •   Question 21

    Perform the following operations on the matrix
    (i) Add the third row to the second row
    (ii) Subtract the third column from the first column.
    The determinant of the resultant matrix is __________.






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     Correct answer is :0

     Solution :
      Determinant is unaltered by the operations (i) and (ii)
    => Determinant of the resultant matrix = Determinant of the given matrix

  •   Question 22

    Which one of the following well formed formulae is tautology?



    A : A only
    B : B only
    C : C only
    D : D only


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     Correct answer is :C


  •   Question 23

    A graph is self-complementary if it is isomorphic to its complement For all selfcomplementary graphs on n vertices, n is

    A : A multiple of 4
    B : Even
    C : Odd
    D : Congruent to 0 mod 4, or, 1 mod 4


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     Correct answer is :D

     Solution :
      An n vertex self complementary graph has exactly half number of edges of the complete graph i.e., n(n-1)/4 edges . Since n(n-1) must be divisible by 4 , n must be congruent to 0 or 1 module 4.

  •   Question 24

    The secant method is used to find the root of an equation f(x) = 0 . It is started from two distinct estimates, a b x and x for the root. It is an iterative procedure involving linear interpolation to a root. The iteration stops if f(xb) is very small and then xb is the solution. The procedure is given below. Observe that there is an expression which is missing and is marked by? Which is the suitable expression that is to be put in place of ? so that it follows all steps of the secant method?
    Secant
    Initialize: xa, xb, ,N // ε = convergence indicator and N = maximum no. of iterations
    fb= -f(xb)
    i=0
    while(i ( ε) do
    i = i + 1 //update counter
    xt = ? //missing expression for
    xa=xb // intermediate value and reset xa
    xb=xt //reset xb
    fb = f(xb) // function value at new xb
    end while
    if |fb| > ε then //loop is terminated with i=N
    write “Non-convergence”
    else

    A : xb - (fb - f(xa ) fb / (xb - xa)
    B : xa - (fb - f(xa ) fa / (xb - xa)
    C : xb - (xb - xa)f(xb ) / fb / (fb -
    D : xa - (xb - xa)f(xa ) / fb / (fb -


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     Correct answer is :D

     Solution :
      Secant method direct formula

  •   Question 25

    Let X and Y denote the sets containing 2and 20 distinct objects respectively and F denote the set of all possible functions defined from X to Y. let f be randomly chosen from F .The probability of f being one-to-one is _______.



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     Correct answer is :0.95

     Solution :
      |X| =2 , |Y| =20
    Number of functions from X to Y is 202i.e., 400 and number of one-one functions from X to Y is 20P2 i.e 20 * 19 =380
    Probability of a function f being one-one is 380/400 = 0.95

  •   Question 26

    In a room there are only two types of people, namely Type 1 and Type 2. Type 1 people always tell the truth and Type 2 people always lie. Your give a fair coin to a person in that room, without knowing which type he is from and tell him to loss it and hide the result from you till you ask for it. Upon asking, the person replies the following.
    “The result of the toss is head if and only if I am telling the truth.”
    Which of the following options are correct?


    A : The result is head
    B : The result is tail
    C : If the person is of Type 2, then the result is tail
    D : If the person is of Type 1, then the result is tail


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     Correct answer is :C


  •   Question 27

    Suppose U is the power set of the set S = {1,2,3,4,5,6}. For any T # U, let |T| denote the number of element in T and T' denote the complement of T. For any T,R # U, let T\R be the set of all elements in T which are not in R. Which one of the following is true?

    A : ∀ X ∈ U ( |X| = |X|' )
    B : ∃ X isin;U ∃ Y∈U ( |X| =5, |Y|=5 and X ∩ Y = ∅
    C : ∀X ∈ U ∀ Y∈ U ( |X| =2, |Y| =3 and 3 and X \ Y =∅
    D : ∀X ∈ U ∀Y ∈ U(X \ Y = Y'\ X')


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     Correct answer is :D


  •   Question 28

    The value of given equation is



    A : 0
    B : 1/2
    C : 1
    D :


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     Correct answer is :A


  •   Question 29

    The number of 4 digit numbers having their digits in non-decreasing order (from left to right) constructed by using the digits belonging to the set {1,2,3} is _____ .



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     Correct answer is :15

     Solution :
      4-digit numbers with first digit ‘1’
    1111, 1112, 1113, 1122, 1123, 1133, 1222, 1223, 1233, 1333 i.e., 10
    4 digit numbers with first digit 2 : 2222, 2223, 2233, 2333 i.e, 4
    4 digit numbers with first digit 3: 3333 i.e, 1

  •   Question 30

    In the given matrix one of the eigen values is 1. The eigenvectors corresponding to the eigen value 1 are



    A : B :
    C : D :


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     Correct answer is :B


  •   Question 31

    Consider the equality Σ i=0n = X and the following choices for X
    I. θ (n4)
    II. θ ( n5)
    III.O(n5)
    IV. Ω (n3)
    The equality above remains correct if X is replaced by


    A : only I
    B : only II
    C : I or III or IV but not II
    D : II or III or IV but not I


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     Correct answer is :C

     Solution :
      X= sum of the cubes of first n natural numbers = (n2(n+1)2)/4

  •   Question 32

    In for non-zero x, af(X) + bf(1/x) = 1/x - 25 where a then 12 f(x)dx is

    A : (1/a2 - b2) [a(ln 2-25) + 47b/2 ]
    B : (1/a2 - b2) [a(2ln 2-25) - 47b/2 ]
    C : (1/a2 - b2) [a(2ln 2-25) + 47b/2 ]
    D : (1/a2 - b2) [a(ln 2-25) + 47b/2 ]


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     Correct answer is :A


  •   Question 33

    Suppose Xi for i=1,2,3 ar independent and identically distributed random variables whose probability mass functions are Pr [Xi = 0] = Pr [Xi =1] =1/ 2 for i =1,2,3. Define another random variable Y = X1 X2 X , where denotes XOR.Then Pr [Y = 0| X3 = 0] = _____ .



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     Correct answer is :0.75


  •   Question 34

    The velocity v (in kilometer/minute) of a motorbike which start from rest, is given at fixed intervals of time t (in minutes as follows.
    t2468101214161820
    10v182529322011520

    The approximate distance (in kilometers) rounded to two places of decimals covered in 20 minutes using Simpson’s 1/3rd rule is ________.




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     Correct answer is :309.33

     Solution :
      Let ‘S’ be the distance covered in 20 minutes, then by simpson’s 1/3rd rule
    = 2/3[(0+0)+4(10+25+32+11+2)+2(18+29+20+5)]
    =309.33km (Here length of each of the subinterval is h = 2)

  •   Question 35

    Let G be a connected undirected graph of 100 vertices and 300 edges. The weight of a minimum spanning tree of G is 500. When the weight of each edge of G is increased by five, the weight of a minimum spanning tree becomes __________.



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     Correct answer is :995

     Solution :
      G has 100 verticles => spanning tree contain 99 edges given, weight of a minimum spanning tree of G is 500 since, each edge of G is increased by five
    Weight of a minimum spanning tree becomes 500 + 5 × 99 = 995

  •   Question 36

    If the following system has non-trivial solution.
    px + qy + rz =0
    qx + ry + pz = 0
    rx + py + qz =0
    Then which one of the following options is TRUE?


    A : p ? q + r = 0 or p = q = ?r
    B : p + q ? r = 0 or p = ?q = r
    C : p + q + r = 0 or p = q = r
    D : p ? q + r = 0 or p = ?q = ?r


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     Correct answer is :A


  •   Question 37

    Let R be a relation on the set of ordered pairs of positive integers such that ( ( p , q ) , ( r , s ) ) ∈ R if and only if p - s = q - r. Which one of the following is true about R?

    A : Both reflexive and symmetric
    B : Reflexive but not symmetric
    C : Not reflexive but symmetric
    D : Neither reflexive nor symmetric


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     Correct answer is :C

     Solution :
      Since p- q q-p
    (p,q)R(p,q)
    R is not reflective
    Let (p,q) R (r,s) then p- s= q- r
    r-q=s-p
    (r,s) R (p,q)
    R is symmetric

  • MY REPORT
    TOTAL = 37
    ANSWERED =
    CORRECT / TOTAL = /37
    POSITIVE SCORE =
    NEGATIVE SCORE =
    FINAL SCORE =