ICAR AICE-JRF/SRF (PhD) — Food Processing Technology, Food Safety & Quality, Processing and Food Engineering · India · ICAR through the National Testing Agency
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 ICAR AICE-JRF/SRF spread across the units of Mathematics for Food Science & Engineering. Difficulty: 0 easy, 7 medium, 8 hard.
| Numerical integration and Euler's method for ODEs | 3 | |
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 2 | |
| Partial differential equations: one-dimensional heat equation and Laplace equation by separation of variables | 2 | |
| Applications in food science: first-order kinetics, exponential decay and logarithmic survivor curves | 2 | |
| Limits, continuity, mean value theorems and Taylor's theorem | 1 | |
| Partial derivatives, maxima and minima and Lagrange multipliers | 1 | |
| Definite, improper and double integrals and their applications | 1 | |
| Sequences, series, power series and Fourier series | 1 | |
| Complex numbers, analytic functions and Cauchy-Riemann equations | 1 | |
| First-order and second-order ordinary differential equations | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
In principal component analysis of NIR spectra, the eigenvalues of the covariance matrix are 6.0, 3.0, 0.6 and 0.4. What fraction of total variance do the first two components explain?
D. 90%
Total variance is the sum of the eigenvalues (the trace): 6.0 + 3.0 + 0.6 + 0.4 = 10.0. The first two explain (6.0 + 3.0)/10 = 90%. 60% counts only the first component; 75% divides 9 by 12; 95% adds the third component.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
Why do the z-value model and the Arrhenius model of thermal death agree over a narrow temperature range but diverge over a wide one?
D. Near a reference temperature, 1/T is close to linear in T, so the two exponentials nearly coincide
Arrhenius makes ln k linear in 1/T; the z model makes log D linear in T. A first-order Taylor expansion of 1/T about a reference temperature is linear in T, so the models match locally but not over a wide range. The units are the other way round (Arrhenius needs kelvin); both describe first-order kinetics; and the predictions do diverge in principle.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
The density of a cylindrical sample is calculated as ρ = m/(πr²h). If the radius has a relative error of 1% and the mass and height are exact, what is the relative error in the density?
C. 2%
From the total differential, dρ/ρ = dm/m − 2dr/r − dh/h. The radius appears squared, so a 1% error in r gives 2% in ρ. 1% ignores the power; 0.5% halves instead of doubling; 3% adds the error as if r were cubed.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
In thermal process calculation, what does the integral of 10^((T − T_ref)/z) dt over the heating and cooling time give?
C. The lethality F, as equivalent minutes at the reference temperature
The integrand is the lethal rate, the minutes at T_ref equivalent to one minute at T. Summing it over the process gives F (F₀ when T_ref = 121.1 °C and z = 10 °C). D and z are kinetic properties of the organism, inputs to the calculation; the integral says nothing about the heat supplied.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
Why does the semi-logarithmic heat-penetration curve of a canned food become a straight line after an initial lag?
B. The higher terms of the Fourier-series solution decay much faster, leaving only the first exponential term
The conduction solution is a series of terms decaying as exp(−λₙ²αt/L²) with λₙ rising rapidly, so after a while only the first term remains, and ln(temperature difference) falls linearly. The other explanations describe effects that do not produce the straight line and are not generally true.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
A sauce cools from 90 °C to 50 °C in 10 min in a room at 20 °C, following Newton's law of cooling. How long from the start does it take to reach 30 °C?
B. 23.0 min
T − 20 = 70e^(−kt). At 10 min: 30 = 70e^(−10k), so k = ln(70/30)/10 = 0.0847 per min. For T = 30: 10 = 70e^(−kt), so t = ln 7/k = 23.0 min. 20 min assumes a linear fall; 15 min and 30 min come from using temperatures rather than temperature differences above the room.
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
A slab of food takes 40 min to reach a target centre temperature. Under the same surface conditions and properties, how long will a slab of twice the thickness take?
A. 160 min
Time enters through the Fourier number Fo = αt/L², so equal Fo needs t ∝ L². Doubling L multiplies the time by 4: 4 × 40 = 160 min. 80 min assumes time proportional to thickness; 320 min uses L³. (With a finite Biot number the scaling is only approximate, because Bi also changes with L.)
Further reading: Kreyszig, Advanced Engineering Mathematics; Grewal, Higher Engineering Mathematics
In the explicit finite-difference solution of the one-dimensional heat equation, which condition on the mesh Fourier number αΔt/Δx² keeps the scheme stable?
B. It must not exceed 1/2
For the explicit (forward-time, centred-space) scheme, stability requires αΔt/Δx² ≤ 1/2; larger time steps make the solution oscillate and grow. Exceeding 1 is unstable; 1/6 is a special value that reduces truncation error but is not the limit; 2 is far beyond the limit.
Further reading: Chapra and Canale, Numerical Methods for Engineers; Kreyszig, Advanced Engineering Mathematics
A cold-point record gives 111.1 °C, 121.1 °C and 121.1 °C at 0, 1 and 2 min (z = 10 °C, reference 121.1 °C). What F₀ does the trapezoidal rule give for these 2 min?
A. 1.55 min
Lethal rates L = 10^((T − 121.1)/10): 0.1, 1 and 1. Trapezoid with h = 1: 0.5[0.1 + 2(1) + 1] = 1.55 min. 2.00 min treats every minute as fully lethal; 1.10 min adds the rates of only two points; 1.00 min uses only the last interval.
Further reading: Chapra and Canale, Numerical Methods for Engineers; Kreyszig, Advanced Engineering Mathematics
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