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GATE EE practice questions

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Question 1Engineering Mathematics

The eigenvalues of the matrix [[2, 1], [1, 2]] are

  1. 1 and 3 Correct answer.
  2. −1 and 5
  3. 2 and 2
  4. 0 and 4

Explanation

For a 2×2 matrix the eigenvalues satisfy λ² − (trace)λ + (determinant) = 0. Here trace = 4 and determinant = 3, so λ² − 4λ + 3 = 0, giving λ = 1 and λ = 3.

Question 2Engineering Mathematics

If A is a 3×3 real matrix with Aᵀ = −A, then the determinant of A equals

  1. −1
  2. 1
  3. the trace of A
  4. 0 Correct answer.

Explanation

A skew-symmetric matrix of odd order satisfies det(A) = det(Aᵀ) = det(−A) = (−1)³det(A) = −det(A), so det(A) = 0. Hence A is singular whenever its order is odd.

Question 3Engineering Mathematics

The value of the limit lim(x→0) (sin 3x)/(tan 5x) is

  1. 3/5 Correct answer.
  2. 5/3
  3. 1
  4. 0

Explanation

Using sin 3x ≈ 3x and tan 5x ≈ 5x for small x, the ratio tends to 3/5. Equivalently, apply L'Hôpital's rule once to get 3cos 3x / (5sec²5x) evaluated at x = 0, which is 3/5.

Question 4Engineering Mathematics

The function f(x) = x³ − 3x has a local maximum at

  1. x = −1 Correct answer.
  2. x = 3
  3. x = 1
  4. x = 0

Explanation

Setting f′(x) = 3x² − 3 = 0 gives x = ±1. The second derivative f″(x) = 6x is negative at x = −1, so that point is a local maximum, while x = 1 is a local minimum.

Question 5Engineering Mathematics

The partial derivative ∂²f/∂x∂y of f(x, y) = x²y³ + e^(xy) equals

  1. 6xy² + xy·e^(xy)
  2. 6x²y + e^(xy)(1 + xy)
  3. 6xy² + e^(xy)(1 + xy) Correct answer.
  4. 2xy³ + y²e^(xy)

Explanation

First differentiate with respect to y: ∂f/∂y = 3x²y² + x·e^(xy). Then differentiate with respect to x: 6xy² + e^(xy) + xy·e^(xy), which equals 6xy² + e^(xy)(1 + xy).

Question 6Engineering Mathematics

The series Σ(n=1 to ∞) 1/n^p converges if and only if

  1. p < 1
  2. p > 1 Correct answer.
  3. p ≥ 1
  4. p > 0

Explanation

This is the p-series, whose convergence is governed by the integral test: ∫₁^∞ x^(−p) dx converges exactly when p > 1. At p = 1 the harmonic series diverges, and for p < 1 the terms decay too slowly.

Question 7Engineering Mathematics

The Laplace transform of t·e^(−2t) for t ≥ 0 is

  1. 1/(s+2)² Correct answer.
  2. 2/(s+2)²
  3. 1/(s²+4)
  4. 1/(s²−4)

Explanation

The transform of e^(−2t) is 1/(s+2), and multiplication by t corresponds to negating the derivative with respect to s. Differentiating gives −1/(s+2)², so the transform of t·e^(−2t) is 1/(s+2)².

Question 8Engineering Mathematics

The order and degree of the differential equation (d²y/dx²)³ + (dy/dx)⁴ + y = 0 are respectively

  1. 4 and 2
  2. 3 and 2
  3. 2 and 3 Correct answer.
  4. 2 and 4

Explanation

Order is the highest derivative present, which is the second derivative, so order = 2. Degree is the power of the highest-order derivative once the equation is polynomial in derivatives, giving degree = 3.

Question 9Engineering Mathematics

The general solution of dy/dx = y/x for x ≠ 0 is

  1. y = Ce^x
  2. y = Cx Correct answer.
  3. y = C/x
  4. y = x + C

Explanation

Separating variables gives dy/y = dx/x, so ln|y| = ln|x| + c, hence y = Cx. Substituting back confirms dy/dx = C = y/x.

Question 10Engineering Mathematics

The value of the integral ∮ z² dz taken counterclockwise around the unit circle |z| = 1 is

  1. 0 Correct answer.
  2. 2πi
  3. πi
  4. 2π

Explanation

The integrand z² is analytic everywhere, so by Cauchy's integral theorem its contour integral over any closed path is zero. Equivalently, the residue at the only candidate point z = 0 is the coefficient of 1/z, which is absent.

Question 11Engineering Mathematics

The residue of f(z) = (z + 1)/(z² − 1) at z = 1 is

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

Explanation

Factor the denominator as (z−1)(z+1), so f(z) = 1/(z−1) for z ≠ −1. The function has a simple pole at z = 1 with residue equal to the coefficient of 1/(z−1), namely 1/2? Careful: (z+1)/((z−1)(z+1)) = 1/(z−1), giving residue 1.

Question 12Engineering Mathematics

The Fourier series of an odd function defined on [−L, L] contains

  1. only cosine terms
  2. only the constant term
  3. only sine terms Correct answer.
  4. both sine and cosine terms

Explanation

For an odd function, the products f(x)cos(nπx/L) are odd and integrate to zero over the symmetric interval, so all cosine coefficients vanish. The sine coefficients generally survive because f(x)sin(nπx/L) is even.

Question 13Engineering Mathematics

A fair die is rolled twice. Given that the sum of the two outcomes is 8, the probability that the first roll showed 3 is

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

Explanation

The sums equal to 8 arise from the five pairs (2,6), (3,5), (4,4), (5,3), (6,2). Exactly one of these has first roll 3, so the conditional probability is 1/5.

Question 14Engineering Mathematics

If X is a Poisson random variable with mean 2, then P(X = 0) equals

  1. e^(−2) Correct answer.
  2. 2e^(−2)
  3. 1 − e^(−2)
  4. e^(−1)

Explanation

For a Poisson variable with parameter λ, P(X = k) = e^(−λ)λ^k/k!. Setting k = 0 and λ = 2 gives e^(−2)·1/1 = e^(−2).

Question 15Engineering Mathematics

The mean and variance of a binomial distribution with n = 10 and p = 0.4 are

  1. 2.4 and 4
  2. 10 and 0.4
  3. 4 and 2.4 Correct answer.
  4. 4 and 4

Explanation

For a binomial variable the mean is np = 10 × 0.4 = 4 and the variance is np(1−p) = 4 × 0.6 = 2.4. The variance is always smaller than the mean when p < 0.5.

Question 16Engineering Mathematics

The rank of the matrix [[1, 2, 3], [2, 4, 6], [1, 1, 1]] is

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

Explanation

The second row is twice the first, so the rows are linearly dependent and the rank is at most 2. The first and third rows are not proportional, so the rank is exactly 2.

Question 17Engineering Mathematics

The system of equations x + y + z = 3, x + 2y + 3z = 4, 2x + 3y + 4z = 7 has

  1. exactly two solutions
  2. infinitely many solutions Correct answer.
  3. no solution
  4. a unique solution

Explanation

The third equation is the sum of the first two, so the coefficient matrix is singular and the augmented system is consistent. A consistent singular system in three unknowns has a one-parameter family of solutions.

Question 18Engineering Mathematics

For the matrix [[4, 1], [2, 3]], the sum of the eigenvalues equals

  1. 7 Correct answer.
  2. 2
  3. 10
  4. 5

Explanation

The sum of the eigenvalues equals the trace, which is the sum of the diagonal entries 4 + 3 = 7. The product would equal the determinant 12 − 2 = 10.

Question 19Engineering Mathematics

The directional derivative of f(x, y) = x² + y² at the point (1, 2) in the direction of the unit vector (3/5, 4/5) is

  1. 10
  2. 2
  3. 22/5 Correct answer.
  4. 6/5

Explanation

The gradient at (1, 2) is (2x, 2y) = (2, 4). The directional derivative is the dot product (2, 4)·(3/5, 4/5) = 6/5 + 16/5 = 22/5.

Question 20Engineering Mathematics

The function f(x) = |x| on [−1, 1] is

  1. differentiable everywhere on [−1, 1]
  2. differentiable only at x = 0
  3. discontinuous at x = 0
  4. continuous but not differentiable at x = 0 Correct answer.

Explanation

The absolute value function is continuous everywhere, but at x = 0 the left derivative is −1 while the right derivative is +1. Since these one-sided derivatives disagree, f is not differentiable at the origin.

Question 21Engineering Mathematics

The value of ∫₀^∞ e^(−3x) dx is

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

Explanation

An antiderivative is −e^(−3x)/3, and evaluating from 0 to ∞ gives 0 − (−1/3) = 1/3. The exponential decay ensures the improper integral converges.

Question 22Engineering Mathematics

The Maclaurin series of ln(1 + x) converges for

  1. −1 ≤ x < 1
  2. −1 < x ≤ 1 Correct answer.
  3. all real x
  4. |x| < 1 only

Explanation

The series x − x²/2 + x³/3 − … has radius of convergence 1. At x = 1 it becomes the alternating harmonic series, which converges, while at x = −1 it becomes the divergent harmonic series.

Question 23Engineering Mathematics

The function u(x, y) = x² − y² is the real part of an analytic function f(z). The imaginary part v(x, y) is

  1. x²y − y³ + C
  2. 2xy + C Correct answer.
  3. −2xy + C
  4. x² + y² + C

Explanation

The Cauchy–Riemann equations give ∂v/∂y = ∂u/∂x = 2x and ∂v/∂x = −∂u/∂y = 2y. Integrating yields v = 2xy + C, consistent with f(z) = z² + iC.

Question 24Engineering Mathematics

The value of the contour integral ∮ e^z/(z − 2) dz taken counterclockwise around |z| = 3 is

  1. 2πi·e^(−2)
  2. 2πi·e² Correct answer.
  3. 2πi
  4. 0

Explanation

The pole z = 2 lies inside the circle |z| = 3, and the integrand has a simple pole there with residue e². By Cauchy's residue theorem the integral equals 2πi times that residue.

Question 25Engineering Mathematics

The eigenvalues of a real symmetric matrix are always

  1. purely imaginary
  2. real Correct answer.
  3. distinct
  4. positive

Explanation

For a real symmetric matrix, eigenvectors corresponding to distinct eigenvalues are orthogonal, and a short argument with the Hermitian inner product shows every eigenvalue must equal its own conjugate. Hence all eigenvalues are real, though they may repeat or be negative.

Question 26Engineering Mathematics

A function f(z) is analytic inside and on a simple closed contour C, and z₀ is a point strictly inside C. The value of the contour integral ∮_C f(z)/(z − z₀) dz taken counterclockwise is

  1. f′(z₀)
  2. 0
  3. 2πi·f(z₀) Correct answer.
  4. πi·f(z₀)

Explanation

Cauchy's integral formula states that for f analytic inside and on a simple closed contour C traversed counterclockwise, the integral of f(z)/(z − z₀) equals 2πi times f(z₀) for any z₀ inside C. The only singularity of the integrand inside C is the simple pole at z₀, whose residue is f(z₀), so the residue theorem gives 2πi·f(z₀).

Question 27General Aptitude

In a GATE-style technical paragraph, a sentence reads: "The transient response of the circuit was analysed by the engineers, and the results were documented carefully." Which grammatical role does "by the engineers" perform in this sentence?

  1. It is the direct object of the verb
  2. It is a subject complement
  3. It is the agent phrase of a passive construction Correct answer.
  4. It is an adverbial clause of condition

Explanation

The verb phrase "was analysed" is passive, and "by the engineers" names the doer of the action, which is the agent phrase. Passive constructions shift the logical subject into a by-phrase introduced by the preposition "by".

Question 28General Aptitude

A student writes: "Neither the voltage readings nor the current measurement were consistent across trials." Which revision keeps subject-verb agreement correct under the proximity rule?

  1. Neither the voltage readings nor the current measurement were consistent across trials.
  2. Neither the voltage readings nor the current measurement was consistent across trials. Correct answer.
  3. Neither the voltage readings or the current measurement were consistent across trials.
  4. Neither the voltage reading nor the current measurements was consistent across trials.

Explanation

With "neither...nor", the verb agrees with the nearer subject, which is the singular "current measurement", so the singular verb "was" is required. The original plural "were" agrees with the farther plural subject instead, which the proximity rule does not permit.

Question 29General Aptitude

Which word is the most precise choice to fill the blank? "The measured efficiency of the prototype was _____ higher than that of the reference design, exceeding it by nearly two percentage points."

  1. randomly
  2. significantly Correct answer.
  3. marginally
  4. negligibly

Explanation

The clause "exceeding it by nearly two percentage points" signals a substantial difference, so "significantly" is the precise adverb. The other adverbs describe small or meaningless differences and contradict the stated magnitude.

Question 30General Aptitude

The word "attenuate" is used in a sentence describing a signal passed through a long cable. Which meaning best fits that technical context?

  1. To increase in frequency
  2. To convert into a digital form
  3. To reduce in magnitude or strength Correct answer.
  4. To reject as invalid

Explanation

In signal and circuit contexts, "attenuate" means to weaken or reduce the amplitude of a quantity. The other meanings belong to different operations such as digitisation or frequency shifting.

Question 31General Aptitude

A candidate claims: "Every student who solved the numerical problems passed the aptitude test." Which statement is the logical contrapositive of this claim?

  1. Every student who did not pass the aptitude test did not solve the numerical problems. Correct answer.
  2. Some students who solved the numerical problems did not pass the aptitude test.
  3. Every student who did not solve the numerical problems did not pass the aptitude test.
  4. Every student who passed the aptitude test solved the numerical problems.

Explanation

For a conditional of the form "if P then Q", the contrapositive is "if not Q then not P", which is logically equivalent. Here P is solving the numerical problems and Q is passing, so the contrapositive denies Q first and then denies P.

Question 32General Aptitude

A workshop has 40 students. Exactly 25 have solved circuit problems, exactly 18 have solved signal problems, and exactly 10 have solved both. How many students have solved neither type of problem?

  1. 15
  2. 7 Correct answer.
  3. 10
  4. 3

Explanation

By inclusion-exclusion, the number who solved at least one type is 25 + 18 - 10 = 33. Subtracting from the total of 40 leaves 7 students who solved neither.

Question 33General Aptitude

A train of length 200 m crosses a platform of length 300 m in 25 seconds. What is its speed in km/h?

  1. 60
  2. 80
  3. 90
  4. 72 Correct answer.

Explanation

The train must cover its own length plus the platform length, a total of 500 m, in 25 s. The speed is 500/25 = 20 m/s, and multiplying by 3.6 gives 72 km/h.

Question 34General Aptitude

A sum of money invested at simple interest doubles in 8 years. In how many years will it become four times the original amount at the same rate?

  1. 20
  2. 16
  3. 32
  4. 24 Correct answer.

Explanation

Doubling means the interest earned equals the principal in 8 years, so the annual rate is 100/8 = 12.5 percent. Becoming four times requires interest of three times the principal, which takes 3 times 8 = 24 years.

Question 35General Aptitude

The average of five numbers is 30. When one number is removed, the average of the remaining four becomes 28. What is the removed number?

  1. 40
  2. 30
  3. 34
  4. 38 Correct answer.

Explanation

The total of the five numbers is 5 times 30 = 150, and the total of the remaining four is 4 times 28 = 112. The removed number is the difference, 150 minus 112 = 38.

Question 36General Aptitude

A bag contains 4 red and 6 blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?

  1. 4/25
  2. 2/15 Correct answer.
  3. 3/20
  4. 1/5

Explanation

The probability of a red on the first draw is 4/10, and given that, the probability of a red on the second draw is 3/9. Multiplying gives 12/90 = 2/15.

Question 37General Aptitude

Read the short passage: "Automated grading tools are fast, but speed alone does not guarantee fairness. A tool trained on biased historical data may reproduce that bias at scale. Therefore, deployment should be preceded by an audit of the training data." Which statement best captures the main conclusion of the passage?

  1. Automated grading tools should be audited for data bias before deployment. Correct answer.
  2. Historical data is never useful for training grading tools.
  3. Automated grading tools are always faster than human graders.
  4. Speed is the most important feature of an automated grading tool.

Explanation

The passage concedes speed but argues that biased training data can scale unfairness, so the concluding recommendation is a pre-deployment audit. The other statements either distort the passage or state claims it does not make.

Question 38General Aptitude

Choose the pair that best expresses a relationship similar to the given pair: THERMOMETER : TEMPERATURE.

  1. Oscilloscope : voltage supply
  2. Ammeter : current Correct answer.
  3. Resistor : resistance
  4. Voltmeter : power

Explanation

A thermometer is an instrument that measures temperature, so the analogy requires an instrument paired with the quantity it measures. An ammeter measures current, which matches that relationship exactly.

Question 39General Aptitude

In a certain code, if ENGINEER is written as FGOHJEFS, how is CIRCUIT written using the same rule?

  1. DJSDUJV Correct answer.
  2. DJSDUKV
  3. DJSDUJU
  4. CJSDUJV

Explanation

Each letter of the original word is shifted forward by one position in the alphabet: E to F, N to O, and so on. Applying the same shift to CIRCUIT gives DJSDUJV.

Question 40General Aptitude

Pointing to a photograph, a woman says, "He is the only son of my mother's only daughter." How is the man in the photograph related to the woman?

  1. Her brother
  2. Her son Correct answer.
  3. Her nephew
  4. Her cousin

Explanation

The woman's mother's only daughter is the woman herself, since she is a daughter of that mother. The only son of that daughter is therefore the woman's own son.

Question 41General Aptitude

A statement is followed by two assumptions. Statement: "The institute has decided to conduct weekly mock tests for all GATE aspirants." Assumption I: Weekly practice tests may improve candidates' exam readiness. Assumption II: All aspirants are already fully prepared for the exam. Which assumption is implicit?

  1. Only Assumption I is implicit Correct answer.
  2. Neither assumption is implicit
  3. Both assumptions are implicit
  4. Only Assumption II is implicit

Explanation

A decision to run weekly mock tests presupposes that such practice is useful for readiness, so Assumption I is implicit. Assumption II contradicts the very need for practice tests and is therefore not implicit.

Question 42General Aptitude

In a sequence, the first term is 3, and each subsequent term is obtained by multiplying the previous term by 2 and adding 1. What is the fourth term?

  1. 15
  2. 27
  3. 31 Correct answer.
  4. 23

Explanation

Applying the rule step by step: 3 becomes 7, 7 becomes 15, and 15 becomes 31. The fourth term is therefore 31.

Question 43General Aptitude

Six aspirants sit in a row facing north. P sits at one end. Q sits immediately to the right of P. R sits third to the left of S. T sits at the other end. U sits between Q and R. Who sits at the extreme right end?

  1. T Correct answer.
  2. P
  3. S
  4. U

Explanation

With P at one end and Q immediately to its right, the left end is P's position. Placing T at the other end and fitting U between Q and R forces R and S into the remaining slots so that R is third to the left of S, leaving T at the extreme right.

Question 44General Aptitude

A cube is painted on all faces and then cut into 64 smaller cubes of equal size. How many of the smaller cubes have exactly two faces painted?

  1. 32
  2. 8
  3. 16
  4. 24 Correct answer.

Explanation

A 4 by 4 by 4 cut gives 12 edges, and each edge contributes 2 cubes with exactly two painted faces, since the corner cubes have three. Multiplying gives 12 times 2 = 24 such cubes.

Question 45General Aptitude

Choose the word that is most nearly opposite in meaning to "redundant" as used in a technical description of a circuit.

  1. Repeated
  2. Excessive
  3. Essential Correct answer.
  4. Duplicate

Explanation

In technical usage, "redundant" describes something duplicated or surplus to requirement, so its antonym is "essential", meaning necessary. The other options are near-synonyms of redundant rather than opposites.

Question 46General Aptitude

Which sentence uses the modifier correctly, avoiding a dangling modifier?

  1. Having calibrated the instrument, the readings were accurate.
  2. Having calibrated the instrument, it was time to record the readings.
  3. Having calibrated the instrument, the engineer recorded the readings. Correct answer.
  4. Having calibrated the instrument, the readings were recorded by the engineer.

Explanation

A participial modifier must attach to the noun that performs the action. Only the first sentence makes the engineer the one who calibrated the instrument, so the modifier is properly attached.

Question 47General Aptitude

A paragraph states: "Solar panels convert sunlight into electricity. Their output depends on irradiance and temperature. Dust accumulation on the panel surface reduces the received irradiance." Which inference is best supported?

  1. Dust increases the irradiance reaching the cells.
  2. Solar panels work best at night.
  3. Cleaning dusty panels can raise their electrical output. Correct answer.
  4. Temperature has no effect on panel output.

Explanation

The passage links reduced irradiance from dust to lower output, so removing dust should restore irradiance and improve output. The other options contradict information stated in the paragraph.

Question 48General Aptitude

Two statements are given: All amplifiers are devices. Some devices are cheap. Which conclusion follows necessarily?

  1. All amplifiers are cheap
  2. Some amplifiers are cheap
  3. No amplifier is cheap
  4. No necessary conclusion about cheap amplifiers follows Correct answer.

Explanation

The second premise places some devices in the cheap category without saying whether any amplifier is among them. Since the overlap is undetermined, no conclusion about cheap amplifiers follows necessarily.

Question 49General Aptitude

A number is increased by 20 percent and then decreased by 20 percent. What is the net change from the original number?

  1. 2 percent decrease
  2. No change
  3. 4 percent increase
  4. 4 percent decrease Correct answer.

Explanation

Multiplying by 1.20 and then by 0.80 gives 0.96, which is 96 percent of the original. The net effect is therefore a 4 percent decrease, not zero.

Question 50General Aptitude

A pipe fills a tank in 12 hours, and a second pipe fills it in 18 hours. A drain empties the full tank in 36 hours. With all three open, how long will the tank take to fill?

  1. 12 hours
  2. 8 hours
  3. 10 hours
  4. 9 hours Correct answer.

Explanation

The rates are 1/12, 1/18 and minus 1/36 per hour. Summing them over a common denominator of 36 gives 3/36 plus 2/36 minus 1/36, equal to 4/36 or 1/9 per hour, so the tank fills in 9 hours.