# Mechanical Engineering - GATE 2008 Paper Solution

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

Given that $\ddot x+3x=0,$ and $x\left(0\right)=1,\dot x\left(0\right)=0$, what is x(1)?

The value of $\underset{x\to 8}{\mathrm{lim}}\frac{{x}^{1}{3}}-2}{\left(x-8\right)}$ is

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

The matrix  $\left[\begin{array}{ccc}1& 2& 4\\ 3& 0& 6\\ 1& 1& p\end{array}\right]$ has one eigen value equal to 3. The sum of the other two eigenvalues is

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

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

A straight rod of length L(t), hinged at one end and freely extensible at the other end, rotates through an angle about the hinge. At time t, L(t) =1 m,$\dot L\left(t\right)$ = 1 m/s, $\theta \left(t\right)=\frac{\mathrm{\pi }}{4}$ rad and $\dot\theta\left(t\right)$= 1 rad/s. The magnitude of the velocity at the other end of the rod 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 counterweight respectively. The mass of the cantilever part is 75 kg. The mass of the counterweight, for static balance, is

A planner mechanism has 8 links and 10 rotary joints. The number of degrees of freedom of the mechanism, using Gruebler’s criterion, 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 become 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 Pa.s ; specific heat at constant pressure = 1kj/kg.K;
thermal conductivity = 1 W/mk.
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

For the continuity equation given by $\overrightarrow\nabla\bullet\overrightarrow V=0$ to be valid, where $\overrightarrow V$ is the velocity vector, which one of the following is a necessary condition?

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

In an M/M/1 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 $\frac{{e}^{-\lambda T}{\left(\lambda T\right)}^{n}}{n!}$). The probability density function f(t) of the inter-arrival time is given by

A set of 5 jobs is to be processed on a single machine. The processing time (in days)is given 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