Page 1 GATE CS - 1993 SECTION - A 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 ( ) 0 0 0 0 0 , 0 is are 0 0 0 a a ? ? ? ? ? ? ? ? ? ? ? (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation 2 2 sin 0 d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives 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 2p) defined by ( ) 0, is , 0 p x f x x x p - < < ? = ? < < ? ( ) 2 1 1 cos 1 cos cos sin 4 p n nx n nx n n p p p ? ? + - - ? ? ? ? ? But putting x = p, we get the sum of the series. 2 2 2 1 1 1 1 is 3 5 7 + + + +K (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? Page 2 GATE CS - 1993 SECTION - A 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 ( ) 0 0 0 0 0 , 0 is are 0 0 0 a a ? ? ? ? ? ? ? ? ? ? ? (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation 2 2 sin 0 d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives 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 2p) defined by ( ) 0, is , 0 p x f x x x p - < < ? = ? < < ? ( ) 2 1 1 cos 1 cos cos sin 4 p n nx n nx n n p p p ? ? + - - ? ? ? ? ? But putting x = p, we get the sum of the series. 2 2 2 1 1 1 1 is 3 5 7 + + + +K (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? GATE CS - 1993 (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 1 cos 2 x dx x - ? 1.7 The function ( ) 2 , 3 2 has f x y x y xy y x = - + + (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 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 ( ) ( ) ( ) 0 1 2 cos 1 lim is _______ 1 cos x x x e x x x ? - + - - 2.2 The radius of convergence of the power series ( ) ( ) 3 3 3 ! ! m m x m 8 ? is: ______________ 2.3 If the linear velocity V ur is given by $ 2 2 , V x yi xyzj yz k = + - ur The angular velocity ? ur at the point (1, 1, -1) is ________ 2.4 Given the differential equation, y x y ' = - with the initial condition ( ) 0 0 y = . The value of ( ) 0.1 y 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 5.0 * 3 1 2 **2 is _____ *3 4 X X = - + + 2.6 The value of the double integral 1 1 2 0 0 is ______ 1 x x dxdy y + ? ? Page 3 GATE CS - 1993 SECTION - A 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 ( ) 0 0 0 0 0 , 0 is are 0 0 0 a a ? ? ? ? ? ? ? ? ? ? ? (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation 2 2 sin 0 d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives 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 2p) defined by ( ) 0, is , 0 p x f x x x p - < < ? = ? < < ? ( ) 2 1 1 cos 1 cos cos sin 4 p n nx n nx n n p p p ? ? + - - ? ? ? ? ? But putting x = p, we get the sum of the series. 2 2 2 1 1 1 1 is 3 5 7 + + + +K (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? GATE CS - 1993 (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 1 cos 2 x dx x - ? 1.7 The function ( ) 2 , 3 2 has f x y x y xy y x = - + + (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 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 ( ) ( ) ( ) 0 1 2 cos 1 lim is _______ 1 cos x x x e x x x ? - + - - 2.2 The radius of convergence of the power series ( ) ( ) 3 3 3 ! ! m m x m 8 ? is: ______________ 2.3 If the linear velocity V ur is given by $ 2 2 , V x yi xyzj yz k = + - ur The angular velocity ? ur at the point (1, 1, -1) is ________ 2.4 Given the differential equation, y x y ' = - with the initial condition ( ) 0 0 y = . The value of ( ) 0.1 y 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 5.0 * 3 1 2 **2 is _____ *3 4 X X = - + + 2.6 The value of the double integral 1 1 2 0 0 is ______ 1 x x dxdy y + ? ? GATE CS - 1993 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? the matrix 4 , A calculated by the use of Cayley-Hamilton theorem or otherwise, is _________ 2.8 Given $ 2 2 2 cos sin z V x yi x e j z yk = + + ur $ 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 $ 1 . . s V ndS ?? ur is _______ 2.9 The differential equation 0 n y y + = is subjected to the boundary conditions. ( ) 0 0 y = ( ) 0 y ? = In order that the equation has non-trivial solution(s), the general value of ? is __________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below, i.e., ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 otherwise t n t n n f t p p ? - = = = = ? ? K is ___________ 0 p 2p 3p 4p 5p 6p 7p 8p t f(t) Page 4 GATE CS - 1993 SECTION - A 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 ( ) 0 0 0 0 0 , 0 is are 0 0 0 a a ? ? ? ? ? ? ? ? ? ? ? (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation 2 2 sin 0 d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives 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 2p) defined by ( ) 0, is , 0 p x f x x x p - < < ? = ? < < ? ( ) 2 1 1 cos 1 cos cos sin 4 p n nx n nx n n p p p ? ? + - - ? ? ? ? ? But putting x = p, we get the sum of the series. 2 2 2 1 1 1 1 is 3 5 7 + + + +K (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? GATE CS - 1993 (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 1 cos 2 x dx x - ? 1.7 The function ( ) 2 , 3 2 has f x y x y xy y x = - + + (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 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 ( ) ( ) ( ) 0 1 2 cos 1 lim is _______ 1 cos x x x e x x x ? - + - - 2.2 The radius of convergence of the power series ( ) ( ) 3 3 3 ! ! m m x m 8 ? is: ______________ 2.3 If the linear velocity V ur is given by $ 2 2 , V x yi xyzj yz k = + - ur The angular velocity ? ur at the point (1, 1, -1) is ________ 2.4 Given the differential equation, y x y ' = - with the initial condition ( ) 0 0 y = . The value of ( ) 0.1 y 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 5.0 * 3 1 2 **2 is _____ *3 4 X X = - + + 2.6 The value of the double integral 1 1 2 0 0 is ______ 1 x x dxdy y + ? ? GATE CS - 1993 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? the matrix 4 , A calculated by the use of Cayley-Hamilton theorem or otherwise, is _________ 2.8 Given $ 2 2 2 cos sin z V x yi x e j z yk = + + ur $ 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 $ 1 . . s V ndS ?? ur is _______ 2.9 The differential equation 0 n y y + = is subjected to the boundary conditions. ( ) 0 0 y = ( ) 0 y ? = In order that the equation has non-trivial solution(s), the general value of ? is __________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below, i.e., ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 otherwise t n t n n f t p p ? - = = = = ? ? K is ___________ 0 p 2p 3p 4p 5p 6p 7p 8p t f(t) GATE CS - 1993 SECTION II – A INSTRUCTIONS: There are THREE questions in this Section. Question 6 has 8 parts, 7 has 10 parts, and 8 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. 6. 6.1. 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 6.2. If the state machine described in figure, should have a stable state, the restriction on the inputs is given by (a) . 1 ab = (b) 1 a b + = (c) 0 a b + = (d) . 1 ab = (e) 1 a b + = 6.3. 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 x y A f(x,y) J Q K Q J Q 2 K Q J Q K Q Clock Page 5 GATE CS - 1993 SECTION - A 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 ( ) 0 0 0 0 0 , 0 is are 0 0 0 a a ? ? ? ? ? ? ? ? ? ? ? (a) ( ) 0,0,a (b) ( ) ,0,0 a (c) ( ) 0,0,1 (d) ( ) 0, ,0 a 1.2 The differential equation 2 2 sin 0 d y dy y dx dx + + = is: (a) linear (b) non-linear (c) homogeneous (d) of degree two 1.3 Simpson’s rule for integration gives 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 2p) defined by ( ) 0, is , 0 p x f x x x p - < < ? = ? < < ? ( ) 2 1 1 cos 1 cos cos sin 4 p n nx n nx n n p p p ? ? + - - ? ? ? ? ? But putting x = p, we get the sum of the series. 2 2 2 1 1 1 1 is 3 5 7 + + + +K (a) 2 4 p (b) 2 6 p (c) 2 8 p (d) 2 12 p 1.6 Which of the following improper integrals is (are) convergent? GATE CS - 1993 (a) 1 0 sin 1 cos x dx x - ? (b) 0 cos 1 x dx x 8 + ? (c) 2 0 1 x dx x 8 + ? (d) 1 5 0 1 cos 2 x dx x - ? 1.7 The function ( ) 2 , 3 2 has f x y x y xy y x = - + + (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 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 ( ) ( ) ( ) 0 1 2 cos 1 lim is _______ 1 cos x x x e x x x ? - + - - 2.2 The radius of convergence of the power series ( ) ( ) 3 3 3 ! ! m m x m 8 ? is: ______________ 2.3 If the linear velocity V ur is given by $ 2 2 , V x yi xyzj yz k = + - ur The angular velocity ? ur at the point (1, 1, -1) is ________ 2.4 Given the differential equation, y x y ' = - with the initial condition ( ) 0 0 y = . The value of ( ) 0.1 y 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 5.0 * 3 1 2 **2 is _____ *3 4 X X = - + + 2.6 The value of the double integral 1 1 2 0 0 is ______ 1 x x dxdy y + ? ? GATE CS - 1993 2.7 If 1 0 0 1 0 1 0 1 0 0 0 0 0 A i i i ? ? ? ? - - ? ? = ? ? ? ? ? ? - ? ? the matrix 4 , A calculated by the use of Cayley-Hamilton theorem or otherwise, is _________ 2.8 Given $ 2 2 2 cos sin z V x yi x e j z yk = + + ur $ 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 $ 1 . . s V ndS ?? ur is _______ 2.9 The differential equation 0 n y y + = is subjected to the boundary conditions. ( ) 0 0 y = ( ) 0 y ? = In order that the equation has non-trivial solution(s), the general value of ? is __________ 2.10 The Laplace transform of the periodic function ( ) f t described by the curve below, i.e., ( ) ( ) ( ) sin if 2 1 2 1,2,3, 0 otherwise t n t n n f t p p ? - = = = = ? ? K is ___________ 0 p 2p 3p 4p 5p 6p 7p 8p t f(t) GATE CS - 1993 SECTION II – A INSTRUCTIONS: There are THREE questions in this Section. Question 6 has 8 parts, 7 has 10 parts, and 8 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. 6. 6.1. 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 6.2. If the state machine described in figure, should have a stable state, the restriction on the inputs is given by (a) . 1 ab = (b) 1 a b + = (c) 0 a b + = (d) . 1 ab = (e) 1 a b + = 6.3. 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 x y A f(x,y) J Q K Q J Q 2 K Q J Q K Q Clock GATE CS - 1993 6.4. 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 6.5. Convert the following numbers in the given bases into their equivalents in the desired bases. (a) ) ) 2 10 110.101 x = (b) ) ) 10 1118 y H = 6.6. 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 × number of bits) that is required to accommodate the Truth table is M words × N bits. Write the values of M and N. 6.7. 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. 6.8. 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. Arrival of interrupt input 10µ sec Main program execution 10µ sec 80µ sec 10µ sec Saving of CPU state Interrupt Service Execution Restoration of CPU state Main program executionRead More

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