List of practice Questions

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).

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?

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?

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?

Which of the following functions represents a cumulative distribution function?

A cutting tool provides a tool life of 60 minutes while machining with the cutting speed of 60 m/min. When the same tool is used for machining the same material, it provides a tool life of 10 minutes for a cutting speed of 100 m/min. If the cutting speed is changed to 80 m/min for the same tool and work material combination, the tool life computed using Taylor's tool life model is ____minutes (rounded off to 2 decimal places).

If the plotting accuracy of a map is 0.25 mm and the scale of the same map is 1:100000, what will be the minimum ground distance that can be plotted on the map?

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.

The consolidated data of a spot speed study for a certain stretch of a highway is given in the table.

Speed Range (kmph)	Number of observations
0-10	7
10-20	31
20-30	76
30-40	129
40-50	104
50-60	78
60-70	29
70-80	24
80-90	13
90-100	9

The "upper speed limit" (in kmph) for the traffic sign is

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

A horizontal curve of radius 1080 m (with transition curves on either side) in a Broad Gauge railway track is designed and constructed for an equilibrium speed of 70 kmph. However, a few years after construction, the Railway Authorities decided to run express trains on this track. The maximum allowable cant deficiency is 10 cm.
The maximum restricted speed (in kmph) of the express trains running on this track is __________ (rounded off to the nearest integer).

Mass of the moon is \(\frac{1}{100}\) times the mass of a planet. Its diameter is \(\frac{1}{16}\) the diameter of the planet if the escape velocity of the planet is \(V\) then the escape velocity of the moon will be

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)

Consider a joint probability density function of two random variables X and Y
\(f_{x,y}(x,y) = \left\{ \begin{array}{ll} 2xy, & 0\lt x \lt2 , & 0\lt y\lt x\\ 0 & \text{otherwise } \end{array} \right.\)
Then, E[Y | X = 1.5] is ______.

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 2 m × 1.5 m tank of 6 m height is provided with a 100 mm diameter orifice at the center of its base. The orifice is plugged and the tank is filled up to 5 m height. Consider the average value of discharge coefficient as 0.6 and acceleration due to gravity (g) as 10 m/s². After unplugging the orifice, the time (in seconds) taken for the water level to drop from 5 m to 3.5 m under free discharge condition is _________ (rounded off to 2 decimal places).

Four equal masses m are kept at corners of a square of side a. If net gravitational force on a mass is given by \(\bigg(\frac{2\sqrt2+1}{32}\bigg) \frac{GM^2}{L^2}\) Find the value of a in the terms of \(L\).

If \(f(x)=\frac {4x+3}{6x-4}\), \(x≠\frac 23 \)and \((fof)(x)=g(x)\), where \(g:R-[\frac 23→R→{\frac 23}]\). Then \((gogog)(4)\) is equal to

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?