GATE Engineering Mathematics Mock Test 2027 – 10 Tough Questions with Detailed Solutions

 

GATE Engineering Mathematics Advanced Mock Test 2027

Engineering Mathematics is one of the most important subjects for GATE aspirants. A strong understanding of mathematical concepts can help students solve complex problems with accuracy and confidence. To improve your preparation, The GATEverse brings you an Advanced Engineering Mathematics Mock Test 2027, specially designed for students who want to practice challenging GATE-level questions.

This mock test contains 10 carefully designed questions based on important topics such as Linear Algebra, Real Analysis, Differential Equations, Complex Analysis, Probability, Numerical Methods, Fourier Series, and Partial Differential Equations. The questions focus on conceptual understanding, mathematical calculations, and problem-solving skills rather than basic formulas.

You will get 30 minutes to complete the entire test, just like a timed examination practice session. You can submit your test before the timer ends, or the test will automatically submit when the 30-minute time limit is over. Once you submit the test, your result, correct answers, and detailed step-by-step solutions will become available.

GENERAL APTITUDE MOCK TEST - https://www.thegateverse.in/2026/11/mock%20te.html

Each solution is designed to help you understand the mathematical approach behind the question. Instead of simply checking whether your answer is correct or incorrect, you can learn the concepts and methods required to solve similar questions in the actual GATE examination.

Whether you are preparing for GATE 2027 or strengthening your Engineering Mathematics fundamentals, this advanced mock test can help you evaluate your preparation, improve time management, and identify topics that need more practice.

Start your test, manage your time wisely, and challenge your Engineering Mathematics preparation with The GATEverse.

GATE ENGINEERING MATHEMATICS

Advanced Mathematics Challenge

GATE Level • Conceptual Problems • Mathematical Reasoning

TIME REMAINING

Automatic submission at 00:00

30:00

EXAM INSTRUCTIONS
• Total Questions: 10
• Duration: 30 Minutes
• Question Type: Single Correct MCQ
• Select one option for every question.
• You can submit the test before the timer ends.
• The test automatically submits when time expires.
• Correct answers and detailed mathematical solutions are revealed only after submission.

YOUR CHALLENGE BEGINS HERE

Are You Ready?

This mock test contains 10 advanced Engineering Mathematics problems designed to challenge your conceptual understanding, calculations and problem-solving ability.

30 Minutes • 10 Questions • Advanced GATE Practice

01

Linear Algebra

Let A be a nonsingular 3 × 3 matrix whose characteristic polynomial is

p(λ)=λ³−6λ²+11λ−6.

The value of det(A⁻¹+I) is:

A3 B4 C6 D8

DETAILED SOLUTION — Q1

Step 1: Find the eigenvalues.

The characteristic polynomial is:

λ³−6λ²+11λ−6=0
=(λ−1)(λ−2)(λ−3)=0

Therefore, the eigenvalues of A are 1, 2 and 3.

Step 2: Determine the eigenvalues of A⁻¹+I.

If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A⁻¹. Consequently, the eigenvalues of A⁻¹+I are:

1+1/1=2,   1+1/2=3/2,   1+1/3=4/3.

Step 3: Calculate the determinant.

The determinant is the product of the eigenvalues:

det(A⁻¹+I)=2 × 3/2 × 4/3
=4.

Correct Answer: B — 4

GATE Concept: Eigenvalues of inverse matrices, matrix transformations and determinant properties.

02

Real Analysis

Consider the sequence of functions defined on [0,1]:

fₙ(x)= nx/(1+n²x²).

Which of the following statements is correct?

Afₙ converges uniformly to zero. Bfₙ converges pointwise but not uniformly to zero. Cfₙ does not converge pointwise on [0,1]. Dfₙ converges uniformly to a nonzero function.

DETAILED SOLUTION — Q2

Step 1: Find the pointwise limit.

At x=0, fₙ(0)=0 for every n.

For any fixed x in (0,1]:

fₙ(x)=nx/(1+n²x²)
= (1/nx)/(1+1/n²x²).

As n approaches infinity, this expression approaches zero.

Hence fₙ(x) converges pointwise to f(x)=0 throughout [0,1].

Step 2: Test uniform convergence.

Uniform convergence to zero requires:

sup₀≤x≤₁ |fₙ(x)| → 0.

Differentiate fₙ with respect to x:

fₙ'(x)=n(1−n²x²)/(1+n²x²)².

The maximum occurs at x=1/n, which belongs to [0,1] for every positive integer n.

fₙ(1/n)=n(1/n)/(1+n²/n²)
=1/2.

Thus:

sup₀≤x≤₁ |fₙ(x)|=1/2.

This does not approach zero. Therefore convergence is not uniform.

Correct Answer: B

GATE Concept: Pointwise convergence, uniform convergence and supremum-based tests.

03

Multivariable Calculus

Let x,y,z be real numbers satisfying:

x+y+z=1
xy+yz+zx=0.

The value of x²+y²+z² is:

A0 B1 C2 D3

DETAILED SOLUTION — Q3

Step 1: Use the algebraic identity.

(x+y+z)²=x²+y²+z²+2(xy+yz+zx).

Step 2: Substitute the given conditions.

We know that:

x+y+z=1
xy+yz+zx=0.

Therefore:

1²=x²+y²+z²+2(0).

Step 3: Obtain the required value.

x²+y²+z²=1.

Correct Answer: B — 1

GATE Concept: Symmetric polynomial identities and algebraic constraints.

04

Differential Equations

Consider the initial value problem:

y''−2y'+y=eˣx²,
y(0)=0,   y'(0)=0.

The value of y(1) is:

Ae/6 Be/12 Ce/4 De/3

DETAILED SOLUTION — Q4

Step 1: Apply an exponential substitution.

Since the differential operator is (D−1)², choose:

y=eˣv.

Then:

y'=eˣ(v+v'),
y''=eˣ(v+2v'+v'').

Substituting into the given equation:

eˣ(v+2v'+v'')−2eˣ(v+v')+eˣv=eˣx².

After cancellation:

v''=x².

Step 2: Integrate twice.

v'=x³/3+C₁,
v=x⁴/12+C₁x+C₂.

Step 3: Apply initial conditions.

From y(0)=0, we obtain v(0)=0, so C₂=0.

Also:

y'(0)=v(0)+v'(0)=0.

Since v(0)=0, C₁=0.

Therefore:

y=eˣx⁴/12.

Step 4: Evaluate at x=1.

y(1)=e/12.

Correct Answer: B — e/12

GATE Concept: Repeated differential operators, exponential substitution and initial value problems.

05

Complex Analysis

Evaluate the contour integral over the positively oriented circle |z|=2:

∮ (z²+1)/[z(z−1)²] dz.

A0 Bπi C2πi D4πi

DETAILED SOLUTION — Q5

Step 1: Identify the singularities.

The integrand is:

f(z)=(z²+1)/[z(z−1)²].

Its poles are z=0 and z=1. Both lie inside |z|=2.

Step 2: Calculate the residue at z=0.

Res(f,0)=lim(z→0) zf(z)
=lim(z→0) (z²+1)/(z−1)²
=1.

Step 3: Calculate the residue at the double pole z=1.

Res(f,1)=d/dz[(z²+1)/z] at z=1.

Simplifying the expression:

(z²+1)/z=z+1/z
d/dz(z+1/z)=1−1/z².

At z=1:

Res(f,1)=1−1=0.

Step 4: Apply the residue theorem.

∮f(z)dz=2πi[1+0]
=2πi.

Correct Answer: C — 2πi

GATE Concept: Residue theorem, simple poles and repeated poles.

06

Probability

Let X and Y be independent exponential random variables, each having parameter λ=1. The probability P(X>Y+1) is:

Ae⁻¹ Be⁻¹/2 C1−e⁻¹ D1/2

DETAILED SOLUTION — Q6

Step 1: Write the joint probability density.

Since X and Y are independent exponential random variables with parameter 1:

fX(x)=e⁻ˣ, x≥0,
fY(y)=e⁻ʸ, y≥0.

Independence gives:

fX,Y(x,y)=e⁻ˣ⁻ʸ.

Step 2: Express the required probability.

The condition X>Y+1 means x>y+1. Hence:

P(X>Y+1)
=∫₀∞ ∫ᵧ₊₁∞ e⁻ˣ⁻ʸ dx dy.

Step 3: Integrate with respect to x.

∫ᵧ₊₁∞ e⁻ˣ⁻ʸ dx
=e⁻ʸ[e⁻⁽ʸ⁺¹⁾]
=e⁻¹e⁻²ʸ.

Step 4: Integrate with respect to y.

P=e⁻¹∫₀∞e⁻²ʸdy
=e⁻¹[1/2]
=e⁻¹/2.

Correct Answer: B — e⁻¹/2

GATE Concept: Joint probability density, independence and continuous random variables.

07

Numerical Methods

Apply the Newton–Raphson method to

f(x)=x³−2x−5

with initial approximation x₀=2. The value of x₂, correct to four decimal places, is:

A2.0946 B2.1000 C2.0952 D2.0900

DETAILED SOLUTION — Q7

Step 1: Write the iteration formula.

xₙ₊₁=xₙ−f(xₙ)/f'(xₙ).

Here:

f(x)=x³−2x−5,
f'(x)=3x²−2.

Step 2: Calculate x₁.

At x₀=2:

f(2)=8−4−5=−1,
f'(2)=12−2=10.

x₁=2−(−1/10)=2.1.

Step 3: Calculate x₂.

At x₁=2.1:

f(2.1)=2.1³−2(2.1)−5
=9.261−4.2−5
=0.061.

f'(2.1)=3(2.1)²−2
=13.23−2=11.23.

Therefore:

x₂=2.1−0.061/11.23
≈2.1−0.005432
≈2.094568.

Correct to four decimal places:

x₂=2.0946.

Correct Answer: A — 2.0946

GATE Concept: Iterative numerical methods and convergence of Newton–Raphson method.

08

Fourier Series

Let f(x)=x for −π<x<π, extended periodically with period 2π. Using Parseval's identity, evaluate:

Σₙ₌₁∞ 1/n².

Aπ²/4 Bπ²/6 Cπ²/8 Dπ²/12

DETAILED SOLUTION — Q8

Step 1: Determine the Fourier coefficients.

The function f(x)=x is odd. Therefore:

a₀=0,   aₙ=0.

The sine coefficients are:

bₙ=(1/π)∫₋ππ x sin(nx)dx
=(2/π)∫₀π x sin(nx)dx.

Integrating by parts:

bₙ=(2/π)[−x cos(nx)/n + sin(nx)/n²]₀π
=−2(−1)ⁿ/n.

Thus the Fourier series is:

x=2Σₙ₌₁∞ [(-1)ⁿ⁺¹ sin(nx)/n].

Step 2: Apply Parseval's identity.

For this Fourier series:

(1/π)∫₋ππ f(x)²dx
=Σₙ₌₁∞(aₙ²+bₙ²).

Since aₙ=0:

(1/π)∫₋ππ x²dx
=4Σₙ₌₁∞1/n².

Step 3: Evaluate the integral.

(1/π)[x³/3]₋ππ
=(1/π)(2π³/3)
=2π²/3.

Therefore:

4Σₙ₌₁∞1/n²=2π²/3
Σₙ₌₁∞1/n²=π²/6.

Correct Answer: B — π²/6

GATE Concept: Fourier coefficients and Parseval's identity.

09

Partial Differential Equations

Consider the heat equation:

uₜ=uₓₓ,   0<x<π, t>0,

with boundary conditions u(0,t)=u(π,t)=0 and initial condition:

u(x,0)=sin x+3sin 3x.

The value of u(π/2,t) is:

Ae⁻ᵗ+3e⁻⁹ᵗ Be⁻ᵗ−3e⁻⁹ᵗ Ce⁻ᵗ−3e⁻³ᵗ D3e⁻ᵗ−e⁻⁹ᵗ

DETAILED SOLUTION — Q9

Step 1: Recall the separated solution.

For the heat equation with zero boundary conditions on (0,π), the eigenfunctions are:

Xₙ(x)=sin(nx).

The corresponding time-dependent factors are e⁻ⁿ²ᵗ. Therefore the general Fourier solution is:

u(x,t)=Σₙ₌₁∞ bₙe⁻ⁿ²ᵗsin(nx).

Step 2: Use the initial condition.

u(x,0)=sin x+3sin 3x.

Comparing with the Fourier expansion gives b₁=1, b₃=3 and all other coefficients zero.

Hence:

u(x,t)=e⁻ᵗsin x+3e⁻⁹ᵗsin 3x.

Step 3: Substitute x=π/2.

sin(π/2)=1,
sin(3π/2)=−1.

Therefore:

u(π/2,t)=e⁻ᵗ−3e⁻⁹ᵗ.

Correct Answer: B

GATE Concept: Separation of variables, eigenfunction expansion and heat equation.

10

Improper Integrals

Evaluate the improper integral:

I=∫₀∞ dx/[√x(1+x)²].

Aπ/4 Bπ/2 Cπ D2π

DETAILED SOLUTION — Q10

Step 1: Write the integral in Beta-function form.

I=∫₀∞ x⁻¹ᐟ²/(1+x)² dx.

Recall the standard Beta integral:

B(a,b)=∫₀∞ xᵃ⁻¹/(1+x)ᵃ⁺ᵇ dx.

Comparing the exponents:

a−1=−1/2 ⇒ a=1/2,
a+b=2 ⇒ b=3/2.

Thus:

I=B(1/2,3/2).

Step 2: Apply the Gamma-function identity.

B(a,b)=Γ(a)Γ(b)/Γ(a+b).

Therefore:

I=Γ(1/2)Γ(3/2)/Γ(2).

Step 3: Use standard Gamma values.

Γ(1/2)=√π,
Γ(3/2)=Γ(1/2)/2=√π/2,
Γ(2)=1.

Substituting:

I=√π × √π/2
=π/2.

Correct Answer: B — π/2

GATE Concept: Improper integrals, Beta functions and Gamma functions.

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