List of top Questions asked in GATE CE- 2026

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?
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
Periodic function \(f(x)\) is given below.
\[ f(x)=\begin{cases}-1, & -\pi < x < 0 \\ 1, & 0 < x < \pi\end{cases}; \quad f(x+2\pi)=f(x) \]
The CORRECT option representing the Fourier series expansion of \(f(x)\) is:
The eigenvalues of \( [A] = \begin{bmatrix} 2 & -3.5 & 6 \\ 3.5 & 5 & 2 \\ 8 & 1 & 8.5 \end{bmatrix} \) are
\( \lambda_1 = -1.547, \lambda_2 = 12.330 \), and \( \lambda_3 = 4.711 \).
The absolute value of the determinant of matrix A is ______ (rounded off to two decimal places).
A partial differential equation is given below.
\[ \frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} = 0 \]
Possible solution(s) is/are:
Consider differential equation \(\dfrac{dy}{dx}+xy=x\) with the condition as \(y=0\) at \(x=0\). The value of \(y\) at \(x=1.0\) is ______ (rounded off to two decimal places).
The analysis of major cations and anions in a water sample collected from a city's water supply is given below. Ions present in minor concentrations are not given.

Anions\(Cl^-\)\(SO_4^{2-}\)\(HCO_3^-\)\(CO_3^{2-}\)
Concentration (mM)1.50.51.00.01
Cations\(Na^+\)\(Ca^{2+}\)\(Mg^{2+}\)\(K^+\)
Concentration (mM)20.50.250.02


\[ HCl \rightleftharpoons H^+ + Cl^- \qquad pK=-3 \]
\[ H_2SO_4 \rightleftharpoons 2H^+ + SO_4^{2-} \qquad pK=-3 \]
\[ H_2CO_3 \rightleftharpoons H^+ + HCO_3^- \qquad pK=6.3 \]
\[ HCO_3^- \rightleftharpoons H^+ + CO_3^{2-} \qquad pK=10.3 \]
\[ HOCl \rightleftharpoons H^+ + OCl^- \qquad pK=7.5 \]

\(HOCl\) (in %) present in the total free chlorine in the water is (rounded off to the nearest integer).
'People are crowding around pit into which elephant has fallen. I have never seen an elephant looking more bewildered miserable. Here it is in a most undignified position, thrust into a pit and made to look up a vast, curiosity-stricken crowd.'

Choose the option with the correct sequence of words to fill the blanks.
Matrix A has the eigenvalues 1, 2, and 3. The Trace of \(A^2\) is
'The team more than 300 runs in 20 overs rains. However, some players needed to improve their batting skills.'

Choose the option with the correct sequence of words to fill the blanks.
Anaerobic bacteria are being utilized for the destruction of biodegradable organic content of wastewater measured in terms of Chemical Oxygen Demand (COD). The yield coefficient (Y) is 0.06 gram Volatile Suspended Solids (VSS) per gram of COD. Bacterial biomass forms the sludge during the process. The proximate chemical formula of anaerobic bacteria is \(C_6H_7NO_2\).

The COD (in %) converted into bacterial biomass is (rounded off to the nearest integer).
The ultimate bearing capacity of a 1 m wide strip footing is 532.80 kPa, when it is embedded at 1 m depth in dry cohesionless soil. The soil has a unit weight of 18 kN/m3. The ultimate bearing capacity is 864 kPa when the depth of embedment becomes 2 m.

Neglecting the effect of the depth factor, the bearing capacity factor \(N_q\) is (rounded off to one decimal place).
In an industry, a cyclone collector with 80% efficiency is installed for air purification, followed by an electrostatic precipitator with 50% efficiency. The concentration of particles entering the cyclone collector is 10 mg/m\(^3\).

The concentration (in mg/m\(^3\)) of particles exiting the precipitator is (rounded off to two decimal places).
Let \( f(x) = \begin{vmatrix} x^3 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix} \) where \(p\) is a constant.
The value of \( \dfrac{d^3}{dx^3} f(x) \) at \(x = 0\) is
The next figure (indicated by '?') in the sequence is
Column I presents common air pollutants, and column II presents treatment technologies commonly employed to control air pollutants.
Air pollutantTreatment technology
1Dust (size 5-25 microns)PCyclone separator
2Carbon monoxide (CO)QWet lime-water scrubber
3Sulfur dioxide (SO2)RElectrostatic precipitator
4Fly ash (size < 2.5 microns)SCatalytic converter
The option that CORRECTLY matches the air pollutant with its treatment technology is:
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).
Vector field \(\vec{V}\) is defined as
\[ \vec{V} = 3x^2yz\,\hat{i} - 5xy\,\hat{j} + 6yz^2\,\hat{k} \] The curl of \(\vec{V}\) at point \((2,-1,1)\) is
Consider a string P of length \(l\) that is laid out as a straight-line segment. Another string K is laid out as a semicircular arc with string P as its diameter, as represented in Figure (i). When both the strings are shortened by a length \(x\) they can be re-arranged such that the shortened string K forms a full circle with the shortened string P as its diameter, as represented in Figure (ii). The value of \(x/l\) is ____________
The grown algae are useful in the operation of
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&deg;C.
The BOD5 (in mg/l) of the raw sewage is
'All the mangoes in the basket are good.'
If the above statement is false, then which one of the following statements is necessarily true?
If a positive real \(x\) satisfies the following equation
\[ \log_2 x + \log_{\sqrt{2}} x = 48, \]
then the value of \(x\) is
The table lists the unit selling price of five products P, Q, R, S, and T. On a particular day, 250 items were sold with the average selling price of Rs. 60. The following observations were made:
(i) The quantity of S sold was twice that of T.
(ii) The quantity of R sold was thrice that of T.
(iii) The quantity of Q sold was four times that of T.

ProductPQRST
Unit selling price (Rs.)10050406060

What is the quantity of product P sold on that day?
Consider the following statements about four numbers:
(S1) The average of the four numbers is 25
(S2) Each number is at most 40
(S3) Each number is at least 20

Choose the option that is necessarily correct.