Skip to content
Parth Ajmera

GATE BT 2026 — Solved Paper

#gate-bt#2026#past-paper#solved
1.0

Equations

AllVery important (14)

Pigeonhole principle

In notes →
at least one group has≥⌈nk⌉items\text{at least one group has} \ge \left\lceil \frac{n}{k} \right\rceil \text{items}
  • nn — total number of items
  • kk — number of groups the items are split into

Permutations (ordered selection, no repetition)

In notes →
P(n,r)=n!(n−r)!P(n,r) = \frac{n!}{(n-r)!}
  • nn — total number of items available
  • rr — number chosen and arranged (order matters)

Inclusion–exclusion (two sets)

In notes →
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|
  • ∣A∣|A|, ∣B∣|B| — count of items with property AA, with property BB
  • ∣A∩B∣|A \cap B| — count of items with both properties

External tangency of two circles

In notes →
∣O1O2∣=r1+r2|O_1O_2| = r_1 + r_2
  • O1,O2O_1, O_2 — centers of the two circles
  • r1,r2r_1, r_2 — their radii (consistent length units)

Eigenvalues of a matrix (characteristic equation)

In notes →
det⁡(A−λI)=0\det(A - \lambda I) = 0
  • AA — the square matrix
  • λ\lambda — an eigenvalue
  • II — the identity matrix of the same size as AA

PI controller transfer function

In notes →
GC(s)=KC(1+1τIs)G_C(s) = K_C\left(1 + \frac{1}{\tau_I s}\right)
  • KCK_C — controller gain
  • τI\tau_I — integral time constant (s\text{s} or min\text{min})
  • ss — Laplace variable

Catalytic efficiency

Very important
In notes →
Catalytic efficiency=kcatKM\text{Catalytic efficiency} = \frac{k_{cat}}{K_M}
  • kcatk_{cat} — turnover number (s−1\text{s}^{-1})
  • KMK_M — Michaelis constant (mol L−1\text{mol L}^{-1})

Characteristic equation (2nd-order linear ODE)

In notes →
d2ydx2+pdydx+qy=0,y=eAx  ⇒  A2+pA+q=0\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0, \quad y=e^{Ax} \;\Rightarrow\; A^2+pA+q=0
  • p,qp, q — constant coefficients of the ODE
  • AA — a root of the characteristic equation (an exponent of a valid solution)

Mean of an arithmetic series

In notes →
xˉ=a1+an2\bar{x} = \frac{a_1+a_n}{2}
  • a1a_1 — first term of the series
  • ana_n — last term of the series

Turnover number (k_cat)

Very important
In notes →
kcat=Vmax[E]totalk_{cat} = \frac{V_{max}}{[E]_{total}}
  • VmaxV_{max} — maximum reaction velocity
  • [E]total[E]_{total} — total enzyme concentration

Viable cell count from a dilution plate (CFU/mL)

In notes →
CFU mL−1=colonies countedvolume plated (mL)×dilution factor\text{CFU mL}^{-1} = \frac{\text{colonies counted}}{\text{volume plated (mL)}} \times \text{dilution factor}
  • dilution factor\text{dilution factor} — reciprocal of the dilution fraction (e.g. 100 for a 100-fold dilution)

Overall reaction free energy from half-reactions

Very important
In notes →
ΔGrxn=ΔGdonor (oxidation)+ΔGacceptor (reduction)\Delta G_{rxn} = \Delta G_{donor\,(oxidation)} + \Delta G_{acceptor\,(reduction)}
  • ΔGdonor\Delta G_{donor} — the electron-donor half-reaction, written as an oxidation
  • ΔGacceptor\Delta G_{acceptor} — the electron-acceptor half-reaction, written as a reduction

Specific activity

In notes →
Specific activity=Activity (IU)mass of enzyme\text{Specific activity} = \frac{\text{Activity (IU)}}{\text{mass of enzyme}}
  • IU\text{IU} — international unit, 1 µmol of substrate converted per minute

First-order process step response

Very important
In notes →
y(t)=MKp(1−e−t/τp)y(t) = MK_p\left(1-e^{-t/\tau_p}\right)
  • MM — step input magnitude
  • KpK_p — process gain
  • τp\tau_p — process time constant

Empirical formula from elemental composition

In notes →
ni=massiatomic weighti,subscripti=ninreferencen_i = \frac{\text{mass}_i}{\text{atomic weight}_i}, \qquad \text{subscript}_i = \frac{n_i}{n_{reference}}
  • nin_i — moles of element ii per basis mass
  • nreferencen_{reference} — moles of the element chosen to normalize against (usually N)

Determinant of a 3×3 matrix (cofactor expansion)

In notes →
det⁡[abcdefghi]=a(ei−fh)−b(di−fg)+c(dh−eg)\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = a(ei-fh)-b(di-fg)+c(dh-eg)
  • a,b,c,d,e,f,g,h,ia,b,c,d,e,f,g,h,i — matrix entries, read row by row

Newtonian fluid constitutive law

Very important
In notes →
τ=μγ˙\tau = \mu\dot\gamma
  • τ\tau — shear stress
  • μ\mu — viscosity
  • γ˙\dot\gamma — shear rate

Reynolds number

Very important
In notes →
Re=ρvDμRe = \frac{\rho v D}{\mu}
  • ρ\rho — fluid density
  • vv — velocity
  • DD — pipe diameter
  • μ\mu — viscosity

Equilibrium constant from Gibbs free energy

Very important
In notes →
Keq=e−ΔG/RTK_{eq} = e^{-\Delta G/RT}
  • ΔG\Delta G — standard Gibbs free energy change
  • RR — gas constant
  • TT — absolute temperature

Trapezoidal rule (numerical integration)

In notes →
∫abf(x) dx≈h2[f(x0)+2∑i=1n−1f(xi)+f(xn)]\int_a^b f(x)\,dx \approx \frac{h}{2}\Big[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\Big]
  • hh — step size between points
  • x0,…,xnx_0,\dots,x_n — equally spaced points from aa to bb

Condition for two lines to be parallel

In notes →
A1A2=B1B2for A1x+B1y=C1, A2x+B2y=C2\frac{A_1}{A_2} = \frac{B_1}{B_2} \quad \text{for } A_1x+B_1y=C_1,\ A_2x+B_2y=C_2
  • A1,B1,C1A_1,B_1,C_1 — coefficients of the first line
  • A2,B2,C2A_2,B_2,C_2 — coefficients of the second line

Circle equation and distance between two points

In notes →
(x−a)2+(y−b)2=r2,d=(x2−x1)2+(y2−y1)2(x-a)^2+(y-b)^2=r^2, \qquad d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
  • (a,b)(a,b) — circle center
  • rr — circle radius

Newton's law of cooling

Very important
In notes →
T(t)=Tenv+(T0−Tenv)e−ktT(t) = T_{env} + (T_0-T_{env})e^{-kt}
  • TenvT_{env} — ambient/final temperature
  • T0T_0 — initial temperature
  • kk — cooling rate constant

Monod kinetics with endogenous decay

Very important
In notes →
μnet=μmaxSKS+S−kd\mu_{net} = \mu_{max}\frac{S}{K_S+S} - k_d
  • μmax\mu_{max} — maximum specific growth rate
  • KSK_S — Monod (half-saturation) constant
  • SS — limiting substrate concentration
  • kdk_d — endogenous decay rate

Mean cell retention time (MCRT)

Very important
In notes →
θc=V XQ Xe\theta_c = \frac{V\,X}{Q\,X_e}
  • VV — reactor volume
  • XX — in-reactor biomass concentration
  • QQ — flow rate of the stream leaving the system
  • XeX_e — biomass concentration in that leaving stream

Degree of reduction (electron equivalents)

In notes →
γ=4nC+nH−2nO\gamma = 4n_C + n_H - 2n_O
  • nC,nH,nOn_C, n_H, n_O — number of carbon, hydrogen, oxygen atoms per molecule
  • γ\gamma — electron equivalents per mole

First-order thermal death (sterilization design)

Very important
In notes →
N(t)=N0 e−kdtN(t) = N_0\,e^{-k_dt}
  • N0N_0 — initial viable count
  • kdk_d — death rate constant
  • tt — time

Zero-order reaction kinetics

Very important
In notes →
C(t)=C0−kt,tcomplete=C0kC(t) = C_0 - kt, \qquad t_{complete} = \frac{C_0}{k}
  • C0C_0 — initial concentration
  • kk — zero-order rate constant

Chemostat washout condition

Very important
In notes →
Dcrit=μmaxS0KS+S0D_{crit} = \mu_{max}\frac{S_0}{K_S+S_0}
  • DcritD_{crit} — critical (washout) dilution rate
  • S0S_0 — feed substrate concentration

Exponential growth by doubling

In notes →
N(t)=N0×2t/tdN(t) = N_0 \times 2^{t/t_d}
  • N0N_0 — starting cell number
  • tdt_d — doubling time

Superficial vs. interstitial velocity (packed bed)

Very important
In notes →
vinterstitial=vsuperficialε,vsuperficial=QAv_{interstitial} = \frac{v_{superficial}}{\varepsilon}, \qquad v_{superficial}=\frac{Q}{A}
  • ε\varepsilon — bed void fraction
  • QQ — volumetric flow rate
  • AA — empty-column cross-sectional area
31 formulas