Consider the following C function.

float f(float x, int y)
{
  float p, s; int i;
  for (s=1, p=1, i=1; i < y; i ++)
  {
    p*= x/i;
    s+=p;
  }
  return s;
}  

For large values of y, the return value of the function f best approximates

Assume the following C variable declaration

int *A [10], B[10][10]; 

Of the following expressions

I. A[2]     II. A[2][3]     III. B[1]     IV. B[2][3]  

which will not give compile-time errors if used as left hand sides of assignment statements in a C program?

Let P(E) denote the probability of the event E. Given P(A) = 1, P(B) = 1/2, the values of P(A|B) and P(B|A) respectively are

Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each is sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A?

n couples are invited to a party with the condition that every husband should be accompanied by his wife. However, a wife need not be accompanied by her husband. The number of different gatherings possible at the party is

Let T(n) be the number of different binary search trees on n distinct elements. Then , where x is

Consider the set ∑* of all strings over the alphabet ∑ = {0, 1}. ∑* with the concatenation operator for strings

Let G be an arbitrary graph with n nodes and k components. If a vertex is removed from G, the number of components in the resultant graph must necessarily lie between

Assuming all numbers are in 2's complement representation, which of the following numbers is divisible by 11111011?

For a pipelined CPU with a single ALU, consider the following situations

I. The j + 1-st instruction uses the result of the j-th instruction as an operand
II. The execution of a conditional jump instruction
III. The j-th and j + 1-st instructions require the ALU at the same time 

Which of the above can cause a hazard?

Consider an array multiplier for multiplying two n bit numbers. If each gate in the circuit has a unit delay, the total delay of the multiplier is

Ram and Shyam have been asked to show that a certain problem Π is NP-complete. Ram shows a polynomial time reduction from the 3-SAT problem to Π, and Shyam shows a polynomial time reduction from Π to 3-SAT. Which of the following can be inferred from these reductions?

Nobody knows yet if P = NP. Consider the language L defined as follows:

Which of the following statements is true?

The regular expression 0*(10*)* denotes the same set as

If the strings of a language L can be effectively enumerated in lexicographic (i.e., alphabetic) order, which of the following statements is true?

Which of the following suffices to convert an arbitrary CFG to an LL(1) grammar?

Assume that the SLR parser for a grammar G has n₁ states and the LALR parser for G has n₂ states. The relationship between n₁ and n₂ is:

In a bottom-up evaluation of a syntax directed definition, inherited attributes can

Suppose the numbers 7, 5, 1, 8, 3, 6, 0, 9, 4, 2 are inserted in that order into an initially empty binary search tree. The binary search tree uses the usual ordering on natural numbers. What is the in-order traversal sequence of the resultant tree?

Consider the following three claims

I. (n + k)m = Θ(nm), where k and m are constants
II. 2n+1 = O(2n)
III. 22n+1 = O(2n) 

Which of these claims are correct?

Consider the following graph

Among the following sequences:

(I) a b e g h f    (II) a b f e h g    (III) a b f h g e    (IV) a f g h b e 

Which are depth first traversals of the above graph?

The usual Θ(n²) implementation of Insertion Sort to sort an array uses linear search to identify the position where an element is to be inserted into the already sorted part of the array. If, instead, we use binary search to identify the position, the worst case running time will

In a heap with n elements with the smallest element at the root, the 7th smallest element can be found in time

Which of the following statements is FALSE?

Using a larger block size in a fixed block size file system leads to:

In a system with 32 bit virtual addresses and 1 KB page size, use of one-level page tables for virtual to physical address translation is not practical because of

Which of the following assertions is FALSE about the Internet Protocol (IP)?

Which of the following functionalities must be implemented by a transport protocol over and above the network protocol?

Which of the following scenarios may lead to an irrecoverable error in a database system?

Consider the following SQL query

   select distinct al, a2,........., an
   from rl, r2,........, rm
   where P 

For an arbitrary predicate P, this query is equivalent to which of the following relational algebra expressions?

Let (S, ≤) be a partial order with two minimal elements a and b, and a maximum element c. Let P: S → {True, False} be a predicate defined on S. Suppose that P(a) = True, P(b) = False and P(x) ⇒ P(y) for all x, y ∈ S satisfying x ≤ y, where ⇒ stands for logical implication. Which of the following statements CANNOT be true?

Which of the following is a valid first order formula? (Here α and β are first order formulae with x as their only free variable)

Consider the following formula a and its two interpretations I₁ and I₂

α: (∀x)[Px ⇔ (∀y)[Qxy ⇔ ¬Qyy]] ⇒ (∀x)[¬Px]
I1: Domain: the set of natural numbers
    Px ≡ 'x is a prime number'
    Qxy ≡ 'y divides x'
I2: same as I1 except that Px = 'x is a composite number'.

Which of the following statements is true?

m identical balls are to be placed in n distinct bags. You are given that m ≥ kn, where, k is a natural number ≥1. In how many ways can the balls be placed in the bags if each bag must contain at least k balls?

Consider the following recurrence relation

T(1) = 1 
T(n+1) = T(n) + ⌊√n+1⌋ for all n≥1 

The value of T(m²) for m ≥ 1 is

How many perfect matching are there in a complete graph of 6 vertices ?

Let f : A → B be an injective (one-to-one) function. Define g: 2^A→2^B as: g(C) = {f(x)|x ∈ C}, for all subsets C of A.
Define h: 2^B → 2^A as: h(D) = {x|x ∈ A, f(x) ∈ D}, for all subsets D of B.

Which of the following statements is always true?

Consider the set {a, b, c} with binary operators + and × defined as follows:

+	a	b	c		×	a	b	c
a	b	a	c		a	a	b	c
b	a	b	c		b	b	c	a
c	a	c	b		c	c	c	b 

For example, a + c = c, c + a = a, c × b = c and b × c = a. Given the following set of equations:

(a × x) + (a × y) = c
(b × x) + (c × y) = c 

The number of solution(s) (i.e., pair(s) (x, y)) that satisfy the equations is:

Let ∑ = (a, b, c, d, e) be an alphabet. We define an encoding scheme as follows:

g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11.

Let p_i denote the i-th prime number (p₁=2).

For a non-empty string s=a₁ ... a_n, where each a_i∈Σ, define
For a non-empty sequence 〈s_j...s_n〉 of strings from ∑⁺, define

Which of the following numbers is the encoding h of a non-empty sequence of strings?

A graph G = (V,E) satisfies |E|≤ 3|V|-6. The min-degree of G is defined as . Therefore, min-degree of G cannot be

Consider the following system of linear equation

Notice that the second and the third columns of the coefficient matrix are linearly dependent. For how many values of α, does this system of equations have infinitely many solutions?

A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).

If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be

The following is a scheme for floating point number representation using 16 bits.

Let s, e, and m be the numbers represented in binary in the sign, exponent, and mantissa fields respectively. Then the floating point number represented is:

What is the maximum difference between two successive real numbers representable in this system?

A 1-input, 2-output synchronous sequential circuit behaves as follows:

Let z_k, n_k denote the number of 0's and 1's respectively in initial k bits of the input (z_k + n_k = k). The circuit outputs 00 until one of the following conditions holds.

•   zk - nk = 2. In this case, the output at the k-th and all subsequent clock ticks is 10.
•   nk - zk = 2. In this case, the output at the k-th and all subsequent clock ticks is 01. 

What is the minimum number of states required in the state transition graph of the above circuit?

The literal count of a boolean expression is the sum of the number of times each literal appears in the expression. For example, the literal count of (xy + xz') is 4. What are the minimum possible literal counts of the product-of-sum and sum-of-product representations respectively of the function given by the following Karnaugh map ? Here, X denotes "don't care"

Consider the ALU shown below.

If the operands are in 2's complement representation, which of the following operations can be performed by suitably setting the control lines K and C₀ only (+ and - denote addition and subtraction respectively)?

Consider the following circuit composed of XOR gates and non-inverting buffers.

The non-inverting buffers have delays δ₁ = 2 ns and δ₂ = 4 ns as shown in the figure. Both XOR gates and all wires have zero delay. Assume that all gate inputs, outputs and wires are stable at logic level 0 at time 0. If the following waveform is applied at input A, how many transition(s) (change of logic levels) occur(s) at B during the interval from 0 to 10 ns?

Consider the following assembly language program for a hypothetical processor. A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.

     MOV B, # 0   	;   B ← 0
     MOV C, # 8	        ;   C ← 8
Z :  CMP C, # 0	        ;   compare C with 0
     JZX	        ;   jump to X if zero flag is set
     SUB C, # 1	        ;   C ← C - 1
     RRC A, # 1 	;   right rotate A through carry by one bit. Thus:
                        ;   if the initial values of A and the carry flag are a7...a0 and
                        ;   c0 respectively, their values after the execution of this
                        ;   instruction will be c0a7...a1 and a0 respectively.
     JC Y	        ;   jump to Y if carry flag is set
     JMP Z	        ;   jump to Z
Y :  ADD B, # 1	        ;   B ← B + 1
     JMP Z	        ;   jump to Z
X :	

If the initial value of register A is A₀ the value of register B after the program execution will be

Consider the following assembly language program for a hypothetical processor. A, B, and C are 8 bit registers. The meanings of various instructions are shown as comments.

     MOV B, # 0   	;   B ← 0
     MOV C, # 8	        ;   C ← 8
Z :  CMP C, # 0	        ;   compare C with 0
     JZX	        ;   jump to X if zero flag is set
     SUB C, # 1	        ;   C ← C - 1
     RRC A, # 1 	;   right rotate A through carry by one bit. Thus:
                        ;   if the initial values of A and the carry flag are a7...a0 and
                        ;   c0 respectively, their values after the execution of this
                        ;   instruction will be c0a7...a1 and a0 respectively.
     JC Y	        ;   jump to Y if carry flag is set
     JMP Z	        ;   jump to Z
Y :  ADD B, # 1	        ;   B ← B + 1
     JMP Z	        ;   jump to Z
X :	

Which of the following instructions when inserted at location X will ensure that the value of register A after program execution is the same as its initial value?

Consider the following deterministic finite state automaton M.

Let S denote the set of seven bit binary strings in which the first, the fourth, and the last bits are 1. The number of strings in S that are accepted by M is

Let G = ({S},{a,b},R,S) be a context free grammar where the rule set R is S → aSb | SS | ε Which of the following statements is true?

Consider two languages L₁ and L₂ each on the alphabet ∑. Let f: ∑ → ∑ be a polynomial time computable bijection such that (∀ x)[x ∈ L₁ iff f(x) ∈ L₂]. Further, let f^-1 be also polynomial time computable.

Which of the following CANNOT be true?

A single tape Turing Machine M has two states q₀ and q₁, of which q₀ is the starting state. The tape alphabet of M is {0, 1, B} and its input alphabet is {0, 1}. The symbol B is the blank symbol used to indicate end of an input string. The transition function of M is described in the following table.

The table is interpreted as illustrated below. The entry (q₁, 1, R) in row q₀ and column 1 signifies that if M is in state q₀ and reads 1 on the current tape square, then it writes 1 on the same tape square, moves its tape head one position to the right and transitions to state q₁.

Which of the following statements is true about M?

Define languages L₀ and L₁ as follows:

L0 = {〈M, w, 0〉 | M halts on w}
L1 = {〈M, w, 1〉 | M does not halts on w} 

Here 〈M, w, i〉 is a triplet, whose first component. M, is an encoding of a Turing Machine, second component, w, is a string, and third component, i, is a bit.

Let L = L₀∪L₁. Which of the following is true?

Consider the NFA M shown below.

Let the language accepted by M be L. Let L₁ be the language accepted by the NFA M₁, obtained by changing the accepting state of M to a non-accepting state and by changing the non-accepting state of M to accepting states. Which of the following statements is true?

Consider the grammar shown below

S → i E t S S' | a
S' → e S | ε
E → b 

In the predictive parse table. M, of this grammar, the entries M[S', e] and M[S', $] respectively are

Consider the grammar shown below.

S → C C
C → c C | d

The grammar is

Consider the translation scheme shown below.

S → T R
R → + T {print ('+');} R|ε
T → num {print(num.val);}

Here num is a token that represents an integer and num.val represents the corresponding integer value. For an input string '9 + 5 + 2', this translation scheme will print

Consider the syntax directed definition shown below.

S → id := E           {gen (id.place = E.place;);}
E → E1 + E2           {t = newtemp ( ); 
                      gen(t = E1.place + E2.place;); 
                      E.place = t}
E → id                {E.place = id.place;}

Here, gen is a function that generates the output code, and newtemp is a function that returns the name of a new temporary variable on every call. Assume that ti's are the temporary variable names generated by newtemp. For the statement 'X: = Y + Z', the 3-address code sequence generated by this definition is

A program consists of two modules executed sequentially. Let f₁(t) and f₂(t) respectively denote the probability density functions of time taken to execute the two modules. The probability density function of the overall time taken to execute the program is given by:

In a permutation a₁...a_n of n distinct integers, an inversion is a pair (a_i, a_j) such that i < j and a_i > a_j.

If all permutations are equally likely, what is the expected number of inversions in a randomly chosen permutation of 1...n ?

In a permutation a₁...a_n of n distinct integers, an inversion is a pair (a_i, a_j) such that ii > a_j.

What would be the worst case time complexity of the Insertion Sort algorithm, if the inputs are restricted to permutations of 1...n with at most n inversions?

A data structure is required for storing a set of integers such that each of the following operations can be done in O(log n) time, where n is the number of elements in the set.

I. Deletion of the smallest element
II. Insertion of an element if it is not already present in the set

Which of the following data structures can be used for this purpose?

Let S be a stack of size n≥1. Starting with the empty stack, suppose we push the first n natural numbers in sequence, and then perform n pop operations. Assume that Push and pop operation take X seconds each, and Y seconds elapse between the end of one such stack operation and the start of the next operation. For m≥1, define the stack-life of m as the time elapsed from the end of Push(m) to the start of the pop operation that removes m from S. The average stack-life of an element of this stack is

Consider the following 2-3-4 tree (i.e., B-tree with a minimum degree of two) in which each data item is a letter. The usual alphabetical ordering of letters is used in constructing the tree.

What is the result of inserting G in the above tree?

The cube root of a natural number n is defined as the largest natural number m such that m³≤n. The complexity of computing the cube root of n (n is represented in binary notation) is:

Let G = (V, E) be an undirected graph with a subgraph G₁ = (V₁, E₁). Weights are assigned to edges of G as follows:

A single-source shortest path algorithm is executed on the weighted graph (V,E,w) with an arbitrary vertex ν₁ of V₁ as the source. Which of the following can always be inferred from the path costs computed?

What is the weight of a minimum spanning tree of the following graph ?

The following are the starting and ending times of activities A, B, C, D, E, F, G and H respectively in chronological order: "a_sb_sc_sa_ed_sc_ee_sf_sb_ed_eg_se_ef_eh_sg_eh_e" Here, x_s denotes the starting time and x_e denotes the ending time of activity X. We need to schedule the activities in a set of rooms available to us. An activity can be scheduled in a room only if the room is reserved for the activity for its entire duration. What is the minimum number of rooms required?

Let G = (V,E) be a directed graph with n vertices. A path from v_i to v_j in G is sequence of vertices (v_i, v_i+1, ......., v_j) such that (v_k, v_k+1) ∈ E for all k in i through j-1. A simple path is a path in which no vertex appears more than once. Let A be an nxn array initialized as follow.

Consider the following algorithm.

for i = 1 to n
   for j = 1 to n
      for k = 1 to n
         A [j,k] = max (A[j,k] (A[j,i] + A [i,k]);

Which of the following statements is necessarily true for all j and k after terminal of the above algorithm?

Consider the following logic program P

A(x) <- B(x, y), C(y) 
     <- B(x,x) 

Which of the following first order sentences is equivalent to P?

The following resolution rule is used in logic programming.

Derive clause (P ∨ Q) from clauses (P ∨ R), (Q ∨ ¬R) 

Which of the following statements related to this rule is FALSE?

The following program fragment is written in a programming language that allows variables and does not allow nested declarations of functions.

global int i = 100, j = 5;
void P(x)
{
    int i = 10;
    print(x + 10);
    i = 200;
    j = 20;
    print(x);
}
main()
{
    P(i + j);
}

If the programming language uses static scoping and call by need parameter passing mechanism, the values printed by the above program are

The following program fragment is written in a programming language that allows variables and does not allow nested declarations of functions.

global int i = 100, j = 5;
void P(x)
{
    int i = 10;
    print(x + 10);
    i = 200;
    j = 20;
    print(x);
}
main()
{
    P(i + j);
}

If the programming language uses dynamic scoping and call by name parameter passing mechanism, the values printed by the above program are:

Consider the following class definitions in a hypothetical Object Oriented language that supports inheritance and uses dynamic binding. The language should not be assumed to be either Java or C++, though the syntax is similar.

Class P {
    void f(int i) {
        print(i);
    }
}

Class Q subclass of P {
    void f(int i) {
        print(2*i);
    }
} 

Now consider the following program fragment:

Px = new Q();
Qy = new Q();
Pz = new Q();
x.f(1); ((P)y).f(1); z.f(1); 

Here ((P)y) denotes a typecast of y to P. The output produced by executing the above program fragment will be

Which of the following is NOT an advantage of using shared, dynamically linked libraries as opposed to using statically linked libraries?

A uni-processor computer system only has two processes, both of which alternate 10ms CPU bursts with 90ms I/O bursts. Both the processes were created at nearly the same time. The I/O of both processes can proceed in parallel. Which of the following scheduling strategies will result in the least CPU utilization (over a long period of time) for this system?

A processor uses 2-level page tables for virtual to physical address translation. Page tables for both levels are stored in the main memory. Virtual and physical addresses are both 32 bits wide. The memory is byte addressable. For virtual to physical address translation, the 10 most significant bits of the virtual address are used as index into the first level page table while the next 10 bits are used as index into the second level page table. The 12 least significant bits of the virtual address are used as offset within the page. Assume that the page table entries in both levels of page tables are 4 bytes wide. Further, the processor has a translation look-aside buffer (TLB), with a hit rate of 96%. The TLB caches recently used virtual page numbers and the corresponding physical page numbers. The processor also has a physically addressed cache with a hit rate of 90%. Main memory access time is 10 ns, cache access time is 1 ns, and TLB access time is also 1 ns.

Assuming that no page faults occur, the average time taken to access a virtual address is approximately (to the nearest 0.5 ns)

A processor uses 2-level page tables for virtual to physical address translation. Page tables for both levels are stored in the main memory. Virtual and physical addresses are both 32 bits wide. The memory is byte addressable. For virtual to physical address translation, the 10 most significant bits of the virtual address are used as index into the first level page table while the next 10 bits are used as index into the second level page table. The 12 least significant bits of the virtual address are used as offset within the page. Assume that the page table entries in both levels of page tables are 4 bytes wide. Further, the processor has a translation look-aside buffer (TLB), with a hit rate of 96%. The TLB caches recently used virtual page numbers and the corresponding physical page numbers. The processor also has a physically addressed cache with a hit rate of 90%. Main memory access time is 10 ns, cache access time is 1 ns, and TLB access time is also 1 ns.

Suppose a process has only the following pages in its virtual address space: two contiguous code pages starting at virtual address 0x00000000, two contiguous data pages starting at virtual address 0×00400000, and a stack page starting at virtual address 0×FFFFF000. The amount of memory required for storing the page tables of this process is:

Suppose we want to synchronize two concurrent processes P and Q using binary semaphores S and T. The code for the processes P and Q is shown below.

Process P:
while (1) {
W:
   print '0';
   print '0';
X:
}
	
Process Q:
while (1) {
Y:
   print '1';
   print '1';
Z:
}

Synchronization statements can be inserted only at points W, X, Y and Z.

Which of the following will always lead to an output staring with '001100110011'?

Suppose we want to synchronize two concurrent processes P and Q using binary semaphores S and T. The code for the processes P and Q is shown below.

Process P:
while (1) {
W:
   print '0';
   print '0';
X:
}
	
Process Q:
while (1) {
Y:
   print '1';
   print '1';
Z:
}

Synchronization statements can be inserted only at points W, X, Y and Z.

Which of the following will ensure that the output string never contains a substring of the form 01ⁿ0 or 10ⁿ1 where n is odd?

The subnet mask for a particular network is 255.255.31.0. Which of the following pairs of IP addresses could belong to this network?

A 2 km long broadcast LAN has 10⁷ bps bandwidth and uses CSMA/CD. The signal travels along the wire at 2×10⁸ m/s. What is the minimum packet size that can be used on this network?

Host A is sending data to host B over a full duplex link. A and B are using the sliding window protocol for flow control. The send and receive window sizes are 5 packets each. Data packets (sent only from A to B) are all 1000 bytes long and the transmission time for such a packet is 50 µs. Acknowledgement packets (sent only from B to A) are very small and require negligible transmission time. The propagation delay over the link is 200 µs. What is the maximum achievable throughput in this communication?

Consider the following functional dependencies in a database:

  Data_of_Birth → Age
  Age → Eligibility
  Name → Roll_number
  Roll_number → Name
  Course_number → Course_name
  Course_number → Instructor
  (Roll_number, Course_number) → Grade 

The relation (Roll_number, Name, Date_of_birth, Age) is:

Consider the set of relations shown below and the SQL query that follows.

  Students: (Roll_number, Name, Date_of_birth)
  Courses: (Course number, Course_name, Instructor)
  Grades: (Roll_number, Course_number, Grade)
  select distinct Name
  from Students, Courses, Grades
  where Students. Roll_number = Grades.Roll_number
      and Courses.Instructor = Korth
      and Courses.Course_number = Grades.Course_number
      and Grades.grade = A 

Which of the following sets is computed by the above query?

Consider three data items D1, D2 and D3 and the following execution schedule of transactions T1, T2 and T3. In the diagram, R(D) and W(D) denote the actions reading and writing the data item D respectively.

Which of the following statements is correct?

In the following C program fragment, j, k n and TwoLog_n are interger variables, and A is an array of integers. The variable n is initialized to an integer ≥3, and TwoLog_n is initialized to the value of 2*⌈log₂(n)⌉

   for (k = 3; k < = n; k++)
               A[k] = 0;
   for (k = 2; k < = TwoLog_n; k++)
          for (j = k + 1; j < = n; j++)
                    A[j] = A[j] || (j%k);
   for (j = 3; j < = n; j++)
          if (!A[j]) printf("%d", j); 

The set of numbers printed by this program fragment is

Consider the C program shown below.

#include 
#define print(x) printf("%d", x)
int x;
void Q(int z)
{
    z += x;
    print(z);
}
void P(int *y)
{
    int x = *y + 2;
    Q(x);
    *y = x - 1;
    print(x);
}
main(void)
{
    x = 5;
    P(&x);
    print(x);
} 

The output of this program is

Consider the function f defined below.

    struct item {
        int data;
        struct item * next;
};
int f(struct item *p) {
    return ((p == NULL) || (p->next == NULL) ||
            ((P->data <= p->next->data) &&
            f(p->next)));
} 

For a given linked list p, the function f returns 1 if and only if