List of practice Questions

If the system of linear equations, \( x - ay - z = 0, ax - y - z = 0, x + y - z = 0 \), has infinite number of solutions, then the possible values of a are
If a random variable \( X \) has the probability density function \[ f(x) = \begin{cases} \frac{5}{32}x^4 & \text{if } 0 \le x \le 2 \\ 0 & \text{otherwise} \end{cases} \] and if \( Y = X^2 \), then the expected value of \( Y \) is .................... (round off to one decimal place)

A rectangular barge is freely floating in a drydock as shown in the following figure. For longitudinal strength analysis which of the following are TRUE? 

 

The buoyancy curve variation of a ship floating in still water and in waves is shown in the following figure. The total area under each curve is the same. The cases 'X' and 'Y' correspond to 

 

A closed system undergoing a thermodynamic cycle consisting of two reversible isothermal and two reversible adiabatic processes is shown in the following figure. If \(\delta Q\) is the infinitesimal heat transfer and T is the instantaneous temperature, then the value of the contour integral \( \oint \frac{\delta Q}{T} \) is 

The time series of rudder angle (\(\delta\)) and heading angle (\(\psi\)) during a ship's maneuver are shown in the following figure. Identify the maneuver and the associated parameters (p, q, r and s) 

 

In the following "GZ (righting lever arm)" versus "angle of heel" curve, the point 'X' indicates 

 

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line. 
The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ (anti-diagonal) and MN (main diagonal) form lines of symmetry? (The figure is representative) 

 

In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and PV = 5 cm. What is the length of RS in cm? (The diagram is representative.) 

 

A tsunami that originated off the Indonesian coast has propagated towards the east-coast of India. It enters the continental shelf at 150 km away from the coast of Chennai. If the average water depth is 80 m from the coast to the continental shelf and 20 minutes is the tsunami period, the time taken by the tsunami to reach the coast of Chennai on entering the continental shelf is .................... hours (round off to two decimal places)
A buoy of virtual mass 30 kg oscillates in a fluid medium as a single degree of freedom system. If the total damping in the system is set as 188.5 N-s/m, such that the oscillation just ceases to occur, then the natural period of the system is .................... s (round off to one decimal place)
For a marine screw propeller, the open water characteristics at J = 0.6 are \(K_T = 0.1336\) and \(10K_Q = 0.2010\). The open water propeller efficiency \(\eta_o\), is .................... (round off to two decimal places)
Saturated liquid water (m = 1 kg) initially at 0.101 MPa and 100 °C is heated at constant pressure until the temperature increases to 500 °C. Assume a constant \(C_p\) of steam = 1.9 kJ/kg-K, and enthalpy of vaporization, \(h_{fg}\) = 2257 kJ/kg at 0.101 MPa. The change in entropy of the water is .................... kJ/K (round off to two decimal places)
The stress field, \(\sigma_x = 4x^3 + 3x^2y + 5xy^2\) \(\sigma_y = -x^3 + 6x^2y - 7xy^2\) \(\tau_{xy} = -5x^2y - 3xy^2\) would satisfy the strain compatibility condition if
If y(x) is the solution of the differential equation \[ (1+x^2)y'' - 2xy' = 0 \] satisfying y(0) = 0 and y'(0) = 3, then y(1) equals ....................
For a ship of length L = 100 m, the distance between the bow and stern pressure system is 0.942L. Assume g = 10 m/s². The ship velocity corresponding to the prismatic hump of the wave making resistance curve is .................... m/s (round off to one decimal place)
For a given time varying load applied on a single degree of freedom system, the dynamic response amplitude is always less than the static response amplitude if
An insulated nozzle has an inlet cross-sectional area of 314 cm². Air flows through the nozzle with an inlet temperature of 300 K at a steady rate of 1.256 m³/s. The velocity at the exit is greater than that at the inlet by 210 m/s. Assume a constant \(C_p = 1.004\) kJ/kg-K. The temperature (in K) of air at the exit of the nozzle lies between
The heave natural frequencies of a Jacket structure, FPSO and a semi-submersible are \(\omega_J\), \(\omega_F\) and \(\omega_S\) respectively. Each one of them has a pay load capacity of 10000 tonnes. Which of the following is TRUE?
A vertical frictionless piston-cylinder arrangement contains air of mass 1 kg. During a process, 50 J of heat is transferred from outside to the system such that the piston is raised slowly by 0.1 m from its initial equilibrium position. The mass of the piston is 1 kg, and the diameter is 0.1 m. Assume that g = 9.81 m/s², and Patm = 100 kPa. The change in internal energy of the air in J (round off to two decimal places) lies between
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For a 2D ideal flow, let \(\varphi\) be the velocity potential and \(\psi\) be the stream function. Which one of the following is TRUE?
The velocity vector of a 2D flow field is given by \(\vec{V} = 2y^2 \hat{i} + x^2t \hat{j}\).
The acceleration is
Note: There's a typo in the OCR of the question (î twice) and options. The solution assumes the standard form V = uî + vĵ.
Water is flowing with a free stream velocity of 0.25 m/s around a submerged flat plate of 2 m length (in the direction of flow) and 1 m width. The local shear stress at a distance \(x\) from the leading edge of the plate is given by \[ \tau = \frac{0.332\rho u^2}{\sqrt{Re_x}} \] where \(\rho = 1000\) kg/m³ is the density of the water, \(u\) is the free stream velocity and \(Re_x\) is the Reynolds number at \(x\). Assume that the flow is laminar, and the kinematic viscosity of water is \(10^{-6}\) m²/s. The drag force (in Newton) acting on one side of the plate lies between
A ship of length 125 m has a design speed of 25 knots (1 knot = 0.5144 m/s). A 5.0 m long geometrically similar model with wetted surface area of 4 m\(^2\) has a coefficient of residuary resistance of \(1.346 \times 10^{-3}\) at the corresponding speed. The ship's residuary resistance in kN (in sea water of density 1025 kg/m\(^3\)), and the model speed in knots (round off to the nearest integer) respectively are
A solid cantilever shaft of diameter 0.1 m and length 2 m is subjected to a torque of 10 kN-m at the free end (shear modulus is 82 GPa). The maximum induced shear stress is .................... N/mm\(^2\) (round off to the nearest integer).