List of top Questions asked in GATE ME

Equal sized circular regions are shaded in a square sheet of paper of 1 cm side length. Two cases, case M and case N, are considered as shown in the figures below. In the case M, four circles are shaded in the square sheet and in the case N, nine circles are shaded in the square sheet as shown.
What is the ratio of the areas of unshaded regions of case M to that of case N?

Given \[ \int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}. \] If $a$ and $b$ are positive integers, the value of 

\(\int_{-\infty}^{\infty} e^{-a(x+b)^2}\, dx \text{ is} \)______

Wien's law is stated as \( \lambda_m T = C \), where \( C = 2898\ \mu\text{mK} \) and \( \lambda_m \) is the wavelength at which the emissive power of a black body is maximum for a given temperature T. The spectral hemispherical emissivity curve shows a maximum at \( \lambda_m \approx 6000\ \text{\AA} \). The temperature at which the total hemispherical emissivity is highest is ________________ K (nearest integer).
If the sum and product of eigenvalues of a $2 \times 2$ real matrix $\begin{bmatrix} 3 & p \\ p & q \end{bmatrix}$ are 4 and $-1$ respectively, then $|p|$ is ____________ (in integer).
Given $z = x + iy$, $i = \sqrt{-1}$. $C$ is a circle of radius $2$ centred at the origin and traversed anticlockwise. The value of \[ \frac{1}{2\pi i} \int_C \frac{1}{(z-i)(z+4i)}\, dz \] is ____________ (round off to one decimal place).
At steady state, 500 kg/s of steam enters a turbine with specific enthalpy \(h_1 = 3500\ \text{kJ/kg}\) and entropy \(s_1 = 6.5\ \text{kJ . kg}^{-1}\text{K}^{-1}\). It expands reversibly in the turbine at 500 K. Exit properties: \(h_2 = 2500\ \text{kJ/kg}\), \(s_2 = 6.3\ \text{kJ . kg}^{-1}\text{K}^{-1}\). Find turbine work output (MW, integer).
Consider steady one-dimensional heat conduction in a slab of thickness $2L$ (with $L=1$ m). Thermal conductivity varies as $k = C T$, with $C = 2$ W·m$^{-1}$·K$^{-2}$, and the slab generates heat at $1280$ kW/m$^3$. Both faces are held at 600 K. The temperature at $x=0$ is __________ K (in integer).
Saturated vapor at 200°C condenses to saturated liquid at 150 kg/s on the shell side of a heat exchanger (latent heat $h_{fg} = 2400$ kJ/kg). A fluid with $c_p = 4$ kJ·kg$^{-1}$·K$^{-1}$ enters the tube side at 100°C. Effectiveness = 0.9. The required tube-side mass flow rate is __________ kg/s (in integer).
A hydrodynamically and thermally fully-developed fluid of 1 kg/s flows in a uniformly heated pipe (diameter 0.1 m, length 40 m). A constant heat flux of 15000 W/m$^2$ is applied. The fluid enters at 200$^\circ$C. Given: Re = 85000, Pr = 5, $k = 0.08$ W/m·K, $c_p = 2600$ J/kg·K. Correlation: \[ \text{Nu} = 0.023\, \text{Re}^{0.8} \text{Pr}^{0.4} \] Find the pipe surface temperature at the exit (nearest integer).
The best size wire is fitted in a groove of a metric screw such that the wire touches the flanks of the thread on the pitch line. The pitch $p = 4$ mm and the included angle of the thread is $60^\circ$. The diameter of the best size wire is ____________ mm (round off to 2 decimal places).
In a DC arc welding process, open-circuit voltage is 100 V and short-circuit current is 1000 A. Arc voltage varies as $V = 10 + 5l$, where $l$ is arc length in mm. The maximum available arc power during the process is ____________ kVA (in integer).
A cylindrical billet of diameter 100 mm and length 100 mm is directly extruded into an L-section. The extrusion pressure is given by \[ p = K_s\, \sigma_m \left[0.8 + 1.5\ln(r) + \frac{2l}{d_0}\right] \] where \( \sigma_m = 50\ \text{MPa} \), \( K_s = 1.05 \), \( d_0 = 100\ \text{mm} \). Find the maximum force at the start of extrusion (in kN).
A project consists of five activities (A, B, C, D, E). Each activity duration follows beta distribution. Three-time estimates are given and expected completion time is __________ weeks (in integer).
A rigid tank (8 m$^3$) is filled with air from a pipeline at 600 kPa and 306 K. Heat loss during filling = 1000 kJ. Final tank pressure equals pipeline pressure. Final temperature is __________ K (rounded to nearest integer).
A uniform wooden rod (specific gravity = 0.6, diameter = 4 cm, length = 8 m) is hinged at A at waterline. A lead sphere (specific gravity = 11.4) is attached at the free end to keep the rod submerged horizontally. Find the radius of the lead ball (in cm, round to 2 decimals).
A shaft of length \(L\) is made of two materials—an inner core and an outer rim—perfectly bonded with no slip. The inner core has diameter \(d_i\), the outer diameter is \(d_o\), and the rigidities are \(G_i\) and \(G_o\) respectively. Given: \( d_o = 2 d_i \) and \( G_i = 3 G_o \). Under an applied torque, the maximum shear stresses in the outer rim and inner core are \( \tau_o \) and \( \tau_i \). Find the ratio \( \tau_i / \tau_o \) (round off to 2 decimals).
A rigid beam AD of length $3a = 6$ m is hinged at A and supported by two strings as shown. A force $F = 9$ kN is applied at C. Assuming small deflection and linear elastic strings, the tension in the string at C is __________ kN (round off to 2 decimal places).
In the four-bar mechanism shown, the $2$ cm input link rotates at $\omega_2 = 5$ rad/s. Given the geometry, the magnitude of angular velocity $\omega_4$ of the 4 cm link is __________ rad/s (round off to 2 decimal places).
A shaft carries a thin pulley of radius \( r = 0.4\ \text{m} \) at end C. Taut and slack belt tensions are \( T_1 = 300\ \text{N} \) and \( T_2 = 100\ \text{N} \). Allowable shear stress in the shaft is 80 MPa. The shaft length segments are \( AB = BC = L = 0.5\ \text{m} \). Neglecting shock and fatigue and using maximum shear stress theory, find the minimum shaft diameter (in mm).
A straight–teeth horizontal slab milling cutter has 4 teeth and diameter $D = 200$ mm. Rotational speed = 100 rpm, Feed = 1000 mm/min, Workpiece width $w = 100$ mm. Cutting force/tooth = $F = K\, t_c\, w$, $K = 10$ N/mm$^2$. Depth of cut is $d = D/2$, and the maximum cutting force is __________ kN (round off to one decimal place).
In an orthogonal machining operation, the cutting and thrust forces are equal. Uncut chip thickness = 0.5 mm, shear angle = \(15^\circ\), rake angle = \(0^\circ\), width of cut = 2 mm. The work material is perfectly plastic with yield shear strength \(500\ \text{MPa}\). Find the cutting force (nearest integer).
Parts P1--P7 are machined first on a milling machine and then polished at a separate machine. Using the information in the following table, the minimum total completion time required for carrying out both the operations for all 7 parts is ____________________ hours.
A manufacturing unit produces two products P1 and P2. For each piece of P1 and P2, the table below provides quantities of materials M1, M2, and M3 required, and also the profit earned. The maximum quantity available per day for M1, M2 and M3 is also provided. The maximum possible profit per day is ₹ ________________.
A tube of uniform diameter $D$ is immersed in a steady flowing inviscid liquid stream of velocity $V$, as shown in the figure. Gravitational acceleration is $g$. The volume flow rate through the tube is ________________.
The steady velocity field in an inviscid fluid of density 1.5 is given to be $\vec{V} = (y^{2} - x^{2})\hat{i} + (2xy)\hat{j}.$ Neglecting body forces, the pressure gradient at $(x = 1, y = 1)$ is ________________.