GATE 1994
Question 1 
FORTRAN implementation do not permit recursion because
they use static allocation for variables  
they use dynamic allocation for variables  
stacks are not available on all machines  
it is not possible to implement recursion on all machines 
→ Recursion requires dynamic allocation of data.
Question 2 
Let A and B be real symmetric matrices of size n × n. Then which one of the following is true?
AA′ = 1  
A = A^{1}  
AB = BA  
(AB)' = BA 
Question 3 
Backward Euler method for solving the differential equation dy/dx = f(x,y) is specified by, (choose one of the following).
y_{n+1} = y_{n} + hf(x_{n}, y_{n})  
y_{n+1} = y_{n} + hf(x_{n+1}, y_{n+1})  
y_{n+1} = y_{n1} + 2hf(x_{n}, y_{n})  
y_{n+1} = (1 + h) f(x_{n+1}, y_{n+1}) 
With initial value y(x_{0}) = y_{0}. Here the function f and the initial data x_{0} and y_{0} are known. The function y depends on the real variable x and is unknown. A numerical method produces a sequence y_{0}, y_{1}, y_{2}, ....... such that y_{n} approximates y(x_{0} + nh) where h is called the step size.
→ The backward Euler method is helpful to compute the approximations i.e.,
y_{n+1} = y_{n} + hf(x _{n+1}, y_{n+1})
Question 4 
Let A and B be any two arbitrary events, then, which one of the following is true?
P(A∩B) = P(A)P(B)  
P(A∪B) = P(A) + P(B)  
P(AB) = P(A∩B)P(B)  
P(A∪B) ≤ P(A) + P(B) 
(B) Happens when A and B are mutually exclusive.
(C) Not happens.
(D) P(A∪B) ≤ P(A) + P(B) is true because P(A∪B) = P(A) + P(B)  P(A∩B).
Question 5 
An unrestricted use of the “goto” statement is harmful because
it makes it more difficult to verify programs  
it increases the running time of the programs  
it increases the memory required for the programs
 
it results in the compiler generating longer machine code 
Question 6 
The number of distinct simple graphs with upto three nodes is
15  
10  
7  
9 
Question 7 
The recurrence relation that arises in relation with the complexity of binary search is:
T(n) = T(n/2) + k, k a constant  
T(n) = 2T(n/2) + k, k a constant  
T(n) = T(n/2) + log n  
T(n) = T(n/2) + n 
∴ T(n) = 2T(n/2) + k, k a constant
Question 8 
The logic expression for the output of the circuit shown in figure below is:
None of the above. 
Question 10 
Some group (G,o) is known to be abelian. Then, which one of the following is true for G?
g = g^{1} for every g ∈ G  
g = g^{2} for every g ∈ G  
(goh)^{2} = g^{2}oh^{2} for every g,h ∈ G  
G is of finite order 
For Abelian group, commutative also holds
i.e., (aob) = (boa)
Consider option (C):
(goh)^{2} = (goh)o(gog)
= (hog)o(goh)
= (ho(gog)oh)
= ((hog^{2})oh)
= (g^{2}oh)oh
= g^{2}o(hoh)
= g^{2}oh^{2} [True]
Question 11 
In a compact single dimensional array representation for lower triangular matrices (i.e all the elements above the diagonal are zero) of size n × n, nonzero elements (i.e elements of the lower triangle) of each row are stored one after another, starting from the first row, the index of the (i, j)^{th} element of the lower triangular matrix in this new representation is:
i + j  
i + j  1  
j + i(i1)/2  
i + j(j1)/2 
If we assume array index starting from 1 then, i^{th} row contains i number of nonzero elements. Before i^{th} row there are (i1) rows, (1 to i1) and in total these rows has 1+2+3......+(i1) = i(i1)/2 elements.
Now at i^{th} row, the j^{th} element will be at j position.
So the index of (i, j)^{th} element of lower triangular matrix in this new representation is
j = i(i1)/2
Question 12 
Generation of intermediate code based on an abstract machine model is useful in compilers because
it makes implementation of lexical analysis and syntax analysis easier  
syntaxdirected translations can be written for intermediate code generation  
it enhances the portability of the front end of the compiler  
it is not possible to generate code for real machines directly from high level language programs 
Question 13 
A memory page containing a heavily used variable that was initialized very early and is in constant use is removed when
LRU page replacement algorithm is used  
FIFO page replacement algorithm is used  
LFU page replacement algorithm is used  
None of the above 
In LRU which can eliminate (or) removed which is least recently used.
In LFU the frequency of the page is more. So it is in constant use so cannot be replaced.
Question 14 
Which of the following permutations can be obtained in the output (in the same order) using a stack assuming that the input is the sequence 1, 2, 3, 4, 5 in that order?
3, 4, 5, 1, 2  
3, 4, 5, 2, 1  
1, 5, 2, 3, 4  
5, 4, 3, 1, 2 
→ Remaining options are not possible.
Question 15 
The number of substrings (of all lengths inclusive) that can be formed from a character string of length n is
n  
n^{2}  
n(n1)/2  
n(n+1)/2 
n = 1
(n1) = 2
(n2) = 3
So, Total = n(n+1)/2
Question 16 
Which of the following conversions is not possible (algorithmically)?
Regular grammar to context free grammar  
Nondeterministic FSA to deterministic FSA  
Nondeterministic PDA to deterministic PDA  
Nondeterministic Turing machine to deterministic Turing machine 
Question 17 
Linked lists are not suitable data structures of which one of the following problems?
Insertion sort  
Binary search  
Radix sort  
Polynomial manipulation

Question 18 
Which of the following features cannot be captured by contextfree grammars?
Syntax of ifthenelse statements  
Syntax of recursive procedures  
Whether a variable has been declared before its use  
Variable names of arbitrary length 
Syntactic rules not checking the meaningful things such as if a variable is declared before it use (or) not.
Like this, things are handled by semantic analysis phase.
Question 19 
Which of the following algorithm design techniques is used in the quicksort algorithm?
Dynamic programming  
Backtracking  
Divide and conquer  
Greedy method 
Question 20 
In which one of the following cases is it possible to obtain different results for callby reference and callbyname parameter passing methods?
Passing a constant value as a parameter  
Passing the address of an array as a parameter  
Passing an array element as a parameter  
Passing an array following statements is true 
{ ........
a[ ] = {1, 2, 3, 4}
i = 0
fun(a[i]);
print a[0];
}
fun(int x)
{
int i = 1;
x = 8;
}
O/p:
Callbyreference = 8
Callbyvalue = 1
Question 21 
Which one of the following statements is true?
Macro definitions cannot appear within other macro definitions in assembly language programs  
Overlaying is used to run a program which is longer than the address space of computer  
Virtual memory can be used to accommodate a program which is longer than the address space of a computer  
It is not possible to write interrupt service routines in a high level language 
Question 22 
Which one of the following statements is false?
Optimal binary search tree construction can be performed efficiently using dynamic programming.  
Breadthfirst search cannot be used to find connected components of a graph.  
Given the prefix and postfix walks over a binary tree, the binary tree cannot be uniquely constructed.  
Depthfirst search can be used to find connected components of a graph. 
Question 23 
Consider the following two functions:
Which of the following is true?
g_{1}(n) is O(g_{2}(n))  
g_{1} (n) is O(^{3})  
g_{2} (n) is O(g_{1} (n))  
g_{2} (n) is O(n)  
Both A and B 
Growth rate of g_{1} is less than that of g_{2} i.e., g_{1}(n) = O(g_{2}(n)) = O(n).
Question 24 
Consider the following heap (figure) in which blank regions are not in use and hatched region are in use.
The sequence of requests for blocks of size 300, 25, 125, 50 can be satisfied if we use
either first fit or best fit policy (any one)  
first fit but not best fit policy  
best fit but first fit policy  
None of the above 
So, request for 300 will be satisfied by 350 size block reducing the free size to 50.
Request for 25, satisfied by 125 size block, reducing it to 125.
Request for 125 satisfied by the 125 size block.
And request for 50 satisfied by 50 size block.
So, all requests can be satisfied.
In best fit strategy, a block request is satisfied by the smallest block in which can fit it. So, request for 300 will be satisfied by 350 size block reducing the free size to 50.
Request for 25, satisfied by 50 size block as its the smallest size that fits 25, reducing it to 25.
Request for 125, satisfied by 150 size block, reducing it to 25.
Now, request for 50 cannot be satisfied as the two 25 size blocks are not contiguous.
Question 25 
The number of flipflops required to construct a binary modulo N counter is __________.
⌈log_{2} N⌉ 
Question 26 
On the set N of nonnegative integers, the binary operation __________ is associative and noncommutative.
fog 
(fog)(x) = f(g(x))
It is associative, (fog)oh = fo(goh), but its usually not commutative. fog is usually not equal to gof.
Note that if fog exists then gof might not even exists.
Question 27 
Amongst the properties {reflexivity, symmetry, antisymmetry, transitivity} the relation R = {(x,y) ∈ N^{2}  x ≠ y } satisfies __________.
symmetry 
It is symmetric as if xRy then yRx.
It is not antisymmetric as xRy and yRx are possible and we can have x≠y.
It is not transitive as if xRy and yRz then xRz need not be true. This is violated when x=x.
So, symmetry is the answer.
Question 28 
The number of subsets {1, 2, ... n} with odd cardinality is __________.
2^{n1} 
And so, no. of subsets with odd cardinality is half of total no. of subsets = 2^{n} /n = 2^{n1}
Question 29 
The number of edges in a regular graph of degree d and n vertices is _________.
d*n/2 
d * n = 2 * E
∴ E = d*n/2
Question 30 
The probability of an event B is P_{1}. The probability that events A and B occur together is P_{2} while the probability that A and occur together is P_{3}. The probability of the event A in terms of P_{1}, P_{2} and P_{3} is __________.
P_{2} + P_{3} 
P_{3} = P(A)  P_{2}
P(A) = P_{2} + P_{3}
Question 31 
Consider nbit (including sign bit) 2’s complement representation of integer number. The range of integer values, N, that can be represented is _________ ≤ N ≤ _________
2^{n1} to 2^{n1}  1 
Question 32 
Let A, B and C be independent events which occur with probabilities 0.8, 0.5 and 0.3 respectively. The probability of occurrence of at least one of the event is __________
0.93 
Since all the events are independent, so we can write
P(A∪B∪C) = P(A) + P(B) + P(C)  P(A)P(B)  P(B)P(C)  P(A)P(C) + P(A)P(B) P(C)
= 0.8 + 0.5 + 0.3  0.4  0.5  0.24 + 0.12
= 0.93
Question 33 
The Hasse diagrams of all the lattices with up to four elements are __________ (write all the relevant Hasse diagrams).
We can't draw lattice with 1 element.
For 2 element:
For 3 element:
For 4 element:
Question 34 
The regular expression for the language recognized by the finite state automaton of figure is __________
L = 0*1* 
L contains all binary strings where a 1 is not followed by a 0.
Question 35 
State True or False with one line explanation
Multiplexing of address/data lines in 8085 microprocessor reduces the instruction
execution time.
True  
False 
The major reason of multiplexing address and data bus is to reduce the number of pins for address and data and dedicate those pins for other several functions of microprocessor.
Question 36 
State True or False with one line explanation
Expanding opcode instruction formats are commonly employed in RISC. (Reduced Instruction Set Computers) machines.
True  
False 
Now the challenge is: How to fit multiple sets of instructions types into limited or fixed size instruction format.
Here comes expanding opcode into the picture, So RISC system uses expanding opcode technique to have fixed size instructions.
Question 37 
State True or False with one line explanation
A FSM (Finite State Machine) can be designed to add two integers of any arbitrary length (arbitrary number of digits).
True  
False 
→ q_{0}: Start state to represent carry bit is 0.
→ q_{1}: State to represent carry bit is 1.
The inputs to the FA will be pairs of bits, i.e., 00, 01, 10, 11.
The FA starts in state 1 (since carry is 0) and inputs a pair of bits. If the pair is 11, the FA outputs a '0' and switches to state 2 (since the carry is 1), where the next pair of bits is input and is added to a carry bit of 1.
Question 38 
Match the following items
(i)  (b), (ii)  (c), (iii)  (d), (iv)  (a) 
Question 39 
Match the following items
(i)  (d), (ii)  (a), (iii)  (b), (iv)  (c) 
Yacc (Yet Another Compiler Compiler) is a computer program for the UNIX operating system. It is a LALR parser generator, generating a parser, the part of a compiler that tries to make syntactic sense of the source code, specially a LALR parser, based on an analytic grammar. Yacc is written in portable C.
Question 40 
State True or False with reason
There is always a decomposition into BoyceCodd normal form (BCNF) that is
lossless and dependency preserving.
True  
False 
Question 41 
An instance of a relational scheme R(A, B, C) has distinct values for attribute A.
Can you conclude that A is a candidate key for R?
Yes  
No 
Question 42 
Give a relational algebra expression using only the minimum number of operators from (∪, −) which is equivalent to R ∩ S.
Out of syllabus (For explanation see below) 
→ No need of using Union operation here. → In question they gave (∪, −) but we don't use both.
→ And also they are saying that only the minimum number of operators from (∪, −) which is equivalent to R ∩ S.
So, the expression is minimal.
Question 43 
Every subset of a countable set is countable.
State whether the above statement is true or false with reason.
True  
False 
Question 44 
Match the following items
(i)  (a), (ii)  (b), (iii)  (d), (iv)  (c) 
Question 45 
State True or False with reason
Logical data independence is easier to achieve than physical data independence
True  
False 
Question 46 
Find the inverse of the matrix
λ^{3} + 2λ^{2}  2 = 0
Using CayleyHamiltonian theorem
A^{3} + 2A^{2}  2I = 0
So, A^{1} = 1/2 (2A  A^{2})
Solving we get,
Question 47 
Let p and q be propositions. Using only the truth table decide whether p ⇔ q does not imply p → q is true or false.
True  
False 
So, "imply" is False making "does not imply" True.
Question 48 
(a) Let * be a Boolean operation defined as
If C = A * B then evaluate and fill in the blanks:
(i) A * A = _______
(ii) C * A = _______
(b) Solve the following boolean equations for the values of A, B and C:
Theory Explanation. 
Question 49 
A 3ary tree is a tree in which every internal node has exactly three children. Use induction to prove that the number of leaves in a 3ary tree with n interval nodes is 2(n1)+3.
Theory Explanation. 
Question 50 
What function of x, n is computed by this program?
Function what (x, n:integer): integer: Var value : integer; begin value:=1 if n>0 then begin if n mod 2 = 1 then value:=value*x; value:=value*what(x*x, n div 2); end; what:value end;
Theory Explanation. 
Question 51 
An array A contains n integers in locations A[0],A[1], …………… A[n1]. It is required to shift the elements of the array cyclically to the left by K places, where 1≤K≤n1. An incomplete algorithm for doing this in linear time, without using another is given below. Complete the algorithm by filling in the blanks. Assume all variables are suitably declared.
min:=n; i=0; while _____do begin temp:=A[i]; j:=i; while _____do begin A[j]:=_____; j:=(j+K) mod n; if j
Theory Explanation. 
Question 52 
A rooted tree with 12 nodes has its nodes numbered 1 to 12 in preorder. When the tree is traversed in postorder, the nodes are visited in the order 3, 5, 4, 2, 7, 8, 6, 10, 11, 12, 9, 1.
Reconstruct the original tree from this information, that is, find the parent of each node, and show the tree diagrammatically.
Theory Explanation. 
Question 53 
Following 7 bit single error correcting Hamming coded message is received. (figure below):
Determine if the message is correct (assuming that at most 1 bit could be corrupted). If the message contains an error find the bit which is erroneous and gives the correct message.
Theory Explanation. 
Question 54 
Write a program in 8085 Assembly language to Add two 16bit unsigned BCD(8421 Binary Coded Decimal) number. Assume the two input operands are in BC and DE Register pairs. The result should be placed in the register pair BC. (Higher order register in the register pair contains higher order digits of operand)
Theory Explanation. 
Question 55 
Find the contents of the flipflop Q_{2}, Q_{1} and Q_{0} in the circuit of figure, after giving four clock pulses to the clock terminal. Assume Q_{2}Q_{1}Q_{0} = 000 initially.
Theory Explanation. 
Question 56 
(a) Assume that a CPU has only two registers R_{1} and R_{2} and that only the following instruction is available XOR R_{i}, R_{j}; {R_{j} ← R_{i} ⊕ R_{j}, for i,j = 1,2}
Using this XOR instruction, find an instruction sequence in order to exchange
the contents of the registers R_{1} and R_{2}.
(b) The line p of the circuit shown in figure has stuck at 1 fault. Determine an input test to detect the fault.
Theory Explanation. 
Question 57 
Consider the following relational schema:
COURSES (cno, cname) STUDENTS (rollno, sname, age, year) REGISTERED FOR (cno, rollno)
(a) Write a relational algebra query to
Print the roll number of students who have registered for cno 322.
(b) Write a SQL query to
Print the age and year of the youngest student in each year.
Theory Explanation. 
Question 58 
Consider B+ − tree of order d shown in figure? (A) B^{+} − tree of order d contains between d and 2d keys in each node. (a) Draw the resulting B^{+} − tree after inserted in the figure.
(b) For a B^{+} − tree of order d with n leaf nodes, the number of nodes accessed during a search is 0().
Theory Explanation. 
Question 59 
(a) Use the patterns given to prove that
(You are not permitted to employ induction)
(b) Use the result obtained in (a) to prove that
Theory Explanation. 
Question 60 
Every element a of some ring (R,+,0) satisfies the equation aoa = a.
Decide whether or not the ring is commutative.
Theory Explanation. 
Question 61 
State whether the following statements are True or False with reasons for your answer:
(a) Coroutine is just another name for a subroutine.
(b) A two pass assembler uses its machine opcode table in the first pass of assembly.
Theory Explanation. 
Question 62 
State whether the following statements are True or False with reasons for your answer
(a) A subroutine cannot always be used to replace a macro in an assembly language program.
(b) A symbol declared as ‘external’ in assembly language is assigned an address outside the program by the assembler itself.
Theory Explanation. 
Question 63 
(a) Given a set
S = {x there is an xblock of 5's in the decimal expansion of π}
(Note: xblock is a maximal block of x successive 5’s)
Which of the following statements is true with respect to S? No reasons need to be given for the answer.
 (i) S is regular
(ii) S is recursively enumerable
(iii) S is not recursively enumerable
(iv) S is recursive
(b) Given that a language L_{1} is regular and that the language L_{1} ∪ L_{2} is regular, is the language L_{2} always regular? Prove your answer.
Theory Explanation. 
Question 64 
A grammar G is in ChomskyNormal Form (CNF) if all its productions are of the form A → BC or A → a, where A, B and C, are nonterminals and a is a terminal. Suppose G is a CFG in CNF and w is a string in L(G) of length, then how long is a derivation of w in G?
Theory Explanation. 
Question 65 
Consider the following recursive function:
function fib (1:integer);integer; begin if (n=0) or (n=1) then fib:=1 else fib:=fib(n1) + fib(n2) end;
The above function is run on a computer with a stack of 64 bytes. Assuming that only return address and parameter and passed on the stack, and that an integer value and an address takes 2 bytes each, estimate the maximum value of n for which the stack will not overflow. Give reasons for your answer.
Theory Explanation. 
Question 66 
Consider the program below:
Program main; var r:integer; procedure two; begin write (r) end; procedure one; var r:integer; begin r:=5 two; end begin r:=2; two; one; two; end.
What is printed by the above program if
(i) Static scoping is assumed for all variables;
(ii) Dynamic scoping is assumed for all variables.
Give reasons for your answer.
Theory Explanation. 
Question 67 
Suppose we have a computer with a single register and only three instructions given below:
LOAD addren ; load register ; from addren STORE addren ; store register ; at addren ADD addren ; add register to ; contents of addren ; and place the result ; in the register
Consider the following grammar:
A → id :=E E → E + TT T → (E)id
Write a syntax directed translation to generate code using this grammar for the computer described above.
Theory Explanation. 
Question 68 
An independent set in a graph is a subset of vertices such that no two vertices in the subset are connected by an edge. An incomplete scheme for a greedy algorithm to find a maximum independent set in a tree is given below:
V: Set of all vertices in the tree; I:=φ; While V ≠ φdo begin select a vertex u; ∈ V such that V:= V – {u}; if u is such that then 1:= I ∪ {u} end; output(I);
(a) Complete the algorithm by specifying the property of vertex u in each case
(b) What is the time complexity of the algorithm.
Theory Explanation. 
Question 69 
An array a contains n integers in nondecreasing order, A[1] ≤ A[2] ≤ ... ≤ A[n]. Describe, using Pascal like pseudo code, a linear time algorithm to find i, j, such that A[i] + A[j] = a given integer M, if such i, j exist.
Theory Explanation. 
Question 70 
A queue Q containing n items and an empty stack S are given. It is required to transfer all the items from the queue to the stack, so that the item at the front of queue is on the top of the stack, and the order of all other items is preserved. Show how this can be done in O(n) time using only a constant amount of additional storage. Note that the only operations which can be performed on the queue and stack are Delete, Insert, Push and Pop. Do not assume any implementation of the queue or stack.
Theory Explanation. 
Question 71 
(a) Draw a precedence graph for the following sequential code. The statements are numbered from S_{1} to S_{6}
S_{1} read n S_{2} i:=1 S_{3} if i>n goto next S_{4} a(i):=i+1 S_{5} i:=i+1 S_{6} next : Write a(i)
(b) Can this graph be converted to a concurrent program using parbeginparend construct only?
Theory Explanation. 
Question 72 
Consider the resource allocation graph given in the figure.
(a) Find if the system is in a deadlock state. (b) Otherwise, find a safe sequence.
Theory Explanation. 