# Mechanical Engineering - GATE 2009 Paper Solution

For a matrix [M]= $\left[\begin{array}{cc}\frac{3}{5}& \frac{4}{5}\\ x& \frac{3}{5}\end{array}\right]$, the transpose of the matrix is equal to the inverse of the matrix [M]T = [M]-1. The value of x is given by

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

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

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

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

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 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 block weighing 981 N is resting on a horizontal surface. The coefficient of friction between the block and the horizontal surface is µ = 0.2.A vertical cable attached to the block provides partial support as shown. A man can pull horizontally with a force of 100 N. What will be the tension, T (in N) in the cable if the man is just able to move the block to the right? If the principal stresses in a plane stress problem are ${\sigma }_{1}$ = 100 MPa,${\sigma }_{2}$ = 40 MPa, the magnitude of the maximum shear stress (in MPa) will be

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 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

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

Friction at the tool-chip interface can be reduced by

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

The expected time (${t}_{e}$) of a PERT activity in terms of optimistic time (${t}_{o}$), pessimistic time (${t}_{p}$) and most likely time (${t}_{l}$) is given by