List of practice Questions

Rotational spectrum of a diatomic molecule consists of lines of equal spacing with an interval of \(20.0\text{ cm}^{-1}\). Its moment of inertia is found to be \(I_0 \times 10^{-47}\text{ kg.m}^2\), the value of \(I_0\) (rounded off to one decimal place) is ______.
(\(h = 6.6 \times 10^{-34}\text{ J.s}\), speed of light in vacuum \(c = 3 \times 10^8\text{ m.s}^{-1}\))

Copper has an electron number density of \(8.3 \times 10^{28}\text{ m}^{-3}\). Its Fermi energy in eV (rounded off to one decimal place) is ______.
(\(\hbar = 1.06 \times 10^{-34}\text{ J.s}\), mass of electron \(m_e = 9.10\times 10^{-31}\text{ kg}\), charge of electron \(= 1.60\times 10^{-19}\text{ C}\))

Two 1 kg blocks are connected to two massless springs of spring constants \(8\text{ N.m}^{-1}\) and \(4\text{ N.m}^{-1}\). The system is kept on a frictionless horizontal floor with one end of a spring attached to a wall (see figure below). They are performing oscillatory motion along the x-axis with the normal mode frequencies \(\omega_H\) and \(\omega_L\) (\(\omega_H > \omega_L\)). The ratio \(\dfrac{\omega_H}{\omega_L}\) (rounded off to two decimal places) is ______

A 15 cm long scale is held horizontally with one of its ends on the edge of a 1 m high table and the other end resting on one's index finger. As the finger is removed (see figure below), the scale starts rotating about its end on the table. After 0.1 s, during which it has rotated by a negligibly small angle but has gained a rotational speed, it leaves the table and falls vertically towards the ground. When its centre of mass has fallen by 0.5 m, it has rotated by an angle \(\theta\). The value of \(\theta\) in degrees (rounded off to one decimal place) is ______
(\(g = 9.8\text{ m.s}^{-2}\))

Expedite, Hasten, Hurry, __________
Fill the blank by choosing a word with a meaning similar to that of the words given above.

A black square PQRS has been cut into two parts. One part of it is shown in Panel I. Which one of the shapes in Panel II is the other part?

A day can only be cloudy or sunny. The probability of a day being cloudy is 0.5, independent of the condition on other days. What is the probability that in any given four days, there will be three cloudy days and one sunny day?

The values of Stock A and Stock B on a particular day are Rs. 50 and Rs. 80, respectively. An investor invests Rs. 100 in Stock A and Rs. 80 in Stock B. He sells all the stocks the next day when the value of Stock A is Rs. 55 and Stock B is Rs. 70. The profit made by the investor is Rs. __________

'When it is raining, peacocks dance.'
Based only on this sentence, which one of the following options is necessarily true?

Water : P :: Food : Q
Choose the P and Q combination from the options below to form a meaningful analogy.

Two tiles are missing in Panel I. Which one of the options in Panel II is the appropriate choice for the missing tiles?

Panel I is a 3 by 3 arrangement of tiles, where each tile is itself a 3 by 3 array of filled and hollow circles; the bottom two tiles in the rightmost column are missing and marked with a question mark. Panel II shows four candidate two-tile stacks labelled (i), (ii), (iii) and (iv) that could fill the missing space.

Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the line segments between them represent intercity highways.
A salesperson needs to make a trip. She needs to start from a city, visit each of the remaining cities exactly once, and finally return to the same city from which she started.
Which one of the following options is then true?

The figure in Panel I below is a grid of cells with four rows and four columns. The numbers on the top and on the left represent the number of cells that are to be shaded in that column and row, respectively. Which one of the options shown in Panel II below represents the grid shaded correctly?

An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice in succession and the number on the top face is recorded each time. The probability that the sum of the two recorded numbers is a prime number is ________

Two identical cubic dice are rolled simultaneously. Which of the following is the probability that at least one of the face values of the two dice is greater than 3?

Which of the following options is CORRECT for the eigenvalues \(\lambda\) of the given matrix?
\[ \begin{bmatrix} 8 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 7 \end{bmatrix} \]

For a given matrix \(A = \begin{bmatrix} 4 & 4 \\ 0 & 4 \\ -4 & 4 \end{bmatrix}\), and \(A^T\) representing its transpose, which of the following options is the CORRECT representation of \(A^T A\)?

Given two vectors \(\vec{U}\) and \(\vec{V}\), which of the following options is CORRECT for the expansion of divergence, \(\nabla \cdot (\vec{U} + \vec{V})\)?

Which of the following is the basic mechanism of hydrocyclones and centrifuges used for solids control in a drilling operation?

Which of the following is the role of stabilizer subs in the rotary drilling process?

For a single-phase dry gas reservoir having a well producing only dry gas, \(E\) represents gas expansion factor and \(B_g\) represents gas formation volume factor. Which of the following options is CORRECT?

Choose the correct option by matching the entries in Group X with their description in Group Y.

Group X	Group Y
(P) Gas hydrate	(I) Gas is mostly present in adsorbed state
(Q) Oil shale	(II) Sedimentary rock contains kerogen
(R) Coal bed methane	(III) Oil recovered from shale
(S) Shale oil	(IV) Gas is present inside the cages formed by water molecules

Which of the following statements is NOT CORRECT in the context of offshore operational environment?

Which of the following has the most proppant carrying capacity during hydraulic fracturing process?

A circular shaped reservoir has an external radius \(r_e\). A well of radius \(r_w\) lies at the centre of the reservoir. The part of the reservoir around the wellbore, up to a radius \(r_s\), is damaged due to drilling and completion operations, where \(r_w < r_s < r_e\). This formation damage causes an additional pressure drop, \(\Delta p(r)\), that varies with the radial distance \(r\) from the well. Which of the following statements is TRUE about the additional pressure drop at the radial location \(r_s\) due to the formation damage?