GATE — Environmental Science and Engineering (ES) · India · IITs and IISc on rotation (2026: IIT Guwahati)
Original practice questions written to the published syllabus · not past papersThe mathematics and statistics needed for every numerical question in this track: calculus and differential equations for growth and decay, probability and risk, regression and hypothesis tests, and sampling designs for field data. Experimental design and survey sampling in depth live in the Agricultural Statistics track; this subject covers the environmental applications.
In the exams: GATE ES Section 1 (Mathematics Foundation); ICAR AICE subject 71 Unit 7; UGC NET Environmental Sciences Unit 9
How our questions for GATE spread across the units of Mathematics & Statistics for Environmental Science. Difficulty: 20 easy, 41 medium, 25 hard.
| Hypothesis tests: t, F and chi-square | 9 | |
| Linear algebra: matrices, linear systems, eigenvalues | 8 | |
| Calculus: limits, derivatives, maxima and minima, integrals | 8 | |
| Ordinary differential equations and first-order decay and growth | 8 | |
| Laplace transforms | 8 | |
| Descriptive statistics: central tendency, dispersion, skewness, kurtosis | 8 | |
| Probability, conditional probability and Bayes' theorem | 8 | |
| Risk and reliability; return periods and exceedance probability | 8 | |
| Probability distributions: binomial, Poisson, normal, log-normal | 7 | |
| Correlation, simple and multiple regression | 7 | |
| Environmental sampling designs: random, stratified, systematic | 7 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
Principal component analysis of water quality data finds components as
D. covariance-matrix eigenvectors
PCA finds the eigenvectors of the covariance matrix (or of the correlation matrix when variables are standardised), ordered by eigenvalue, the variance each explains. Determinants and regression residuals play no part in defining the components.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
A pollutant decays by dC/dt = −kC with k = 0.1 per day and C0 = 100 mg/L. What is C after 10 days?
A. 36.8 mg/L
C = 100e^(−1) ≈ 36.8. Linear decay would give 0 or 90; 10 confuses e^(−1) with 0.1.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
What is the half-life of a first-order process with k = 0.1 per day?
C. 6.93 days
t½ = ln 2/k = 0.693/0.1 = 6.93 days.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
Which average is most appropriate for summarising log-normally distributed bacterial counts?
A. Geometric mean
Geometric means suit multiplicative, skewed data; regulations often use them for coliforms.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
Adding an outlier of 30 to the data 4, 8, 6, 5, 7 changes the median from 6 to
B. 6.5
Ordered 4, 5, 6, 7, 8, 30: median = (6 + 7)/2 = 6.5; the mean jumps to 10, showing the median's robustness.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
What does the central limit theorem state?
D. Means of large samples are near-normally distributed
The central limit theorem: whatever the parent distribution, means of sufficiently large samples are approximately normally distributed, with standard error σ/√n. It says nothing about the raw data themselves being normal.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
What is the probability that a 100-year flood occurs at least once during a 50-year design life, assuming independent years?
D. About 39 per cent
R = 1 − 0.99⁵⁰ ≈ 0.39. 50% assumes linear accumulation; 1% is the annual probability.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
What return period must a structure be designed for to keep the risk of exceedance to 10 per cent over a 50-year life?
C. About 475 years
Solve 1 − (1 − 1/T)⁵⁰ = 0.1: 1/T = 1 − 0.9^(1/50), T ≈ 475. 500 years is a rough rounding of 50/0.1; 50 equals the design life.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
With slope 0.6, mean x = 3 and mean y = 4, what is the intercept of the regression line?
C. 2.2
a = ȳ − b x̄ = 4 − 1.8 = 2.2.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
Why should regression predictions not be extrapolated far beyond the range of observed data?
A. The data do not support the fit there
A fitted regression is supported only within the range of the data; outside it the relationship may change form, so extrapolated predictions have no empirical basis. Software can compute them, which is exactly the danger.
Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran
Ten here, 86 in the app. The Laisha app has every practice question we have written to GATE in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
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