Parth Ajmera

Q3 — 1 mark, MCQ

\text{at least one group has} \ge \left\lceil \frac{n}{k} \right\rceil \text{items}

$n$ — total number of items
$k$ — number of groups the items are split into

Q4 — 1 mark, MCQ

P(n,r) = \frac{n!}{(n-r)!}

$n$ — total number of items available
$r$ — number chosen and arranged (order matters)

Q5 — 1 mark, MCQ

|A \cup B| = |A| + |B| - |A \cap B|

$|A|$ , $|B|$ — count of items with property $A$ , with property $B$
$|A \cap B|$ — count of items with both properties

Q10 — 2 marks, MCQ

|O_1O_2| = r_1 + r_2

$O_1, O_2$ — centers of the two circles
$r_1, r_2$ — their radii (consistent length units)

Q12 — 1 mark, MCQ

\det(A - \lambda I) = 0

$A$ — the square matrix
$\lambda$ — an eigenvalue
$I$ — the identity matrix of the same size as $A$

Q15 — 1 mark, MCQ

G_C(s) = K_C\left(1 + \frac{1}{\tau_I s}\right)

$K_C$ — controller gain
$\tau_I$ — integral time constant ( $\text{s}$ or $\text{min}$ )
$s$ — Laplace variable

Q21 — 1 mark, MCQ

Very important

\text{Catalytic efficiency} = \frac{k_{cat}}{K_M}

$k_{cat}$ — turnover number ( $\text{s}^{-1}$ )
$K_M$ — Michaelis constant ( $\text{mol L}^{-1}$ )

Q27 — 1 mark, MSQ

\frac{d^2y}{dx^2} + p\frac{dy}{dx} + qy = 0, \quad y=e^{Ax} \;\Rightarrow\; A^2+pA+q=0

$p, q$ — constant coefficients of the ODE
$A$ — a root of the characteristic equation (an exponent of a valid solution)

Q31 — 1 mark, NAT

\bar{x} = \frac{a_1+a_n}{2}

$a_1$ — first term of the series
$a_n$ — last term of the series

Q32 — 1 mark, NAT

Very important

k_{cat} = \frac{V_{max}}{[E]_{total}}

$V_{max}$ — maximum reaction velocity
$[E]_{total}$ — total enzyme concentration

Q33 — 1 mark, NAT

\text{CFU mL}^{-1} = \frac{\text{colonies counted}}{\text{volume plated (mL)}} \times \text{dilution factor}

$\text{dilution factor}$ — reciprocal of the dilution fraction (e.g. 100 for a 100-fold dilution)

Q34 — 1 mark, NAT

Very important

\Delta G_{rxn} = \Delta G_{donor\,(oxidation)} + \Delta G_{acceptor\,(reduction)}

$\Delta G_{donor}$ — the electron-donor half-reaction, written as an oxidation
$\Delta G_{acceptor}$ — the electron-acceptor half-reaction, written as a reduction

Q35 — 1 mark, NAT

\text{Specific activity} = \frac{\text{Activity (IU)}}{\text{mass of enzyme}}

$\text{IU}$ — international unit, 1 µmol of substrate converted per minute

Q36 — 2 marks, MCQ

Very important

y(t) = MK_p\left(1-e^{-t/\tau_p}\right)

$M$ — step input magnitude
$K_p$ — process gain
$\tau_p$ — process time constant

Q40 — 2 marks, MCQ

n_i = \frac{\text{mass}_i}{\text{atomic weight}_i}, \qquad \text{subscript}_i = \frac{n_i}{n_{reference}}

$n_i$ — moles of element $i$ per basis mass
$n_{reference}$ — moles of the element chosen to normalize against (usually N)

Q41 — 2 marks, MSQ

\det\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix} = a(ei-fh)-b(di-fg)+c(dh-eg)

$a,b,c,d,e,f,g,h,i$ — matrix entries, read row by row

Q45 — 2 marks, MSQ

Very important

\tau = \mu\dot\gamma

$\tau$ — shear stress
$\mu$ — viscosity
$\dot\gamma$ — shear rate

Q47 — 2 marks, MSQ

Very important

Re = \frac{\rho v D}{\mu}

$\rho$ — fluid density
$v$ — velocity
$D$ — pipe diameter
$\mu$ — viscosity

Q50 — 2 marks, MSQ

Very important

K_{eq} = e^{-\Delta G/RT}

$\Delta G$ — standard Gibbs free energy change
$R$ — gas constant
$T$ — absolute temperature

Q51 — 2 marks, NAT

\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]

$h$ — step size between points
$x_0,\dots,x_n$ — equally spaced points from $a$ to $b$

Q52 — 2 marks, NAT

\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

$A_1,B_1,C_1$ — coefficients of the first line
$A_2,B_2,C_2$ — coefficients of the second line

Q53 — 2 marks, NAT

(x-a)^2+(y-b)^2=r^2, \qquad d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

$(a,b)$ — circle center
$r$ — circle radius

Q54 — 2 marks, NAT

Very important

T(t) = T_{env} + (T_0-T_{env})e^{-kt}

$T_{env}$ — ambient/final temperature
$T_0$ — initial temperature
$k$ — cooling rate constant

Q55 — 2 marks, NAT

Very important

\mu_{net} = \mu_{max}\frac{S}{K_S+S} - k_d

$\mu_{max}$ — maximum specific growth rate
$K_S$ — Monod (half-saturation) constant
$S$ — limiting substrate concentration
$k_d$ — endogenous decay rate

Q56 — 2 marks, NAT

Very important

\theta_c = \frac{V\,X}{Q\,X_e}

$V$ — reactor volume
$X$ — in-reactor biomass concentration
$Q$ — flow rate of the stream leaving the system
$X_e$ — biomass concentration in that leaving stream

Q57 — 2 marks, NAT

\gamma = 4n_C + n_H - 2n_O

$n_C, n_H, n_O$ — number of carbon, hydrogen, oxygen atoms per molecule
$\gamma$ — electron equivalents per mole

Q59 — 2 marks, NAT

Very important

N(t) = N_0\,e^{-k_dt}

$N_0$ — initial viable count
$k_d$ — death rate constant
$t$ — time

Q60 — 2 marks, NAT

Very important

C(t) = C_0 - kt, \qquad t_{complete} = \frac{C_0}{k}

$C_0$ — initial concentration
$k$ — zero-order rate constant

Q61 — 2 marks, NAT

Very important

D_{crit} = \mu_{max}\frac{S_0}{K_S+S_0}

$D_{crit}$ — critical (washout) dilution rate
$S_0$ — feed substrate concentration

Q63 — 2 marks, NAT

N(t) = N_0 \times 2^{t/t_d}

$N_0$ — starting cell number
$t_d$ — doubling time

Q65 — 2 marks, NAT

Very important

v_{interstitial} = \frac{v_{superficial}}{\varepsilon}, \qquad v_{superficial}=\frac{Q}{A}

$\varepsilon$ — bed void fraction
$Q$ — volumetric flow rate
$A$ — empty-column cross-sectional area