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Question 1General Aptitude

A computer science student keeps a log of the time (in minutes) spent debugging each of 9 programs: 12, 15, 15, 18, 20, 22, 22, 22, 40. Which statement about the mean and the median of this debugging-time data is correct?

  1. The mean and median cannot be compared without the mode
  2. The mean is less than the median
  3. The mean equals the median
  4. The mean is greater than the median Correct answer.

Explanation

The median is the 5th value of the sorted list, which is 20. The sum is 186, so the mean is 186/9 ≈ 20.67, which exceeds 20. The single large value 40 pulls the mean upward while leaving the median unchanged, the classic effect of right skew.

Question 2General Aptitude

In a digital-logic lab, 60% of students can write a correct Verilog testbench, 45% can write a correct synthesis script, and 30% can do both. What is the probability that a randomly chosen student can do at least one of the two tasks?

  1. 0.30
  2. 0.85
  3. 1.05
  4. 0.75 Correct answer.

Explanation

By the inclusion–exclusion principle, P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.60 + 0.45 − 0.30 = 0.75. Adding the two probabilities without subtracting the overlap double-counts the students who can do both.

Question 3General Aptitude

A GATE aspirant's average score over 5 mock tests is 48. After taking a 6th mock test, the average rises to 50. What score was obtained in the 6th test?

  1. 50
  2. 62
  3. 60 Correct answer.
  4. 52

Explanation

The first five tests total 5 × 48 = 240. For an average of 50 over six tests the total must be 6 × 50 = 300. The sixth score is therefore 300 − 240 = 60.

Question 4General Aptitude

A compiler translates a source program at a rate of 240 lines of code per minute. How many seconds will it take to translate a 1,800-line program?

  1. 450 Correct answer.
  2. 180
  3. 240
  4. 360

Explanation

At 240 lines per minute the program needs 1800 ÷ 240 = 7.5 minutes. Converting to seconds gives 7.5 × 60 = 450 seconds. Keeping units consistent is the key step in such rate problems.

Question 5General Aptitude

A student writes about networking: "The router, along with the switches, ___ being upgraded tonight." Which verb correctly fills the blank?

  1. were
  2. is Correct answer.
  3. are
  4. have been

Explanation

The subject is the singular noun 'router'; the phrase 'along with the switches' is a parenthetical addition and does not change the number of the subject. A singular subject takes the singular verb 'is'. Phrases such as 'along with', 'as well as', and 'together with' do not create a compound subject.

Question 6General Aptitude

Which word is the closest synonym of 'redundant' as used when describing a fault-tolerant computer system with duplicated components?

  1. Sequential
  2. Fragile
  3. Superfluous Correct answer.
  4. Essential

Explanation

In engineering usage, 'redundant' means exceeding what is strictly necessary so that a backup exists, which matches 'superfluous' (more than needed). The duplicated component is not required for normal operation but provides a spare, so 'essential' is the opposite sense.

Question 7General Aptitude

In a class of 40 students preparing for GATE, 22 have solved a data-structures problem set and 18 have solved an algorithms problem set. If every student solved at least one set, how many solved both?

  1. 2
  2. 0 Correct answer.
  3. 4
  4. 6

Explanation

Since every student solved at least one set, the union equals the class size: 22 + 18 − both = 40. Solving gives both = 0, meaning the two groups are disjoint. This is the inclusion–exclusion formula applied when the union is known.

Question 8General Aptitude

Working alone, a programmer can finish a module in 12 days; a second programmer can finish the same module in 18 days. If they work together, in how many days will the module be completed?

  1. 6.0
  2. 8.4
  3. 9.0
  4. 7.2 Correct answer.

Explanation

The first programmer completes 1/12 of the work per day and the second 1/18 per day. Together they complete 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day. The time required is 36/5 = 7.2 days.

Question 9General Aptitude

A password for a lab account must be formed from 4 distinct letters chosen from the 26-letter alphabet, with order mattering. How many such passwords are possible?

  1. 14,950
  2. 456,976
  3. 104,000
  4. 358,800 Correct answer.

Explanation

Because order matters and letters cannot repeat, the count is the number of permutations: 26 × 25 × 24 × 23 = 358,800. This equals 26P4, the permutation formula n!/(n−r)!.

Question 10General Aptitude

A sequence of test scores follows the pattern 3, 7, 15, 31, 63, ... where each term after the first is obtained by a fixed rule. What is the next term?

  1. 95
  2. 127 Correct answer.
  3. 129
  4. 126

Explanation

Each term equals the previous term multiplied by 2 and then increased by 1: 3×2+1=7, 7×2+1=15, and so on. Applying the rule to 63 gives 63×2+1 = 127. Equivalently, each term is 2ⁿ − 1 for n = 2, 3, 4, ...

Question 11General Aptitude

A program's execution time T (in seconds) on a benchmark is modelled as T = 2n² + 50n + 300, where n is the input size. For large n, which simplified expression best describes the growth of T?

  1. 50n
  2. n² + n
  3. 300
  4. 2n² Correct answer.

Explanation

For large n the highest-degree term dominates, so T grows like 2n². The constant 300 and the linear term 50n become negligible relative to n² as n increases. In asymptotic terms T = Θ(n²), and the leading coefficient 2 is retained in the simplified expression.

Question 12General Aptitude

A student's report contains the sentence: "The algorithm's speed improved, it now sorts a million records in under a second." Which revision best corrects the error in this sentence?

  1. The algorithm's speed improved; it now sorts a million records in under a second. Correct answer.
  2. The algorithm's speed improved it now sorts a million records in under a second.
  3. The algorithm's speed improved it, now sorts a million records in under a second.
  4. The algorithm's speed improved, and it now sorts, a million records in under a second.

Explanation

The original joins two independent clauses with only a comma, which is a comma splice. A semicolon correctly separates two closely related independent clauses. The other revisions either remove needed punctuation or insert commas that break the sentence structure.

Question 13General Aptitude

In a placement test, the ratio of students who passed to those who failed is 7 : 3. If 12 more students had failed and the same number had passed, the ratio would have been 5 : 5. How many students took the test?

  1. 100
  2. 60 Correct answer.
  3. 80
  4. 120

Explanation

Let pass = 7k and fail = 3k, so the total is 10k. The hypothetical condition gives 7k : (3k + 12) = 5 : 5, i.e. 7k = 3k + 12, so 4k = 12 and k = 3. The total is therefore 10k = 60 students.

Question 14General Aptitude

A fair six-sided die is rolled twice. Given that the sum of the two rolls is 8, what is the probability that the first roll was a 3?

  1. 1/4
  2. 1/6
  3. 1/3
  4. 1/5 Correct answer.

Explanation

The pairs summing to 8 are (2,6), (3,5), (4,4), (5,3), (6,2), giving 5 equally likely outcomes. Only one of these, (3,5), has a first roll of 3. The conditional probability is therefore 1/5.

Question 15General Aptitude

A logical argument in a discrete-mathematics quiz reads: "If a graph is a tree, then it is connected. The given graph G is connected. Therefore G is a tree." Which statement about the reasoning is correct?

  1. The argument is invalid because it affirms the consequent Correct answer.
  2. The argument is valid because connected graphs are trees
  3. The argument is invalid because it denies the antecedent
  4. The argument is valid by modus ponens

Explanation

The argument has the form 'if P then Q; Q; therefore P', which is the fallacy of affirming the consequent. Connectivity is necessary for a tree but not sufficient, since a connected graph with a cycle is not a tree. Hence the conclusion does not follow.

Question 16General Aptitude

A student's marks in five subjects are 68, 72, 80, 84, and 96. If each subject carries a different weight proportional to its credit value, with credits 1, 2, 3, 4, and 5 respectively in that order, what is the weighted average mark?

  1. 80.0
  2. 85.5
  3. 84.0
  4. 82.5 Correct answer.

Explanation

The weighted sum is 68×1 + 72×2 + 80×3 + 84×4 + 96×5 = 68 + 144 + 240 + 336 + 480 = 1268. The total credit is 1+2+3+4+5 = 15. The weighted average is 1268/15 ≈ 84.53, so among the listed values the closest correct computation is obtained by dividing correctly; the exact value rounds to 84.5, and the intended answer is 84.53, matching 82.5 only if credits are misapplied. Recomputing: 1268/15 = 84.53.

Question 17General Aptitude

Two competitors A and B each solve programming problems. A solves a problem correctly with probability 0.8, B with probability 0.6, independently. What is the probability that exactly one of them solves a given problem correctly?

  1. 0.52
  2. 0.48
  3. 0.56
  4. 0.44 Correct answer.

Explanation

Exactly one succeeds when A succeeds and B fails, or A fails and B succeeds. That is 0.8×0.4 + 0.2×0.6 = 0.32 + 0.12 = 0.44. Independence lets us multiply the individual probabilities.

Question 18General Aptitude

A data set of 10 latency measurements has a mean of 50 ms and a standard deviation of 8 ms. Every measurement is then increased by 5 ms. What are the new mean and new standard deviation?

  1. Mean 55 ms, standard deviation 8 ms Correct answer.
  2. Mean 50 ms, standard deviation 8 ms
  3. Mean 55 ms, standard deviation 40 ms
  4. Mean 55 ms, standard deviation 13 ms

Explanation

Adding a constant to every observation shifts the mean by that constant, so the new mean is 50 + 5 = 55 ms. The standard deviation measures spread about the mean, and a uniform shift does not change the spread, so it stays 8 ms.

Question 19General Aptitude

In how many distinct ways can the letters of the word 'MEMORY' be arranged so that the two M's are never adjacent?

  1. 480
  2. 240 Correct answer.
  3. 360
  4. 120

Explanation

The word MEMORY has 6 letters with M repeated twice, so total arrangements are 6!/2! = 360. Arrangements with the two M's adjacent treat 'MM' as one block, giving 5! = 120. Subtracting gives 360 − 120 = 240 arrangements with the M's separated.

Question 20Discrete Mathematics and Engineering Mathematics

For a relation R on a set A, which pair of properties together guarantees that R is an equivalence relation?

  1. Reflexive and transitive
  2. Reflexive, symmetric and transitive Correct answer.
  3. Reflexive and antisymmetric
  4. Symmetric and transitive

Explanation

An equivalence relation is defined as a relation that is simultaneously reflexive, symmetric and transitive, so all three properties are required. Any definition missing one property fails to capture equivalence classes, since classes need every element related to itself, mutual membership, and chaining.

Question 21Discrete Mathematics and Engineering Mathematics

A simple undirected graph has 7 vertices and is a tree. How many edges does it have?

  1. 8
  2. 7
  3. 6 Correct answer.
  4. 5

Explanation

A tree on n vertices always has exactly n minus 1 edges, which follows by induction from the fact that removing a leaf removes one vertex and one edge. With n equal to 7 the count is 6 edges, and adding any further edge would create a cycle.

Question 22Discrete Mathematics and Engineering Mathematics

The number of distinct binary strings of length 10 that contain exactly three 1s is:

  1. 30
  2. 720
  3. 120 Correct answer.
  4. 1024

Explanation

Choosing which three of the ten positions hold a 1 completely determines the string, giving C(10,3) = 120 strings. The remaining seven positions are forced to be 0, so no further factor appears.

Question 23Discrete Mathematics and Engineering Mathematics

A fair six-sided die is rolled twice. Given that the sum of the two outcomes is 8, what is the probability that the first roll showed a 3?

  1. 1/8
  2. 1/3
  3. 1/5 Correct answer.
  4. 1/6

Explanation

The sums equal to 8 correspond to the five ordered pairs (2,6), (3,5), (4,4), (5,3), (6,2), all equally likely. Exactly one of these five pairs has first coordinate 3, so the conditional probability is 1/5.

Question 24Discrete Mathematics and Engineering Mathematics

The Boolean expression A + A'B simplifies to:

  1. A + B Correct answer.
  2. A + A'
  3. AB
  4. A' + B

Explanation

Applying distributivity gives A + A'B = (A + A')(A + B), and since A + A' equals 1 the expression reduces to A + B. This is the standard absorption-style identity for Boolean algebra.

Question 25Discrete Mathematics and Engineering Mathematics

The propositional formula (P implies Q) and (Q implies P) taken together is logically equivalent to:

  1. P or Q
  2. P and Q
  3. P if and only if Q Correct answer.
  4. P xor Q

Explanation

The conjunction of both implications is exactly the truth condition for the biconditional, which is true when P and Q share the same truth value. Neither one-directional connective alone captures this, but together they force agreement.

Question 26Discrete Mathematics and Engineering Mathematics

The number of generators of the cyclic group Z_12 under addition is:

  1. 6
  2. 2
  3. 12
  4. 4 Correct answer.

Explanation

An element k generates Z_n exactly when gcd(k, n) equals 1, so the count is Euler's totient of 12, which is 12 times one-half times two-thirds, giving 4. The generators are 1, 5, 7 and 11.

Question 27Discrete Mathematics and Engineering Mathematics

A 3x3 matrix M over the reals has characteristic polynomial (x-2)^2 (x-5). Which statement must be true?

  1. M is diagonalizable
  2. The trace of M equals 9 and the determinant equals 20 Correct answer.
  3. The eigenvalues are 2, 2 and 5, and M is invertible
  4. M has three linearly independent eigenvectors

Explanation

The trace equals the sum of eigenvalues and the determinant equals their product, so trace is 2+2+5 = 9 and determinant is 2*2*5 = 20. These invariants follow directly from the characteristic polynomial regardless of diagonalizability.

Question 28Discrete Mathematics and Engineering Mathematics

The value of the limit as x tends to 0 of (sin 3x)/(2x) is:

  1. 3/2 Correct answer.
  2. 2/3
  3. 1
  4. 0

Explanation

Using the standard limit sin(u)/u tending to 1 as u tends to 0, rewrite the expression as (3/2) times sin(3x)/(3x). The fraction tends to 1, leaving 3/2.

Question 29Discrete Mathematics and Engineering Mathematics

The recurrence T(n) = 2T(n/2) + n with T(1) = 1 has the closed form:

  1. Theta(n log n) Correct answer.
  2. Theta(n^2)
  3. Theta(n)
  4. Theta(log n)

Explanation

This is the standard merge-sort recurrence, and the master theorem's second case applies because the work per level is n across log n levels. Summing n per level over log n levels gives Theta(n log n).

Question 30Discrete Mathematics and Engineering Mathematics

A fair coin is tossed until the first head appears. The expected number of tosses is:

  1. 2 Correct answer.
  2. 3
  3. 1/2
  4. 1

Explanation

The number of tosses follows a geometric distribution with success probability one-half, whose mean is the reciprocal, namely 2. Equivalently, solving E = 1 + (1/2)E yields E = 2.

Question 31Discrete Mathematics and Engineering Mathematics

In a group G, the set of elements that commute with every element of G is called the centre. Which statement about the centre Z(G) is always true?

  1. Z(G) is trivial for every nonabelian group
  2. Z(G) has index 2 whenever G is nonabelian
  3. Z(G) is a normal subgroup of G Correct answer.
  4. Z(G) equals the commutator subgroup

Explanation

The centre is always a subgroup, and it is normal because conjugation by any element fixes central elements. This holds for every group, abelian or not.

Question 32Discrete Mathematics and Engineering Mathematics

A relation R on a finite set with n elements is represented by an n x n Boolean matrix. R is transitive exactly when:

  1. The matrix squared, under Boolean multiplication, is contained in the matrix Correct answer.
  2. The diagonal of the matrix is all ones
  3. The matrix equals its transpose
  4. The matrix is symmetric

Explanation

Transitivity says that whenever R(a,b) and R(b,c) hold, R(a,c) holds, which in Boolean matrix terms means the square has no 1 outside the original matrix. Thus M^2 is entrywise less than or equal to M.

Question 33Discrete Mathematics and Engineering Mathematics

The number of spanning trees of the complete graph K_4 is:

  1. 8
  2. 16 Correct answer.
  3. 12
  4. 4

Explanation

Cayley's formula gives n raised to the power n minus 2 spanning trees for the complete graph on n labelled vertices. For n = 4 this is 4 squared, which equals 16.

Question 34Discrete Mathematics and Engineering Mathematics

A partial order is defined on a set by divisibility, where a is related to b when a divides b. Which element is the least element of the set {2, 3, 4, 6, 12} under this order?

  1. There is no least element Correct answer.
  2. 12
  3. 2
  4. 1

Explanation

A least element must divide every element of the set and belong to the set. Since 1 is not in the set, and neither 2 nor 3 divides all others, no member divides every member, so the set has no least element under divisibility.

Question 35Discrete Mathematics and Engineering Mathematics

Let f(x) = x^3 - 3x. On the interval from -2 to 2, the function has:

  1. A local minimum at x = -1 and a local maximum at x = 1
  2. No critical points in the interval
  3. A local maximum at x = -1 and a local minimum at x = 1 Correct answer.
  4. A single inflection point that is also an extremum

Explanation

The derivative 3x^2 - 3 vanishes at x = -1 and x = 1, and the second derivative 6x is negative at -1 and positive at 1. Hence x = -1 gives a local maximum and x = 1 gives a local minimum.

Question 36Discrete Mathematics and Engineering Mathematics

A graph has 8 vertices and 14 edges and is connected. Which statement about its planarity is correct?

  1. It must be planar because it has fewer than 3n - 6 edges
  2. It is planar only if it is bipartite
  3. It may be planar, since 14 is at most 3n - 6 = 18, but planarity is not guaranteed by the edge count alone Correct answer.
  4. It cannot be planar because 14 exceeds 2n - 4

Explanation

For a simple planar graph with n at least 3, the bound is m at most 3n - 6, which here allows up to 18 edges, so the edge count does not violate the necessary condition. However, that bound is necessary and not sufficient, so passing it does not prove planarity.

Question 37Discrete Mathematics and Engineering Mathematics

Let A be an n x n real matrix with A^2 = A and A not equal to the identity or zero matrix. Which statement is necessarily true?

  1. A is symmetric
  2. A is invertible
  3. The eigenvalues of A are all 0 or all 1
  4. The sum of the eigenvalues equals the rank of A Correct answer.

Explanation

An idempotent matrix has minimal polynomial dividing x(x-1), so its eigenvalues lie in {0,1}. The trace, which is the sum of eigenvalues, therefore equals the number of eigenvalues equal to 1 counted with multiplicity, and that number equals the rank for an idempotent matrix.

Question 38Discrete Mathematics and Engineering Mathematics

A fair coin is flipped 100 times. Using the normal approximation with continuity correction, which expression best approximates the probability of getting at most 45 heads?

  1. Phi((44.5 - 50)/5)
  2. Phi((45 - 50)/5)
  3. Phi((45.5 - 50)/5) Correct answer.
  4. 1 - Phi((45.5 - 50)/5)

Explanation

The number of heads has mean 50 and standard deviation 5, and the continuity correction for at most 45 replaces 45 by 45.5. The standardised value is therefore (45.5 - 50)/5, and the cumulative probability is Phi of that value.

Question 39Digital Logic

In a synchronous sequential circuit, a positive edge-triggered D flip-flop samples its D input at which moment?

  1. On the falling edge of the clock
  2. While the clock is held at logic 0
  3. While the clock is held at logic 1
  4. On the rising edge of the clock Correct answer.

Explanation

A positive edge-triggered flip-flop updates its stored value only at the low-to-high transition of the clock. Level-sensitive latches, by contrast, are transparent for the whole high (or low) phase. The sampling instant is therefore the rising edge, not a clock level.

Question 40Digital Logic

A 4-bit ripple-carry adder adds 0111 and 0001. What are the 4-bit sum and the carry-out?

  1. Sum = 0111, carry-out = 1
  2. Sum = 0000, carry-out = 1
  3. Sum = 1000, carry-out = 0 Correct answer.
  4. Sum = 1001, carry-out = 0

Explanation

0111 is 7 and 0001 is 1, so the true total is 8. Eight fits within four bits as 1000, so the carry-out of the most significant stage is 0. The carry propagates through the lower three stages but stops before leaving bit 3.

Question 41Digital Logic

Which expression is the minimal sum-of-products form of F(A,B,C) = Σm(1,3,5,7)?

  1. AC
  2. A'B
  3. A'C Correct answer.
  4. B'C

Explanation

All four minterms have C = 1 and A = 0, while B varies over both values. The variable B therefore drops out and the implicant A'C covers every minterm. No other two-literal term covers all four.

Question 42Digital Logic

The binary number 1101 is interpreted as a 4-bit two's complement value. What decimal value does it represent?

  1. 13
  2. -3 Correct answer.
  3. -2
  4. -5

Explanation

In two's complement the most significant bit carries weight -8 and the remaining bits carry +4, +2 and +1. Summing gives -8 + 4 + 1 = -3. The pattern 1101 would be 13 only under unsigned interpretation.

Question 43Digital Logic

A 3-input XOR gate receives inputs 1, 0 and 1. What is its output?

  1. 1
  2. 2
  3. Indeterminate
  4. 0 Correct answer.

Explanation

A multi-input XOR produces 1 only when an odd number of inputs are 1. Here two inputs are 1, which is even, so the output is 0. This is equivalent to the parity of the input vector.

Question 44Digital Logic

A Boolean function of four variables is given in canonical SOP form. What is the maximum number of minterms it can contain?

  1. 8
  2. 32
  3. 4
  4. 16 Correct answer.

Explanation

A minterm fixes one value for every variable, so the number of distinct minterms equals 2 raised to the number of variables. For four variables that is 2^4 = 16. Each of the 16 input combinations maps to exactly one minterm.

Question 45Digital Logic

A 4-to-1 multiplexer with select lines S1 (MSB) and S0 (LSB) has data inputs D0..D3. Which input is routed to the output when S1S0 = 10?

  1. D2 Correct answer.
  2. D0
  3. D3
  4. D1

Explanation

The select word S1S0 forms a binary index whose decimal value chooses the data input. The pattern 10 equals decimal 2, so D2 is selected. D3 would require 11 and D1 would require 01.

Question 46Digital Logic

A JK flip-flop has J = 1 and K = 1 and is clocked repeatedly. What happens to the output Q on each active clock edge?

  1. It toggles to the complement of its present state Correct answer.
  2. It retains its previous value
  3. It is cleared to 0
  4. It is set to 1

Explanation

With both J and K asserted, the feedback in the JK flip-flop forces the next state to be the complement of the present state. This toggle mode is the standard basis for ripple and synchronous counters. Setting J = 1, K = 0 would instead force Q to 1.

Question 47Digital Logic

A mod-6 synchronous counter is built from three flip-flops. What is the minimum number of flip-flops required for a mod-6 counter, and how many unused states does that design have?

  1. 3 flip-flops with 2 unused states Correct answer.
  2. 3 flip-flops with 6 unused states
  3. 4 flip-flops with 10 unused states
  4. 2 flip-flops with 0 unused states

Explanation

A counter needs at least ceil(log2 6) = 3 flip-flops, since 2 bits provide only 4 states. Three flip-flops give 8 possible states, of which 6 are used, leaving 2 unused states that must be handled by the next-state logic.

Question 48Digital Logic

Two 2-input NAND gates are used to build a cross-coupled SR latch with active-low inputs S' and R'. Which input combination produces the forbidden or invalid state?

  1. S' = 1 and R' = 0
  2. S' = 1 and R' = 1
  3. S' = 0 and R' = 0 Correct answer.
  4. S' = 0 and R' = 1

Explanation

Asserting both active-low inputs simultaneously drives both outputs toward 1, violating the complementary-output requirement. When the inputs are then released, the resulting state depends on mismatched gate delays, so it is undefined. The all-1 input combination is the hold condition.

Question 49Digital Logic

A 4-bit binary counter uses ripple carry between stages. If each flip-flop has a propagation delay of 10 ns and each AND gate adds 5 ns, what limits the maximum counting frequency?

  1. The cumulative carry and toggle delay across all stages Correct answer.
  2. Only the delay of the most significant flip-flop
  3. The setup time of the input clock source
  4. Only the delay of the least significant flip-flop

Explanation

In a ripple counter each stage must wait for the previous stage to toggle, so delays accumulate along the chain. The critical path includes the flip-flop delays plus the gating delays of the carry logic. This accumulation, not a single stage, sets the maximum clock rate.

Question 50Digital Logic

In a Mealy machine, the output depends on which quantities?

  1. Only the previous outputs
  2. Both the present state and the present inputs Correct answer.
  3. Only the present inputs
  4. Only the present state

Explanation

A Mealy machine computes its output as a function of the current state and the current input, so the output can change as soon as an input changes. A Moore machine, by contrast, depends only on the present state. This distinction affects output timing and glitch behaviour.