Given that $\bar{X}$ + 3x = 0, and x (0) = 0, what is x (1)?

The value of $lim_{x\rightarrow8}\frac{x1/3-2}{(x-8)}$ is

A rod of length L and diameter D is subjected to a tensile load P. Which of the following is sufficient to calculate the resulting change in diameter?

The matrix $\begin{bmatrix} \ 1 & 2 & 4 \ \ 3 & 0 & 6 \ \ 1 & 1 & p \ \end{bmatrix}$ has one eigen value equal to 3. The sum of the other two eigen values is

A cantilever type gate hinged at Q is shown in the figure. P and R are the centers of gravity of the cantilever part and the counter weight respectively. The mass of the cantilever part is 75 kg. The mass of the counter weight, for static balance is

A straight rod of length L(T), hinged at one end and freely extensible at the other end, rotates through an angle $\theta$(t) about the hinge. At time t, L(t) = 1 m, L(t) = 1 m/s, $\theta$(t) = $\frac{\pi}{4}$ rad and $\theta$(t) = 1 rad/s The magnitude of the velocity at the other end of the rod is

An axial residual compressive stress due to a manufacturing process is present on the outer surface of a rotating shaft subjected to bending. Under a given bending load, the fatigue life of the shaft in the presence of the residual compressive stress is

2 moles of oxygen are mixed adiabatically with another 2 moles of oxygen in mixing chamber, so that the final total pressure and temperature of the mixture becomes same as those of the individual constituents at their initial states. The universal gas constant is given as R. The change in entropy due to mixing, per mole of oxygen, is given by

For flow of fluid over a heated plate, the following fluid properties are known viscosity = 0.001; specific heat at constant pressure = 1 kJ/kg. k; Thermal conductivity = 1 W/m.k The hydrodynamic boundary layer thickness at a specified location on the plate is 1 mm. The thermal boundary layer thickness at the same location is

The divergence of the vector field (x - y) $\hat{i}$ + (y - x) $\hat{j}$ + (x + y + z) $\hat{k}$ is

For the continuity equation given by ^{$\bar{V}$$\bullet$$\bar{V}$} = 0 to be valid, where ^$\bar{V}$ is the velocity, which one of the following is a necessary condition?

A planner mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler's criterion, is

The transverse shear stress acting in beam of rectangular cross-section, subjected to a transverse shear load, is

The directional derivative of the scalar function f(x, y, z) = x² + 2y² + z at the point P = (1, 1, 2) in the direction of the vector $\bar{a}=3\hat{i}-4\hat{j}$ is

Which one of the following is NOT a necessary assumption for the air standard Otto cycle?

For what value of a, if any, will the following system of equations in x, y and z have a solution? 2x + 3y = 4 x + y + x = 4 x + 2y - z = a

In an M/M1 queuing system, the number of arrivals in an interval of length T is a Poisson random variable (i.e the probability of there being n arrivals in an interval of length T is $\frac{e^{-\lambda T(\lambda T)^n}}{n!}$) . The probability density function f(t) of the inter-arrival time is given by

Consider the shaded triangular region P shown in the figure. What is $\int\limits\int\limits_p\text{x y d x d y}$?

The length of the curve y = $\frac{2}{3}X^{1/2}$ between x = 0 and x = 1 is

Which of the following integrals is unbounded?

The eigenvector of matrix $\begin{bmatrix} \ 1 & 2 \ \ 0 & 2 \ \end{bmatrix}$ are written in the form $\begin{bmatrix} \ 1 \ \ a \ \end{bmatrix}and\begin{bmatrix} \ 1 \ \ b \ \end{bmatrix}$. What is a + b?

It is given that y” + 2y' + y = 0, y(0) = 0, y(1) = 0. What is y (0.5)?

Let f = y^x. What is $\frac{\partial^2f}{\partial x\partial y}$ at x = 2, y = 1?

A solid circular shaft of diameter 100 mm is subjected to an axial stress of 50 MPa. It is further subjected to a torque of 10 kNm. The maximum principal stress experienced on the shaft is closest to

A set of 5 jobs is to be processed on a single machine. The processing time (in days) is given in the table below. The holding cost for each job is Rs. K per day ||| |---|---| | Job | Processing time | | P | 5 | | Q | 2 | | R | 3 | | S | 2 | | T | 1 |

A schedule that minimizes the total inventory cost is

A spur gear has a module of 3 mm, number of teeth 16, a face width of 36 mm and a pressure angle of 20⁰. It is transmitting a power of 3 kW at 20 rev/s. Taking a velocity factor of 1.5, and a form factor of 0.3, the stress in the gear tooth is about

A circular disk of radius R rolls without slipping at a velocity $v$. The magnitude of the velocity at point P (see figure) is

A two dimensional fluid element rotates like a rigid body. At point within the element, the pressure is 1 unit. Radius of the Mohr's circle, characterizing the state of stress at that point, is

A gas expands in a frictionless piston–cylinder arrangement. The expansion process is very slow, and is resisted by an ambient pressure of 100 kPa. During the expansion process, the pressure of the system (gas) remains constant at kPa. The change in volume of the gas is 0.01 m³. The maximum amount of work that could be utilized from the above process is

For the three dimensional object shown in the figure below, five faces are insulated. The sixth face (PQRS), which is not insulated, interacts thermally with the ambient, with a convective heat transfer coefficient of 10 W/m².k The ambient temperature is 30^oC. Heat is uniformly generated inside the object at the rate of 100 W/m³. Assuming the face PQRS to be at uniform temperature, its steady state temperature is

A clutch has outer and inner diameters 100 mm and 40 mm respectively. Assuming a uniform pressure of 2 MPa and coefficient of friction of linear material 0.4, the torque carrying capacity of the clutch is

A rigid, insulated tank is initially evacuated. The tank is connected with a supply line through which air (assumed to be ideal gas with constant, specific heat) passes at 1 MPa, 350⁰C. A valve connected with the supply line is opened and the tank is charged with air until the final pressure inside the tank reaches 1 MPa. The final temperature inside the tank

For the component loaded with a force F as shown in the figure, the axial stress at the corner point P is

A journal bearing has shaft diameter of 40 mm and a length of 40 mm. The shaft is rotating at 20 rad/s and the viscosity of the lubricant is 20 mPa.s. The clearance is 0.020 mm. The loss of torque due to the viscosity of the lubricant is approximately

A balloon containing an ideal gas is initially kept in an evacuated and insulated room. The balloon ruptures and the gas fills up the entire room. Which one of the following statements is TRUE at the end of the above process?

In a cam design, the rise motion is given by a simple harmonic motion (SHM) s = ^{$\frac{h}{2}\bigg(1-cos\frac{\pi \theta}{\beta}\bigg)$}, where h is the total rise, ^$\theta$is camshaft angle and ^$\beta$ is the total angle of the rise interval. The jerk is given by

A thermal power plant operates on a regenerative cycle with a single open feed water heater, as shown in the figure. For the state point shown, the specific enthalpies are : h₁ = 2800 Kj/kg and h₂ = 200 Kj/kg. The bleed to the feed water heater is 20% of the boiler steam generation rate. The specific enthalpy at state 3 is

A hollow enclosure is formed between two infinitely long concentric cylinders of radii 1 m and 2 m, respectively. Radiative heat exchange takes place between the inner surface of the larger cylinder (surface-2) and the outer surface of the smaller cylinder (surface-1). The radiating surfaces are diffuse and the medium in the enclosure is non-participating. The fraction of the thermal radiation leaving the larger surface and striking itself is

For the standard transportation linear programme with m sources and n destinations and total supply equaling total demand, an optimal solution(lowest cost) with the smallest number of non-zero x₀ values (amounts from source I to destination j) is desired. The best upper bound for this number is

Water, having a density of 1000 kg/m³, issues from a nozzle with a velocity of 10 m/s and the jet strikes a bucket mounted on a Pelton wheel. The wheel rotates at 10 rad/s. The mean diameter of the wheel is 1 m. The jet is split into two equal streams by the bucket, such that each stream is deflected by 120⁰, as shown in the figure. Friction in the bucket may be neglected. Magnitude of the torque exerted by the water on the wheel, per unit mass flow rate of the incoming jet, is

A moving average system is used for forecasting weekly demand. F₁(t) and F₂(t) are sequences of forecasts with parameters m₁ and m₂, respectively, where m₁ and m₂(m₁ > m₂) denotes the numbers of weeks over which the moving averages are taken. The actual demands shows a step increase from d₁ to d₂ at a certain time. Subsequently,

A cyclic device operates between three thermal reservoirs, as shown in the figure. Heat is transferred to form the cyclic device. It is assumed that heat transfer between each thermal reservoir and the cyclic device takes place across negligible temperature difference. Intersections between the cyclic device, the respective thermal reservoirs that are shown in the figure are all in the form of heat transfer.

The cyclic device can be

A uniform rigid rod of mass m = 1 kg and length L = 1 m is hinged at its centre and laterally supported at one end by a spring of spring constant k = 300 N/m. The natural frequency $\omega_n$ in rad/s is

In a single pass rolling operation, a 20 mm thick plate with plate width of 100 mm, is reduced to 18 mm. The roller radius is 250 mm and rotational speed is 10 rpm. The average flow stress for the plate material is 300 MPa. The power required for the rolling operation (in kW) is closest to

For the network, the objective is to find the length of the shortest path from node P to node G. Let d_v be the length of directed are from node I to node j. Let s_j be the length of the shortest path from P to node j. Which of the following equations can be used to find S_G?

In arc welding of a butt joint, the welding speed is to be selected such that highest cooling rate is achieved. Melting efficiency and heat transfer efficiency are 0.5 and 0.7, respectively. The area of weld cross section is 5 mm² and the unit energy required to melt the metal is 10 J/mm³. If the welding power is 2 kW, the welding speed in mm/s is closest to

Steady two-dimensional heat conduction takes place in the body shown in the figure below. The normal temperature gradients over surfaces P and Q can be considered to be uniform. The temperature gradient $\frac{\partial T}{\partial X}$ at surface Q is equal to 10 K/m. Surfaces P and Q are maintained at constant temperatures as shown in the figure, while the remaining part of the boundary is insulated. The body has constant thermal conductivity of 0.1 W/m.K. The values of $\frac{\partial T}{\partial X}and\frac{\partial T}{\partial Y}$ at surface P are

The product structure of an assembly P is shown in the figure. Estimated demand for end product P is as follows

Ignore lead times for assembly and sub-assembly. Production capacity (per week) for component R is the bottleneck operation. Starting with zero inventory, the smallest capacity that will ensure a feasible production plan up to week 6 is

The rod PQ of length L and with flexural rigidity EI is hinged at both ends. For what minimum force F is it expected to buckle?

In the figure shown, the system is a pure substance kept in a piston cylinder arrangement. The system is initially a two phase mixture containing 1 kg of liquid and 0.03 kg of vapour at a pressure of 100 kPa. Initially, the piston rest on a set of stops, as shown in the figure. A pressure of 200 kPa is required to exactly balance the weight of the piston and the outside atmospheric pressure. Heat transfer takes place into the system until its volume increases by 50%. Heat transfer to the system occurs in such a manner that the piston, when allowed to move, does so in a very slow (quasi-static / quasi - equilibrium) process. The thermal reservoir from which heat is transferred to the system has a temperature of 400⁰C. Average temperature of the system boundary can be taken as 175⁰C. The heat transfer to the system is 1 kJ during which its entropy increased by 10 J/K.

Specific volume of liquid (v_f) and vapour (v_g) phase, as well as values of saturation temperatures, are given in the table below ||||| |---|---|---|---| | Pressure (kPa) | Saturation temperature, T_sat(⁰C) | V_f(m³/kg) | V_g(m₃/kg) | | 100 | 100 | 0.001 | 0.1 | | 200 | 200 | 0.0015 | 0.002 |

At the end of the process, which one of the following situations will be true?

A cylindrical container of radius R = 1 m, wall thickness 1 mm is filled with water up to a depth of 2 m and suspended along its upper rim. The density of water is 1000 kg/m³ and acceleration due to gravity is 10 m/s². The self weight of the cylinder is negligible. The formula for hoop stress in a thin walled cylinder can be used at all points along the height of the cylindrical container.

If the Young's modulus and Poisson's ratio of the container material are 100 GPa and 0.3 respectively, the axial strain in the cylinder wall at mid-depth is

A pressure of 100 kPa is applied. Initially, the piston rest on a set of stops, as shown in the figure. A pressure of 200 kPa is required to exactly balance the weight of the piston and the outside atmospheric pressure. Heat transfer takes place into the system until its volume increases by 50%. Heat transfer to the system occurs in such a manner that the piston, when allowed to move, does so in a very slow (quasi-static / quasi - equilibrium) process. The thermal reservoir from which heat is transferred to the system has a temperature of 400⁰C. Average temperature of the system boundary can be taken as 175⁰C. The heat transfer to the system is 1 kJ during which its entropy increased by 10 J/K.

Specific volume of liquid (v_f) and vapour (v_g) phase, as well as values of saturation temperatures, are given in the table below ||||| |---|---|---|---| | Pressure (kPa) | Saturation temperature, T_sat(⁰C) | V_f(m³/kg) | V_g(m³/kg) | | 100 | 100 | 0.001 | 0.1 | | 200 | 200 | 0.0015 | 0.002 |

The net entropy generation (considering the system and the thermal reservoir together) during the process is closest to

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and in viscid.

The radial velocity v_r at any radius r, when the gap width is h, is

A cylindrical container of radius R = 1 m wall of thickness 1 mm is filled with water up to a depth of 2 m and suspended along its upper rim. The density of water is 1000 kg/m³ and acceleration due to gravity is 10 m/s². The self weight of the cylinder is negligible. The formula for hoop stress in a thin walled cylinder can be used at all points along the height of the cylindrical container.

The axial and circumferential stress $(\sigma_d,\sigma_c)$ experienced by the cylinder wall at mid-depth (1 m as shown) are

A compression spring is made of music wire of 2 mm diameter having a shear strength and shear modulus of 800 MPa and 80 GPa respectively. The mean coil diameter is 20 mm, free length is 40 mm and the number of active coils is 10. If the mean coil diameter is reduced to 10 mm, the stiffness of the spring is approximately

The gap between a moving circular plate and a stationary surface is being continuously reduced, as the circular plate comes down at a uniform speed V towards the stationary bottom surface, as shown in the figure. In the process, the fluid contained between the two plates flows out radially. The fluid is assumed to be incompressible and in viscid.

The radial component of the fluid acceleration at r = R is

Consider a truss PQR loaded at P with a force F as shown in the figure. The tension in the member QR is

The natural frequency of the spring mass system shown in the figure is closet to

The logarithmic mean temperature difference (LMTD) of a counter flow heat exchanger is 20⁰C. The cold fluid enters at 20⁰C and the hot fluid enters at 100⁰C. Mass flow rate of the cold fluid is twice that of the hot fluid. Specific heat at constant pressure of the hot fluid is twice that of the cold fluid. The exit temperature of the cold fluid is

In the figure shown, the system is a pure substance kept in a piston cylinder arrangement. The system is initially a two phase mixture containing 1 kg of liquid and 0.03 kg of vapour at a pressure of 100 kPa. Initially, the piston rests on a set of stops, as shown in the figure. A pressure of 200 kPa is required to exactly balance the weight of the piston and the outside atmospheric pressure. Heat transfer takes place into the system until its volume increases by 50%. Heat transfer to the system occurs in such a manner that the piston, when allowed to move, does so in a very slow (quasi-static / quasi - equilibrium) process. The thermal reservoir from which heat is transferred to the system has a temperature of 400⁰C. Average temperature of the system boundary can be taken as 175⁰C. The heat transfer to the system is 1 kJ during which its entropy increased by 10 J/K.

Specific volume of liquid (v_f) and vapour (v_g) phase, as well as values of saturation temperatures, are given in the table below ||||| |---|---|---|---| | Pressure (kPa) | Saturation temperature, T_sat(⁰C) | V_f(m³/kg) | V_g(m₃/kg) | | 100 | 100 | 0.001 | 0.1 | | 200 | 200 | 0.0015 | 0.002 |

The work done by the system during the process is

Consider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0. ||||||| |---|---|---|---|---|---| | | -4 | -6 | 0 | 0 | 0 | | s | 3 | 2 | 1 | 0 | 6 | | t | 2 | 3 | 0 | 1 | 6 | | | x | y | s | t | RHS |

After some simplex iterations, the following table is obtained ||||||| |---|---|---|---|---|---| | | 0 | 0 | 0 | 2 | 12 | | s | 5/3 | 0 | 1 | -1/3 | 2 | | y | 2/3 | 1 | 0 | 1/3 | 2 | | | x | y | s | t | RHS |

From this, one can conclude that

Consider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 After introducing slack variables s and t, the initial basic feasible solution is represented by the table below (basic variables are s = 6 and t = 6, and the objective function value is 0. ||||||| |---|---|---|---|---|---| | | -4 | -6 | 0 | 0 | 0 | | s | 3 | 2 | 1 | 0 | 6 | | t | 2 | 3 | 0 | 1 | 6 | | | x | y | s | t | RHS |

After some simplex iterations, the following table is obtained ||||||| |---|---|---|---|---|---| | | 0 | 0 | 0 | 2 | 12 | | s | 5/3 | 0 | 1 | -1/3 | 2 | | y | 2/3 | 1 | 0 | 1/3 | 2 | | | x | y | s | t | RHS |

From this, one can conclude thatConsider the linear programme (LP) Max 4x + 6y Subject to 3x + 2y $\leq$ 6 2x + 3y $\leq$ 6 x, y $\leq$ 0 The dual for the LP is

For generating a Coon's surface we require

While cooling, a cubical casting of side 40 mm undergoes 3%, 4% and 5% volume shrinkage during the liquid state, phase transition and solid state, respectively. The volume of metal compensated from the riser is

In a single point turning tool, the side rake angle and orthogonal rake angle are equal. ^$\varphi$is the principal cutting edge angle and its range is ^{$0^0\leq\varphi\leq90^0$}. The chip flows in the orthogonal plane. The value of ^$\varphi$is closest to

A researcher conducts electrochemical machining (ECM) on binary alloy (density 6000 kg/m³) of iron (atomic weight 56, valency 2) and metal P (atomic weight 24, valency 4). Faraday's constant = 96500 coulomb/mole. Volumetric material removal rate of the alloy is 50 mm³/s at a current of 2000 A. The percentage of the metal P in the alloy is closet to

In the feed drive of a point to point open loop CNC drive, a stepper motor rotating at 200 steps/rev drives a table through a gear box and lead screw nut mechanism (pitch = 4 mm, number of starts = 1). The gear $ratio=\bigg {\frac{\text{output rotational speed} }{\text{input rotational speed}}\bigg }$ is given by U = $\frac{1}{4}$. The stepper motor (driven by voltage pulses from a pulse generator) executes 1 step/pulse of the pulse generator. The frequency of the pulse train from the pulse generator is f = 10,000 pulses per minute.

The basic length unit (BLU), i.e. the table movement corresponding to 1 pulse of the pulse generator, is

The figure shows an incomplete schematic of a conventional lathe to be used for cutting threads with, different pitches. The speed gear box U_r is shown and the feed gear box U_s is to be placed P, Q, R and S denote locations and have no other significance. Change in U_r should NOT affect the pitch of the thread being cut and changes in U_s should NOT affect the cutting speed.

The correct connections and the correct placement of U_s are given by

A displacement sensor (a dial indicator) measures the lateral displacement of a mandrel mounted on the taper hole inside a drill spindle. The mandrel axis is an extension of the drill spindle taper hole axis and the protruding portion of the mandrel surface is perfectly cylindrical. Measurements are taken with the sensor placed at two positions P and Q as shown in the figure. The readings are recorded as R_x = maximum deflection minus minimum deflection, corresponding to sensor position at X, over one rotation.

If R_P = R_Q > 0, which one of the following would be consistent with the observation?

Orthogonal turning is performed on a cylindrical work piece with shear strength of 250 MPa. The following conditions are used cutting velocity is 180 m/min, feed is 0.20 mm/rev, depth of cut is 3 mm, chip thickness ratio = 0.5. The orthogonal rake angle is 7⁰. Apply merchant's theory for analysis.

The shear plane angle (in degrees) and the shear force respectively are

Orthogonal turning is performed on a cylindrical work piece with shear strength of 250 MPa. The following conditions are used cutting velocity is 180 m/min, feed is 0.20 mm/rev, depth of cut is 3 mm, chip thickness ratio = 0.5. The orthogonal rake angle is 7⁰. Apply merchant's theory for analysis.

The cutting and frictional forces, respectively, are

Moist air at a pressure of 100 kPa is compressed to 500 kPa and then cooled to 35⁰C in an after cooler. The air at the entry to the after cooler is unsaturated and becomes just saturated at the exit of the after cooler. The saturation pressure of water at 35⁰C is 5.628 kPa. The partial pressure of water vapour (in kPa) in the moist air entering the compressor is closet to

Air (at atmospheric pressure) at a dry bulb temperature of 40⁰C and wet bulb temperature of 20⁰C is humidified in an air washer operating with continuous water recirculation. The wet bulb depression (i.e. the difference between the dry and wet bulb temperature) at the exit is 25% of that at the inlet. The dry bulb temperature at the exit of the air washer is

One tooth of gear having 4 module and 32 teeth is shown in the figure. Assume that the gear tooth and the corresponding tooth space make equal intercepts on the pitch circumference. The dimensions 'a' and 'b' respectively, are closest to

A steel bar of 10 × 50 mm is cantilevered with two M 12 bolts (P and Q) to support a static load of 4 kN as shown in the figure:

The primary and secondary shear load on bolt P₁ respectively, are

A steel bar of 10 × 50 mm is cantilevered with two M 12 bolts (P and Q) to support a static load of 4 kN as shown in the figure:

The resultant shear stress on bolt is closest to

Match the type of gears with their most approximate description.

In a steady state steady flow process taking place in a device with a single inlet and a single outlet, the work done per unit mass flow rate is given by $w=\int\limits^{outlet}_{inlet}vdp$, where v is the specific volume and p is the pressure. The expression for w given above

Internal gear cutting operation can be performed by

In the feed drive of a Point to point open loop CNC drive, a stepper motor rotating at 200 steps/rev drives a table through a gear box and lead screw nut mechanism (pitch = 4 mm, number of starts = 1). The gear $ratio=\bigg {\frac{\text{output rotational speed}}{\text{input rotational speed}}\bigg }$ is given by U = $\frac{1}{4}$. The stepper motor (driven by voltage pulses from a pulse generator) executes 1 step/pulse of the pulse generator. The frequency of the pulse train from the pulse generator is f = 10,000 pulses per minute.

A customer insists on a modification to change the BLU of the CNC drive to 10 microns without changing the table speed. The modification can be accomplished by

The strain energy stored in the beam with flexural rigidity FL and loaded as shown in the figure is

The integral $\oint\limits f(z)dz$ evaluated around the unit circle on the complex plane for f(z) = $\frac{cosz}{z}$ is

A coin is tossed 4 times. What is the probability of getting heads exactly 3 times?

In the Taylor series, expansion of ex about x = 2, the coefficient of (x - 2)⁴ is

In the deep drawing of cups, blank show a tendency to wrinkle up around the periphery (flange). The most likely cause and remedy of the phenomenon are, respectively,