NCEES FE Other Disciplines · USA · NCEES
Original practice questions written to the published syllabus · not past papersThe algebra, calculus, differential equations and numerical methods used in every numerical question of the degree. GATE AG gives it a fixed block of marks every year.
In the exams: GATE AG Section 1, about 12–13 marks a year (2019–26); ICAR AIEEA-PG Unit I; NCEES FE 8–12 questions; allied subject in PRC ABE
NCEES FE Other Disciplines
About 8–12 mathematics questions answered with the NCEES FE Reference Handbook available on screen. Formulas are given, so the test is choosing and applying the right one quickly: derivatives and integrals, vector operations, differential equations, matrix operations and numerical methods such as Newton's method and the trapezoidal rule. Probability and statistics are a separate section of the exam.
How our questions for NCEES FE Other Disciplines spread across the units of Engineering Mathematics. Difficulty: 22 easy, 24 medium, 12 hard.
| Higher-order linear ordinary differential equations | 5 | |
| Laplace transforms | 5 | |
| Numerical solution of equations: bisection and Newton-Raphson | 5 | |
| Numerical integration: trapezoidal and Simpson's rules | 5 | |
| Limits, continuity and partial derivatives | 4 | |
| Vector calculus: gradient, divergence and curl | 4 | |
| Line, surface and volume integrals: Green, Stokes and Gauss theorems | 4 | |
| Partial differential equations: Laplace, heat and wave equations | 4 | |
| Numerical solution of ODEs: Euler and Runge-Kutta methods | 4 | |
| Matrices, determinants and systems of linear equations | 3 | |
| Maxima and minima of functions of several variables | 3 | |
| Sequences, series and convergence | 3 | |
| Taylor, Maclaurin and Fourier series | 3 | |
| First-order ordinary differential equations | 3 | |
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 2 | |
| Euler's theorem on homogeneous functions | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What is the inverse of the 2×2 matrix [[4, 7], [2, 6]]?
C. (1/10)[[6, −7], [−2, 4]]
det = 4·6 − 7·2 = 10. For [[a, b], [c, d]] the inverse is (1/det)[[d, −b], [−c, a]]: swap the diagonal and negate the off-diagonal. Forgetting to negate b and c gives [[6, 7], [2, 4]]; not swapping a and d gives [[4, −7], [−2, 6]]; adding instead of subtracting in the determinant gives 38.
Further reading: Kreyszig, Advanced Engineering Mathematics
A is a 3×3 matrix with det(A) = 5. What is det(2A⁻¹)?
D. 8/5
det(kA) = kⁿ det(A) for an n×n matrix, and det(A⁻¹) = 1/det(A). So det(2A⁻¹) = 2³ × (1/5) = 8/5. Using k instead of k³ gives 2/5; 40 is det(2A), forgetting the inverse; 10 multiplies 2 by 5.
Further reading: Grewal, Higher Engineering Mathematics
z = x²y with x = t² and y = t³. What is dz/dt at t = 1?
A. 7
Chain rule: dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) = (2xy)(2t) + (x²)(3t²). At t = 1, x = y = 1: 2·2 + 1·3 = 7. Check: z = t⁷, dz/dt = 7t⁶ = 7. Taking only one path gives 4 or 3; 5 and 6 come from adding exponents wrongly.
Further reading: Kreyszig, Advanced Engineering Mathematics
The radius of a circular centre-pivot field is measured as 50 m with a possible error of 0.5 m. Using differentials, what is the approximate percentage error in the computed area?
D. 2%
A = πr², so dA/A = 2 dr/r = 2 × 0.5/50 = 0.02 = 2%. The relative error of rⁿ is n times the relative error of r. 1% forgets the factor 2 from the square; 4% squares the 2 as well.
Further reading: Grewal, Higher Engineering Mathematics
The p-series Σ 1/nᵖ (n = 1 to ∞) converges if and only if
C. p > 1
By the integral test, ∫ dx/xᵖ from 1 to ∞ is finite only for p > 1. At p = 1 it is the harmonic series, which diverges, so p ≥ 1 is wrong. p > 0 only makes the terms tend to zero, which is not enough.
Further reading: Kreyszig, Advanced Engineering Mathematics
Evaluate ∫₀¹ ∫₀ˣ xy dy dx.
A. 1/8
Inner: ∫₀ˣ xy dy = x·x²/2 = x³/2. Outer: ∫₀¹ x³/2 dx = 1/8. 1/4 comes from integrating y over 0 to 1 (wrong inner limit), giving ∫ x/2 dx = 1/4.
Further reading: Kreyszig, Advanced Engineering Mathematics
What is a particular integral of y'' − 3y' + 2y = e^(3x)?
B. e^(3x)/2
For e^(ax) with f(a) ≠ 0, PI = e^(ax)/f(a). f(m) = m² − 3m + 2, f(3) = 9 − 9 + 2 = 2, so PI = e^(3x)/2. 1/20 uses 9 + 9 + 2 (sign slip on −3m).
Further reading: Grewal, Higher Engineering Mathematics
What is the Laplace transform of t²?
D. 2/s³
L{tⁿ} = n!/s^(n+1). For n = 2: 2!/s³ = 2/s³. 1/s³ forgets the n!; 2/s² uses sⁿ instead of s^(n+1).
Further reading: Kreyszig, Advanced Engineering Mathematics
Using the trapezoidal rule with h = 0.5, what is ∫₀¹ x² dx?
C. 0.375
Points x = 0, 0.5, 1 → f = 0, 0.25, 1. T = (h/2)[f₀ + 2f₁ + f₂] = 0.25 × (0 + 0.5 + 1) = 0.375 (exact 1/3; the rule over-estimates a convex function). 0.750 forgets the h/2 and uses h; 0.3333 is the exact value, which Simpson's rule would give.
Further reading: Chapra and Canale, Numerical Methods for Engineers
A storm hydrograph at a watershed outlet is read every hour: 0, 5, 12, 8, 3 and 0 m³/s. Using the trapezoidal rule, what is the runoff volume?
A. 100,800 m³
h = 1 h = 3,600 s. V = (h/2)[Q₀ + Q₅ + 2(Q₁ + Q₂ + Q₃ + Q₄)] = 1,800 × 2 × 28 = 100,800 m³. Forgetting to convert hours to seconds gives 28; converting 1 h to 60 s gives 1,680; using h instead of h/2 doubles the result to 201,600 m³.
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