Engineering Mathematics Gate Questions

Here, we will see Engineering Mathematics GATE Questions from the previous year’s papers and the syllabus for the Engineering Mathematics GATE Exam.

1. Engineering Mathematics in Gate CSE Exam

Engineering Mathematics is a pivotal section in the GATE Computer Science (CSE) exam, testing candidates’ analytical and problem-solving skills. Accounting for 13% of the total marks (15 marks in General Aptitude + 85 in core subjects), it forms the backbone of logical reasoning in topics like algorithms, networks, and system design

2. Engineering Mathematics Syllabus

The syllabus for Engineering Mathematics in GATE CSE includes:

1. Discrete Mathematics: Propositional logic, graphs, combinatorics, and group theory.
2. Linear Algebra: Matrices, eigenvalues, systems of linear equations, and vector spaces.
3. Calculus: Limits, continuity, integration, Fourier series, and multivariable calculus.
4. Probability & Statistics: Distributions (normal, binomial), hypothesis testing, regression, and Bayes’ theorem.
5. Numerical Methods: Solutions for linear equations, interpolation, and differential equations.

Engineering Mathematics Gate Questions

Q. The function has f(x) = x⁵-5x⁴+5x³-1 has

1. one minima and two maxima
2. two minima and one maxima
3. two minima and two maxima
4. one minima and one maxima

Q. A box contains 10 screws, 3 of which are defective. Two screws are drawn at random with replacement. The probability that none of two screws is defective will be

1. 100%
2. 50%
3. 49%
4. None of these.

Q. The following vectors (1,9,9,8),(2,0,0,8),(2,0,0,3) are

1. Linearly dependent
2. Linearly independent
3. Constant
4. None of these

Q. A ladder 13 feet long rests against the side of a house. The bottom of the ladder slides away from the house at a rate of 0.5 ft/s. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 5 feet from the house?

1. 5/24 ft/s
2. 5/12 ft/s
3. -5/24 ft/s
4. -5/12 ft/s

Q. The system of simultaneous equations has
x+2y+z=6
2x+y+2z=6
x+y+z=5

1. unique solution.
2. infinite number of solutions.
3. no solution.
4. exactly two solutions.

Q. In a parallelogram ABCD, AP and BP are the angle bisectors of ∠DAB and ∠ABC. Find ∠APB:

1. 85°
2. 90°
3. 94°
4. 86°

Q. The equation of the plane through the point (-1,3,2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z=0 is

1. 7x-8y+3z+25=0
2. 7x+8y+3z+25=0
3. 7x-8y+3z-25=0
4. 7x-8y-3z-25=0

Q. Find the area bounded by the curve y=√(5-x²) and y=|x-1|

1. 2/0(2√6 – √3) – 5/2
2. 2/3(6√6 + 3√3) = 5/2
3. 2(√6 – √3) – 5
4. 2/3(√6 – √3) +5

Q. The area under the curve y(x)=3e^-5x from x=0 to x=∞ is

1. 3/5
2. -3/5
3. 5
4. 5/3

Q. If A and B are two related events, and P(A|B) represents the conditional probability, Bayes’ theorem states that

1. P(A|B)=P(A)/P(B).P(B|A)
2. P(A|B)=P(A)P(B)P(B|A)
3. P(A|B)=P(A)/P(B)
4. P(A|B)=P(A)+P(B)

Q. In which of the following methods proper choice of initial value is very important?

1. Bisection method
2. False position
3. Newton-Raphson
4. Bairsto method

Q. If x=p, y=q, then which of following are p and q respectively for pair of equations 3x-7y+10=0 and y-2x-3=0:

1. -1,1
2. 1,1
3. 1,0
4. 0,1

Q. 1/2 log₁₀25-2log₁₀3+log₁₀18 equals

1. 18
2. 1
3. log₁₀3
4. None of these

Q. If p(A∩B)=1/2, P(Ā∩B̅)=1/2 and 2P(A)=P(B)=p, then the value of p is given by:

1. 1/4
2. 1/2
3. 1/3
4. 2/3

Q. The function f(x) = (x² -1)/(x-1) at x=1 is

1. Continuous and differentiable
2. Continuous but not differentiable
3. Differentiable but not continuous
4. Neither continuous nor differentiable

Q. Differential equation, d²x/dt² + 10dx/dt + 25x =0 will have a solution of the form, where and are constants.

1. (C₁ + C₂t)e^-5t 
2. C₁e^-2t 
3. C₁e^-5t  + C₂e^5t 
4. C₁e^-5t + C₂e^2t 

Q. If z=cos(x/y) + sin(x/y), then xz_x + yz_y is equal to ______ .log₁₀

1. z
2. 2z
3. 4z
4. 0

Q. Consider three vectors x=[1 2], y=[4 8] z=[3 1]. Which of the following statements is true

1. x and y are linearly independent
2. x and y are linearly dependent
3. x and z are linearly dependent
4. y and z are linearly dependent

Q. For the function (z) = 1/z²(e-1)^z, z=0 is a pole of order:

1. 1
2. 2
3. 3
4. None of these

Q. If  y^α is an integrating factor of the differential equation 2xydx-(3x²-y²)dy=0, then the value of α is

1. -4
2. 4
3. -1
4. 1

Q. The general solution of the differential equation dy/dx = (1+y²)(e^-x2 – 2xtan^-1y) is:

1. e^x2tan^-1y = x+c
2. e^-x2tan y = x+c
3. e^xtan y = x²+c
4. e^-xtan^-1y = x³+c

Q. While solving the differential equation d²y/dx² + 4y = tan2x by the method of variation of parameters, then value of Wronskion (W) is:

1. 1
2. 2
3. 3
4. 4

Q. The general solution of the partial differential equation (D²-D’²-2D+2D’)Z=0 where D=δ/δx and D’=δ/δy:

1. f(y+x) + e^2x g(y-x)
2. e^2x f(y+x) + g(y-x)
3. e^-2x f(y+x) + g(y-x)
4. f(y+x) + e^-2x g(y-x)

Q. Using Green’s theorem in plane, evaluate ∫_c(2x-y)dx + (x+y)dy, , where c is the circle x²+y²=4 in the plane:

1. 2π
2. 4π
3. -4π
4. 8π

Q. If f(x,y) = x³y+e^x, the partial derivatives, δf/δx, δf/δy are

1. 3 x²y+1, x³+1
2. 3 x²y+e^x, x³
3. x³y+xe^x, x³+e^x
4. 2 x²y + e^x/x

Q. If u = f(y-z,z-x,x-y), then δu/δx + δu/δy + δu/δz is equal to:

1. x+y+z
2. 1+x+y+z
3. 1
4. 0

Q. If =f(z)=u(x,y) + iν(x,y) is an analytic function, then dω/dz is:

1. δu/δx – iδu/δy
2. δu/δx + iδν/δy
3. δu/δx – iδν/δx
4. δu/δx + iδu/δy

Q. Two eigenvalues of a 3×3 real matrix P are (2+√-1) and 3. The determinant of P is ________.

1. 0
2. 1
3. 15
4. -1

Q. A solution for the differential equation x'(t) + 2x(t) = δ(t) with initial condition x(0′)=0

1. e^-2t u(t)
2. e^2t u(t)
3. e^-t u(t)
4. e^t u(t)

Q. ∫∫ xy/√(1-y²) dxdy. Over the positive quadrant of the circle x²+y²=1 is ________

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

Q. log(x+3) + log(x+5) = log 35, solve for x:

1. 1
2. 2
3. 3
4. 4

Q. Which of the following is true?

1. log₁₇275=log₁₉375
2. log₁₇175>log₁₉375
3. log₁₇275<log₁₉375
4. None of these

Q. If X,Y and Z are three exhaustive and mutually exclusive events related with any experiment and the P(X)=0.5P(Y) and P(Z)=0.3P(Y). Then P(Y) = _____________ .

1. 0.54
2. 0.66
3. 0.33
4. 0.44

Q. Let u and υ be two vectors in R² whose Eucledian norms satisfy |u| = 2|υ|. What is the value α such that ω = u+αυ bisects the angle between u and υ?

1. 2
2. 1
3. 1/2
4. -2

Q. The function f(x,y) = x²+y²+6x+12 has:

1. minimum value, -3 and maximum value, 0.
2. only minimum value, 3.
3. only maximum value, 12.
4. neither maxima nor minima.

Q. Using bisection method, one root of x⁴-x-1 lies between 1 and 2. After second iteration the root may lie in an interval:

1. (1.25, 1.5)
2. (1, 1.25)
3. (1, 1.5)
4. None of the options

Q. Following marks are obtained by the students in a test:
81,72,90,90,86,85,92,70,71,83,89,95,85,79,62. The range of the marks is

1. 9
2. 17
3. 27
4. 33

Q. Given √(224)_r = (13)_r. The value of the radix r is:

1. 10
2. 8
3. 5
4. 6

Q. Maxima and minimum of the function f(x) = 2x³ – 15x² + 36x + 10 occur; respectively at

1. x=3 and x=2
2. x=1 and x=3
3. x=2 and x=3
4. x=3 and x=4

Q. In how many ways 8 girls and 8 boys can sit around a circular table so that no two boys sit together?

1. (7!)²
2. (8!)²
3. 7!8!
4. 15!

Q. The minimum value of |x² – 5x +2| is

1. -5
2. 0
3. -1
4. -2

Q. Let A,B,C,D be n×n matrices, each with non-zero determinant. If ABCD=1, then B^-1 is:

1. D^-1C^-1A^-1
2. CDA
3. ADC
4. Does not necessarily exist.

Q. If z = cos(x/y)+sin(x/y), then xz_x+yz_y is equal to _________.

1. z
2. 2z
3. 4z
4. 0

Q. If f(x)=k exp, {-(9x² -12x + 13)}, is a p,d,f of a normal distribution (k, being a constant), the mean and standard deviation of the distribution:

1. μ=2/3, σ=1/3√2
2. μ=2, σ=1/√2
3. μ=1/3, σ=1/3√2
4. μ=2/3, σ=1/√3

Q. The probability that top and bottom cards of a randomly shuffled deck are both aces is

1. 4/52*4/52
2. 4/52*3/52
3. 4/52*3/51
4. 4/52*4/51

Q. M is a square matrix of order ‘n’ and its determinant value is 5.If all the elements of M are multiple by 2, its determinant value becomes 40.The value of ‘n’ is

1. 2
2. 3
3. 5
4. 4

Q. If a random coin is tossed 11 times, then what is the probability that for 7th toss head appears exactly 4 times?

1. 5/32
2. 15/128
3. 35/128
4. None of the options

Q. The greatest and the least value of f(x) = x⁴ – 8x³ + 22x² – 24x + 1 in [0,2] are

1. 0,8
2. 0,-8
3. 1,8
4. 1,-8

Q. If A and B are square matrices of size n×n, then which of the following statements is not true?

1. det(AB)=det(A)det(B)
2. det(kA)=kⁿdet(A)
3. det(A+B)=det(A)+det(B)
4. det(A^T)=1/det(A^-1)

Q. If P is risk probability, L is loss, then Risk Exposure (RE) is computed as.

1. RE=P/L
2. RE=P+L
3. RE=P*L
4. RE=2PL

Q. Consider two matrices M1 and M2 with M1*M2=0 and M1 is non singular. Then which of the following is true?

1. M2 is non singular
2. M2 is null matrix
3. M2 is the identity matrix
4. M2 is transpose of M1

Q. What is the maximum value of the function f(x) = 2x² – 2x + 6 in the interval [0,2]?

1. 6
2. 10
3. 12
4. 5,5

Q. The convergence of the bisection method is

1. Cubic
2. Quadratic
3. Linear
4. None

Q. A continuous random variable x is distributed over the interval [0,2] with probability density function f(x)=ax² + bx, where a and b are constants. If the mean of the distribution is 1/2. Find the values of the constants a and 1.

1. a=2, b=-13/6
2. a=-15/8, b=3
3. a=-29/6, b=2
4. a=3, b=-7/2

Q. A bag contains 10 white balls and 5 blue balls. A ball is drawn from the bag and its color is noted. This ball is put back in the bag along with 3 more balls of the same color. A ball is drawn again from the bag at random. The probability that the first ball drawn is blue, given that the second ball drawn is blue, is:

1. 1/3
2. 3/4
3. 8/9
4. 4/9

Q. Choose the most appropriate option.
The Newton-Raphson iteration x_n+1 = x_n/2 + 3/2x_n can be used to solve the equation

1. x²=3
2. x³=3
3. x²=2
4. x³=2