LAISHA
Open Laisha

NCEES FE Environmental Mathematics & Statistics for Environmental Science: practice questions

NCEES FE Environmental · USA · National Council of Examiners for Engineering and Surveying

Original practice questions written to the published syllabus · not past papers
10worked here, free
30written to this exam in this subject
6syllabus units

What this subject covers

The 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

Where the 30 questions fall

How our questions for NCEES FE Environmental spread across the units of Mathematics & Statistics for Environmental Science. Difficulty: 9 easy, 13 medium, 8 hard.

Laplace transforms8
Calculus: limits, derivatives, maxima and minima, integrals6
Ordinary differential equations and first-order decay and growth6
Linear algebra: matrices, linear systems, eigenvalues5
Risk and reliability; return periods and exceedance probability4
Probability distributions: binomial, Poisson, normal, log-normal1

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Calculus: limits, derivatives, maxima and minima, integrals · medium

River flows of 2, 5, 7 and 4 m³/s are measured at hourly intervals. Using the trapezoidal rule, what is the total volume over the 3 hours in (m³/s)·h?

  1. 9
  2. 18
  3. 15
  4. 36
Answer and working

C. 15

(1/2)[2 + 2(5 + 7) + 4] = 15 (m³/s)·h = 54,000 m³. 18 sums all flows; 9 omits doubling interior points; 36 doubles every value.

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

2 · Ordinary differential equations and first-order decay and growth · easy

A pollutant decays by dC/dt = −kC with k = 0.1 per day and C0 = 100 mg/L. What is C after 10 days?

  1. 36.8 mg/L
  2. 0.0 mg/L
  3. 90.0 mg/L
  4. 10.0 mg/L
Answer and working

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

3 · Ordinary differential equations and first-order decay and growth · easy

What is the half-life of a first-order process with k = 0.1 per day?

  1. 5.00 days
  2. 10.00 days
  3. 6.93 days
  4. 0.07 days
Answer and working

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

4 · Ordinary differential equations and first-order decay and growth · hard

Which method solves the linear first-order equation dy/dt + p(t)y = q(t)?

  1. Applying the trapezoidal rule once
  2. Taking the determinant of the coefficient matrix
  3. Multiplying by the integrating factor e^(∫p dt)
  4. Using the characteristic equation am² + bm + c = 0
Answer and working

C. Multiplying by the integrating factor e^(∫p dt)

The integrating factor turns the left side into the derivative of y·e^(∫p dt).

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

5 · Laplace transforms · medium

Using the derivative property, L{f′(t)} equals

  1. sF(s) + f(0)
  2. sF(s) − f(0)
  3. F(s)/s
  4. F′(s)
Answer and working

B. sF(s) − f(0)

Differentiation in time becomes multiplication by s minus the initial value.

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

6 · Laplace transforms · medium

Solving dC/dt + kC = 0 with C(0) = C0 by Laplace transforms gives C(s) =

  1. C0/(s + k)
  2. C0 s/(s + k)
  3. C0/(s − k)
  4. k/(s + C0)
Answer and working

A. C0/(s + k)

sC − C0 + kC = 0, so C(s) = C0/(s + k), whose inverse is C0e^(−kt).

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

7 · Laplace transforms · medium

What is the Laplace transform of sin(at)?

  1. s/(s² + a²)
  2. a/(s² + a²)
  3. 1/(s − a)
  4. a/(s² − a²)
Answer and working

B. a/(s² + a²)

L{sin at} = a/(s² + a²); s/(s² + a²) is cos at.

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

8 · Laplace transforms · hard

What is the Laplace transform of t²?

  1. 2/s²
  2. 2/s³
  3. 1/s³
  4. t²/s
Answer and working

B. 2/s³

L{tⁿ} = n!/sⁿ⁺¹; for n = 2 this is 2!/s³ = 2/s³. 2/s² uses sⁿ instead of sⁿ⁺¹, and 1/s³ forgets the factorial.

Further reading: See the manual sources: Kreyszig; Zar; Gilbert; Gomez & Gomez; Chow et al.; Magurran

9 · Risk and reliability; return periods and exceedance probability · medium

What is the probability that a 100-year flood occurs at least once during a 50-year design life, assuming independent years?

  1. About 1 per cent
  2. About 100 per cent
  3. About 50 per cent
  4. About 39 per cent
Answer and working

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

10 · Risk and reliability; return periods and exceedance probability · hard

What return period must a structure be designed for to keep the risk of exceedance to 10 per cent over a 50-year life?

  1. About 5 years
  2. About 50 years
  3. About 475 years
  4. About 500 years
Answer and working

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

Ten here, 30 in the app. The Laisha app has every practice question we have written to NCEES FE Environmental 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.

Free for ten days, then one subscription of US$10 a month. We would rather say so here than have you find it later.

Practise the full set in Laisha →

Read the subject

Mathematics & Statistics for Environmental ScienceThe full study manual, free NCEES FE EnvironmentalFormat, sections and how it has changed

More NCEES FE Environmental practice

Air & Noise Pollution: Monitoring & Control

Mathematics & Statistics for Environmental Science for other exams

GATE · IndiaUGC NET · IndiaGATE · India

Ask Laisha about this

Opens Laisha's chat with your question ready to send. Sign in or start free.