Page 1 GATE CS - 1996 SECTION - A 1. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 1.1 Let A and B be sets and let and c c A B denote the complements of the sets A and B. the set (a – b) ?(b-a) ?(anb) is equal to. (a) A ? B (b) c c A B ? (c) A n B (d) c c A B n 1.2 Let X = {2,3,6,12,24}, Let = be the partial order defined by X = Y if x divides y. Number of edge as in the Hasse diagram of (X,=) is (a) 3 (b) 4 (c) 9 (d) None of the above 1.3 Suppose X and Y are sets and and X Y are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that (a) 1, 97 X Y = = (b) 97, 1 X Y = = (c) 97, 97 X Y = = (d) None of the above 1.4 Which of the following statements is false? (a) The set of rational numbers is an abelian group under addition. (b) The set of integers in an abelian group under addition. (c) The set of rational numbers form an abelian group under multiplication. (d) The set of real numbers excluding zero in an abelian group under multiplication. 1.5 Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is (a) 1 36 (b) 1 3 (c) 25 36 (d) 11 36 1.6 The formula used to compute an approximation for the second derivative of a function f at a point X 0 is. (a) ( ) ( ) 0 0 2 f x h f x h + + - (b) ( ) ( ) 0 0 2 f x h f x h h + - - (c) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + + + - (d) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + - + - Page 2 GATE CS - 1996 SECTION - A 1. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 1.1 Let A and B be sets and let and c c A B denote the complements of the sets A and B. the set (a – b) ?(b-a) ?(anb) is equal to. (a) A ? B (b) c c A B ? (c) A n B (d) c c A B n 1.2 Let X = {2,3,6,12,24}, Let = be the partial order defined by X = Y if x divides y. Number of edge as in the Hasse diagram of (X,=) is (a) 3 (b) 4 (c) 9 (d) None of the above 1.3 Suppose X and Y are sets and and X Y are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that (a) 1, 97 X Y = = (b) 97, 1 X Y = = (c) 97, 97 X Y = = (d) None of the above 1.4 Which of the following statements is false? (a) The set of rational numbers is an abelian group under addition. (b) The set of integers in an abelian group under addition. (c) The set of rational numbers form an abelian group under multiplication. (d) The set of real numbers excluding zero in an abelian group under multiplication. 1.5 Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is (a) 1 36 (b) 1 3 (c) 25 36 (d) 11 36 1.6 The formula used to compute an approximation for the second derivative of a function f at a point X 0 is. (a) ( ) ( ) 0 0 2 f x h f x h + + - (b) ( ) ( ) 0 0 2 f x h f x h h + - - (c) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + + + - (d) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + - + - GATE CS - 1996 1.7 Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknows. Which of the following is false? (a) The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank. (b) If m < n and b is the zero vector, then the system has infinitely many solutions. (c) If m = n and b is non-zero vector, then the system has a unique solution. (d) The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. 1.8 Which two of the following four regular expressions are equivalent? (e is the empty string). (i) (00) * (e + 0) (ii) (00)* (iii) 0* (iv) 0(00)* (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (iii) and (iv) 1.9 Which of the following statements is false? (a) The Halting problem of Turing machines is undecidable. (b) Determining whether a context-free grammar is ambiguous is undecidbale. (c) Given two arbitrary context-free grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). (d) Given two regular grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). 1.10 Let L ? S* where S = {a, b}. which of the following is true? (a) { } has an equal number of a's and b's is regular L x x = (b) { } 1 is regular n n L a b n = = (c) { } has more a's and b's is regular L x x = (d) { } , 1 is regular m n L a b m n = = = 1.11 Which of the following is false? (a) log 100 log 100 n n n ? ? = ? ? ? ? (b) ( ) log 0 loglog n n = (c) ( ) If 0 then 0 x y x y n n < < = (d) ( ) 2 0 n nk ? Page 3 GATE CS - 1996 SECTION - A 1. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 1.1 Let A and B be sets and let and c c A B denote the complements of the sets A and B. the set (a – b) ?(b-a) ?(anb) is equal to. (a) A ? B (b) c c A B ? (c) A n B (d) c c A B n 1.2 Let X = {2,3,6,12,24}, Let = be the partial order defined by X = Y if x divides y. Number of edge as in the Hasse diagram of (X,=) is (a) 3 (b) 4 (c) 9 (d) None of the above 1.3 Suppose X and Y are sets and and X Y are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that (a) 1, 97 X Y = = (b) 97, 1 X Y = = (c) 97, 97 X Y = = (d) None of the above 1.4 Which of the following statements is false? (a) The set of rational numbers is an abelian group under addition. (b) The set of integers in an abelian group under addition. (c) The set of rational numbers form an abelian group under multiplication. (d) The set of real numbers excluding zero in an abelian group under multiplication. 1.5 Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is (a) 1 36 (b) 1 3 (c) 25 36 (d) 11 36 1.6 The formula used to compute an approximation for the second derivative of a function f at a point X 0 is. (a) ( ) ( ) 0 0 2 f x h f x h + + - (b) ( ) ( ) 0 0 2 f x h f x h h + - - (c) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + + + - (d) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + - + - GATE CS - 1996 1.7 Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknows. Which of the following is false? (a) The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank. (b) If m < n and b is the zero vector, then the system has infinitely many solutions. (c) If m = n and b is non-zero vector, then the system has a unique solution. (d) The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. 1.8 Which two of the following four regular expressions are equivalent? (e is the empty string). (i) (00) * (e + 0) (ii) (00)* (iii) 0* (iv) 0(00)* (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (iii) and (iv) 1.9 Which of the following statements is false? (a) The Halting problem of Turing machines is undecidable. (b) Determining whether a context-free grammar is ambiguous is undecidbale. (c) Given two arbitrary context-free grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). (d) Given two regular grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). 1.10 Let L ? S* where S = {a, b}. which of the following is true? (a) { } has an equal number of a's and b's is regular L x x = (b) { } 1 is regular n n L a b n = = (c) { } has more a's and b's is regular L x x = (d) { } , 1 is regular m n L a b m n = = = 1.11 Which of the following is false? (a) log 100 log 100 n n n ? ? = ? ? ? ? (b) ( ) log 0 loglog n n = (c) ( ) If 0 then 0 x y x y n n < < = (d) ( ) 2 0 n nk ? GATE CS - 1996 1.12 Consider the following statements: (i) First-in-first out types of computations are efficiently supported by STACKS. (ii) Implementing LISTS on linked lists is more efficient than implementing LISTS on an array for almost all the basic LIST operations. (iii) Implementing QUEUES on a circular array is more efficient than implementing QUEUES on a linear array with two indices. (iv) Last-in-first-out type of computations are efficiently supported by QUEUES. (a) (ii) and (iii) are true (b) (i) and (ii) are true (c) (iii) and (iv) are true (d) (ii) and (iv) are true 1.13 An advantage of chained hash table (external hashing) over the open addressing scheme is (a) Worst case complexity of search operations is less? (b) Space used is less (c) Deletion is easier (d) None of the above 1.14 In the balanced binary tree in Fig.1.14 given below, how many nodes will become unbalanced when a node is inserted as a child of the node “g”? (a) 1 (b) 3 (c) 7 (d) 8 1.15 Which of the following sequences denotes the post order traversal sequence of the tree of question 1.14? (a) f e g c d b a (b) g c b d a f e (c) g c d b f e a (d) f e d g c b a 1.16 Relative mode of addressing is most relevant to writing (a) coroutines (b) position – independent code (c) shareable code (d) interrupt handlers 1.17 The pass numbers for each of the following activities (i) object code generation (ii) literals added to literal table (iii) listing printed (iv) address resolution of local symbols that occur in a two pass assembler respectively are (a) 1, 2, 1, 2 (b) 2, 1, 2, 1 (c) 2, 1, 1, 2 (d) 1, 2, 2, 2 a b c g d f e Page 4 GATE CS - 1996 SECTION - A 1. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 1.1 Let A and B be sets and let and c c A B denote the complements of the sets A and B. the set (a – b) ?(b-a) ?(anb) is equal to. (a) A ? B (b) c c A B ? (c) A n B (d) c c A B n 1.2 Let X = {2,3,6,12,24}, Let = be the partial order defined by X = Y if x divides y. Number of edge as in the Hasse diagram of (X,=) is (a) 3 (b) 4 (c) 9 (d) None of the above 1.3 Suppose X and Y are sets and and X Y are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that (a) 1, 97 X Y = = (b) 97, 1 X Y = = (c) 97, 97 X Y = = (d) None of the above 1.4 Which of the following statements is false? (a) The set of rational numbers is an abelian group under addition. (b) The set of integers in an abelian group under addition. (c) The set of rational numbers form an abelian group under multiplication. (d) The set of real numbers excluding zero in an abelian group under multiplication. 1.5 Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is (a) 1 36 (b) 1 3 (c) 25 36 (d) 11 36 1.6 The formula used to compute an approximation for the second derivative of a function f at a point X 0 is. (a) ( ) ( ) 0 0 2 f x h f x h + + - (b) ( ) ( ) 0 0 2 f x h f x h h + - - (c) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + + + - (d) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + - + - GATE CS - 1996 1.7 Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknows. Which of the following is false? (a) The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank. (b) If m < n and b is the zero vector, then the system has infinitely many solutions. (c) If m = n and b is non-zero vector, then the system has a unique solution. (d) The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. 1.8 Which two of the following four regular expressions are equivalent? (e is the empty string). (i) (00) * (e + 0) (ii) (00)* (iii) 0* (iv) 0(00)* (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (iii) and (iv) 1.9 Which of the following statements is false? (a) The Halting problem of Turing machines is undecidable. (b) Determining whether a context-free grammar is ambiguous is undecidbale. (c) Given two arbitrary context-free grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). (d) Given two regular grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). 1.10 Let L ? S* where S = {a, b}. which of the following is true? (a) { } has an equal number of a's and b's is regular L x x = (b) { } 1 is regular n n L a b n = = (c) { } has more a's and b's is regular L x x = (d) { } , 1 is regular m n L a b m n = = = 1.11 Which of the following is false? (a) log 100 log 100 n n n ? ? = ? ? ? ? (b) ( ) log 0 loglog n n = (c) ( ) If 0 then 0 x y x y n n < < = (d) ( ) 2 0 n nk ? GATE CS - 1996 1.12 Consider the following statements: (i) First-in-first out types of computations are efficiently supported by STACKS. (ii) Implementing LISTS on linked lists is more efficient than implementing LISTS on an array for almost all the basic LIST operations. (iii) Implementing QUEUES on a circular array is more efficient than implementing QUEUES on a linear array with two indices. (iv) Last-in-first-out type of computations are efficiently supported by QUEUES. (a) (ii) and (iii) are true (b) (i) and (ii) are true (c) (iii) and (iv) are true (d) (ii) and (iv) are true 1.13 An advantage of chained hash table (external hashing) over the open addressing scheme is (a) Worst case complexity of search operations is less? (b) Space used is less (c) Deletion is easier (d) None of the above 1.14 In the balanced binary tree in Fig.1.14 given below, how many nodes will become unbalanced when a node is inserted as a child of the node “g”? (a) 1 (b) 3 (c) 7 (d) 8 1.15 Which of the following sequences denotes the post order traversal sequence of the tree of question 1.14? (a) f e g c d b a (b) g c b d a f e (c) g c d b f e a (d) f e d g c b a 1.16 Relative mode of addressing is most relevant to writing (a) coroutines (b) position – independent code (c) shareable code (d) interrupt handlers 1.17 The pass numbers for each of the following activities (i) object code generation (ii) literals added to literal table (iii) listing printed (iv) address resolution of local symbols that occur in a two pass assembler respectively are (a) 1, 2, 1, 2 (b) 2, 1, 2, 1 (c) 2, 1, 1, 2 (d) 1, 2, 2, 2 a b c g d f e GATE CS - 1996 1.18 The process state transition diagram in Fig.1.8 is representative of (a) a batch operating system (b) an operating system with a preemptive scheduler (c) an operating system with a non-preemptive scheduler (d) a uni-programmed operating system. 1.19. A critical section is a program segment (a) which should run in a certain specified amount of time (b) which avoids deadlocks (c) where shared resources are accessed (d) which must be enclosed by a pair of semaphore operations, P and V 1.20. Which of the following is an example of spooled device? (a) A line printer used to print the output of a number of jobs. (b) A terminal used to enter input data to a running program. (c) A secondary storage device in a virtual memory sytem. (d) A graphic display device. 1.21. A ROM is sued to store the table for multiplication of two 8-bit unsigned integers. The size of ROM required is (a) 256 × 16 (b) 64 K × 8 (c) 4 K × 16 (d) 64 K × 16 1.22. Number of machine cycles required for RET instruction in 8085 microprocessor is (a) 1 (b) 2 (c) 3 (d) 5 1.23. Both’s algorithm for integer multiplication gives worst performance when the multiplier pattern is (a) 101010 …..1010 (b) 100000 …..0001 (c) 111111 …..1111 (d) 011111 …..1110 TERMINATED RUNNING BLOCKED READY NEW Page 5 GATE CS - 1996 SECTION - A 1. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 1.1 Let A and B be sets and let and c c A B denote the complements of the sets A and B. the set (a – b) ?(b-a) ?(anb) is equal to. (a) A ? B (b) c c A B ? (c) A n B (d) c c A B n 1.2 Let X = {2,3,6,12,24}, Let = be the partial order defined by X = Y if x divides y. Number of edge as in the Hasse diagram of (X,=) is (a) 3 (b) 4 (c) 9 (d) None of the above 1.3 Suppose X and Y are sets and and X Y are their respective cardinalities. It is given that there are exactly 97 functions from X to Y. from this one can conclude that (a) 1, 97 X Y = = (b) 97, 1 X Y = = (c) 97, 97 X Y = = (d) None of the above 1.4 Which of the following statements is false? (a) The set of rational numbers is an abelian group under addition. (b) The set of integers in an abelian group under addition. (c) The set of rational numbers form an abelian group under multiplication. (d) The set of real numbers excluding zero in an abelian group under multiplication. 1.5 Two dice are thrown simultaneously. The probability that at least one of them will have 6 facing up is (a) 1 36 (b) 1 3 (c) 25 36 (d) 11 36 1.6 The formula used to compute an approximation for the second derivative of a function f at a point X 0 is. (a) ( ) ( ) 0 0 2 f x h f x h + + - (b) ( ) ( ) 0 0 2 f x h f x h h + - - (c) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + + + - (d) ( ) ( ) ( ) 0 0 0 2 2 f x h f x f x h h + - + - GATE CS - 1996 1.7 Let Ax = b be a system of linear equations where A is an m × n matrix and b is a m × 1 column vector and X is a n × 1 column vector of unknows. Which of the following is false? (a) The system has a solution if and only if, both A and the augmented matrix [A b] have the same rank. (b) If m < n and b is the zero vector, then the system has infinitely many solutions. (c) If m = n and b is non-zero vector, then the system has a unique solution. (d) The system will have only a trivial solution when m = n, b is the zero vector and rank (A) = n. 1.8 Which two of the following four regular expressions are equivalent? (e is the empty string). (i) (00) * (e + 0) (ii) (00)* (iii) 0* (iv) 0(00)* (a) (i) and (ii) (b) (ii) and (iii) (c) (i) and (iii) (d) (iii) and (iv) 1.9 Which of the following statements is false? (a) The Halting problem of Turing machines is undecidable. (b) Determining whether a context-free grammar is ambiguous is undecidbale. (c) Given two arbitrary context-free grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). (d) Given two regular grammars G 1 and G 2 it is undecidable whether L(G 1 ) = L(G 2 ). 1.10 Let L ? S* where S = {a, b}. which of the following is true? (a) { } has an equal number of a's and b's is regular L x x = (b) { } 1 is regular n n L a b n = = (c) { } has more a's and b's is regular L x x = (d) { } , 1 is regular m n L a b m n = = = 1.11 Which of the following is false? (a) log 100 log 100 n n n ? ? = ? ? ? ? (b) ( ) log 0 loglog n n = (c) ( ) If 0 then 0 x y x y n n < < = (d) ( ) 2 0 n nk ? GATE CS - 1996 1.12 Consider the following statements: (i) First-in-first out types of computations are efficiently supported by STACKS. (ii) Implementing LISTS on linked lists is more efficient than implementing LISTS on an array for almost all the basic LIST operations. (iii) Implementing QUEUES on a circular array is more efficient than implementing QUEUES on a linear array with two indices. (iv) Last-in-first-out type of computations are efficiently supported by QUEUES. (a) (ii) and (iii) are true (b) (i) and (ii) are true (c) (iii) and (iv) are true (d) (ii) and (iv) are true 1.13 An advantage of chained hash table (external hashing) over the open addressing scheme is (a) Worst case complexity of search operations is less? (b) Space used is less (c) Deletion is easier (d) None of the above 1.14 In the balanced binary tree in Fig.1.14 given below, how many nodes will become unbalanced when a node is inserted as a child of the node “g”? (a) 1 (b) 3 (c) 7 (d) 8 1.15 Which of the following sequences denotes the post order traversal sequence of the tree of question 1.14? (a) f e g c d b a (b) g c b d a f e (c) g c d b f e a (d) f e d g c b a 1.16 Relative mode of addressing is most relevant to writing (a) coroutines (b) position – independent code (c) shareable code (d) interrupt handlers 1.17 The pass numbers for each of the following activities (i) object code generation (ii) literals added to literal table (iii) listing printed (iv) address resolution of local symbols that occur in a two pass assembler respectively are (a) 1, 2, 1, 2 (b) 2, 1, 2, 1 (c) 2, 1, 1, 2 (d) 1, 2, 2, 2 a b c g d f e GATE CS - 1996 1.18 The process state transition diagram in Fig.1.8 is representative of (a) a batch operating system (b) an operating system with a preemptive scheduler (c) an operating system with a non-preemptive scheduler (d) a uni-programmed operating system. 1.19. A critical section is a program segment (a) which should run in a certain specified amount of time (b) which avoids deadlocks (c) where shared resources are accessed (d) which must be enclosed by a pair of semaphore operations, P and V 1.20. Which of the following is an example of spooled device? (a) A line printer used to print the output of a number of jobs. (b) A terminal used to enter input data to a running program. (c) A secondary storage device in a virtual memory sytem. (d) A graphic display device. 1.21. A ROM is sued to store the table for multiplication of two 8-bit unsigned integers. The size of ROM required is (a) 256 × 16 (b) 64 K × 8 (c) 4 K × 16 (d) 64 K × 16 1.22. Number of machine cycles required for RET instruction in 8085 microprocessor is (a) 1 (b) 2 (c) 3 (d) 5 1.23. Both’s algorithm for integer multiplication gives worst performance when the multiplier pattern is (a) 101010 …..1010 (b) 100000 …..0001 (c) 111111 …..1111 (d) 011111 …..1110 TERMINATED RUNNING BLOCKED READY NEW GATE CS - 1996 1.24. For the daisy chain scheme of connecting I/O devices, which of the following statements is true? (a) It gives non-uniform priority to various devices. (b) It gives uniform priority to all devices. (c) It is only useful for connecting slow devices to a processor device. (d) It requires a separate interrupt pin on the processor for each device. 1.25 Consider the following floating-point number representation. 31 24 23 0 Exponent Mantissa The exponent is in 2’s complement representation and mantissa is in the sign magnitude representation. The range of the magnitude of the normalized numbers in this representation is (a) 0 to 1 (b) 0.5 to 1 (c) 23 2 to 0.5 - (d) ( ) -23 0.5 to 1-2 2. Write in your answer book the correct or the most appropriate answer to the following multiple choice questions by writing the corresponding letter a, b, c or d against the sub-question number. 2.1 Let R denotes the set of real numbers. Let f:R×R R×R be a bijective function defined by f(x,y)=(x+y,x-y). the inverse function of f is given by (a) ( ) 1 1 1 , , f x y x y x y - ? ? = ? ? + - ? ? (b) ( ) ( ) 1 , , f x y x y x y - = - + (c) ( ) 1 , , 2 2 x y x y f x y - + - ? ? = ? ? ? ? (d) ( ) ( ) ( ) 1 , 2 ,2 f x y x y x y - ? ? = - + ? ? 2.2. Let R be a non-empty relation on a collection of sets defined by A R B if and only if A n B = f. Then, (pick the true statement) (a) R is reflexive and transitive (b) R is symmetric and not transitive (c) R is an equivalence relation (d) R is not reflexive and not symmetricRead More

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