List of practice Questions

Two parallel first-order liquid phase reactions \( A \xrightarrow{k_1} B \) and \( A \xrightarrow{k_2} C \) are carried out in a well-mixed isothermal batch reactor. The initial concentration of A in the reactor is \( 1 \) kmol m\(^{-3}\), while that of B and C is zero. After 2 hours, the concentration of A reduces to half its initial value, and the concentration of B is twice that of C. The rate constants \( k_1 \) and \( k_2 \), in h\(^{-1}\), are, respectively,

Let \( p(\bar{p}, \bar{q}, t) \) be the phase space density of an ensemble of a system. The Hamiltonian of the system is \( H(p, q) \). If \( \{A, B\} \) denotes the Poisson bracket of \( A \) and \( B \), then \( \frac{dp}{dt} = 0 \) implies

Which one of the following metabolic intermediates is common to glycolysis, nucleotide synthesis and glycogen synthesis?

The approximate value of the integral \( \int_{2}^{3} \frac{dx}{x} \) using Simpson's rule with \( h = 0.5 \) is

Consider the following ¹H-NMR (400 MHz, DMSO-d₆) data of a compound :
δ in ppm: 3.85 (s, 6H), 6.73 (t, J = 2.2 Hz, 1H), 7.1 (d, J = 2.2 Hz, 2H), and 13.05 (brs, 1H).
The compound is

GDP_F = Gross Domestic Product at Factor Cost; GDP_M = Gross Domestic Product at Market Price; NNP_F= Net National Product at Factor Cost; C = Consumption; I = Investment; G = Government Expenditure; X = Export; M = Import; T = Tax; S = Saving; D = Depreciation; NIA = Net Income from Abroad
Which of the following expressions is/are CORRECT?

Using the following table,

Year	Population of the Economy	GDP of the Economy (in crore)
2010	20,000	25,000
2020	25,000	40,000

the average growth rate (compounded annually) of per capita GDP in an economy during the period 2010-2020 is ________ (in percent, round off to 2 decimal places).

Consider a Keynesian Cross Model with following features,
Consumption Function: C = C0 + b(Y-T) 
Tax Function: T = T0 + tY 
Income Identity: Y = C + I0 + G0 
Where, C = Consumption; Y = Real Income; T = Tax; I = Investment; 
G = Government Expenditure; b = Parameter; t = Tax Rate 
(The subscript 0 (zero) indicates that the concerned variable is autonomous) 
If b = 0.7 and t = 0.2, value of the Keynesian multiplier is ________ (round off to 2 decimal places).

The co-ordinate system (x,y,z) is transformed to the system (u,v,w), as given by:
𝑢 = 2𝑥 + 3𝑦 − 𝑧 
𝑣 = 𝑥 − 4𝑦 + 𝑧 
𝑤 = 𝑥 + 𝑦 
The Jacobian of the above transformation is ________.

Which of the following is/are example(s) of a lotic ecosystem?

Which of the following functions represents a cumulative distribution function?

Let \( X \) and \( Y \) be continuous random variables having the joint probability density function
\[ f(x, y) = \begin{cases} e^{-x}, & \text{if } 0 \leq y < x < \infty \\0, & \text{otherwise}\end{cases} \]
Then which of the following statements is/are correct?

For \( n \geq 2 \), let \( X_1, X_2, \ldots, X_n \) be a random sample from a distribution with \( E(X_1) = 0 \), \( \text{Var}(X_1) = 1 \), and \( E(X_1^4) < \infty \). Let \[ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i \] and \[ S_n^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X}_n)^2. \]
Then which of the following statements is/are always correct?

Statements: Young adults spend most of their time on social media. Young adults should spend more time making connections in the real world.
Conclusions
I. Young Adults should avoid social media.
II. Young adults cannot make real life connections because they spend most of their time on social media.
Which of the following conclusions can be properly inferred? Choose the most appropriate answer.

In an opencast coal mine, blast vibrations are measured at two locations, A and B simultaneously for a maximum charge per delay (Q) of 1200 kg as given.

Location	Distance from the blast face, D (m)	PPV (mm/s)
A	100	112.5
B	300	20.3

\(PPV=K[\frac{D}{\sqrt{Q}}]^{-B}\)
where, K and ẞ are site constants. The PPV in mm/s, at a distance of 200 m from the blast face, is (round off up to 2 decimals) 

As the package builds up in a roving frame, the component(s) whose speed DOES NOT remain constant, amongst the following, is/are

The following figure shows an experimental liquid-liquid phase diagram of phenol and water at the vapor pressure of the system. The total amount of phenol and water (in mol) present in the phenol-rich phase when 5 mol of water was shaken with 5 mol of phenol at 40 °C is _______.

(rounded off to one decimal place)

Alumina particles with an initial moisture content of 5 kg per kg dry solid are dried in a batch dryer. For the first two hours, the measured drying rate is constant at 2 kg m\(^{-2}\) h\(^{-1}\). Thereafter, in the falling-rate period, the rate decreases linearly with the moisture content. The equilibrium moisture content is 0.05 kg per kg dry solid, and the drying area of the particles is 0.5 m\(^2\) per kg dry solid. The total drying time, in hours, to reduce the moisture content to half its initial value is:

In the given reaction sequence, the amount of R produced (in g) is ______.

(Given: molar mass (in g mol^-1) of H = 1, C = 12, N = 14, O = 16, and S = 32)
(rounded off to two decimal places)

A solid ball of mass 10 kg is subjected to forces as shown. The magnitude of the acceleration in m/s², is____.(round off up to 2 decimals)

Which one of the following failure theories in the most conservative design approach against fatigue failure

A cube is to be cut into 8 pieces of equal size and shape Here each cut should be straight and it should not stop till it reaches the other end of the cube The minimum number of such cuts required?

A typical 24-hour activity of a mobile crusher plant is shown. The utilization, in %, of the plant, is (round off up to 2 decimals)

An extrinsic semiconductor shown in the figure carries a current of 2 mA along its length parallel to the \( +x \) axis.

When the majority charge carrier concentration is \( 12.5 \times 10^{13} \, \text{cm}^{-3} \) and the sample is exposed to a constant magnetic field applied along the \( +z \) direction, a Hall voltage of 20 mV is measured with the negative polarity at the \( y = 0 \) plane. Take the electric charge as \( 1.6 \times 10^{-19} \) C. The concentration of minority charge carriers is negligible. Which of the following statements is/are true?

Which one or more of the following statements is/are correct regarding the transport and retention of proteins in different cell organelles?