GATE CSE 1993 Previous-Year Questions (PYQ)
Section - A
Question 1
In questions 1.1 to 1. 7 below, one or more of the alternatives are correct. Write the code letter(s) a, b, c, d corresponding to the correct alternative(s) in the answer book. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up.
1.1 The eigen vector(s) of the matrix
$$\begin{pmatrix}0&0&\alpha\\0&0&0\\0&0&0\end{pmatrix},\;\alpha\neq0\;is\;(are)$$
(a)$$\begin{pmatrix}0&0&\alpha\end{pmatrix}$$
(b)$$\begin{pmatrix}\alpha&0&0\end{pmatrix}$$
(c)$$\begin{pmatrix}0&0&1\end{pmatrix}$$
(d)$$\begin{pmatrix}0&\alpha&0\end{pmatrix}$$
1.2 The differential equation
$$\frac{d^2y}{dx^2}\;+\;\frac{dy}{dx}\;+\;\sin\;y\;=\;0\;is:$$
(a) linear
(b) non-linear
(c) homogeneous
(d) of degree two
1.3 Simpson’s rule for integration gives an exact result when f (x) is a polynomial of degree
(a) 1 (b) 2 (c) 3 (d) 4
1.4 Which of the following is (are) valid FORTRAN 77 statement(s)?
(a) DO 13 I = 1
(b) A = DIM ***7
(c)READ=15.0
(d) GO TO 3 = 10
1.5 Fourier series of the periodic function (period 2\(\mathrm\pi\)) defined by
$$f(x)\;=\;\left\{\begin{array}{l}0,\;\;\;\;-p<x<\\x,\;\;\;\;\;0<x<p\end{array}\;is\right.\\\frac p4+\sum\left[\frac1{\mathrm{πn}^2}(\cos n\mathrm\pi\;-1)\mathrm{cosnx}\;-\;\frac1{\mathrm n}\mathrm{cosnπ}\;\mathrm{sinnx}\right]$$
But putting \(x\;=\;\mathrm\pi\), we get the sum of the series.
$$1\;+\;\frac1{3^2}\;+\;\frac1{5^2}\;+\;\frac1{7^2}\;+\;……..\;is$$
(a) \(\frac{\mathrm\pi^2}4\)
(b) \(\frac{\mathrm\pi^2}6\)
(c) \(\frac{\mathrm\pi^2}8\)
(d) \(\frac{\mathrm\pi^2}{12}\)
1.6 Which of the following improper integrals is (are) convergent?
(a) \(\int_0^1\frac{\sin x}{1-\cos x}dx\)
(b) \(\int_0^\infty\frac{\cos x}{1+x}dx\)
(c) \(\int_0^\infty\frac x{1+x^2}dx\)
(d) \(\int_0^1\frac{1-\cos x}{\displaystyle\frac{x^5}2}dx\)
1.7 The function f (x, y) = x2y – 3xy + 2y + x has
(a) no local extremum
(b) one local minimum but no local maximum
(c) one local maximum but no local minimum
(d) one local minimum and one local maximum
Question 2
In questions 2.1 to 2.10 below, each blank ( _____) is to be suitably filled in. In the answer book write the question number and the answer only. Do not copy the question. Also, no explanations for the answers are to be given.
2.1 \(\lim_{x\rightarrow0}\frac{x\left(e^x-1\right)\;+\;2\left(\cos x-1\right)}{x\left(1-\cos x\right)}\;\;\;\) is _________
2.2 The radius of convergence of the power series
$$\sum_{}^\infty\frac{\left(3m\right)!}{\left(m!\right)^3}x^{3m}\;\;\;\;is:\_\_\_\_\_\_\_\_\_\_$$
2.3 If the linear velocity \(\overrightarrow V\) is given by
\(\overrightarrow V\) = x2yi + xyz\(\widehat j\)- yz2k,
The angular velocity \(\overrightarrow\omega\) the point (1, 1, -1) is ________
2.4 Given the differential equation, y’ = x – y with the initial condition y (0) = 0. The value of y (0.1) calculated numerically upto the third place of decimal by the second order Runga-Kutta method with step size h = 0.1 is ________
2.5 For x = 4.0, the value of I in the FORTRAN 77 statement
\(1\;=\;-2\ast\ast2\;+\;\frac{5.0\ast x}{x\ast3}\;+\;\frac3{4\;}\) is _______
2.6 The value of the double integral \(\int_0^1\int_0^\frac1x\frac x{1+y^2}dxdy\) is ________
2.7 If \(A=\begin{pmatrix}1&0&0&1\\0&-1&0&-1\\0&0&i&i\\0&0&0&-i\end{pmatrix}\) the matrix A4, calculated by the use of Cayley-Hamilton theorem or otherwise, is ____________
2.8 Given \(\overrightarrow{V\;}=x\cos^2y\widehat i+x^2e^z\widehat j+z\sin^2yk\) and S the surface of a unit cube with one corner at the origin and edges parallel to the coordinate axes, the value of integral
$$\int_{}^1\int_s^{}\overrightarrow V.\widehat ndS\;\;is\;\_\_\_\_\_\_\_\_\_\_\_$$
2.9 The differential equation yn + y = 0 is subjected to the boundary conditions.
y (0) = 0 y(λ) = 0
In order that the equation has non-trivial solution(s), the general value of λ is ___________
2.10 The Laplace transform of the periodic function f(t)described by the curve below, i.e.,
$$f\left(t\right)=\left\{\begin{array}{l}\sin t\;\;\;if\;\left(2n-1\right)\mathrm\pi\leq\mathrm t\leq2\mathrm{nπ}\;\\\;\;\;\;\;\;\;\;\;\;(\mathrm n=1,2,3,…)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\\0\;\;\;\;\;\;otherwise\end{array}\right.$$
is _________
Section II - A
INSTRUCTIONS: There are THREE questions in this Section. Question 3 has 8 parts, 4 has 10 parts, and 5 has 7 parts. Each part of a question carries 2 marks. There may be more than one correct alternative the multiple-choice questions. Credit will be given if only all the correct alternatives have been indicated.
Question 3
3.1 Identify the logic function performed by the circuit shown in figure.
(a) exclusive OR
(b) exclusive NOR
(c) NAND
(d) NOR
(e) None of the above
3.2 If the state machine described in figure, should have a stable state, the restriction on the in puts is given by
(a) a.b=1
(b) a+b=1
(c) \(\overline a+\overline b=0\)
(d) \(\overline{a.b}=0\)
(e)\(\overline{a+b}=1\)
3.3 For the initial state of 000, the function performed by the arrangement of the J-K flip-flops in figure is:
(a) Shift Register
(b) Mod-3 Counter
(c) Mod-6 Counter
(d) Mod-2 Counter
(e) None of the above
3.4 Assume that each character code consists of 8 bits. The number of characters that can be transmitted per second through an asynchronous serial line at 2400 baud rate, and with two stop bits, is
(a) 109
(b) 216
(c) 218
d) 219
(e) 240
3.5 Convert the following numbers in the given bases into their equivalents in the desired bases.
(a) (110.101)2 = (X)10
(b) (1118)10 = (y)H
3.6 A ROM is used to store the Truth table for a binary multiple unit that will multiply two 4-bit numbers. The size of the ROM (number of words \(\times\) number of bits) that is required to accommodate the Truth table is M words \(\times\) N bits. Write the values of M and N.
3.7 A certain moving arm disk storage, with one head, has the following specifications.
Number of tracks/recording surface = 200
Disk rotation speed = 2400 rpm
Track storage capacity = 62,500 bits
The average latency of this device is P msec and the data transfer rate is Q bits/sec.
Write the value of P and Q.
3.8 The details of an interrupt cycle are shown in figure.
Given that an interrupt input arrives every 1 msec, what is the percentage of the total time that the CPU devotes for the main program execution.
Question 4
Note: For sub-questions 4.1 to 4.3, refer to the PASCAL program shown below.
Program PARAM (input, output);
var m, n : integer;
procedure P (var, x, y : integer);
var m : integer;
begin
m : = 1;
x:=y+1
end;
procedure Q (x:integer; vary: integer);
begin
x: =y+1;
end;
begin
m: =0; P(m,m); write (m);
n: =0; Q(n*1,n); write (n)
end
4.1 The value of m, output by the program PARAM is:
(a) 1, because m is a local variable in P
(b) 0, because m is the actual parameter that corresponds to the formal parameter in p
(c) 0, because both x and y are just reference to m, and y has the value 0
(d) 1, because both x and y are just references to m which gets modified in procedure P
(e) none of the above
4.2 The value of n, output by the program PARAM is:
(a) 0, because n is the actual parameter corresponding to x in procedure Q.
(b) 0, because n is the actual parameter to y in procedure Q.
(c) 1, because n is the actual parameter corresponding to x in procedure Q.
(d) 1, because n is the actual parameter corresponding to y in procedure Q.
(e) none of the above
4.3 What is the scope of m declared in the main program?
(a) PARAM, P, Q
(b) PARAM, P
(c) PARAM, Q
(d) P, Q
(e) none of the above
4.4 What does the following code do?
var a, b : integer;
begin
a: =a+b;
b: =a-b;
a: =a-b
end;
(a) exchanges a and b
(b) doubles a and stores in b
(c) doubles band stores in a
(d) leaves a and b unchanged
(e) none of the above
4.5 For the program segment given below, which of the following are true?
program main (output);
type link = ^data;
data = record
d : real;
n : link
end;
var ptr : link;
begin
new (ptr);
ptr:=nil;
.ptr^.d:=5.2;
Write In(ptr)
End.
(a) The program leads to compile time error
(b) The program leads to run time error
(c) The program outputs 5.2
(d) The program produces error relating to nil pointer dereferencing
(e) None of the above
4.6 A simple two-pass assembler does the following in the first pass:
(a) It allocates space for the literals.
(b) It computes the total length of the program
(c) It builds the symbol table for the symbols and their values.
(d) It generates code for all the load and store register instructions.
(e) None of the above.
4.7 A part of the system software, which under all circumstances must reside in the main memory, is:
(a) text editor
(b) assembler
(c) linker
(d) loader
(e) none of the above
4.8 The root directory of a disk should be placed
(a) at a fixed address in main memory
(b) at a fixed location on the disk
(c) anywhere on the disk
(d) at a fixed location on the system disk
(e) anywhere on the system disk
4.9 Consider a system having m resources of the same type. These resources are shared by 3 processes A, B and C, which have peak demands of 3, 4 and 6 respectively. For what value of m deadlock will not occur?
(a) 7
(b) 9
(c) 10
(d) 13
(e) 15
4.10 Assume
that the following jobs are to be executed on a single processor system
The jobs are assumed to have arrived at time 0+ and in the order p, q, r, s, t. Calculate the departure time (completion time) for job p if scheduling is round robin with time slice 1.
(a) 4
(b) 10
(c) 11
(d) 12
(e) None of the above
Question 5
5.1. Consider a simple connected graph G with n vertices and n-edges (n>2). Then, which of the following statements are true?
(a) G has no cycles.
(b) The graph obtained by removing any edge from G is not connected. (c) G has at least one cycle.
(d) The graph obtained by removing any two edges from G is not connected.
(e) None of the above
5.2. The proposition P ∧ (∼ p ∨ q) is:
(a) a tautology
(b) logically equivalent to p ∧ q
(c) logically equivalent to p ∨ q
(d) a contradiction
(e) none of the above
5.3. Let S be an infinite set and S1 ……, Sn be sets such that S1 ∪ S2 ∪ S3 ∪ …….∪ Sn = S. Then,
(a) at least one of the set Si is a finite set
(b) not more than one of the set Si can be finite
(c) at least one of the sets Si is an infinite set
(d) not more than one of the sets Si can be infinite
(e) None of the above
5.4. Let A be a finite set of size n. The number of elements in the power set of A × A is:
(a) \(2^{2^n}\)
(b) \(2^{n^2}\)
(c)\(\left(2^n\right)^2\)
(d)\(\left(2^2\right)^n\)
(e) None of the above
5.5. The less-than relation, <, on reals is
(a) a partial ordering since it is asymmetric and reflexive
(b) a partial ordering since it is antisymmetric and reflexive
(c) not a partial ordering because it is not asymmetric and not reflexive
(d) not a partial ordering because it is not antisymmetric and reflexive
(e) none of the above
5.6. Let A and B be sets with cardinalities m and n respectively. The number of one-one mappings (injections) from A to B, when m < n, is:
(a) \(m^n\)
(b) \({}^nP_m\)
(c) \({}^mC_n\)
(d) \({}^nC_m\)
(e) \({}^mP_n\)
5.7. \(\sum_{1\leq k\leq n}O(n)\), where O(n) stands for order n is:
(a) O(n)
(b) O(n2)
(c) O(n3)
(d) O(3n2)
(e) O(1.5n2)