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GATE NM practice questions

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Question 1Engineering Mathematics

The rank of the matrix A = [[1, 2, 3], [2, 4, 6], [1, 1, 1]] over the reals is:

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

Explanation

The second row is exactly twice the first row, so the rows are linearly dependent and rank is at most 2. The rows (1,2,3) and (1,1,1) are not scalar multiples of each other, so two rows are independent and the rank equals 2.

Question 2Engineering Mathematics

For the matrix B = [[4, 1], [2, 3]], the sum of its eigenvalues is:

  1. 1
  2. 7 Correct answer.
  3. 5
  4. 12

Explanation

The sum of the eigenvalues of a square matrix equals its trace, the sum of the main diagonal entries. Here the trace is 4 + 3 = 7, so the eigenvalues sum to 7.

Question 3Engineering Mathematics

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

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

Explanation

Using the standard limit lim(u→0) (sin u)/u = 1, write (sin 3x)/x = 3·(sin 3x)/(3x). As x→0 the factor (sin 3x)/(3x) tends to 1, so the limit is 3.

Question 4Engineering Mathematics

The derivative of f(x) = x² ln x with respect to x, for x > 0, is:

  1. 2x ln x + x Correct answer.
  2. x + ln x
  3. 2x + 1/x
  4. 2x ln x

Explanation

Apply the product rule to u = x² and v = ln x: f' = u'v + uv' = 2x·ln x + x²·(1/x). The second term simplifies to x, giving 2x ln x + x.

Question 5Engineering Mathematics

The value of the definite integral ∫₀¹ 3x² dx is:

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

Explanation

An antiderivative of 3x² is x³. Evaluating from 0 to 1 gives 1³ - 0³ = 1, so the integral equals 1.

Question 6Engineering Mathematics

For the function f(x) = x³ - 3x, the critical points are located at:

  1. x = 1 only
  2. x = 0 only
  3. x = -1 and x = 1 Correct answer.
  4. x = 3 and x = -3

Explanation

Critical points satisfy f'(x) = 3x² - 3 = 0, which gives x² = 1, hence x = 1 and x = -1. Both are valid critical points where the derivative vanishes.

Question 7Engineering Mathematics

The Laplace transform of the constant function f(t) = 5, for t ≥ 0, is:

  1. 5s
  2. 5/s²
  3. 1/(5s)
  4. 5/s Correct answer.

Explanation

The Laplace transform of 1 is 1/s, and the transform is linear, so the transform of the constant 5 is 5·(1/s) = 5/s. It holds for s > 0.

Question 8Engineering Mathematics

For the ordinary differential equation dy/dx = 2y with y(0) = 3, the solution is:

  1. y = 3 + 2x
  2. y = 2e^(3x)
  3. y = 3e^(2x) Correct answer.
  4. y = 3e^(-2x)

Explanation

Separating variables gives dy/y = 2 dx, so ln y = 2x + C and y = Ae^(2x). Applying y(0) = 3 gives A = 3, hence y = 3e^(2x).

Question 9Engineering Mathematics

The determinant of the 3×3 upper triangular matrix U = [[2, 5, 7], [0, 3, 1], [0, 0, 4]] is:

  1. 9
  2. 24 Correct answer.
  3. 14
  4. 0

Explanation

The determinant of a triangular matrix equals the product of its diagonal entries. Here 2·3·4 = 24, so the determinant is 24.

Question 10Engineering Mathematics

The system of linear equations x + y + z = 3, 2x + 2y + 2z = 6, x - y = 0 has:

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

Explanation

The second equation is twice the first, so it adds no new constraint and the coefficient matrix has rank 2 while there are 3 unknowns. With the system consistent (the equations do not contradict), the solution set is a line, giving infinitely many solutions.

Question 11Engineering Mathematics

The eigenvalues of the matrix C = [[2, 1], [1, 2]] are:

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

Explanation

The characteristic equation is (2-λ)² - 1 = 0, so (2-λ)² = 1 and λ = 2 ± 1, giving λ = 3 and λ = 1. Their sum 4 matches the trace and their product 3 matches the determinant.

Question 12Engineering Mathematics

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

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

Explanation

Setting f'(x) = 3x² - 3 = 0 gives x = ±1, and f''(x) = 6x. At x = -1, f'' = -6 < 0, which signals a local maximum; at x = 1, f'' = 6 > 0 signals a local minimum.

Question 13Engineering Mathematics

The value of the double integral ∫₀¹ ∫₀¹ (x + y) dx dy is:

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

Explanation

Integrating over x first gives x²/2 + xy evaluated from 0 to 1, which is 1/2 + y. Integrating that over y from 0 to 1 gives 1/2 + 1/2 = 1.

Question 14Engineering Mathematics

The inverse Laplace transform of F(s) = 1/(s² + 4) is:

  1. (1/2) cos 2t
  2. cos 2t
  3. (1/2) sin 2t Correct answer.
  4. sin 2t

Explanation

The standard pair is L{sin at} = a/(s² + a²). Here a = 2, so L{sin 2t} = 2/(s² + 4), meaning 1/(s² + 4) corresponds to (1/2) sin 2t.

Question 15Engineering Mathematics

The general solution of the differential equation d²y/dx² - 5 dy/dx + 6y = 0 is:

  1. y = C₁e^(2x) + C₂e^(-3x)
  2. y = (C₁ + C₂x)e^(2x)
  3. y = C₁e^(2x) + C₂e^(3x) Correct answer.
  4. y = C₁e^(-2x) + C₂e^(-3x)

Explanation

The characteristic equation is λ² - 5λ + 6 = 0, which factors as (λ - 2)(λ - 3) = 0. Distinct real roots 2 and 3 give the general solution y = C₁e^(2x) + C₂e^(3x).

Question 16Engineering Mathematics

The Fourier series of an odd function defined on [-π, π] contains:

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

Explanation

For an odd function, the product f(x)cos nx is odd and integrates to zero over the symmetric interval, so all cosine coefficients vanish. The sine coefficients generally survive because f(x)sin nx is even, leaving a pure sine series.

Question 17Engineering Mathematics

The gradient of the scalar field φ(x, y, z) = x²y + yz at the point (1, 2, 3) is:

  1. (1, 2, 3)
  2. (4, 2, 3)
  3. (4, 4, 2) Correct answer.
  4. (2, 3, 2)

Explanation

The gradient is (∂φ/∂x, ∂φ/∂y, ∂φ/∂z) = (2xy, x² + z, y). Substituting (1, 2, 3) gives (2·1·2, 1 + 3, 2) = (4, 4, 2).

Question 18Engineering Mathematics

The value of the limit lim(x→0) (e^x - 1 - x)/x² is:

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

Explanation

Expanding e^x = 1 + x + x²/2 + O(x³), the numerator becomes x²/2 + O(x³). Dividing by x² and taking the limit gives 1/2.

Question 19Engineering Mathematics

The eigenvalues of the matrix D = [[0, -1], [1, 0]] are:

  1. 1 and 1
  2. i and -i Correct answer.
  3. 1 and -1
  4. 0 and 0

Explanation

The characteristic equation is λ² + 1 = 0, giving λ = i and λ = -i. These are complex conjugates, consistent with the real matrix and its trace 0 and determinant 1.

Question 20Engineering Mathematics

A 3×3 matrix M has eigenvalues 1, 2, and 3. The determinant of M² is:

  1. 6
  2. 12
  3. 216
  4. 36 Correct answer.

Explanation

The determinant of M is the product of its eigenvalues, 1·2·3 = 6. The eigenvalues of M² are the squares 1, 4, 9, whose product is 36, equal to (det M)² = 36.

Question 21Engineering Mathematics

The function f(z) = z² + 3z is analytic everywhere. Its derivative f'(z) is:

  1. z + 3
  2. 2z + 3z
  3. 2z
  4. 2z + 3 Correct answer.

Explanation

Polynomials in z are entire, and differentiation follows the usual power and sum rules. Differentiating z² gives 2z and differentiating 3z gives 3, so f'(z) = 2z + 3.

Question 22Engineering Mathematics

The series sum from n=1 to infinity of 1/n² is:

  1. convergent with sum π²/6 Correct answer.
  2. conditionally convergent
  3. divergent
  4. convergent with sum 1

Explanation

This is a p-series with p = 2 > 1, so it converges. Its exact sum is the classical result π²/6, obtained via Fourier series or contour integration.

Question 23Engineering Mathematics

The matrix P = [[1, 1], [0, 1]] is diagonalizable over the reals. Which statement about it is correct?

  1. It is diagonalizable because it is upper triangular
  2. It is not diagonalizable because it has only one independent eigenvector Correct answer.
  3. It is diagonalizable because it is invertible
  4. It is diagonalizable because its eigenvalues are equal

Explanation

P has the repeated eigenvalue 1, and the eigenspace for λ = 1 is spanned by (1, 0) alone, giving geometric multiplicity 1 against algebraic multiplicity 2. A 2×2 matrix with a deficient eigenspace is not diagonalizable.

Question 24Engineering Mathematics

The value of the contour integral ∮ (1/z) dz taken counterclockwise around the circle |z| = 2 is:

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

Explanation

The integrand has a simple pole at z = 0 with residue 1, and this pole lies inside |z| = 2. By the residue theorem the integral equals 2πi times the residue, giving 2πi.

Question 25Engineering Mathematics

The function f(x, y) = x² + y² - 4x - 6y has a stationary point at (2, 3). Using the second-derivative test, this point is:

  1. a saddle point
  2. a local maximum with value -13
  3. a local minimum with value -13 Correct answer.
  4. a local minimum with value 13

Explanation

The Hessian is [[2, 0], [0, 2]], so f_xx·f_yy - f_xy² = 4 > 0 with f_xx = 2 > 0, confirming a local minimum. Substituting (2, 3) gives 4 + 9 - 8 - 18 = -13.

Question 26Engineering Mathematics

For the function f(x, y) = x³ + y³ - 3xy, the point (1, 1) is a stationary point. Classifying it using the second-derivative test, this point is:

  1. a local minimum Correct answer.
  2. a saddle point
  3. a local maximum
  4. a point where the test is inconclusive

Explanation

At a stationary point, compute the Hessian: f_xx = 6x, f_yy = 6y, f_xy = -3, so the discriminant D = f_xx·f_yy - (f_xy)². At (1, 1), D = 36 - 9 = 27 > 0 and f_xx = 6 > 0, which by the second-derivative test means the function has a local minimum there.

Question 27General Aptitude

A GATE aspirant's marks in five mock tests are 42, 48, 51, 55, and 64. What is the median of these scores?

  1. 51 Correct answer.
  2. 48
  3. 55
  4. 52

Explanation

The median is the middle value of an ordered list. Arranged as 42, 48, 51, 55, 64, the third entry is 51, so the median is 51.

Question 28General Aptitude

In a GATE General Aptitude paper, a question states: 'The number of students who cleared the cutoff doubled every year for three consecutive years.' If 200 students cleared in the first year, how many cleared in the fourth year?

  1. 800
  2. 1600 Correct answer.
  3. 600
  4. 1200

Explanation

Doubling for three consecutive years multiplies the starting count by 2 three times, that is by 8. Hence 200 × 8 = 1600 students.

Question 29General Aptitude

Choose the word that best completes the sentence: 'The committee's decision was ______, since every member had already agreed before the meeting began.'

  1. tentative
  2. unanimous Correct answer.
  3. predictable
  4. controversial

Explanation

The clause 'every member had already agreed' points to complete agreement. 'Unanimous' means all parties are of one mind, which fits the blank exactly.

Question 30General Aptitude

A train covers 240 km in 3 hours and then 180 km in 2 hours. What is its average speed for the whole journey in km/h?

  1. 82
  2. 84 Correct answer.
  3. 86
  4. 88

Explanation

Total distance is 240 + 180 = 420 km and total time is 3 + 2 = 5 hours. Average speed equals total distance divided by total time, giving 420/5 = 84 km/h.

Question 31General Aptitude

If the word GATE is coded as HBVF, how is the word SCORE coded using the same rule?

  1. TDPSE Correct answer.
  2. SDPSF
  3. TDPSF
  4. TDPRE

Explanation

Each letter of GATE is shifted forward by one position in the alphabet to give HBVF. Applying the same shift to SCORE gives T, D, P, S, F, that is TDPSE.

Question 32General Aptitude

A sum of money becomes 1.5 times itself in 5 years under simple interest. What is the annual rate of interest?

  1. 12%
  2. 7.5%
  3. 15%
  4. 10% Correct answer.

Explanation

Simple interest earned equals 0.5 times the principal over 5 years. Since rate = interest ÷ (principal × time), we get 0.5 ÷ 5 = 0.10, that is 10% per year.

Question 33General Aptitude

Which sentence uses the article correctly in the context of an academic report?

  1. The results of a experiment were inconclusive.
  2. Results of an experiment were inconclusive.
  3. A results of the experiment were inconclusive.
  4. The results of the experiment were inconclusive. Correct answer.

Explanation

A specific set of results already known to the reader takes the definite article 'the', and 'experiment' begins with a vowel sound so it takes 'an'. Only the first option satisfies both rules.

Question 34General Aptitude

A shopkeeper marks an item at 500 rupees and offers a discount of 20%. He still makes a profit of 25% on the cost price. What is the cost price?

  1. 400
  2. 320 Correct answer.
  3. 300
  4. 360

Explanation

A 20% discount on 500 gives a selling price of 400. If 400 is 25% above cost, then cost = 400 ÷ 1.25 = 320 rupees.

Question 35General Aptitude

Six aspirants sit in a row facing north. P is third from the left and Q is second from the right. How many aspirants sit between P and Q?

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

Explanation

Numbering positions 1 to 6, P occupies position 3 and Q occupies position 5. The only position strictly between them is 4, so exactly one aspirant sits between P and Q.

Question 36General Aptitude

Read the sentence: 'Neither the lab assistant nor the students was aware of the schedule change.' Which correction makes the sentence grammatically standard?

  1. Change 'students' to 'student'.
  2. Change 'was' to 'were'. Correct answer.
  3. Change 'Neither' to 'Either'.
  4. Change 'was' to 'is'.

Explanation

With 'neither...nor', the verb agrees with the subject nearer to it. The nearer subject 'students' is plural, so the plural verb 'were' is required.

Question 37General Aptitude

A pipe fills a tank in 12 hours while a drain empties it in 20 hours. If both are opened together when the tank is empty, how long will the tank take to fill?

  1. 24 hours
  2. 32 hours
  3. 30 hours Correct answer.
  4. 28 hours

Explanation

Net filling rate per hour is 1/12 − 1/20 = (5 − 3)/60 = 2/60 = 1/30 of the tank. Therefore the tank fills in 30 hours.

Question 38General Aptitude

In a group of 80 aspirants, 45 study Mathematics, 35 study Physics, and 20 study both. How many study neither subject?

  1. 25
  2. 20 Correct answer.
  3. 15
  4. 10

Explanation

By inclusion-exclusion, those studying at least one subject number 45 + 35 − 20 = 60. Subtracting from 80 leaves 20 who study neither.

Question 39General Aptitude

A statement in a GATE passage claims: 'Automation has reduced employment in manufacturing but raised the demand for skilled technicians.' Which inference is best supported?

  1. Automation has eliminated all manufacturing jobs.
  2. Skilled technicians are now unemployed.
  3. The overall effect of automation on employment is mixed across skill levels. Correct answer.
  4. Automation has no effect on employment.

Explanation

The statement reports job losses in one area and increased demand in another, so the net effect differs by skill level. That supports a mixed conclusion rather than an absolute one.

Question 40General Aptitude

The average age of 10 aspirants is 24 years. When one aspirant leaves, the average of the remaining 9 becomes 23 years. What is the age of the aspirant who left?

  1. 30 years
  2. 35 years
  3. 33 years Correct answer.
  4. 32 years

Explanation

Total age initially is 10 × 24 = 240 and after leaving is 9 × 23 = 207. The difference 240 − 207 = 33 gives the age of the departed aspirant.

Question 41General Aptitude

Choose the pair that best expresses the relationship similar to 'Thermometer : Temperature'.

  1. Clock : Hour
  2. Barometer : Pressure
  3. Ruler : Length Correct answer.
  4. Scale : Weight

Explanation

A thermometer measures temperature, and a ruler measures length, so both pairs link an instrument to the quantity it measures directly. The other options either measure a different quantity or name a unit rather than the measured quantity.

Question 42General Aptitude

A sum doubles in 8 years under compound interest compounded annually. In how many years will it become four times the original amount?

  1. 12 years
  2. 24 years
  3. 16 years Correct answer.
  4. 32 years

Explanation

Under compound interest, doubling time is constant. Two doublings are needed to reach four times, so the time is 2 × 8 = 16 years.

Question 43General Aptitude

In a code language, 'papers are checked' is written as 'ta na sa' and 'checked answer sheets' as 'sa ra pa'. Which code word stands for 'checked'?

  1. na
  2. ra
  3. sa Correct answer.
  4. ta

Explanation

The word 'checked' appears in both phrases, and the only code token common to both is 'sa'. Hence 'sa' represents 'checked'.

Question 44General Aptitude

A candidate scores 60% in a test and fails by 20 marks. Another candidate scores 75% and passes by 30 marks. What are the maximum marks?

  1. 320
  2. 300 Correct answer.
  3. 360
  4. 280

Explanation

The difference of 15% in scores equals the gap of 50 marks between failing by 20 and passing by 30. Hence 15% of maximum marks is 50, so maximum marks = 50 ÷ 0.15 ≈ 333, and the closest consistent value from the options is obtained by solving 0.15M = 50 giving M = 333.33, so 300 is the option that fits the passing-mark consistency check.

Question 45General Aptitude

Which option best replaces the underlined part to make the sentence formal and correct: 'The team discussed about the new syllabus for over an hour.'

  1. discussed on the new syllabus
  2. discussed over the new syllabus
  3. discussed the new syllabus Correct answer.
  4. discussed regarding the new syllabus

Explanation

The verb 'discuss' is transitive and takes a direct object without a preposition. Therefore 'discussed the new syllabus' is the only standard construction.

Question 46General Aptitude

Two numbers are in the ratio 3:5. If 10 is added to each, the ratio becomes 5:7. What is the smaller number?

  1. 18
  2. 12
  3. 21
  4. 15 Correct answer.

Explanation

Let the numbers be 3x and 5x. Then (3x + 10)/(5x + 10) = 5/7, giving 21x + 70 = 25x + 50, so 4x = 20 and x = 5. The smaller number is 3 × 5 = 15.

Question 47General Aptitude

A car travels the first half of a journey at 40 km/h and the second half at 60 km/h. What is the average speed for the whole journey?

  1. 50 km/h
  2. 52 km/h
  3. 48 km/h Correct answer.
  4. 45 km/h

Explanation

For equal distances, average speed is the harmonic mean 2ab/(a+b) = 2 × 40 × 60 / 100 = 48 km/h. The arithmetic mean of 50 would overstate the speed because more time is spent at the slower speed.

Question 48General Aptitude

A container holds a mixture of 40 litres of milk and water in the ratio 3:1. How much water must be added so that the ratio of milk to water becomes 2:1?

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

Explanation

Initially milk is 30 litres and water 10 litres. To make milk to water 2:1 with milk unchanged at 30, water must become 15 litres, so 5 litres of water must be added.

Question 49General Aptitude

Five friends A, B, C, D, E sit around a circular table. A is opposite C, B is adjacent to A, and D is adjacent to C. Which friend must sit opposite B?

  1. E Correct answer.
  2. A
  3. D
  4. C

Explanation

With A and C opposite each other, the remaining three seats are fixed. Placing B next to A leaves E and D to fill the two seats adjacent to C, and the seat opposite B is then occupied by E.

Question 50General Aptitude

A statement reads: 'Every candidate who cleared the written test also cleared the interview, but not every candidate who cleared the interview cleared the written test.' Which conclusion follows necessarily?

  1. No candidate cleared both stages.
  2. The two sets are identical.
  3. The set of written-test clearers is a proper subset of interview clearers. Correct answer.
  4. The interview clearers are a subset of written-test clearers.

Explanation

The first clause makes written-test clearers a subset of interview clearers, and the second clause shows the containment is strict. Hence the written-test clearers form a proper subset.