GATE — Engineering Sciences (XE), Section G Food Technology · India · IITs and IISc on rotation (2026: IIT Guwahati)
Original practice questions written to the published syllabus · not past papersThe calculus, linear algebra, differential equations and numerical methods behind every food engineering calculation: heat penetration curves, drying rates, reaction kinetics and process optimisation. GATE XE makes this section compulsory for every candidate who takes the Food Technology section.
In the exams: GATE XE-A (compulsory section for XE candidates); foundation for every numerical question in GATE XE-G Section 4 and ICAR AIEEA-PG Code 18 Unit II
How our questions for GATE spread across the units of Mathematics for Food Science & Engineering. Difficulty: 28 easy, 43 medium, 9 hard.
| Matrices, determinants, rank and systems of linear equations | 7 | |
| Limits, continuity, mean value theorems and Taylor's theorem | 7 | |
| Partial derivatives, maxima and minima and Lagrange multipliers | 7 | |
| Definite, improper and double integrals and their applications | 7 | |
| Sequences, series, power series and Fourier series | 7 | |
| First-order and second-order ordinary differential equations | 7 | |
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 6 | |
| Vector calculus: gradient, divergence, curl and Green's theorem | 6 | |
| Complex numbers, analytic functions and Cauchy-Riemann equations | 6 | |
| Numerical methods: Gauss elimination, LU decomposition, interpolation and Newton-Raphson | 6 | |
| Partial differential equations: one-dimensional heat equation and Laplace equation by separation of variables | 5 | |
| Numerical integration and Euler's method for ODEs | 5 | |
| Applications in food science: first-order kinetics, exponential decay and logarithmic survivor curves | 4 |
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 coefficient of x³ in the Maclaurin series of eˣ?
A. 1/6
eˣ = 1 + x + x²/2! + x³/3! + …, so the x³ coefficient is 1/3! = 1/6. 1/3 omits the factorial; 1 treats every coefficient as 1; 1/24 is the x⁴ coefficient.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
For f(x, y) = x²y + 3y², what is ∂f/∂y?
D. x² + 6y
Holding x constant: ∂(x²y)/∂y = x² and ∂(3y²)/∂y = 6y, giving x² + 6y. 2xy is ∂f/∂x; 2xy + 6y mixes the two partials; x² + 3y forgets to differentiate 3y² properly.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
What is the value of the integral of e^(−2x) from 0 to infinity?
B. 0.5
∫₀^∞ e^(−ax) dx = 1/a, so with a = 2 the value is 0.5. 2 is a rather than 1/a; 1 is the case a = 1; the integral converges because the exponent is negative.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
What is the sum of the series 1 + 1/2 + 1/4 + 1/8 + …?
B. 2
A geometric series with a = 1 and r = 1/2 sums to a/(1 − r) = 1/(1/2) = 2. 1 stops at the first term; 1.5 at the second; a geometric series with |r| < 1 converges.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
What is the radius of convergence of the power series Σ xⁿ/3ⁿ?
D. 3
aₙ = 3^(−n), and R = lim|aₙ/aₙ₊₁| = 3. Equivalently it is a geometric series in x/3, converging for |x/3| < 1, i.e. |x| < 3. 1/3 inverts the ratio; 1 is the radius for Σxⁿ; ∞ is for eˣ.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
What is the integrating factor for dy/dx + 2y/x = x?
B. x²
For dy/dx + P(x)y = Q(x), the integrating factor is e^(∫P dx) = e^(2 ln x) = x². e^(2x) treats P as the constant 2; 2 ln x is the exponent, not the factor; 1/x² has the sign of the exponent reversed.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
Which is the condition for M(x, y)dx + N(x, y)dy = 0 to be an exact equation?
C. ∂M/∂y = ∂N/∂x
The equation is exact if there is a function ψ with ψ_x = M and ψ_y = N, which requires ψ_xy = ψ_yx, i.e. ∂M/∂y = ∂N/∂x. The other conditions are not the exactness test; the reversed-sign version resembles a Cauchy-Riemann equation.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
For the same data (y = 1, 4, 9, 16, 25 at x = 1 to 5, h = 1), what does Simpson's one-third rule give for the integral?
B. 41.33
Simpson: (h/3)[y₀ + 4(y₁ + y₃) + 2y₂ + y₄] = (1/3)[1 + 80 + 18 + 25] = 124/3 = 41.33, exact because Simpson's rule integrates polynomials up to degree three exactly. 42.00 is the trapezoidal result; 55.00 is the sum of ordinates; 41.00 rounds the weights wrongly.
Further reading: Chapra and Canale, Numerical Methods for Engineers; Kreyszig, Advanced Engineering Mathematics
How does the global error of Euler's method change when the step size is halved?
D. It roughly halves
Euler's method is first order: global error ∝ h, so halving h roughly halves it. Quartering corresponds to a second-order method such as Heun's; a factor of sixteen to fourth-order Runge-Kutta.
Further reading: Chapra and Canale, Numerical Methods for Engineers; Kreyszig, Advanced Engineering Mathematics
A can holds 10⁴ spores of a spoilage organism with D₁₂₁ = 1.5 min. What F₀ gives a probability of one surviving spore per 10⁵ cans?
B. 13.5 min
From 10⁴ to 10⁻⁵ per can is 9 log reductions: F = 9 × 1.5 = 13.5 min. 6.0 min is only the 4 logs to one spore per can; 7.5 min uses 5 logs; 15.0 min uses 10 logs.
Further reading: Singh and Heldman, Introduction to Food Engineering; Toledo, Fundamentals of Food Process Engineering
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