An analytic function of a complex variable z = x + iy is expressed as f(z) = u(x, y) + iv (x, y) where i = $\sqrt{-1}$ If u = xy, the expression for v should be

Let Δ = $\begin{vmatrix} \ a & a+b & a+b+c \ \ 3a & 4a+3b & 5a+4b+3c \ \ 6a & 9a + 6b & 11a + 9b + 6c \ \end{vmatrix}$, where a = i, b = ^_$\omega$, c = ^_$\omega$2, then Δ is equal to

The inverse Laplace transform of $\frac{1}{(s^2+s)}$is

If a closed system is undergoing an irreversible process, the entropy of the system

A coolant fluid at 30°C flows over a heated flat plate maintained at a constant temperature of 100°C. The boundary layer temperature distribution at a given location on the plate may be approximated as T = 30 + 70exp (y) where y (in m) is the distance normal to the plate and T is in °C. If thermal conductivity of the fluid is 1.0 W/mK, the local convective heat transfer coefficient (in W/m²K) at that location will be

The divergence of vector field 3xz$\hat{i}$ + 2xy$\hat{j}$-yz^2$\hat{k}$at a point (1, 1, 1) is equal to

If the principal stresses in a plane stress problem, are $\sigma_1$= 100MPa, $\sigma_2$= 40MPa, the magnitude of the maximum shear stress (in MPa) will be

A frictionless piston-cylinder device contains a gas initially at 0.8 MPa and 0.015 m³. It expands quasi-statically at constant temperature to a final volume of 0.030 m³. The work output (in kJ) during this process will be

A 20 Kg box is pulled with uniform speed across a horizontal floor by a force directed 30^o above horizontal.The coefficient of friction is 0.25.What is the magnitude of the force?

A compressor undergoes a reversible, steady flow process. The gas at inlet and outlet of the compressor is designated as state 1 and state 2, respectively. Potential and kinetic energy changes are to be ignored. The following notations are used:

V = specific volume and P = pressure of the gas The specific work required to be supplied to the compressor for this gas compression process is

A simple quick return mechanism is shown in the figure. The forward to return ratio of the quick return mechanism is 2:1. If the radius of the crank O1P is 125 mm, then the distance 'd' (in mm) between the crank centre to lever pivot centre point should be

If three coins are tossed simultaneously, the probability of getting at least one head is

The effective number of lattice points in the unit cell of simple cubic, body centered cubic, and face centered cubic space lattices, respectively, are

The rotor shaft of a large electric motor supported between short bearings at both deflection of 1.8 mm in the middle of the rotor. Assuming the rotor to be perfectly balanced and supported at knife edges at both the ends, the likely critical speed (in rpm) of the shaft is

A solid circular shaft of diameter d is subjected to a combined bending moment M and torque, T. The material property to be used for designing the shaft using the relation $\frac{16}{\pi d^3} \sqrt{M^2 + T^2}$is

Which of the following forecasting methods takes a fraction of forecast error into account for the next period forecast?

The expected time (t_e) of a PERT activity in terms of optimistic time (t₀), pessimistic time (t_p) and most Likely time (t_l) is given by

The distance between the origin and the point nearest to it on the surface z² = 1 + xy is

Which of the following is the correct data structure for solid models?

A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y) ² on path AB traversed in a counter-clockwise sense is

The area enclosed between the curves y² = 4x and x² = 4y is

The standard deviation of a uniformly distributed random variable between 0 and 1 is

The solution of x$\frac{dy}{dx}$+ y = x⁴ with the condition y(1) = $\frac{6}{5}$ is

In an air-standard Otto cycle, the compression ratio is 10. The condition at the beginning of the compression process is 100 kPa and 27°C. Heat added at constant volume is 1500 kJ/kg, while 700 kJ/kg of heat is rejected during the other constant volume process in the cycle. Specific gas constant for air = 0.287 kJ/kgK. The mean effective pressure (in kPa) of the cycle is

Consider steady, incompressible and irrotational flow through a reducer in a horizontal pipe where the diameter is reduced from 20 cm to 10 cm. The pressure in the 20 cm pipe just upstream of the reducer is 150 kPa. The fluid has a vapour pressure of 50 kPa and a specific weight of 5 kN/m³. Neglecting frictional effects, the maximum discharge (in m³/s) that can pass through the reducer without causing cavitation is

An irreversible heat engine extracts heat from a high temperature source at a rate of 100 kW and rejects heat to a sink at a rate of 50 kW. The entire work output of the heat engine is used to drive a reversible heat pump operating between a set of independent isothermal heat reservoirs at 17°C and 75°C. The rate (in kW) at which the heat pump delivers heat to its high temperature sink is

You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of the two-dimensional velocity fields in the xy-plane, are made available. P. u = 2y, v = - 3x Q. u = 3xy, v = 0 R. u = 2x, v = 2y Which flow(s) should be recommended when the application requires the flow to be incompressible and irrotational?

Water at 25°C is flowing through a 1.0 km long G.I pipe of 200 mm diameter at the rate of 0.07 m³/s. If value of Darcy friction factor for this pipe is 0.02 and density of water is 1000 kg/m³, the pumping power (in kW) required to maintain the flow is

A vehicle suspension system consists of a spring and a damper. The stiffness of the spring is 3.6 kN/m and the damping constant of the damper is 400 Ns/m. If the mass is 50 kg, the damping factor(d) and damped natural frequency (f_n), respectively, are

In a parallel flow heat exchanger operating under steady state, the heat capacity rates (product of specific heat at constant pressure and mass flow rate) of the hot and cold fluid are equal. The hot fluid, flowing at 1 kg/s with Cp = 4 kJ/kgK, enters the heat exchanger at 102°C while the cold fluid has an inlet temperature of 15°C. The overall heat transfer coefficient for the heat exchanger is estimated to be 1 kW/m²K and the corresponding heat transfer surface area is 5m². Neglect heat transfer between the heat exchanger and the ambient. The heat exchanger is characterized by the following relation: 2$\epsilon$= 1 - exp (-2NTU) The exit temperature (in °C) for the cold fluid is

Consider steady-state heat conduction across the thickness in a plane composite wall (as shown in the figure) exposed to convection conditions on both sides.

Given: h_i = 20W / m² k; h₀ = 55 W / m²K; $T_{\infty, 0}$= 2⁰C; k₁ = 20W / mk; k₂ = 500 W / mk; L₁ = 0.30 m and L₂ = 0.15 m. Assuming negligible contact resistance between the wall surfaces, the interface temperature, T (in °C), of the two walls will be

A solid shaft of diameter, d and length L is fixed at both the ends. A torque, T₀ is applied at a distance L/4 from the left end as shown in the figure given below:

The maximum shear stress in the shaft is

An epicyclic gear train is shown schematically in the adjacent figure. The sun gear 2 on the input shaft is a 20 teeth external gear. The planet gear 3 is a 40 teeth external gear. The ring gear 5 is a 100 teeth internal gear. The ring gear 5 is fixed and the gear 2 is rotating at 50 rpm ccw (ccw = counter-clockwise and cw = clockwise) The arm 4 attached to the output shaft will rotate at

Match the approaches given below to perform stated kinematics / dynamics analysis of machine.

A forged steel link with uniform diameter of 30 mm at the centre is subjected to an axial force that varies from 40 kN in compression to 160 kN in tension. The tensile (S_u), yield (S_y) and corrected endurance (S_e) strengths of the steel material are 600 MPa, 420 MPa and 240 MPa respectively. The factor of safety against fatigue endurance as per Soderberg's criterion is

An automotive engine weighing 240 kg is supported on four springs with linear characteristics. Each of the front two springs have a stiffness of 16 MN/m while the stiffness of each rear spring is 32 MN/m. The engine speed (in rpm), at which resonance is likely to occur, is

The velocity profile of a fully developed laminar flow in a straight circular pipe, as shown in the figure, is given by the expression u(r) = -$\frac{R^2}{4 \mu} \Big(\frac{dp}{dx}\Big) \Big(1-\frac{r^2}{R^2}\Big)$, where $\frac{dp}{dx}$ is a constant. The average velocity of fluid in the pipe is

A uniform rigid rod of mass M and length L is hinged at one end as shown in the adjacent figure. A force P is applied at a distance of 2L/3 from the hinge so that the rod swings to the right. The reaction at the hinge is

A company uses 2555 units of an item annually. Delivery lead time is 8 days. The recorder point (in number of units) to achieve optimum inventory is

A frame of two arms of equal length L is shown in the adjacent figure. The flexural rigidity of each arm of the frame is EI. The vertical deflection at the point of application of load P is

Six jobs arrived in a sequence as given below:

Average flow time (in days) for the above jobs using Shortest Processing Time rule is

Match the items in Column I and Column II.

The inlet and the outlet conditions of stream for an adiabatic steam turbine are indicated in the notations as usually followed.

If mass flow rate of steam through the turbine is 20 kg/s, the power output of the turbine (in MW) is

The inlet and the outlet conditions of stream for an adiabatic steam turbine are indicated in the notations as usually followed.

Assume the above turbine to be part of a simple Rankine cycle. The density of water at the inlet to the pump is 1000 kg/m³. Ignoring kinetic and potential energy effects, the specific work (in kJ/kg) supplied to the pump is

Consider the following Linear Programming Problem (LPP): Maximize z = 3x₁ + 2x₂ Subject to x₁ $\leq$ 4 x₂ $\leq$ 6 3x₁ + 2x₂ $\leq$ 18 x_{1 $\geq$}0, x₂ $\geq$ 0

Radiative heat transfer is intended between the inner surfaces of two very large isothermal parallel metal plates. While the upper plate (designated as plate 1) is a black surface and is the warmer one being maintained at 727؛C, the lower plate (plate 2) is a diffuse and gray surface with an emissivity of 0.7 and is kept at 227؛C. Assume that the surfaces are sufficiently large to form a two-surface enclosure and steady state conditions to exist. Stefan Boltzmann constant is given as 5.67×10^-8 W/m²K⁴.

The irradiation (in kW/m²) for the upper plate (plate 1) is

Radiative heat transfer is intended between the inner surfaces of two very large isothermal parallel metal plates. While the upper plate (designated as plate 1) is a black surface and is the warmer one being maintained at 727؛C. The lower plate (plate 2) is a diffuse and gray surface with an emissivity of 0.7 and is kept at 227؛C. Assume that the surfaces are sufficiently large to form a two-surface enclosure and steady state conditions to exist. Stefan Boltzmann constant is given as 5.67×10^-8W/m²K⁴.

If plate 1 is also a diffuse and gray surface with an emissivity value of 0.8, the net radiation heat exchange (in kW/m²) between plate 1 and plate 2 is

Consider the following PERT network:

The optimistic time, most likely time and pessimistic time of all the activities are given in the table below:

The standard deviation of the critical path is

Consider the following PERT network:

The optimistic time, most likely time and pessimistic time of all the activities are given in the table below:

The critical path duration of the network (in days) is

Friction at the tool-chip interface can be reduced by

Minimum shear strain in orthogonal turning with a cutting tool of zero rake angle is

Two streams of liquid metal, which are not hot enough to fuse properly, result into a casting defect known as

Electrochemical machining is performed to remove material from an iron surface of 20 mm x 20 mm under the following conditions: Inter electrode gap 0.2mm = 0.2 mm Supply voltage DC 12 V = 12 V Specific resistance of electrolyte = 2 $\Omega$cm Atomic weight of Iron = 55.85 Valency of Iron = 2 Faraday 's constant = 96540 coulombs The material removal rate (in g/s) is

Match the following:

What are the upper and lower limits of the shaft represented by 60 f₈? Use the following data: Diameter 60 lies in the diameter step of 50-80 mm Fundamental tolerance unit, i, in $\mu$m = 0.45D^1/3+0.001D, where D is the representative size in mm; Tolerance value for IT8=25i. Fundamental deviation for 'f' shaft = -5.5D^0.41

In a machining experiment, tool life was found to vary with the cutting speed in the following manner:

The exponent (n) and constant (k) of the Taylor's tool life equation are

In a machining experiment, tool life was found to vary with the cutting speed in the following manner:

What is the percentage increase in tool life when the cutting speed is halved?

In an ideal vapour compression refrigeration cycle, the specific enthalpy of refrigerant (in kJ/kg) at the following states is given as: Inlet of condenser: 283 Exit of condenser: 116 Exit of evaporator: 232 The COP of this cycle is

A 20؛ full depth involute spur pinion of 4 mm module and 21 teeth is to transmit 15 kW at 960 rpm. Its face width is 25mm. The tangential force transmitted (in N) is

A 20؛ full depth involute spur pinion of 4 mm module and 21 teeth is to transmit 15 kW at 960 rpm. Its face width is 25 mm. Given that the tooth geometry factor is 0.32 and the combined effect of dynamic load and allied factors intensifying the stress is 1.5. The minimum allowable stress (in MPa) for the gear material is