What topics are covered on Page 1 of GATE 2026 Civil Engineering Set 2?

Page 1 contains Questions 1 to 13. Topics include matrix eigenvalues and trace, numerical differentiation, Fourier series, partial differential equations, determinants, vector calculus, differential equations, normal distribution, air pollution control, sludge properties, oxidation ponds and biochemical oxygen demand.

What is the trace of A squared in GATE 2026 Civil Engineering Set 2?

If matrix A has eigenvalues 1, 2 and 3, the eigenvalues of A squared are 1, 4 and 9. Their sum is 14, so the trace of A squared is 14.

What is the height h in the normal distribution question of GATE 2026 Civil Set 2?

For the given normally distributed age and height populations, the height corresponding to the same upper-tail probability as an age greater than 50 years is 171.67 cm.

What is the BOD5 of the raw sewage in GATE 2026 Civil Set 2?

For 2.4 ml of raw sewage diluted to 240 ml with dissolved oxygen falling from 8 mg/l to 6 mg/l after five days, the BOD5 of the raw sewage is 200 mg/l.

Civil Engineering Gate 2026 Set-2 Questions with Answer

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Ques 1 GATE 2026 SET-2

Matrix A has the eigenvalues 1, 2, and 3. The Trace of A² is

(b) is the correct answer.

Ques 2 GATE 2026 SET-2

A fifth-degree polynomial in x is defined for x > 0. All coefficients of the polynomial are positive. The first derivative of the polynomial is obtained numerically at a point by using the first-order forward as well as the first-order backward difference methods. Identical step lengths are used for both the methods.
Following statements are made.
(I) Forward difference method underestimates the true derivative.
(II) Backward difference method overestimates the true derivative.
Which one of the following options is CORRECT?

Both statements (I) and (II) are FALSE.

Both statements (I) and (II) are TRUE.

Statement (I) is TRUE and statement (II) is FALSE.

Statement (I) is FALSE and statement (II) is TRUE.

(b) is the correct answer.

Ques 3 GATE 2026 SET-2

Periodic function f(x) is given below.
f(x) = { −1, when −π < x < 0; 1, when 0 < x < π }; f(x+2π) = f(x)
The CORRECT option representing the Fourier series expansion of f(x) is:

f(x) = (4/π)[sin x + sin3x/3 + sin5x/5 + ⋯]

f(x) = −(4/π)[sin x + sin3x/3 + sin5x/5 + ⋯]

f(x) = 1 + (4/π)[sin x + sin3x/3 + sin5x/5 + ⋯]

f(x) = 1 − (4/π)[sin x + sin3x/3 + sin5x/5 + ⋯]

(a) is the correct answer.

Ques 4 GATE 2026 SET-2

A partial differential equation is given below.
∂²u/∂x² − ∂²u/∂y² = 0
Possible solution(s) is/are:

(x + y)⁵

(x − 2y)³

cos(x + y)

sin(x − 2y)

(a,b,c,d) is the correct answer.

Ques 5 GATE 2026 SET-2

The eigenvalues of [A] = [[2, −3.5, 6], [3.5, 5, 2], [8, 1, 8.5]] are
λ₁ = −1.547, λ₂ = 12.330, and λ₃ = 4.711.
The absolute value of the determinant of matrix A is ______ (rounded off to two decimal places).

(89.03) is the correct answer.

Ques 6 GATE 2026 SET-2

Let where p is a constant.
The value of d³/dx³ f(x) at x =

6p³

p + p²0 is

p + p³

independent of p

(a) is the correct answer.

Ques 7 GATE 2026 SET-2

Vector field V⃗ is defined as
V⃗ = 3x²yz î − 5xy ĵ + 6yz² k̂
The curl of V⃗ at point (2,−1,1) is

6î − 12ĵ − 7k̂

−12î − 10ĵ − 12k̂

−34

(a) is the correct answer.

Ques 8 GATE 2026 SET-2

Consider differential equation dy/dx + xy = x with the condition as y = 0 at x = 0.
The value of y at x = 1.0 is ______ (rounded off to two decimal places).

(0.63) is the correct answer.

Ques 9 GATE 2026 SET-2

The age (in years) of a population is normally distributed with a mean of 36 and standard deviation of 12. The height (in cm) of the same population is also normally distributed with a mean of 160 and standard deviation of 10.
If the probability of age greater than 50 years is equal to the probability of height greater than h, the value of h (in cm) is ______ (rounded off to two decimal places).

(171.67) is the correct answer.

Ques 10 GATE 2026 SET-2

Column I presents common air pollutants, and column II presents treatment technologies commonly employed to control air pollutants.

Air Pollutant	Treatment Technology
1. Dust (size 5–25 µm)	P. Cyclone separator
2. Carbon monoxide (CO)	S. Catalytic converter
3. Sulfur dioxide (SO₂)	Q. Wet lime-water scrubber
4. Fly ash (size < 2.5 µm)	R. Electrostatic precipitator

The option that CORRECTLY matches the air pollutant with its treatment technology is:

1–P; 2–S; 3–Q; 4–R

1–P; 2–R; 3–S; 4–Q

1–R; 2–Q; 3–P; 4–S

1–R; 2–Q; 3–S; 4–P

(d) is the correct answer.

Ques 11 GATE 2026 SET-2

One-third of the solid matter in a sludge containing 90 % water is composed of fixed mineral solids with specific gravity 2.5, and two-third is composed of volatile solids with specific gravity 1.0.
Specific gravity of all solids lies between

1.2 and 1.3

1.5 and 1.6

1.7 and 1.8

2.0 and 2.1

(a) is the correct answer.

Ques 12 GATE 2026 SET-2

The grown algae are useful in the operation of

oxidation pond.

slow sand filter.

cyclone separator.

septic tank.

(a) is the correct answer.

Ques 13 GATE 2026 SET-2

The 2.4 ml of raw sewage is diluted to 240 ml. The Dissolved Oxygen (DO) of the diluted sample at the beginning of Biochemical Oxygen Demand (BOD) test was 8 mg/l and it was 6 mg/l after 5-day incubation at 20 °C.
The BOD₅ (in mg/l) of the raw sewage is

200

100

250

150

(a) is the correct answer.

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