Question 1

Organic fraction of municipal solid waste (OFMSW) with bulk density of 315 kg/m³ and water content of 30% is mixed with municipal sludge of bulk density 700 kg/m³ and water content of 70%, such that the water content of the mixture is 40%. The amount (in kg) of sludge to be mixed per kg of

Accepted Answer

0.33 kg of sludge added per kg of OFMSW. The correct option is accurate because the calculations for the mixture's water content and the proportions of OFMSW and sludge lead to the conclusion that 0.33 kg of sludge is needed for every kg of OFMSW to achieve the desired water content of 40%.

Question 2

A probability distribution with right skew is shown in the figure. The correct statement for the probability distribution is

Accepted Answer

Mean is greater than median and mode. In a right-skewed distribution, the tail on the right side pulls the mean to the right, making it greater than both the median and the mode, which are closer to the peak of the distribution.

Question 3

The probability density function of a continuous random variable distributed uniformly between x and y (for y > x) is

Accepted Answer

1/(y - x). The probability density function for a uniform distribution is defined as the reciprocal of the range of the distribution, which is (y - x). This ensures that the total area under the density function equals 1, satisfying the properties of a probability distribution.

Question 4

The dimension of dynamic viscosity is:

Accepted Answer

M L⁻¹ T⁻¹. Dynamic viscosity measures a fluid's resistance to flow, which is defined as the force per unit area (pressure) divided by the velocity gradient. This results in the dimensional formula of mass per unit length per unit time, represented as M L⁻¹ T⁻¹.

Question 5

A single-lane highway has a traffic density of 40 vehicles/km. If the time-mean speed and space-mean speed are 40 kmph and 30 kmph, respectively, the average headway (in seconds) between the vehicles is

Accepted Answer

3.00. The average headway is calculated as the inverse of the traffic density multiplied by the time-mean speed. With a density of 40 vehicles/km and a time-mean speed of 40 km/h, the headway comes out to be 1/(40 vehicles/km) * (3600 seconds/hour) = 3 seconds, confirming option A as correct.

Question 6

A fair (unbiased) coin was tossed four times in succession and resulted in the following outcomes: (i) H head, (ii) H head, (iii) H head, (iv) H head. The probability of obtaining a 'Tail' when the coin is tossed again is

Accepted Answer

1/2. The outcome of previous tosses does not affect the probability of future tosses with a fair coin; each toss is independent. Therefore, the probability of getting a 'Tail' remains 1/2 regardless of the previous results.

Question 7

If {x} is a continuous, real valued random variable defined over the interval (-∞, +∞) and its occurrence is defined by the density function given as: f(x) = 1/sqrt(2π+b) e^(-1/2((x-a)/b)²) where 'a' and 'b' are the statistical attributes of the random variable {x}. The value of the integr

Accepted Answer

1. The integral of a probability density function over its entire range must equal 1, as it represents the total probability of all possible outcomes. In this case, the given function is a normalized Gaussian (normal) distribution, which is defined to have an integral of 1.

Question 8

Probability density function of a random variable X is given below
f(x) = { 0.25 if 1 ≤ x ≤ 5
 { 0 otherwise
P(X ≤ 4) is

Accepted Answer

3/4. The probability P(X ≤ 4) is calculated by integrating the probability density function from 1 to 4, which gives the area under the curve. Since the function is constant at 0.25 over the interval from 1 to 5, the area from 1 to 4 is 0.25 times the length of the interval (3), resulting in 0.75 or 3/4.

Question 9

X and Y are two random independent events. It is known that P(X) = 0.4 and P(X ∪ Y^c) = 0.7. Which one of the following is the value of P(X ∪ Y)?

Accepted Answer

0.7. With independence, 0.7=0.4+(1-p)-0.4(1-p)=0.4+0.6(1-p), giving 1-p=0.5 so P(Y)=0.5. Then P(X u Y)=0.4+0.5-0.4*0.5=0.7. Stored answer 0.5 is wrong; correct is 0.7.

Question 10

Consider a permutation sampled uniformly at random from the set of all permutations of {1,2,3,...,n} for some n ≥ 4. Let X be the event that 1 occurs before 2 in the permutation, and Y be the event that 3 occurs before 4. Which one of the following statements is TRUE?

Accepted Answer

The events X and Y are independent. Events X and Y are independent because the relative positions of 1 and 2 do not affect the relative positions of 3 and 4 in a random permutation. Each pair's ordering is determined independently, leading to a probability of 1/2 for each event occurring.

Question 11

A shop has 4 distinct flavors of ice-cream. One can purchase any number of scoops of any flavor. The order in which the scoops are purchased is inconsequential. If one wants to purchase 3 scoops of ice-cream, in how many ways can one make that purchase?

Accepted Answer

20. The problem can be solved using the stars and bars combinatorial method, where the 3 scoops (stars) are distributed among 4 flavors (bins). The formula for this is given by the combination of (n+k-1) choose (k-1), where n is the number of scoops and k is the number of flavors, resulting in 20 ways.

Question 12

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement: “Not every graph is connected”?

Accepted Answer

∀x (Graph(x) ⇒ ¬Connected(x)). The option states that for every graph, it is not connected, which implies that all graphs are disconnected. This contradicts the original statement that not every graph is connected, as it suggests that every graph fails to be connected.

Question 13

Suppose we uniformly and randomly select a permutation from the 20! permutations of 1,2,3,...,20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?

Accepted Answer

1/10. Among the 10 even numbers {2,4,...,20}, by symmetry each is equally likely to appear first within that subset. So the probability that 2 precedes all other evens is 1/10, not 1/2.

Question 14

Statement for Linked Answer Questions 84 & 85:
Suppose that a robot is placed on the Cartesian plane. At each step it is allowed to move either one unit up or one unit right, i.e., if it is at (i, j) then it can move to either (i+1, j) or (i, j+1).
How many distinct paths are there for the

Accepted Answer

(20 10). The correct option, (20 10), represents the number of ways to arrange 10 moves up and 10 moves right in a sequence of 20 total moves, which is calculated using the binomial coefficient formula. This accounts for the distinct paths the robot can take to reach the point (10,10) from (0,0).

Question 15

Aishwarya studies either computer science or mathematics everyday. If she studies computer science on a day, then the probability that she studies mathematics the next day is 0.6. If she studies mathematics on a day, then the probability that she studies computer science the next day is 0.

Accepted Answer

0.4. P(CS->CS)=0.4 and P(M->CS)=0.4. Tuesday: P(CS)=0.4, P(M)=0.6. Wednesday CS = 0.4*0.4 + 0.6*0.4 = 0.16+0.24 = 0.40.

Question 16

Let G = (V, E) be a graph. Define ξ(G) = ∑ d_i × d_i, where r_i is the number of vertices of degree d_i in G. If S and T are two different trees with ξ(S) = ξ(T), then

Accepted Answer

|S| = |T|. The value ξ(G) represents the sum of the squares of the degrees of the vertices in the graph. Since both S and T are trees and have the same number of vertices, their degree distributions must also be equivalent, leading to the conclusion that |S| equals |T|.

Question 17

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Accepted Answer

17/(2e³). The probability of observing fewer than 3 cars in a Poisson distribution can be calculated by summing the probabilities of observing 0, 1, and 2 cars. Given the mean of 3, the calculations yield a total probability of 17/(2e³) for these events.

Question 18

Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?

Accepted Answer

17/(2e³). The correct option is derived from the Poisson distribution, which models the number of events in a fixed interval. Given the mean of 3 cars per minute, the probability of observing fewer than 3 cars (0, 1, or 2 cars) can be calculated, and the sum of these probabilities equals 17/(2e³).

Question 19

Consider a set U of 23 different compounds in a Chemistry lab. There is a subset S of U of 9 compounds, each of which reacts with exactly 3 compounds of U. Consider the following statements:
I. Each compound in U S reacts with an odd number of compounds.
II. At least one compound in U S re

Accepted Answer

Only II. Model 'reacts with' as an undirected graph; the total degree sum is even. The 9 compounds in S contribute 9*3=27 (odd) to that sum, so the degrees of the other 14 compounds (U\S) must sum to an odd number, forcing at least one of them to have odd degree. Hence statement II is always true.

Question 20

Let N be the set of natural numbers. Consider the following sets.
P: Set of Rational numbers (positive and negative)
Q: Set of functions from {0,1} to N
R: Set of functions from N to {0,1}
S: Set of finite subsets of N.
Which of the sets above are countable?

Accepted Answer

P, Q and S only. The set of rational numbers (P) is countable because it can be listed in a sequence, while the set of functions from a finite set to natural numbers (Q) is also countable since it has a finite domain. The set of finite subsets of natural numbers (S) is countable as well, as each finite subset can be represented by a finite sequence of natural numbers.

Question 21

If E denotes expectation, the variance of a random variable X is given by

Accepted Answer

E[X²] - E²[X]. The variance of a random variable is defined as the expected value of the squared deviation from the mean, which mathematically is expressed as E[X²] minus the square of the expected value E[X]. This captures the spread of the random variable around its mean.

Question 22

An examination consists of two papers, Paper 1 and Paper 2. The probability of failing in Paper 1 is 0.3 and that in Paper 2 is 0.2. Given that a student has failed in Paper 2, the probability of failing in Paper 1 is 0.6. The probability of a student failing in both the papers is

Accepted Answer

0.12. P(fail both)=P(F1|F2)*P(F2)=0.6*0.2=0.12. The stored 0.06 is incorrect; the answer is 0.12.

Question 23

A memoryless source emits n symbols each with a probability p. The entropy of the source as a function of n

Accepted Answer

increases as log n. The entropy of a memoryless source is related to the number of symbols it can emit, and as the number of symbols increases, the uncertainty or information content also increases, which is logarithmic in nature, hence it grows as log n.

Question 24

Silicon is doped with boron to a concentration of 4×10¹⁷ atoms/cm³. Assume the intrinsic carrier concentration of silicon to be 1.5×10¹⁰/cm³ and the value of kT/q to be 25 mV at 300 K. Compared to undoped silicon, the Fermi level of doped silicon

Accepted Answer

goes down by 0.427 eV. Boron is an acceptor, so silicon becomes p-type and the Fermi level moves toward the valence band (down). The shift is kT*ln(Na/ni) = 0.025*ln(4e17/1.5e10) = 0.025*17.1 = 0.427 eV. So it goes down by 0.427 eV (option C), not down by 0.13 eV.

Question 25

The ratio of the mobility to the diffusion coefficient in a semiconductor has the units

Accepted Answer

(A) V⁻¹. By the Einstein relation mu/D = q/(kT), and kT/q has units of volts, so mu/D has units of 1/volt, i.e. V^-1 (option 0). The stored choice cm.V^-1 is incorrect.

Question 26

Consider two independent random variables X and Y with identical distributions. The variables X and Y take values 0, 1 and 2 with probabilities 1/2, 1/4 and 1/4 respectively. What is the conditional probability P(X + Y = 2 | X − Y = 0)?

Accepted Answer

1/6. X-Y=0 means X=Y, with P=1/4+1/16+1/16=3/8. Within that, X+Y=2 requires X=Y=1 with P=1/16. The conditional probability is (1/16)/(3/8)=1/6.

Question 27

A discrete random variable X takes values from 1 to 5 with probabilities as shown in the table. A student calculates the mean of X as 3.5 and her teacher calculates the variance of X as 1.5. Which of the following statements is true?

k: 1 2 3 4 5
P(X=k): 0.1 0.2 0.4 0.2 0.1

Accepted Answer

Both the student and the teacher are wrong. The mean of the discrete random variable X is calculated as the sum of each value multiplied by its probability, which results in a mean of 3. The variance is calculated using the squared differences from the mean, leading to a variance of 2. Therefore, both the student's mean and the teacher's variance are incorrect.

Question 28

A communication channel with AWGN operating at a signal to noise ratio SNR >> 1 and bandwidth B has capacity C1. If the SNR is doubled keeping B constant, the resulting capacity C2 is given by

Accepted Answer

C2 = C1 + B. By Shannon, C = B*log2(1+SNR). For SNR>>1 this is approximately B*log2(SNR). Doubling SNR raises capacity by B*log2(2) = B, so C2 = C1 + B.

Question 29

The Nyquist sampling rate for the signal x(t) = (sin(500πt))/(πt) × (sin(700πt))/(πt) is given by

Accepted Answer

1200 Hz. sin(500 pi t)/(pi t) has bandwidth 250 Hz and sin(700 pi t)/(pi t) has 350 Hz. Their product convolves the spectra, giving total bandwidth 600 Hz, so the Nyquist rate is 2*600=1200 Hz, not 1400 Hz.

Question 30

An analog signal is band-limited to 4 kHz, sampled at the Nyquist rate and the samples are quantized into 4 levels. The quantized levels are assumed to be independent and equally probable. If we transmit two quantized samples per second, the information rate is

Accepted Answer

4 bits/sec. Four equiprobable levels carry log2(4)=2 bits per sample. Transmitting 2 samples per second gives an information rate of 2*2 = 4 bits/sec, which is option 4 (index 3), not 2 bits/sec.

Question 31

A fair dice is tossed two times. The probability that the second toss results in a value that is higher than the first toss is

Accepted Answer

5/12. Of 36 equally likely ordered pairs, 15 have the second value greater than the first (second>first symmetric with first>second, ties excluded). So the probability is 15/36 = 5/12, option C, not 1/2.

Question 32

Let U and V be two independent zero mean Gaussian random variables of variances 1/4 and 1/9 respectively. The probability P(3V < 2U) is

Accepted Answer

1/2. The inequality 3V < 2U can be transformed into a standard normal variable comparison, leading to a symmetric distribution around zero. Since both U and V are independent Gaussian variables, the probability of one being greater than the other is 1/2.

Question 33

Consider two identically distributed zero-mean random variables U and V. Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

Accepted Answer

(F(x) ≥ G(x)) ∀ x ≥ 0. For x>=0, G(x)=P(2V<=x)=P(V<=x/2)<=P(U<=x)=F(x) since x/2<=x and U,V are identically distributed. Thus F(x)>=G(x) for all x>=0, so the stored F<=G is wrong.

Question 34

Let U and V be two independent and identically distributed random variables such that P(U < 0) = P(V < 0) = 1/2. The entropy H(U | V) in bits is

Accepted Answer

1. The entropy H(U | V) is equal to H(U) since U and V are independent. Given that both U and V are identically distributed with P(U < 0) = 1/2, the entropy of a binary random variable with equal probabilities is 1 bit.

Question 35

What is the chance that a leap year, selected at random, will contain 53 Saturdays?

Accepted Answer

2/7. A leap year has 52 full weeks plus 2 extra days forming one of 7 equally likely consecutive pairs. Two of those pairs (Fri-Sat and Sat-Sun) include a Saturday, so the probability of 53 Saturdays is 2/7, not 3/7.

Question 36

Let U and V be two independent and identically distributed random variables such that P(U = 0) = P(V = 0) = 1/2. The entropy H(U ⊕ V) in bits is

Accepted Answer

1. The random variable U ⊕ V represents the XOR operation between U and V, which results in 0 if both are the same and 1 if they are different. Since U and V are independent and identically distributed with equal probabilities of being 0 or 1, the resulting distribution of U ⊕ V is uniform with equal probabilities for 0 and 1, leading to an entropy of 1 bit.

Question 37

A silicon bar is doped with donor impurities N_D = 2.25 × 10¹⁵ atoms/cm³. Given the intrinsic carrier concentration of silicon at T = 300 K is n_i = 1.5 × 10¹⁰ cm⁻³. Assuming complete impurity ionization, the equilibrium electron and hole concentrations are

Accepted Answer

n₀ = 2.25 × 10¹⁵ cm⁻³, p₀ = 1 × 10⁵ cm⁻³. The correct option reflects that the doping concentration of donor impurities significantly increases the electron concentration (n₀) to 2.25 × 10¹⁵ cm⁻³, while the hole concentration (p₀) decreases due to charge neutrality and is calculated using the mass action law, resulting in a much lower value of 1 × 10⁵ cm⁻³.

Question 38

An unbiased coin is tossed an infinite number of times. The probability that the fourth head appears at the tenth toss is

Accepted Answer

0.082. The 4th head at the 10th toss means exactly 3 heads in the first 9 tosses and a head on the 10th: C(9,3)*(1/2)^9*(1/2)=84/1024=0.082. The stored 0.073 is incorrect. Answer: 0.082.

Question 39

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

Accepted Answer

Poisson. The Poisson distribution is appropriate here because it models the number of events (calls) occurring in a fixed interval of time when these events happen independently and at a constant average rate.

Question 40

If the signal x(t) = sin(t)/(πt) * sin(t)/(πt) with * denoting the convolution operation, then x(t) is equal to

Accepted Answer

sin(t)/(πt). sin(t)/(pi t) is an ideal lowpass (sinc) whose Fourier transform is a unit-height rectangle over |w|<1. Convolution in time multiplies the transforms, and rect*rect=rect, so the result is the same sinc: sin(t)/(pi t). Stored answer is wrong.

Question 41

Let H(X) denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?

Accepted Answer

H(X) ≤ log₂ K bits. The entropy H(X) measures the uncertainty in a random variable X, and it is maximized when all K outcomes are equally likely. Therefore, the maximum entropy is log₂ K bits, which means H(X) cannot exceed this value.

Question 42

Suppose X and Y are independent and identically distributed random variables that are distributed uniformly in the interval [0,1]. The probability that X ≥ Y is _______.

Accepted Answer

1/2. Since X and Y are uniformly distributed over the same interval and are independent, the area where X is greater than or equal to Y is exactly half of the total area of the unit square formed by their possible values. Thus, the probability that X is greater than or equal to Y is 1/2.

Question 43

Consider a continuous-time, real-valued signal f(t) whose Fourier transform F(ω) = ∫_(-∞)^(∞) f(t) exp(−j ω t) dt exists. Which one of the following statements is always TRUE?

Accepted Answer

|F(ω)| ≤ ∫_(-∞)^(∞) |f(t)| dt. The correct option is true due to the triangle inequality in the context of integrals, which states that the absolute value of the integral of a function is less than or equal to the integral of the absolute value of that function. This means that the magnitude of the Fourier transform |F(ω)| cannot exceed the total integral of the absolute value of the signal f(t).

Question 44

The random variable X takes values in {-1,0,1} with probabilities P(X = -1) = P(X = 1) = α and P(X = 0) = 1 - 2α, where 0 < α < 1/2. Let g(α) denote the entropy of X (in bits), parameterized by α. Which of the following statements is/are TRUE?

Accepted Answer

g(0.3) > g(0.4). The entropy function is concave, meaning that as the probabilities become more uniform, the entropy increases. Since g(α) is decreasing for values of α in the specified range, g(0.3) is greater than g(0.4) because 0.3 is less than 0.4, leading to a more uniform distribution.

Question 45

X is a uniformly distributed random variable that takes values between 0 and 1. The value of E[X³] will be

Accepted Answer

1/4. For X uniform on (0,1), E[X^3] = integral_0^1 x^3 dx = 1/4. The stored value 1/8 is E[X^3] computed incorrectly; the correct answer is 1/4.

Question 46

A box contains 4 white balls and 3 red balls. In succession, two balls are randomly selected and removed from the box. Given that the first removed ball is white, the probability that the second removed ball is red is

Accepted Answer

1/2. Given the first ball drawn is white, the box has 3 white and 3 red left (6 total). P(second is red)=3/6=1/2. Stored answer 3/7 is wrong; correct is 1/2.

Question 47

A zero mean random signal is uniformly distributed between limits -a and +a and its mean square value is equal to its variance. Then the r.m.s value of the signal is

Accepted Answer

a/√3. The root mean square (r.m.s) value of a uniformly distributed signal between -a and +a is calculated as the square root of the mean of the squares of the values. For a uniform distribution, this results in an r.m.s value of a/√3, which is derived from the integral of the square of the distribution over its range.

Question 48

Common Data for Questions 48 and 49:
The input voltage given to a converter is
v_i = 100√(2) sin(100π t) V
The current drawn by the converter is
i_i = (10√(2) sin(100π t - π/3) + 5√(2) sin(300π t + π/4) + 2√(2) sin(500π t - π/6)) A
Q.48 The input power factor of the converter is

Accepted Answer

0.44. The input power factor is calculated by taking the ratio of the real power to the apparent power. In this case, the phase difference between the voltage and the fundamental component of the current leads to a power factor of 0.44, indicating that the converter is not operating at maximum efficiency.

Question 49

Two independent random variables X and Y are uniformly distributed in the interval [-1,1]. The probability that max[X,Y] is less than 1/2 is

Accepted Answer

9/16. The probability that both X and Y are less than 1/2 can be calculated by considering the area of the square formed by their uniform distributions. Since both variables are independent and uniformly distributed over the interval [-1, 1], the area of the region where both are less than 1/2 is (1/2 + 1)(1/2 + 1) = (3/2)(3/2) = 9/16 of the total area of the square, which confirms that the correct answer is 9/16.

Question 50

A continuous random variable X has a probability density function f(x) = e^(−x), 0 ≤ x < ∞. Then P{X > 1} is

Accepted Answer

0.368. For the exponential density f(x)=e^{-x}, P(X>1)=integral from 1 to infinity of e^{-x} dx = e^{-1} = 0.368. (0.632 is P(X<1).)

GATE Engineering Mathematics: Probability & Statistics questions with solutions

Questions