List of top Engineering Sciences Questions asked in GATE XE

A bar of length \( L = 1 \, {m} \) is fixed at one end. Before heating its free end has a gap of \( \delta = 0.1 \, {mm} \) from a rigid wall as shown in the figure. Now the bar is heated resulting in a uniform temperature rise of \( 10^\circ {C} \). The coefficient of linear thermal expansion of the material is \( 20 \times 10^{-6} / \degree C \) and the Young’s modulus of elasticity is 100 GPa. Assume that the material properties do not change with temperature. 
The magnitude of the resulting axial stress on the bar is .......... MPa (in integer). 

A massless cantilever beam, with a tip mass \( m \) of 10 kg, is modeled as an equivalent spring-mass system as shown in the figure. The beam is of length \( L = 1 \, {m} \), with a circular cross-section of diameter \( d = 20 \, {mm} \). The Young’s modulus of the beam material is 200 GPa.
The natural frequency of the spring-mass system is ............ Hz (rounded off to two decimal places).

A simply-supported beam has a circular cross-section with a diameter of 20 mm, area of 314.2 mm\(^2\), area moment of inertia of 7854 mm\(^4\), and a length \( L \) of 4 m. A point load \( P = 100 \, {N} \) acts at the center and an axial load \( Q = 20 \, {kN} \) acts through the centroidal axis as shown in the figure.
The magnitude of the offset between the neutral axis and the centroidal axis, at \( L/2 \) from the left, is ............ mm (rounded off to one decimal place).

A simply-supported beam, with a point load \( P = 150 \, {kN} \) at a distance of \( L/3 \) from the left end, is shown in the figure. The elastic-strain energy \( U \) of the beam is given by the following expression:
\[ U = \frac{2}{243} \frac{P^2 L^3}{EI}, \] where the section modulus, \( EI \), is \( 16.66 \times 10^5 \, {Nm}^2 \) and the length of the beam \( L \) is 1 m.
The deflection at the loading point is ............ mm (rounded off to two decimal places).

The figure shows the stress distribution across an internal surface of a rectangular beam of height 30 mm and depth 10 mm. The normal stress distribution is given by the expression \( \sigma_{xx} = 200y + 500 \, {N/mm}^2 \), where \( y \) is the distance in mm from the centroidal axis of the beam. Assume that there is no variation in the stress distribution along the z-direction.

An infinitesimal square element PQRS is shown in the figure. The x and y axes are also marked in the figure. The strains on the element are given by \( \varepsilon_{xx} = 500 \times 10^{-6}, \, \varepsilon_{yy} = 100 \times 10^{-6} \) and \( \varepsilon_{xy} = 0 \). 
Which of the following statements is/are CORRECT?

The frame shown in the figure is loaded at S with a force of 2000 N. The reactions at T are denoted by \( T_x \) and \( T_y \), while the reaction at W is \( W_y \). Neglect the weight of the members. Which one of the following options for the magnitudes of the forces (in N), \( T_x \), \( T_y \), and \( W_y \), is CORRECT?

Consider a spring-mass system with mass \( m \) and spring stiffness \( k \) as shown in the illustration. At time \( t = 0 \), the mass is displaced by \( P \) units and the velocity of the mass is zero. The displacement of the mass, \( x(t) \), is measured from the equilibrium position. Which one of the following functions represents \( x(t) \)? 

Corrosion of pure iron takes place in an acidic electrolyte by forming \( {Fe}^{2+} \) ions at ambient condition. The corrosion current density is measured to be \( 2 \times 10^{-4} \, {A cm}^{-2} \). The corrosion rate (in mm per year) of iron is (rounded off to one decimal place) ............

An aluminum transmission line of 7 km length is designed to carry 100 A current with no more than 2 MW power loss. The required minimum diameter (in mm) of the transmission line is (rounded to the two decimal places) ........... 

On applying 10 V across the two ends of a 100 cm long copper wire, the average drift velocity (in cm s\(^{-1}\)) in the wire is (rounded off to two decimal places).............

For a pure element with a BCC crystal structure, the surface energies per unit area of \( \{100\} \) and \( \{110\} \) free surfaces are \( S_{100} \) and \( S_{110} \), respectively. The ratio, \( \frac{S_{100}}{S_{110}} \), is (rounded off to one decimal place): 

A diffraction pattern is obtained from a powdered sample of a pure element, which has FCC crystal structure. If \( x \) and \( y \) are the Bragg angles of the first and the third peaks, respectively, then the ratio, \( \frac{\sin y}{\sin x} \), is (rounded off to one decimal place)........... 

The triple point \( (T_t, P_t) \) is shown in a schematic phase diagram (pressure (P) – temperature (T) plot) for one component system. \( G_S, G_L, G_V \) are the free energies of solid, liquid, and vapor, respectively. At a constant pressure, \( P_t \), the correct free energy \( G \) versus temperature \( T \) plot is:

Match the detector for a scanning electron microscope (SEM) in Column I with the resulting output in Column II. 

In an FCC crystal with lattice parameter \( a \), consider the reaction of two leading partial dislocations, AB and CD, at the line of intersection of their slip planes \( (111) \), respectively, as shown in the figure below. Dislocations AB and CD, have Burgers vectors \( \mathbf{b_1} \) and \( \mathbf{b_2} \), respectively, as given in the figure. Which one of the following options for the slip plane and the Burgers vector of the resulting dislocation is correct? 

The figure below shows a plane PQR in a unit cell. The Miller indices of the plane PQR is: 

A fixed control volume has four one-dimensional boundary sections (1, 2, 3, and 4). For a steady flow inside the control volume, the flow properties at each section are tabulated below: 

The rate of change of energy of the system which occupies the control volume at this instant is \( E \times 10^6 \, {J/s} \). The value of \( E \) (rounded off to 2 decimal places) is ........

A liquid flows under steady and incompressible flow conditions from station 1 to station 4 through pipe sections P, Q, R, and S as shown in the figure. Consider, \( d \), \( V \), and \( h \) represent the diameter, velocity, and head loss, respectively, in each pipe section with subscripts ‘P’, ‘Q’, ‘R’, and ‘S’. \( \Delta h \) represents the head difference between the inlet (station 1) and outlet (station 4). All the pipe sections are placed on the same horizontal plane for which the figure shows the top view. 
(Insert diagram here, if possible)