The GATE 2025 PI Question paper with Solution PDF is available to download here. GATE 2025 was conducted by IIT Roorkee. As per the updated exam pattern, the exam consists of 65 questions totaling 100 marks, with 10 questions from General Aptitude and 55 questions covering Engineering Mathematics and Production & Industrial Engineering topics.
The difficulty level of GATE 2025 PI was moderate.
GATE 2025 PI Question Paper with Solutions PDF
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GATE 2025 PI Question Paper with Solutions
Courage : Bravery :: Yearning : _______
Select the most appropriate option to complete the analogy.
We ______ tennis in the lawn when it suddenly started to rain.
Select the most appropriate option to complete the above sentence.
A \(4 \times 4\) digital image has pixel intensities (\(U\)) as shown in the figure. The number of pixels with \(U \le 4\) is:
In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of \(P\), \(Q\), and \(R\)?
A rectangle has a length \(L\) and a width \(W\), where \(L > W\). If the width \(W\) is increased by \(10%\), which one of the following statements is correct for \emph{all values of \(L\) and \(W\)?
Column–I has statements made by Shanthala; Column–II has responses by Kanishk. Identify the option that has the correct match between Column–I and Column–II.
\begin{tabular{p{0.42\linewidth p{0.42\linewidth
Column–I & Column–II
P. This house is in a mess. & 1. Alright, I won’t bring it up during our conversations.
Q. I am not happy with the marks given to me. & 2. Well, you can easily look it up.
R. Politics is a subject I avoid talking about. & 3. No problem, let me clear it up for you.
S. I don’t know what this word means. & 4. Don’t worry, I will take it up with your teacher.
\end{tabular
Weight of a person can be expressed as a function of their age. Suppose this function is identical for two brothers and it monotonically increases till the age of 50 years and then monotonically decreases. Let \(a_1\) and \(a_2\) (in years) denote the ages of the brothers and \(a_1 < a_2\). Which one of the following statements is correct about their ages on the day when they attain the same weight?
A regular dodecagon (12-sided polygon) is inscribed in a circle of radius \(r\). The side of the dodecagon is \(d\). All the 12 isosceles triangles (numbered \(1\) to \(12\)) are used to form squares of side \(r\), with each triangle used only once. The number of squares that can be formed and the number of triangles required to form each square are:
If a real variable \(x\) satisfies \(3^{x^2}=27\times 9^x\), then the value of \(\dfrac{2^{x^2}}{(2^x)^2}\) is:
The number of patients per shift \((X)\) consulting Dr.\ Gita in her past 100 shifts is shown in the figure. If the amount she earns is \(Rs.\,1000\,(X-0.2)\), what is the \emph{average amount (in Rs.) she has earned per shift in the past 100 shifts?
\emph{(From the graph: 20 shifts had \(X=5\), 40 shifts had \(X=6\), 30 shifts had \(X=7\), and 10 shifts had \(X=8\).)
The eigenvalues of the matrix \(\begin{bmatrix}0 & -1
1 & 0\end{bmatrix}\) are:
If \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are orthogonal unit vectors in Cartesian \(x\)–\(y\)–\(z\) system, the curl of the vector \(\mathbf{F}=-2y\,\mathbf{i}+x\,\mathbf{j}\) is:
If \(F(s)\) denotes the Laplace transform of some function \(f(t)\), then the Laplace transform of \(e^{bt}f(t)\), where \(b\) is a real constant, is:
Which one of the following equations is a linear differential equation?
A bag contains 5 red, 7 green and 3 blue balls. Two balls are drawn at random from the bag one-by-one. The probability of the \emph{second} drawn ball being red is
Exit-hole occurrence is common in
An aircraft has two engines, each having a reliability \(R\). The aircraft will crash only when both engines stop working. The reliability of the aircraft flying without crash is:
The proper sequence of design of a product is:
In a work sampling, out of \(n\) observations, a worker was sitting idle in \(x\) observations. The standard deviation of the mean proportion of idle time is given by
Atomic packing factor of a body-centered cubic (BCC) structure is closest to
Which one of the following statements is FALSE with respect to the injection molding of polymer composite?
A through hole of \(8\,mm\) diameter is to be drilled in a \(30\,mm\) thick mild steel plate. Which process is the most appropriate to achieve \emph{high dimensional accuracy with less processing time?
Which one of the following casting defects is caused due to the supply of the molten metal through two gates?
Match the following with reference to the CNC machine and its minimum number of axes available in the machine.
\begin{tabular{|c|c|
\hline
Type of CNC Machine & Minimum number of axes
\hline
P: Turning center (CNC Lathe) & i) 3
Q: Machining center (CNC Vertical milling) & ii) 2
& iii) 4
\hline
\end{tabular
A simply supported beam \(AB\) of span \(L\) is shown in the figure. A moment \(M\) is applied at point \(C\) (located at a distance \(a\) from \(A\)). The magnitude of the reaction force at point \(A\) is:
The relationship between the hoop stress \(\sigma_{1}\) and the longitudinal stress \(\sigma_{2}\) of a closed cylindrical thin-walled pressure vessel is:
The starting simplex table of a linear programming problem is given below, where \(S_1, S_2, S_3\) and \(S_4\) are the slack variables. The objective is to maximize \(z=6x_1+4x_2\). The leaving variable among the basic variables is:
For an ideal Diesel cycle, the heat addition is an
Three principal stresses at a point in a material are \(300\) MPa, \(250\) MPa and \(100\) MPa. If yielding just starts at that point, the yield strength (in MPa) of the material as per Tresca criterion is \underline{\hspace{1.5cm. (Answer in integer)
In an orthogonal straight turning process, the feed is \(0.1\) mm/rev and the depth of cut is \(0.5\) mm. In ASA system, the side cutting edge angle of the cutting tool is \(0^\circ\). The width (in mm) of the chip is \underline{\hspace{1.5cm. (Rounded off to one decimal place)
The pitch of the single-start lead screw of a lathe is \(6\) mm. It is used to cut a double-start thread of \(3\) mm pitch on a cylindrical workpiece. During thread cutting, the spindle rotates at \(400\) RPM. The speed (in RPM) of the lead screw is \underline{\hspace{1.5cm. (Answer in integer)
Two options are available to meet the annual demand of batteries in a toy company. In option 1, batteries are manufactured in the plant having fixed cost of Rs.~2,00,000 and a variable cost of Rs.~70 per unit. Option 2 consists of buying batteries from the market at a price of Rs.~90 per unit. The annual demand (in number of batteries) at which the company should switch from buying to making the batteries in the plant is \hspace{1.5cm}. (Answer in integer)
A company estimates the demand of 2000 bulbs for the next year. The ordering cost is Rs.~300 per order and the annual carrying cost per bulb is Rs.~30. The economic order quantity (number of bulbs) is \hspace{1.5cm}. (Answer in integer)
While inspecting final assembly of automobile gear-boxes, 15 features were considered critical-to-quality (CTQ). During the last quarter, 40{,}000 gear boxes were produced among which 1{,}500 defects were found of the CTQ features. The defects per million opportunities (DPMO) is \hspace{1.5cm}. (Answer in integer)
The hole and the shaft dimensions (in mm) are given as
Hole dimension \(= 30^{+0.04}_{-0.02}\)
Shaft dimension \(= 30^{+0.06}_{-0.03}\)
The maximum possible clearance (in mm) is \hspace{1.5cm}. (Rounded off to two decimal places)
The solution of the linear differential equation \(\dfrac{dy}{dx} + y = e^{x}\), when \(y(0)=0\), is:
Which one of the following functions is analytic, given \(i=\sqrt{-1}\)?
Match the following with reference to the machining process and its feature:
\[ \begin{array}{|c|c|} \hline \textbf{Process} & \textbf{Feature}
\hline P \; EDM & 1 \; Loss of dimensional accuracy due to under cutting
Q \; LBM & 2 \; Cutting edible items
R \; CHM & 3 \; Machining of a deep square blind hole on a mild steel plate
S \; WJM & 4 \; High speed profile cutting on a thin mild steel plate
\hline \end{array} \]
Match the operation/phenomenon in a grinding process with the corresponding definition listed in the table.
\[ \begin{array}{|c|c|} \hline \textbf{Operation/phenomenon} & \textbf{Definition}
\hline P \; Loading & 1 \; Regenerating the sharpness of the grinding wheel
Q \; Glazing & 2 \; Filling of grinding chips in the space between the abrasive grits
R \; Dressing & 3 \; Restoring the geometry/shape of the grinding wheel
S \; Truing & 4 \; Condition of dull grinding wheel with worn-out grains
\hline \end{array} \]
A CNC vertical milling machine is cutting a straight-line slot in the \(x\)–\(y\) plane. The cutter starts at point \(P(10,\,5)\) (all coordinates in mm). The slope of the straight line created by the cutter is \(\left(\dfrac{dy}{dx}\right)=1.25\). The \(x\)-axis feed rate is \(120\ mm/min\). What is the new position of the cutter after \(20\ s\)?
The network diagram of eight activities (A to H) with their durations (in days) is shown. Find the critical path of the project.
The benefit(s) of product standardization is/are
The value of the integral \(\displaystyle \int_{1}^{3} (x^{2}-2x)\,dx\) obtained using Simpson’s \(1/3\) rule with 4 subintervals is equal to \(\dfrac{n}{3}\). The value of \(n\) is ______. (Answer in integer)
If \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are the orthogonal unit vectors in Cartesian \(x\)-\(y\)-\(z\) coordinates, the rate of change of the function \(f(x,y,z)=x^{2}+2y^{2}+z\) at point \((1,1,1)\) in the direction of \(3\mathbf{i}+4\mathbf{k}\) is ______. (Answer in integer)
In a wire drawing of a perfectly plastic material with flow stress of \(300\,MPa\), the back tension is zero and the front tension is \(200\,MPa\). Assuming ideal deformation with zero friction, the percentage reduction of the cross-sectional area of the wire is ______. (Rounded off to one decimal place)
In a cold rolling process without front and back tensions, the required minimum coefficient of friction is \(0.04\). Assume large rolls. If the draft is doubled and roll diameters are halved, then the required minimum coefficient of friction is ______. (Rounded off to two decimal places)
In a direct current arc welding, the voltage \(V\) (in volt) is related to the arc length \(l\) (in cm) as \(V=30+30l\). The open-circuit voltage is \(80\) V. The maximum possible arc length (in cm) is ______. (Rounded off to two decimal places)
A worker is allowed half an hour personal time in a normal 8-hour shift. If the normal time for manufacturing a product is \(5\) minutes, the standard time (in second) is ______. (Answer in integer)
A product has to be manufactured in a single-line layout by carrying out the six tasks in a sequence. The time (in minute) of the six sequential tasks are 37, 8, 19, 34, 36 and 17. These tasks cannot be further sub-divided. For minimizing the cycle time, the number of stations to be used is ................ (Answer in integer)
A through hole of 10 mm diameter is to be drilled in a mild steel plate of 30 mm thickness. The selected spindle speed and feed for drilling hole are 600 revolutions per minute (RPM) and 0.3 mm/rev, respectively. Take initial approach and breakthrough distances as 3 mm each. The total time (in minute) for drilling one hole is ................ (Rounded off to two decimal places)
In the iron-carbon equilibrium phase diagram, the eutectoid reaction occurs at 723 °C with the eutectoid composition of 0.83 weight % carbon. Ferrite and cementite phases are considered to contain 0.022 weight % carbon and 6.67 weight % carbon, respectively. If a steel specimen with 0.7 weight % carbon is cooled from 950 °C to below 723 °C, the fraction of eutectoid ferrite is .................. (Rounded off to two decimal places)
During orthogonal cutting with a tool of 10° rake angle, the cutting and thrust forces are 900 N and 275 N, respectively. The coefficient of friction on the rake surface of the tool is ................. (Rounded off to two decimal places)
In casting a cube of 80 mm side, the volumetric shrinkages due to solidification and solid contraction are 4.5% and 2%, respectively. Assume uniform cooling in all directions. The side (in mm) of the cubical pattern for getting the required size casting is ....... (Rounded off to two decimal places)
The solidification of a casting starts at 10 AM. However, the solidification of the molten metal at the center-line of the mold starts at 10:03 AM and ends at 10:10 AM. The casting is considered solidified completely when the solidification is completed at the center-line of the mold. The center-line feeding resistance (CFR) in percentage is ........... (Answer in integer)
A link \(OA\) of length 200 mm is rotating counter clockwise about \(O\) in \(x\!-\!y\) plane with a constant angular velocity of 100 rad/s, as shown in the figure. The absolute value of the \(x\)-component of the linear velocity (in m/s) of point \(A\) at the instant shown in the figure is ................ (Rounded off to one decimal place)
A force of 1000 N is acting at point \(A\) on a bracket fixed at point \(B\) as shown in the figure. The magnitude of the moment of the force about \(B\) (in N·m) is ................. (Rounded off to one decimal place)
A steel plate is fastened to a channel using three identical bolts as shown in the figure. The bolts are made of carbon steel of permissible yield strength in shear as 400 N/mm\(^2\). The plate is subjected to a force of 12 kN. Neglect the weight of the plate. The magnitude of the resultant shear force (in N) on bolt 2 is ................ (Answer in integer)
The annual profit of a company depends on its annual marketing expenditure. The information of preceding 3 years’ annual profit and marketing expenditure is given in the table. Based on linear regression, the estimated profit (in units) of the 4th year at a marketing expenditure of 5 units is .............. (Rounded off to two decimal places)
\begin{tabular{|c|c|c|
\hline
Year & Expenditure for marketing (units) & Annual profit (units)
\hline
1 & 3 & 22
2 & 4 & 27
3 & 6 & 36
\hline
\end{tabular
Three plants \(P_1, P_2\) and \(P_3\) produce \(6,\;1,\;9\) thousand liters of fruit juice, respectively. The produced juice is to be transported to three distribution centers \(D_1,\;D_2,\;D_3\) with requirements \(7,\;5,\;4\) thousand liters, respectively. The transportation cost (in \emph{hundreds of Rupees) from each plant to each distribution center is:
\[ \begin{array}{c|ccc|c} & D_1 & D_2 & D_3 & Supply
\hline P_1 & 2 & 3 & 11 & 6
P_2 & 1 & 0 & 6 & 1
P_3 & 5 & 8 & 15 & 9
\hline Demand & 7 & 5 & 4 & \end{array} \]
Find the total transportation cost (in \emph{hundreds of Rupees) of the \underline{initial basic feasible solution using Vogel’s Approximation Method (VAM). (Answer in integer)
A company purchases items in bulk with a quantity discount. Annual demand \(D=5000\) units; ordering cost \(S=Rs. 400\) per order. Annual inventory carrying cost \(=\) \(30%\) of unit price. Price breaks: \[ \begin{array}{c|c} Quantity interval Q & Unit price p(Rs.)
\hline 0\le Q<1200 & 10
1200\le Q<2000 & 8
2000\le Q & 7 \end{array} \]
Find the \emph{optimal order size \(Q^*\) (Answer in integer).
The zero line of the Vernier scale lies between divisions 20 and 21 of the main scale. The 4th Vernier scale division exactly coincides with a main scale division. The 5 divisions of the Vernier scale are equal to 4 divisions of the main scale. If one main scale division is 1 mm, the measured value (in mm) is ............... (Rounded off to one decimal place)
A broaching machine makes key slots with a mean dimension of \(10.56\ mm\) and a standard deviation of \(0.05\ mm\). The upper control limit for mean of sample size \(5\) calculated using \( \bar{X} \) chart is ............. (Rounded off to two decimal places)
The table shows the data of running a machine for five years. The original machine cost is Rs. 70{,000. In order to minimize the \emph{average total cost per year for running the machine, after how many years should the machine be replaced? (Answer in integer)
% Requires: \usepackage{graphicx
\begin{table[h]
\centering
\begin{tabular{|c|c|c|c|c|c|
\hline
& 1st year & 2nd year & 3rd year & 4th year & 5th year
\hline
Resale value (Rupees) & 40000 & 30000 & 25000 & 22000 & 20000
\hline
Maintenance cost (Rupees) & 19100 & 20300 & 23500 & 30500 & 40000
\hline
\end{tabular
\caption{Resale Value and Maintenance Cost Over 5 Years
\label{tab:resale_maintenance
\end{table
Water flows through a smooth circular pipe of diameter \(10\ cm\) and length \(10\ m\). The pressure drop across the length of the pipe is \(0.2\ Pa\). Kinematic viscosity and density of water are \(1\times10^{-6}\ m^2/s\) and \(1000\ kg/m^3\), respectively. Assuming laminar and fully developed flow throughout the pipe, the velocity of water (in mm/s) at the \emph{center of the pipe is .............. (Rounded off to one decimal place)
The left-hand side of a \(20\ cm\) thick wall is maintained at \(25^\circC\). The right-hand side of the wall is exposed to hot air at \(50^\circC\). There is no heat generation in the wall; its thermal conductivity is \(k=100\ W/m·K\). The convective heat transfer coefficient on the right side is \(h=50\ W/m^2\!\cdot\!K\). Under steady state, the temperature (in \(^{\circ}C\)) of the \emph{right-hand side surface of the wall is ............... (Rounded off to one decimal place)
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