List of practice Questions

Suppose the polynomial $p(x)=a+bx+cx^{2}+dx^{3}$ interpolates the data $(-1,1),\ (0,3),\ (1,2),\ (2,4)$. Then which one of the following statements is correct?

Let $C$ be the positively oriented boundary of the domain bounded by the curves $y=2x^{2}$ and $y^{2}=4x$. Then the value of the line integral \[ \oint_{C} (2y^{2}+2xy+4y)\,dx + (x^{2}+4xy+8x)\,dy \] is equal to:

Let $y(x)$ be the solution of the initial value problem, \[ x^{2}y'' + xy' - y = 0, x>0, \] with $y(1)=0$, $y'(1)=2$. Then the value of $y'\!\left(\tfrac{1}{2}\right)$ is equal to (Answer in integer):

In the context of the given figure, which one of the following options correctly represents the entries in the blocks labelled (i), (ii), (iii), and (iv), respectively? \[ \begin{array}{|c|c|c|c|} \hline N & U & F & (i) \\ 21 & 14 & 9 & 6 \\ \hline H & L & (ii) & O \\ 12 & (iv) & 15 & (iii) \\ \hline \end{array} \]

A bag contains Violet (V), Yellow (Y), Red (R), and Green (G) balls. On counting them, the following results are obtained: (i) The sum of Yellow balls and twice the number of Violet balls is 50.
(ii) The sum of Violet and Green balls is 50.
(iii) The sum of Yellow and Red balls is 50.
(iv) The sum of Violet and twice the number of Red balls is 50.
Which one of the following Pie charts correctly represents the balls in the bag?

Passage:
“His life was divided between the books, his friends, and long walks. A solitary man, he worked at all hours without much method, and probably courted his fatal illness in this way. To his own name there is not much to show; but such was his liberality that he was continually helping others, and fruits of his erudition are widely scattered, and have gone to increase many a comparative stranger’s reputation.” (From E.V. Lucas’s “A Funeral”)
Question: Based only on the information provided in the above passage, which one of the following statements is true?

For the clock shown in the figure, if $O^{*} = O\ Q\ S\ Z\ P\ R\ T$ and $X^{*} = X\ Z\ P\ W\ Y\ O\ Q$, then which one among the given options is most appropriate for $P^{*}$?

Consider a five-digit number $PQRST$ that has distinct digits $P, Q, R, S,$ and $T$, and satisfies the following conditions: \[ P<Q, S>P>T, R<T \] If integers 1 through 5 are used to construct such a number, the value of $P$ is:

A business person buys potatoes of two different varieties $P$ and $Q$, mixes them in a certain ratio and sells them at ₹ 192 per kg. The cost of the variety $P$ is ₹ 800 for 5 kg.
The cost of the variety $Q$ is ₹ 800 for 4 kg.
If the person gets 8% profit, what is the $P:Q$ ratio (by weight)?

Even though I had planned to go skiing with my friends, I had to ________________ at the last moment because of an injury.

The President, along with the Council of Ministers, ________________ to visit India next week.

An electricity utility company charges ₹ 7 per kWh (kilo watt-hour). If a 40-watt desk light is left on for 10 hours each night for 180 days, what would be the cost of energy consumption? If the desk light is on for 2 more hours each night for the 180 days, what would be the percentage-increase in the cost of energy consumption?

The eigenvalues of the matrix \[ A = \begin{bmatrix} 3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3 \end{bmatrix} \] are $\lambda_1, \lambda_2, \lambda_3$. The value of $\lambda_1 \lambda_2 \lambda_3 (\lambda_1 + \lambda_2 + \lambda_3)$ is:

A stationary tank is cylindrical in shape with two hemispherical ends and is horizontal, as shown in the figure. $R$ is the radius of the cylinder as well as of the hemispherical ends. The tank is half filled with an oil of density $\rho$ and the rest of the space in the tank is occupied by air. The air pressure, inside the tank as well as outside it, is atmospheric. The acceleration due to gravity ($g$) acts vertically downward. The net horizontal force applied by the oil on the right hemispherical end (shown by the bold outline in the figure) is:

If a function $f(x)$ is continuous in the closed interval $[a, b]$ and the first derivative $f'(x)$ exists in the open interval $(a, b)$, then according to the Lagrange's Mean Value Theorem: \[ \frac{f(b) - f(a)}{b - a} = f'(c) \] If $a = 0, b = 1.5$, and $f(x) = x(x - 1)(x - 2)$, then the value(s) of $c$ in $[a, b]$ is/are:

Which of the following logging tool(s) underestimate(s) porosity in a gas-bearing formation?

The drainage oil-water capillary pressure data for a core retrieved from a homogeneous isotropic reservoir is listed in the table. The reservoir top is at 4000 ft from the surface and the water-oil contact (WOC) depth is at 4100 ft. \[ \begin{array}{|c|c|} \hline \textbf{Water saturation (\%)} & \textbf{Capillary pressure (psi)} \\ \hline 100.0 & 0.0 \\ \hline 100.0 & 5.5 \\ \hline 99.0 & 5.6 \\ \hline 89.2 & 6.4 \\ \hline 81.8 & 6.9 \\ \hline 44.2 & 11.2 \\ \hline 29.7 & 17.1 \\ \hline 25.1 & 36.0 \\ \hline \end{array} \] Assume the densities of water and oil at reservoir conditions are 1.04 g/cc and 0.84 g/cc, respectively. The acceleration due to gravity is 980 cm/s2. The interfacial tension between oil and water is 35 dynes/cm and the contact angle is 0 degree. The depth of free-water level (FWL) is at ________ ft (rounded off to one decimal place).

The effect of pressure on various properties of black oil is shown in the figure. The bubble point pressure is $P_b$.

Which of the following option(s) is/are CORRECT?

Which of the following definition(s) related to fire and explosion is/are CORRECT?

The formation resistivity factor $(F)$ is related to the formation porosity $(\phi)$ in a water-bearing carbonate formation by the following correlation: \[ F = 0.9 \phi^{-2} \] where $\phi$ is in fraction. The resistivity of the invaded zone of the formation obtained by the Microspherically Focused Log (MSFL) is $4.5 \, \Omega m$, and the resistivity of the mud-filtrate is $0.05 \, \Omega m$. The formation porosity is ________ % (rounded off to one decimal place).

Which of the following option(s) is/are CORRECT for well testing analysis of a reservoir?

In a capillary rise experiment with a capillary tube of length $l_1$, water rises to a height $h$ such that $h \lt l_1$.

If the capillary tube is cut to a length $l_2$ such that $l_2 \lt h$, and the experiment is repeated, which of the following statements is/are CORRECT?

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2) =$

The porosity of a formation with matrix density of $2.65 \, g/cc$ and fluid density of $1.0 \, g/cc$ is $0.15$. The formation has shear modulus of $30 \, GPa$ and bulk modulus of $36 \, GPa$. The compressional wave velocity in the formation is $\underline{\hspace{1cm}} \times 10^3 \, m/s$ (rounded off to two decimal places).

A vertical well is drilled up to a depth of 4000 ft. Further drilling starts with 10 ppg of fresh mud and $50000 \, lbf$ weight on bit (WOB). An equivalent circulation density (ECD) of $10.75 \, ppg$ was recorded. The total circulation pressure loss is estimated to be 110 psi. The still density is $65.5 \, ppg$. The decrease in hook load is ________ lbf (rounded off to one decimal place). (Note: 1 ppg mud is equivalent to $0.052 \, psi/ft$).