List of top Questions asked in GATE PI

In the composite filament winding process as shown in the figure, the mandrel diameter is 700 mm and is rotating at a speed, \( N = 6 \) rev/min.
If a \( 45^\circ \) helical winding angle is needed, the axial velocity, \( v_c \), of the slider should be ______ m/s (rounded off to two decimal places).
Assume \( \pi = 22/7 \). Ignore the thickness of the composite layers already wound on the mandrel.

Note: Figure is not to scale.
Consider an assembly line producing an item through a process involving three activities in the following order: cutting, welding, and polishing. There are three workers available: one is assigned only for cutting, one is assigned only for welding, and one is assigned only for polishing.
The activity time per item is given in the table.
Activity nameCuttingWeldingPolishing
Activity time/item (in minutes)304025

Assuming that the process is running in steady state, the total number of items coming out of the assembly line in six hours is ______ (in integer).
Note: Assume that workers, items, and machines/tools are available as and when needed.
A time study was carried out for a job broken into two elements. For each element, three iterations of task were observed, and observed time was recorded. The observed times (in seconds) and the rating factor for each element are given in the table.
Work elementObserved time (seconds)Rating factor
A20; 25; 21105%
B19; 17; 1890%

If a further 20% needs to be added for allowances, then the standard output of the worker is ______ units per hour (in integer).
In a hole-shaft assembly, the following are the dimensions and tolerances (all in mm) provided for the hole and shaft:
Hole: \( 22^{+0.018}_{+0.010} \)
Shaft: \( 22^{+0.075}_{+0.004} \)
The magnitude of the clearance when the shaft is at the minimum material condition and hole is at the maximum material condition is ______ mm (to three decimal places).

If \( \frac{dy}{dt} = 2y \), and the value of \( y \) at \( t = 0 \) is 2, then the value of \( y \) at \( t = 1 \) is ______ (rounded off to two decimal places).

The computer centre (CC) in an institute manages the central server for which researchers queue to run their computer programs. CC processes programs only one at a time. Four computer programs are available at 10:00 am on a given day. The time required to run each program on the central server and their respective promised time of completion (i.e., due date) on that same day are given in the table.

Program1234
Processing time (in minutes)604090100
Promised time (i.e., due date)12 noon11:30 am1:00 pm1:30 pm

If the CC follows shortest processing time (SPT) rule to sequence the computer programs, then number of tardy jobs is ______.

Note: Ignore program change-over times.

The number of soldering defects that occur in a semiconductor device follows a discrete Poisson distribution with a probability mass function
\[ p(x) = \frac{e^{-\lambda}\lambda^x}{x!} \]
The average number of soldering defects per semiconductor device is 3.
The probability that a randomly selected semiconductor device will have at least two soldering defects is ______ (rounded off to two decimal places).

A company has a fixed cost of INR 3,00,000 and a variable cost of INR 150 per unit for manufacturing of a product. The company sells 5,000 units of that product making a profit equivalent to 20% of the total sales revenue.

The break-even quantity for that product is _____ units (rounded off to the nearest integer).

A custom pinion, with a width equal to 10 times the module, has 20 full depth teeth and a pressure angle of 20 degrees is being designed. It should transmit a torque of 95 N-m to a corresponding spur gear.

Considering the safe bending stress to be 180 MPa and form factor to be 0.342, the module of the pinion is ______ mm (rounded off to one decimal place).

Note: The Lewis equation gives the tangential force as \( F_t = \sigma b Y m \), where \( \sigma \) is the safe bending stress, \( b \) is the width, \( Y \) is the form factor, and \( m \) is the module.

A company must allocate oil produced from their two plants to meet all the demands of two markets. The cost per litre of allocation from plant \( i \in \{1,2\} \) to market \( j \in \{1,2\} \) is denoted by \( C_{ij} \). The market demand \( D_j \), plant production capacity \( K_i \) and \( C_{ij} \) values are given in the table.

Market 1 (j = 1)Market 2 (j = 2)Capacity \( K_i \) (litres)
Plant 1 (i = 1)\( C_{11} = 250 \)\( C_{12} = 280 \)\( K_1 = 500 \)
Plant 2 (i = 2)\( C_{21} = 150 \)\( C_{22} = 180 \)\( K_2 = 600 \)
Demand \( D_j \) (litres)\( D_1 = 300 \)\( D_2 = 400 \)

The company hired an intern to formulate an optimization model to decide on the quantity (\( X_{ij} \)) to be allocated from plant \( i \) to market \( j \), and the formulation is given below:

Minimize \( Z = \sum_{i=1}^{2} \sum_{j=1}^{2} C_{ij} X_{ij} \)

Subject to:
\( \sum_{j=1}^{2} X_{ij} \le K_i \quad \forall i \in \{1,2\} \)
\( \sum_{i=1}^{2} X_{ij} \le D_j \quad \forall j \in \{1,2\} \)
\( X_{ij} \ge 0 \)

The optimal value of the objective function of the linear programming problem formulated by the intern is ______ (in integer).

A cylindrical metal component of 10 mm diameter is subjected to uniform uniaxial tension during operation. It was observed that a force of 14 kN produces a uniform reduction of \(3 \times 10^{-3}\) mm in diameter. Assume that material behaviour is homogeneous, isotropic, and linear elastic.

If its Young's modulus is 150 GPa, the Poisson's ratio of the material is ______ (rounded off to two decimal places).

Note: Assume \(\pi = 3.14\).
Which ONE or MORE among the following options is/are TRUE for the ideal operating characteristics (OC) curve given in the figure?

Note: \( p_0 \) represents the acceptable quality level (AQL) and \( p_1 \) represents the lot tolerance percent defective (LTPD).

In laser beam machining, the time (\(t_m\)) required for the material to attain the melting temperature from a room temperature (\(\theta_0\)) of 32°C is expressed by the following expression:

\[ t_m = \frac{\pi}{\alpha} \left( \frac{(\theta_m - \theta_0) k}{2H} \right)^2 \]

where \(\alpha\) is thermal diffusivity, \(\theta_m\) is melting temperature, \(k\) is thermal conductivity, \(H\) is heat flux.

If a uniformly distributed 1 kW power laser beam with a beam diameter of 0.1 mm is used for machining tungsten carbide, and 10% of beam absorption is assumed, the time \(t_m\) is ______ \(\mu s\) (rounded off to one decimal place).

Note: Thermal properties of tungsten carbide: melting temperature = 3400°C; thermal conductivity = 2.15 W/cm-°C; diffusivity = 0.79 cm\(^2\) s\(^{-1}\); assume \(\pi\) = 3.14.
A linear slot is to be milled in a single pass from the point (0, 0) to (180, 180) on XY plane by a CNC machine. The actual velocity along the Y-axis is 5% less than the intended value. The programmed feed rate along the intended slot is 150 mm/min.

The magnitude of the positional error along the Y-axis when the X-coordinate reaches 180 mm, is ______ mm (in integer).

Note: All coordinates are in mm.
The operation chart of a method of assembling nuts and bolts for a left-handed operator has been illustrated in the figure.

Which ONE or MORE among the following options is/are CORRECT?

In the measurement of surface roughness using a 2D stylus profilometer, a surface profile measured over a length of 0.8 mm was recorded on a graph paper. During recording, vertical magnification of 10,000 and horizontal magnification of 100 were used. The areas in the as-recorded graph above and below the datum line are as follows:

Above (mm\(^2\)): 140, 60, 150, 50
Below (mm\(^2\)): 70, 50, 120, 160

The average surface roughness (\(R_a\)) of the surface is ______ \(\mu m\) (rounded off to two decimal places).
Which ONE or MORE among the following work materials is/are NOT commonly machined using diamond cutting tools?
In a rolling operation, a 200 mm wide strip of 23 mm thickness needs to be reduced to 20 mm in a single pass. The roll diameter is 200 mm. Four lubricants P, Q, R, and S with coefficient of friction 0.05, 0.1, 0.2, and 0.25, respectively, are available for use at the roll-strip interface. Assume that the strip width remains constant throughout the process.

Which ONE or MORE among the following lubricants will enable the rolling process to achieve the desired final thickness of the strip in a single pass?
In the theory of orthogonal machining, assuming that the shear plane will take up an angle that minimizes energy, the shear angle can be shown to be

\[ \phi = C - \frac{\beta}{2} + \frac{\alpha}{2} \]

where \( \phi \) is the shear angle, \( \alpha \) the rake angle, and \( \beta \) the friction angle.

Which ONE or MORE among the following is/are TRUE?
Which ONE or MORE among the following homogeneous representations in robotics involve(s) rotation about Y-axis by an angle \( \theta \)?
In an M/M/1 queuing system, the customers arrive at a rate of 3 per minute, and the average number of customers in the system is \( n \) at steady state. Assume that the server utilization is less than 1.
Which ONE of the following is the average time a customer spends in the system (in minutes) at steady state?
A glass tube of 2.5 mm diameter is immersed vertically in a fluid.
Assume contact angle is zero. The surface tension of the fluid is 0.1 N/m, density of the fluid is \( 1000\ kg/m^3 \) and acceleration due to gravity is \( 10\ m/s^2 \).
The approximate capillary rise is ______ mm.
A steel cube of side 10 cm is made by the sand-casting process. A cylindrical side-riser with diameter, \( d \), and height, \( h \), needs to be used.
Assume:
  • No surface sharing between the riser and casting.
  • \( d = h \).
  • The connecting link between the riser and casting does not freeze before the casting.
  • All the surfaces of the riser and casting are subjected to identical cooling conditions.
In this situation, which ONE or MORE among the following values of \( d \) (in cm) can theoretically fully compensate for shrinkage during casting?
A sheet metal drawing is considered feasible if the following conditions are met:
  • Reduction ratio < 0.5
  • Thickness to diameter ratio > 1%
A sheet metal drawing process is being planned on a sheet metal with a thickness of 2 mm with a punch of diameter 100 mm with various blanks of diameter \( D_b \).
Which ONE of the following blank diameters \( D_b \) (in mm) will result in feasible drawing?
Iron powder produced by gas atomization has a tap density and an apparent density of 3.26 g/cm\(^3\) and 2.66 g/cm\(^3\), respectively. The density of solid iron is 7.85 g/cm\(^3\).
Which ONE of the following options is the closest approximation to the degree of densification (in percentage) of this iron powder?