GATE AG cheat sheet

Soil water content on mass basis: θm = (Ww − Wd)/Wd; on volume basis: θv = θm × (bulk density / water density).

Bulk density ρb = oven-dry soil mass / total volume; particle density ρs ≈ 2.65 g/cm³ for most mineral soils; porosity = 1 − ρb/ρs.

Infiltration is described by Kostiakov (i = a·t^n) or Philip (i = S·t^0.5 + A·t); Horton uses exponential decay from initial to final rate.

Reference evapotranspiration by Penman–Monteith: ET0 = [0.408Δ(Rn−G) + γ(900/(T+273))u2(es−ea)] / [Δ + γ(1+0.34u2)].

Irrigation water requirement = ETc − effective rainfall; ETc = Kc × ET0, where Kc is the crop coefficient for the growth stage.

Drainage spacing by Hooghoudt: S² = (4Ka·h² + 8Ka·d·h) / q, with q the drain discharge rate and d the equivalent depth to the impermeable layer.

Tractor power take-off power = (2π N T)/60, with N in rpm and torque T in N·m, giving watts; drawbar power = draft × forward speed.

Sieve analysis and Stokes' law give settling velocity v = (ρs − ρf)·g·d²/(18μ), the basis for sedimentation and soil particle-size analysis.

Boolean identity set: A + A'B = A + B; De Morgan: (A·B)' = A' + B', (A + B)' = A'·B'.

Op-amp inverting gain = −Rf/Rin; non-inverting gain = 1 + Rf/R1; ideal op-amp assumes zero input current and zero differential input voltage.

Successive-approximation ADC resolves 1 LSB = Vref/2^n; conversion time ≈ n clock periods for n bits.

Nyquist criterion: sample at more than twice the highest signal frequency to avoid aliasing; anti-alias filter before the sampler.

Wheatstone bridge balances when R1/R2 = R3/R4; a strain gauge in one arm gives ΔV/V ≈ (ΔR/R)/4 for small changes.

PID controller output u = Kp·e + Ki∫e dt + Kd·(de/dt); integral action removes steady-state offset, derivative action adds damping.

Sensor linearity, sensitivity, resolution, hysteresis, and drift are the five static characteristics used to specify an instrument.

In a data-acquisition chain, total error is combined as root-sum-square of individual uncertainty components when they are independent.

Roof pitch and overhang control rain shedding: minimum slope for metal sheeting is typically 1 in 5 (about 11°), steeper for corrugated asbestos or tiles.

Ventilation rate for livestock housing is set per animal: ridge opening area ≈ 1/5 of total inlet area for natural cross-ventilation design.

Psychrometric chart: dry-bulb, wet-bulb, dew point, relative humidity, and enthalpy are read together; RH = 100% at saturation.

Structural design uses limit-state checks: factored load effect ≤ design resistance; dead, live, wind, and snow loads are combined per code.

Concrete mix by volume 1:2:4 (cement:sand:aggregate) gives roughly M15 grade; water-cement ratio governs strength and durability.

Grain storage structures must keep moisture below the safe limit (typically 12–14% for cereals) and control temperature to prevent mould.

Greenhouse orientation: a north–south ridge maximises winter light capture; east–west is better for summer cooling in hot climates.

Soil pH affects nutrient availability; most crops grow best at pH 6.0–7.0, with micronutrients like iron and zinc less available above pH 7.

Cation exchange capacity (CEC) is measured in cmol(+)/kg; higher CEC means greater nutrient-holding capacity and buffering.

Law of the minimum: growth is limited by the nutrient in shortest supply relative to requirement, not by total nutrient abundance.

Photosynthesis light response: net assimilation = gross assimilation − respiration; light compensation point occurs when the two are equal.

Nitrogen fixation by legumes requires the nitrogenase enzyme, which is inhibited by oxygen and by high soil nitrate.

Standard deviation of the mean: SE = s/√n; a 95% confidence interval uses t or 1.96 × SE depending on sample size.

Coefficient of variation CV = (standard deviation / mean) × 100%; it compares variability across datasets with different units.

Randomised complete block design removes field gradient effects by grouping plots into blocks; analysis uses two-way ANOVA.