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1 No part of this publication may be reproduced or distributed in any form or any means, electronic, mechanical, photocopying, or otherwise without the prior permission of the author. GATE SOLVED PAPER Computer Science Engineering Digital Logic Copyright By NODIA & COMPANY Information contained in this book has been obtained by authors, from sources believes to be reliable. However, neither Nodia nor its authors guarantee the accuracy or completeness of any information herein, and Nodia nor its authors shall be responsible for any error, omissions, or damages arising out of use of this information. This book is published with the understanding that Nodia and its authors are supplying information but are not attempting to render engineering or other professional services. NODIA AND COMPANY B-8, Dhanshree Tower Ist, Central Spine, Vidyadhar Nagar, Jaipur Ph :

2 YEAR 2001 Q. 1 Given the following Karnaugh map, which one of the following represents the minimal sum-of-products of the map? (A) xy + y' z (B) wx'' y + xy + xz (C) wx ' + yz ' + xy (D) xz + y Q. 2 Consider the following circuit with initial state Q0= Q1= 0. The D flip-flops are positive edged triggered and have set up times 20 nanosecond and hold times 0. Consider the following timing diagrams of X and C ; the clock of C \$ 40 nanosecond. Which one is the correct plot of Y

3 Q. 3 The 2 s complement representation of (- 539) 10 is hexadecimal is (A) ABE (B) DBC (C) DE5 (D) 9E7 Q. 4 Consider the circuit shown below. The output of a 2:1 Mux is given by the function ( ac' + bc). Which of the following is true? (A) f = x1' + x2 (B) f = x1' x2+ x1x2' (C) f = x1x2 + x1' x2' (D) f = x1+ x2 Q. 5 Consider the circuit given below the initial state Q0= 1, Q1= Q2= 0. The state of the circuit is given by the value 4Q + 2Q + Q Which one of the following is the correct state sequence of the circuit? (A) 1, 3, 4, 6, 7, 5, 2 (B) 1, 2, 5, 3, 7, 6, 4 (C) 1, 2, 7, 3, 5, 6, 4 (D) 1, 6, 5, 7, 2, 3, 5 YEAR 2002 Q. 6 Minimum sum of product expression for fwxyz (,,, ) shown in Karnaugh-map below is (A) xz + y' z (B) xz' + zx' (C) xy ' + zx' (D) None of the above

4 Q. 7 The decimal value of 0.25 (A) is equivalent to the binary value 0.1 (B) is equivalent to the binary value 0.01 (C) is equivalent to the binary value (D) cannot be represented precisely in binary. Q. 8 The 2 s complement represent representation of the decimal value - 15 is (A) 1111 (B) (C) (D) Q. 9 Sign extension is a step in (A) floating point multiplication (B) signed 16 bit integer addition (C) arithmetic left shift (D) converting a signed integer from one size to another. Q. 10 In 2 s complement addition, overflow (A) Relational algebra is more powerful than relational calculus (B) Relational algebra has the same power as relational calculus. (C) Relational algebra has the same power as safe relational calculus. (D) None of the above. Q. 11 Consider the following logic circuit whose inputs are functions f1, f2, f3 and output is f Given that f1 (,,) x y z = S(,,,) 0135 f2 (,,) x y z = S(,) 67, and fxyz (,,) = S(,,) 145 f 3 is (A) S (,,) 145 (B) S(,) 67 (C) S (,,,) 0135 (D) None of the above Q. 12 Consider the following multiplexor where 10, 11, 12, 13 are four data input lines selected by two address line combinations AA 10 = 00, 01, 10, 11 respectively and f is the output of the multiplexor. EN is the Enable input.

5 The function fxyz (,,) implemented by the above circuit is (A) xyz ' (B) xy + z (C) x+ y (D) None of the above Q. 13 Let fab (, ) = A' + B. Simplified expression for function ffx (( + yy, ),) z is (A) x' + z (B) xyz (C) xy' + z (D) None of the above Q. 14 What are the states of the Auxiliary Carry (AC) and Carry Flag (CY) after executing the following 8085 program? MIV H, 5DH MIV L, 6BH MOV A, H ADD L (A) AC = 0 and CY = 0 (B) AC = 1 and CY = 1 (C) AC = 1 and CY = 0 (D) AC = 0 and CY = 1 Q. 15 The finite state machine described by the following state diagram with A as starting state, where an arc label is x and x stands for 1-bit input and y stands y for 2-bit output. (A) Outputs the sum of the present and the previous bits of the input. (B) Outputs 01 whenever the input sequence contains 11 (C) Outputs 00 whenever the input sequence contains 10 (D) None of the above. YEAR 2003 Q. 16 Assuming all numbers are in 2 s complement representation, which of the following number is divisible by ? (A) (B) (C) (D)

6 YEAR 2003 TWO MARKS Q. 17 The following is a scheme for floating point number representation using 16 bits. Let sc, and m be the number represented in binary in the sign, exponent, and mantissa fields respectively. Then the flouting point number represented id 2-9 e-31 (-1) (1 + m # 2 )2, if the exponent ) 0 otherwise What is the maximum difference between two successive real numbers representable in this system? (A) 2-40 (B) 2-9 (C) 2 22 (D) 2 31 Q. 18 A 1-input, 2-output synchronous sequential circuit behaves as follows. Let zk, nk denote the number of 0 s and 1 s respectively in initial k bits of the input ( zk+ nk = k). The circuit outputs 00 until one of the following conditions holds. 1. nk- nk = 2. In this case, the output at the k -th and all subsequency clock ticks is nk- zk = 2. In this case, the output at the k -th and all subsequent clock ticks is 01. What in the minimum number of states required in the state transition graph of the above circuit? (A) 5 (B) 6 (C) 7 (D) 8 Q. 19 The literal count of a boolean expression is the sum of the number of times each literal appears in the expression. For example, the literal count of ( xy + xz) is 4. What are the minimum possible literal counts of the product-of-sum and sum-ofproduct representations respectively of the function given by the following karnaugh map? Here, denotes don t care (A) (11,9) (B) (9,13) (C) (9,10) (D) (11,11)

7 Q. 20 Consider the following circuit composed of XOR gates and non-inverting buffers. The non-inverting buffers have delays d = 2ns and d = 4ns as shown in the 1 2 figure. both XOR gates and al wires have zero delay. Assume that all gate inputs, outputs and wires are stable at logic level 0. If the following waveform is applied at input. A, how many transition (s) (change of logic levels) occur (s) at B during the interval from 0 to 10 ns? (A) 1 (B) 2 (C) 3 (D) 4 YEAR 2004 Q. 21 The Boolean function xy '' + xy+ xy ' is equivalent to (A) x' + y' (B) x+ y (C) x+ y' (D) x' + y Q. 22 In an SR latch made by cross-coupling two NAND gates, if both S and R inputs are set to 0, then it will result in (A) Q = 0, Q' = 1 (B) Q = 1, Q' = 0 (C) Q = 1, Q' = 1 (D) Indeterminate states Q. 23 If 73 x (in base-x number system) is equal to 54, (in base-y number system), the possible values of x and y are (A) 8, 16 (B) 10, 12 (C) 9, 13 (D) 8, 11 Q. 24 What is the result of evaluating the following two expressions using three-digit floating point arithmetic with rounding? ( ) ( ) (A) 9.51 and 10.0 respectively (B) 10.0 and 9.51 respectively (C) 9.51 and 9.51 respectively (D) 10.0 and 10.0 respectively YEAR 2004 TWO MARKS Q. 25 A circuit outputs a digit in the form of 4 bits. 0 is represented by 0000, 1 by 0001,...9 by A combinational circuit is to be diesigned which takes these 4 bits as input and outputs 1 if the digit \$ 5, and 0 otherwise. If only AND, OR and NOT gates may be used, what is the minimum number of gates required? (A) 2 (B) 3 (C) 4 (D) 5

8 Q. 26 Which are the essential prime implicates of the following Boolean function? fabc (,, ) = ac ' + ac' + bc ' (A) ac ' and ac ' (B) ac ' and bc ' (C) ac ' only (D) ac ' and bc' Q. 27 Consider the partial implementation fo a 2-bit counter using T flip flops following the sequence , as shown below To complete the circuit, the input X should be (A) Q 2 ' (B) Q + Q 2 1 (C) ( Q 5 Q)' (D) Q 5 Q Q. 28 A 4-bit carry look ahead adder, which adds two 4-bit numbers, is designed using AND, OR, NOT, NAND, NOR gates only. Assuming that all the inputs are available in both complemented and uncompensated forms and the delay of each gate is one time unit, what is the overall propagation delay of the adder? Assume that the carry network has been implemented using two-level AND-OR logic. (A) 4 time units (B) 6 time units (C) 10 time units (D) 12 time units Q. 29 Let A = and B be two 8-bit 2 s complement numbers. Their product in 2 s complement is (A) (B) (C) (D) YEAR 2005 Q. 30 Consider the following circuit. Which one of the following is TRUE? (A) f is independent of X (B) f is independent of Y (C) f is independent of Z (D) None of XYZ,, is redundant

9 Q. 31 The range of integers that can be represented by an a bit 2 s complement number system is n-1 n-1 n-1 n-1 (A) -2 to (2-1) (B) -(2-1) to (2-1) n 1 n 1 (C) -2 - to 2 - n- (D) (2 1 n ) to (2 1-1) Q. 32 The hexadecimal representation of is (A) 1AF (B) D78 (C) D71 (D) 32F Q. 33 The switching expression corresponding to fabcd (,,, ) = / ( ,,,,, ) is (A) BC' D' + A' C' D + AB' D (B) ABC' + ACF + B' C' D (C) ACD' + A' BC' + AC' D' (D) A' BD + ACD' + BCD' YEAR 2005 Q. 34 Consider the following circuit involving a positive edge triggered D-FF. TWO MARKS Consider the following timing diagram. Let Ai represent the logic level on the line A in the i- th clock period. Let A represent the complement of A. The correct output sequence on Y over the clock perids 1 through 5 is (A) AAA 0 1 1' AA 3 4 (B) AAA 0 1 2' AA 3 4 (C) AAA 1 2 2' AA 3 4 (D) AA' AAA Q. 35 The following diagram represents a finite state machine which takes as input a binary number from the least significant bit Which one of the following is TRUE? (A) It computes 1 s complement of the input number (B) It computes 2 s complement of the input number (C) It increments the input number (D) It decrements the input number

10 Q. 36 Consider the following circuit The flip-flops are positive edge triggered D FFs. Each state is designated as a two bit string Q 0, Q 1. Let the initial state be 00. The state transition sequence is (A) 00 " 11 " 01 (B) 00 " 11 ABBBBBBB C ABB BC (C) 00 " 10 " 01 " 11 (D) 00 " 11 " 01 " 10 ABBBBBBBBBB C ABBBBBBBBBB C Common Data For Q. 37 & 38 Solve the problems and choose the correct answers. Consider the following floating point format Mantissa is a pure fraction is sign-magnitude form. Q. 37 The decimal number # 2 13 has the following hexadecimal representation without normalization and rounding off (A) 0D 24 (B) 0D 4D (C) 4D 0D (D) 4D 3D Q. 38 The normalized representation for the above format is specified as follows. The mantissa has an implicit 1 preceding the binary (radix) point. Assume that only 0 s are padded in while shifting a field. The normalized representation of the above number ( # 2 13 ) is (A) 0A 20 (B) (C) 4DD0 (D) 4AE8 YEAR 2006 Q. 39 You are given a free running clock with a duty cycle of 50% and a digital waveform f which changes only at the negative edge of the clock. Which one of the following circuits (using clocked D flip flops) will delay the phase of f by 180c?

11 YEAR 2006 TWO MARKS Q. 40 Consider the circuit below. Which one of the following options correctly represents (,,) fxyz? - - (A) xz + xy + yz - - (B) xz + xy + yz - - (C) xz + xy + yz -- (D) xz + xy + yz Q. 41 Given two three bit numbers a2aa 1 0 and b2bb 1 0 and c, the carry in, the function that represents the carry generate function when these two numbers are added is (A) a2b2+ aab aaab aabb abb aabb abbb (B) a2b2+ abb aabb aab b1+ aab aabb aabb (C) a + b + ( a 5b)[ a + b + ( a 5b)( a + b )] (D) a2b2+ aab aaab aabb abbaabb abbb Q. 42 Consider a boolean function fwxyz. (,,, ) Suppose that exactly one of its inputs is allowed to change at a time. If the function happens to be true for two input vectors i1+ < w1, x1, y1, x1> and i2+ < w2, x2, y2, z2>, we would like the function to remain true as the input changes from i 1 to i2( i1 and i 2 differ in exactly one bit position), without becoming false momentarily. Let fwxyz (,,, ) = / ( 5,, 711, 12, 13, 15). Which of the following cube covers of f will ensure that the required property is satisfied? (A) wxz, wxy, xyz, xyz, wyz (B) wxy, wxz, wyz (C) wxyz, xz, wxyz (D) wzy, wyz, wxz, wwxz, xyz, xyz

12 Q. 43 We consider addition of two 2 s complement numbers bn-1bn b0 and an-1an a0. A binary adder for adding unsigned binary numbers is used to add the two numbers. The sum is denoted by cn-1cn c0 and the carryout by c out. Which one of the following options correctly identifies the overflow condition? (A) cout ( an -15 bn - 1) (B) an-1bn-1cn-1+ an-1bn-1cn-1 (C) c 5 c - (D) a 5b 5c out n 1 n-1 n-1 n-1 Q. 44 Consider number represented in 4-bit gray code. Let h3hhh be the gray code representation of a number n and let g3ggg be the gray code of ( n + 1) (modulo 16) value of the number. Which one of the following functions is correct? (A) g ( h h h h ) = ( ,,,,,,, ) / (B) g1( h1h2h1h0 ) = / (4,9,10,11,12,,13,14,15) (C) g2( h1h2h1h0 ) = / ( 2,4, 5, 6, 7,12,,13,15) (D) g3( h1h2h1h0 ) = / ( 0, 1,6,7, 10, 11,12,,13,) YEAR 2007 Q. 45 What is the maximum number of different Boolean functions involving n Boolean variables? (A) n 2 (B) 2 n (C) 2 2n (D) 2 n2 Q. 46 How many 3-to-8 line decoders with an enable input are needed to construct a 6-to-64 line decoder without using any other logic gates? (A) 7 (B) 8 (C) 9 (D) 10 Q. 47 Consider the following Boolean function of four variables fwx (,,, yz, ) = / ( ,,,,,,, ) The function is (A) independent of one variable (B) independent of two variables (C) independent of three variables (D) dependent on all the variables YEAR 2007 TWO MARKS Q. 48 Let fwxyz (,,, ) = / ( ,,,,,,, ). Which of the following expressions are NOT equivalent to f? (A) x'' y z + w' xy' + wy' z + xz (B) wyx ''' + wxy '' + xz (C) w''' y z + wx'' y + xyz + xy' z (D) xyz '' + wxy '' + wy ' Q. 49 Define the connective* for the boolean variable X and Y as: X* Y = XY + X' Y' Let Z = X* Z Consider the following expression PQ, and R. P : X = Y* ZQ: Y = X* Z R : X* Y* Z = 1 Which of the following is TRUE? (A) only P and Qare valid (B) Only Qand Rare valid (C) Only P and Rare valid (D) All P, Q,Rare valid

13 Q. 50 Suppose only one multiplexer and one inverter are allowed to be used to implement any Boolean function of nvariables. What is the minimum size of the multiplexer needed? (A) 2 n line to 1 line (B) 2 n+ 1 line to 1 line (C) 2 n-1 line to 1 line (D) 2 n-2 line to 1 line Q. 51 In a look-ahead carry generator, the carry generate function G i and the carry propagate function P i for inputs, A i and B i are given by Pi = Ai5 Bi and Gi = AiBi The expressions for the sum bit S and carry bit C i+ 1 of the look ahead carry adder are given by Si+ Pi5 Ci and Ci+ 1 Gi+ PC i i, Where C 0 is the input carry. Consider a two-level logic implementation of the look-ahead carry generator.. Assume that all P i and G i are available for the carry generator circuit and that the AND and OR gates can have any number of inputs. The number of AND gates and OR gates needed to implement the look-ahead carry generator for a 4-bit adder with S3, S2, S1, S0 and C 4 as its outputs are respectively (A) 6,3 (B) 10,4 (C) 6,4 (D) 10,5 Q. 52 The control signal functions of 4-bit binary counter are given below (where X is don t care ) Clear Clock Load Count Function 1 X X X Clear to 0 0 X 0 0 No change 0-1 X Load input Count next The counter is connected as follows Assume that the counter and gate delays are negligible. If the counter starts at 0, then it cycles through the following sequence (A) 0,3,4 (B) 0,3,4,5 (C) 0,1,2,3,4 (D) 0,1,2,3,4,5 YEAR 2008 Q. 53 In the IEEE floating point representation the hexadecimal value 0x corresponds to (A) the normalized value (B) the normalized value (C) the normalized value + 0 (D) the special value + 0

14 Q. 54 In the karnaugh map shown below, X denoted a don t care term. What is the nominal form of the function represented by the karnaugh map (A) bd. + ad (C) bd. + abd (B) ab. + bd. + abd (D) ab. + bd. + ad. Q. 55 Let a denote number system radix. The only value(s) of r that satisfy the equation , is/are (A) decimal 10 (B) decimal 11 (C) decimal 10 and 11 (D) any value> 2 Q. 56 Give f 1, f 3 and f in canonical sum of products form (in decimal) for the circuit f 1 = /m(4,5,6,7,8) f 3 = /m(1,6,15) f = /m(1,6,8,15) Then f 2 is (A) / m(,) 46 (B) /m(4, 8) (C) / m(,) 68 (D) /m( 4,6,8) YEAR 2008 TWO MARKS Q. 57 If P, Q, R are Boolean variables, ( P+ Q - - ) ( PQ. + PR. ) ( PR Q - ) simplifies to (A) PQ. - (B) PR. - - (C) PQ. + R - (D) PR. + Q YEAR 2009 Q. 58 ( 1217) 8 is equivalent to (A) ( 1217) 16 (B) ( 028F) 16 (C) ( 2297) 10 (D) ( 0B17) 16 Q. 59 What is the minimum number of gates required to implement the Boolean function ( AB + C) if we have to use only 2-input NOR gates? (A) 2 (B) 3 (C) 4 (D) 5

15 YEAR 2010 Q. 60 The minterm expansion of f( P, Q, R) = PQ + QR + PR is (A) m2+ m4+ m6+ m1 (B) m + m + m + m (C) m0+ m1+ m6+ m1 (D) m + m + m + m Q. 61 P is a 16-bit signed integer. The 2 s complement representation of P is ( F B) The 2 s complement representation of 8 ) P is (A) ( C3D8) 16 (B) ( 187B) 16 (C) ( F878) 16 (D) ( 987B) 16 Q. 62 The Boolean expression for the output f of the multiplexer shown below is (A) P5Q5 R (B) P5Q5R (C) P+ Q+ R (D) P+ Q+ R YEAR 2010 TWO MARKS Q. 63 What is the boolean expression for the output f of the combinational logic circuit of NOR gates given below? (A) Q+ R (B) P+ Q (C) P+ R (D) P+ Q+ R Q. 64 In the sequential circuit shown below, if the initial value of the output Q 1 Q 0 is 00, what are the next four values of Q 1 Q 0? (A) 11, 10, 01, 00 (B) 10, 11, 01, 00 (C) 10, 00, 01, 11 (D) 11, 10, 00, 01

16 YEAR 2011 Q. 65 Which one of the following circuits is NOT equivalent to a 2-input XNOR (exclusive NOR) gate? Q. 66 The minimum number of D flip-flops needed to design a mod-258 counter is (A) 9 (B) 8 (C) 512 (D) 258 YEAR 2011 TWO MARKS Common Data For Q. 67 and 68 : Consider the following circuit involving three D-type flip-flops used in a certain type of counter configuration Q. 67 If a some instance prior to the occurrence of the clock edge, P, Q and R have a value 0, 1 and 0 respectively, what shall be the value of PQR after the clock edge? (A) 000 (B) 001 (C) 010 (D) 011

17 Q. 68 If all the flip-flops were reset to 0 at power on, what is the total number of distinct outputs (states) represented by PQR generated by the counter? (A) 3 (B) 4 (C) 5 (D) 6 YEAR 2012 Q. 69 The truth table X Y fxy ^, h represents the Boolean function (A) X (B) X+ Y (C) X5 Y (D) Y YEAR 2012 TWO MARKS Q. 70 What is the minimal form of the Karnaugh map shown below? Assume that X denotes a don t care term. (A) bd (B) bd + bc (C) bd + abcd (D) bd + bc + cd **********

18 ANSWER KEY Digital Logic (A) (A) (C) (C) (B) (B) (B) (D) (A) (B) (A) (A) (C) (C) (A) (A) (C) (C) (C) (C) (D) (D) (D) (A) (B) (A) (D) (B) (A) (D) (A) (A) (A) (A) (B) (D) (D) (D) (B) (A) (D) (A) (C) (C) (C) (B) (B) (D) (D) (C) (B) (C) (D) (A) (D) (C) (A) (B) (B) (A) (A) (B) (A) (A) (D) (A) (D) (B) (A) (B)

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