GATE 4 A Brief Analysis (Based on student test experiences in the stream of CS on st March, 4 - Second Session) Section wise analysis of the paper Mark Marks Total No of Questions Engineering Mathematics 6 9 5 Data Structures and Algorithms 5 5 DBMS 3 5 Theory of Computation 4 Compiler Design Computer Organization 3 4 Computer Networks 3 5 Digital Logic 3 Operating systems 3 5 Software Engineering and Web Technologies Verbal Ability 4 Numerical Ability 3 3 6 3 35 65 written permission. Discuss this question paper at www.gatementor.com.
Types of questions asked from each section Engineering Mathematics Data Structures and Algorithms DBMS Theory of Computation Compiler Design Computer Organization Computer Networks Digital Logic Operating Systems SEWT There were questions from Graph, Calculus, Functions, Probability, Matrices and Predicate Logic Questions from C-programs, Time Complexity, NP Complete, Quick sort, Analyzing Algorithms Questions from finding candidate keys, Normal Forms, Minimal Cover, SQL, Conflict Serializability Questions from Regular Expression, Finite Automata, Closure Properties Questions from SR and RR Conflicts, Code Optimization Questions from addressing mode, Cache Organization Questions from Network Security, Routing Protocols, Efficiency of SR Protocol Questions from Multiplexer, SOP, Radix Questions from SRTF, Optimal Page Replacement, Mutli threading Matching question on software models written permission. Discuss this question paper at www.gatementor.com.
Question s & Solutions for nd March ( nd Session Paper ) www.gateforum.com. Consider minterm expansion of the function F F(P,Q,R,S) =,,5,7,8,,3,5 Where minterms, 7, 8, 3 are do not care terms. The minimal Sum of Product form of F is (A) QS Solution: (B) QS (B) QS QS (C) QRS QRS QRS QRS (D) PQS PQS PQS PQS RS PQ X X Minimal SOP form QS Q S X X. Let X, Y be finite sets and F: XY be a function. Which of the following statement is TRUE? (A) For any subsets A and B of X, f A B f A f B (B) For any subsets A and B of X, f A B f A f B (C) For any subsets A and B of X, f A B min f A, f B (D) For any subsets S and T of Y, f S T f S f T Solution: (D) 3. Consider the following four pipeline systems P: Four pipeline stages with delay,,, P: Four pipeline stages with delay,.5,.5,.5 P3: Five pipeline stages with delay.5,,,.6, P4: Five pipeline stages with delay.5,.5,,,. Which of the following has peak clock cycle rate? Solution: (A) (A) P (B) P (C) P3 (D) P4 4. Which of the following statement is true about every nxn matrix with only real eigen values? (A) If trace of matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative (B) If the trace of the matrix is positive, all its eigen values are positive (C) If the determinant of the matrix is positive, all its eigen values are positive (D) If the product of the trace and determinant of the matrix is positive, all its eigen values are positive written permission. Discuss this question paper at www.gatementor.com. 3
Solution: (A) www.gateforum.com If either the trace or determinant is positive, there is at least one positive eigen value. Trace of matrix is positive and the determinant of the matrix is negative, this is possible only when there is odd number of negative eigen values. Hence at least one eigen value is negative. 5. If x sin x dx K, then the value of K is Solution: x sin x dx K x sin x dx x sin x dx K x sin x dx x sin x dx K x cos x sin x x cos x sin x K 4 K K 4 6. Consider the following combinational function block involving four Boolean variable x, y, a, b where x, a, b are inputs and y is the output. F(x, y, a, b) { } If (x is ) y = a; else y = b; Which of the following digital block is most suitable for implementing the function? (A) Full Adder (C) Multiplexer (B) Priority Encoder (D) Flip flop b a I I MUX y x written permission. Discuss this question paper at www.gatementor.com. 4
7. Let G be a group with 5 elements. Let L be a subgroup of G. It is known that LG and that the size of L is at least 4. The size of L is Solution: Order of subgroup divides order of group (Lagrange s theorem). 3, 5 and 5 can be the order of subgroup. As subgroup has at least 4 elements and it is not equal to the given group, also order of subgroup can t be 3 and 5. Hence it is 5. 8. The smallest length of the string not generated by the given regular expression is a * b * (ba) * a * Solution: Smallest length is 3 and string is bab. 9. Consider the following statements: P: Good mobile phones are not cheap Q: Cheap mobile phones are not good L: P implies Q M: Q implies P N: P is equivalent to Q Solution: (D) (A) Only L is true (C) Only N is true (B) Only M is true (D) L, M, N are true g : mobile is good c : mobile is cheap P : Good mobile phones are not cheap g c g c Q a b a b Q : Cheap mobile phones are not good c g c g Both P and Q are equivalent which means P and Q imply each other. If V and V are four dimensional subspace of a six dimensional vector space V then the smallest possible dimension of V V is. Consider = {a,b} and * denotes all possible string that can be generated by {a,b}, and (A) (B) (C) (D) * is the power set of *. Which of the following is true? * is countable * is uncountable * is countable * is countable * is uncountable * is countable * is uncountable * is uncountable With diagonalization we can prove that * is uncountable and * is countable as one to one mapping is possible with natural numbers. written permission. Discuss this question paper at www.gatementor.com. 5
. Consider the set of all functions, f: {,,., 4}{,,., 4} such that f(f(i))= i i4. Solution: (B) Consider the following statements: P: For each such function it must be the case that for every i, f(i) = i Q: For each such function it must be the case that for some i, f(i) = i R: Each such function must be onto. Which of the following is CORRECT? (A) P, Q and R are true (C) Only P and Q are true P : For example, let f and f f f f and f f f f f i i but f i i P is not true. (B) Only Q and R are true (D) Only R is true For some i, f(i) = i since there are odd number of elements in the set on which function is defined. 3. The value of the integral given below is x cos xdx Solution: (A) (A) - (B) (C) - (D) x cos x dx x sin x x cos x sin x O sin cos sin. o 4. Consider a basic block a = b+c c = a+d d = b+c e = d-b a = e+b Maximum number of nodes and edges in DAG generated for the above block (A) 6, 6 (B) 9, (C) 8, (D) 4, 4 written permission. Discuss this question paper at www.gatementor.com. 6
c d e a a d c b 5. Consider a decision problem CNFSAT = { is satisfiable propositional formula in CNF with at most literals per clauses} For example Solution: (B) = (x x ) (x x 3 The decision problem CNFSAT is (A) NP- complete ) (x x 4 ) a Boolean formula CNFSAT (B) Solvable in P time by reduction to directed graph reachability problem (C) Solves in constant time if every input sequence is satisfiable (D) NP hard but not NP complete 6. With respect to the numerical evaluation of the definite integral, K b x dx a, where a and b are given, which of the following is/are TRUE? I. The value of K obtained using Trapezoidal rule is always greater than or equal to the exact value of the definite integral. II. The value of K obtained using Simpson s rule is always equal to the exact value of the definite integral. (A) I only (B) II only (C) Both I and II (D) Neither I nor II Let us consider 3 x x dx.33333 up to 5decimals 3 3 o Let n = 4 written permission. Discuss this question paper at www.gatementor.com. 7
x.5.5.75 y x.65.5.565 y y y y y 3 4 h x dx y y 4 y y y3 (By trapezoidal rule).5.65.5.565.34375 Clearly the value is greater than.33333, also for greater values of n there will be slight difference, hence given integral is always greater than.33333 Using Simpson s rd 3 rule h x dx y y 4 y 4 y y3 o 3.5.5 4.65.565 3.33333 up to 5 decimals, which is equal to exact value. 7. J Q J Q J Q C C C K Q K Q K Q The above synchronous sequential circuit built using JK flip flops is initialized with Q Q Q =. The state sequence for this circuit for the next three clock cycles is (A),, (B),, (C),, (D),, P.S. FF inputs Q Q Q J K J K J K Q Q Q Q Q Q N.S. Q Q Q written permission. Discuss this question paper at www.gatementor.com. 8
8. Which of the following decision problem is undecidable? Solution: (A) (A) Ambiguity problem of context free grammar (B)If the string generated by the context free grammar (C) If context free grammar generates empty language (D) If context free grammar generates finite language www.gateforum.com There is no algorithm for determining the ambiguity of a given CFG, hence it is undecidable. 9. A cache memory has memory access time for read operation as ns if cache hit occurs and 5ns for read operation if cache miss occurs. Memory access time for write operation is ns if cache hit occurs and ns for write operation if cache miss occurs. Suppose instruction fetch operations, 6 memory operand read operations and 4 memory operand write operations are being performed. The cache hit rate is.9, what is the average execution time for the instruction? Solution: For fetch operation 9 ns + 5ns = 4ns Memory operand read operation 54ns + 3ns = 84 ns Memory operand write operation 7ns+4ns = ns Total time for instruction is 4 +84+ = 336ns Average time is 3.36ns.. Consider the five stage pipeline system with Instruction Fetch (IF), Instruction Decode and Register Fetch (ID/RF), Execute (EX), Memory access( MEM) and Write Back (WB) operations. The respective stages are taking time ns,.ns, ns, ns,.75ns. A modification is proposed on this pipeline system and ID/RF stage is further divided in ID, RF and RF taking./3ns each, further EX stage is also divided in EX and EX each taking equal time. If the instruction is a branch instruction, next instruction pointer will be available at the end of EX stage of first pipeline and after EX for the second pipeline. All instruction other than branch instruction have average CPI as. % instructions are branch instruction. If time average access time for first pipeline is P and for second is Q, the ratio of P & Q is Solution: T avg = (+ stall frequency Stall cycle) T clock TP avg = (+. ).ns = 3.8ns TQ avg = (+. 5) ns = ns IF ID/RF stall EX stall IF For first pipeline stall cycles P.54 Q IF ID stall RF stall RF EX EX stall stall stall IF For second pipeline 5 stall cycles. If G is a forest with n vertices and k connected components, how may edges does G have? (A) n k (B) n k (C) n k (D) n k written permission. Discuss this question paper at www.gatementor.com. 9
www.gateforum.com Forest is collection of trees and in a forest each tree will form one connected component. Assume that there are k connected components where each is having n, n, n3.. nk vertices respectively. Total number of edges in a forest with n vertices and k connected components will be as follows n n n... n n n n... n...k times 3 k 3 k n k. Let denotes the minimum degree of vertex in a graph. For all planar graphs on n vertices with 3, which of the following is TRUE? (A) In any planar embedding, the number of faces is at least n (B) In any planar embedding, the number of faces is less than n (C) There is a planar embedding in which the number of faces is less than n (D) There is a planar embedding in which the number of faces is at most n 3. The correct formula for the sentence not all rainy days are cold is (A) d Rainy d ~ Cold d (B) d ~ Rainy d Cold d (C) d ~ Rainy d Cold d (D) d Rainy d ~ Cold d Solution: (D) not all rainy days are cold:~ d Rainy d Cold d ~ d ~ Rainy d Cold d d Rainy d ~ Cold d 4. Consider the following language defined over = {,,c} L = { n n n } L = {wcw r w {,}*} L 3 = {ww r w {,}*} Which of the above language can be recognized by deterministic pushdown automata? (A) None of the language (B) Only L (C) L & L (D) All the languages L and L are deterministic CFLs and so those can be recognized by deterministic PDA where as L 3 is not deterministic. written permission. Discuss this question paper at www.gatementor.com.
5. Let S be a sample space and two mutually exclusive events A and B be such that AB = S. If P(.) denotes the probability of the event, the maximum value of P(A)P(B) is Solution: A B S Q P A B P S P A P B P A B P A P B [ A & B are M.E.] Let P A x P B x y P A P B x x dy d y d y x x ; ; dx dx dx y has maximum at x ymax.5 4 x 6. Let denotes the exclusive XOR operator. Let, denote binary constants. Consider following Boolean expression for F over two variables P and Q: F P, Q P P Q P Q Q The equivalent expression for F is (A) P Solution: (D) Q (B) P Q (C) P Q (D) P Q F P, Q P P Q P Q Q P PQ PQ PQ PQ Q PQ PQ PQ PQ P PQ P Q P PQ PQ PQ PQ Q PQ PQ Q Q P PQ PQ P Q 7. There are two elements x, y in a group (G,*) such that every element in the group can be written as a product of some number of x s and y s in some order. It is known that x*x = y*y=x*y*x*y=y*x*y*x=e where e is the identity element. The maximum number of elements in such a group is written permission. Discuss this question paper at www.gatementor.com.
8. The table below has question wise data on the performance of students in an examination. The marks for each question are also listed. There is no negative or partial marking in the examination. Question Number Marks Answered Correctly Answered Wrongly Not attempted 7 6 3 5 7 3 3 8 3 What is the average of the marks obtained by the class in the examination? (A).34 (B).74 (C) 3. (D) 3.9 3 5 3 33 Average marks : 3. 44 44 9. While trying to collect an envelop I was losing consciousness. IV from under the table II, Mr. X fell down III Which of the above underlined parts of the sentences is NOT appropriate? (A) I (B) II (C) III (D) IV and Solution: (D) 3. If she how to calibrate the instrument, she done the experiment. (A) knows, will have (C) had known, could have (B) knew, had (D) should have known, would have 3. Choose the word that is opposite in meaning to the word coherent. (A) sticky (B) well-connected (C) rambling (D) friendly 3. Which number does not belong to the series below?, 5,, 7, 6, 37, 5, 64 (A) 7 (B) 37 (C) 64 (D) 6 written permission. Discuss this question paper at www.gatementor.com.
33. Consider the equation: (756) 8 - (Y) 8 =(4364) 8, where (X) N stands for X to the base N. Find Y. (A) 634 (B) 737 (C) 34 (D) 36 7 5 6 3 4 4 3 6 4 8 8 8 34. By the beginning of th century, several hypothesis were being proposed, suggesting a paradigm shift in our understanding of the universe. However the clinching evidence was provided by experimental measurements of the positions of a star which was directly behind our sun. Which of the following inference(s) may be drawn from the above paragraph? i. Our understanding of the universe changes based on the position of stars ii. Paradigm shift usually occur at the beginning of centuries iii. Stars are important objects in the universe iv. Experimental evidence was important in confirming this paradigm shift (A) i, ii & iv (B) iii only (C) i & iv (D) iv only 35. The Gross Domestic Product (GDP) in Rupees grew at 7% during -3. For international comparison, the GDP is compared in US dollars (USD) after conversion based on the market exchange rate. During the period -3 the exchange rate for the USD increased from Rs. 5/USD to Rs. 6/USD. India s GDP in USD during the period -3 (A) increased by 5% (B) decreased by 3% (C) decreased by % (D) decreased by % Solution: (B) 36. The ratio of Male to Female students in a college for 5 years is plotted in the following line graph. If the number of female students in and is equal, what is the ratio of male students in to male students in? Ratio of Male to Female students 3.5 3.5.5.5 8 9 (A) : (B) : (C).5: (D).5: written permission. Discuss this question paper at www.gatementor.com. 3
37. A dance program is scheduled for. am. Some students are participating in the program and they need to come hour earlier than the start of the event. These students should be accompanied by a parent. Other students and parents should come in time for the program. The instruction you think that is appropriate for this is Solution: (B) (A) Students should come at 9. am and parents should come at. am (B) Participating students should come at 9. am accompanied by a parent and other parents and students should come by. am (C) Students who are not participation should come by. am and they should not bring their parents. Participating students should come at 9. am. (D) Participating students should come before 9. am. Parents who accompany them should come at 9. am. All other should come at. am written permission. Discuss this question paper at www.gatementor.com. 4