GATE ME

1. In order to numerically solve the ordinary differential equation dy/dt = -y for t > 0, with an initial condition y(0) = 2. The value of the surface integralwhere S is the external surface of the sphere x2 + y2 + z2 = R2 is 3. Let f(z) be an analytic function, where z = x + iy . If the real part of f(z) is cosh x cos y , and the imaginary 4. Consider the system of linear equationsx + 2y + z = 52x + ay + 4z = 122x + 4y + 6z = bThe values of a and b such that th 5. Let $f(.)$ be a twice differentiable function from $\mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0}\in \ 7. If the value of the double integral$\int_{x=3}^{4} \int_{y=1}^{2} \frac{dydx}{(x + y)^2}$is $\log_e(\frac{a}{24})$, then 8. If $x(t)$ satisfies the differential equation$t \frac{dx}{dt} + (t - x) = 0$subject to the condition $x(1) = 0$, then th 9. Let $X$ be a continuous random variable defined on $[0, 1]$ such that its probability density function $f(x) = 1$ for $0 10. A ram in the form of a rectangular body of size $l = 9 \text{ m}$ and $b = 2 \text{ m}$ is suspended by two parallel rop 11. A rigid massless tetrahedron is placed such that vertex O is at the origin and the other three vertices A, B, and C lie 12. A three-hinge arch ABC in the form of a semi-circle is shown in the figure. The arch is in static equilibrium under vert 13. The velocity field of a two-dimensional, incompressible flow is given by $\overrightarrow{V} = \ 2sin{h}\,x\,\hat{i} + 14. In the pipe network shown in the figure, all pipes have the same cross-section and can be assumed to have the same frict 15. Steady, compressible flow of air takes place through an adiabatic converging-diverging nozzle, as shown in the figure. F 16. A liquid fills a horizontal capillary tube whose one end is dipped in a large pool of the liquid. Experiments show that 17. A plane, solid slab of thickness L, shown in the figure, has thermal conductivity k that varies with the spatial coordin 18. Consider incompressible laminar flow over a flat plate with freestream velocity of $u_{\infty}$. The Nusselt number corr 19. Consider a hydrodynamically fully developed laminar flow through a circular pipe with the flow along the axis (i.e., z d 20. Consider a slab of 20 mm thickness. There is a uniform heat generation of $ \dot{q} = 100 \text{ MW/m}^3 $ inside the sl 21. A condenser is used as a heat exchanger in a large steam power plant in which steam is condensed to liquid water. The co 22. Consider a hemispherical furnace of diameter $ D = 6 \text{ m} $ with a flat base. The dome of the furnace has an emissi 23. A set of jobs $U, V, W, X, Y, Z$ arrive at time $t = 0$ to a production line consisting of two workstations in series. E 24. A queueing system has one single server workstation that admits an infinitely long queue. The rate of arrival of jobs to 25. A company orders gears in conditions identical to those considered in the economic order quantity (EOQ) model in invento 26. At the current basic feasible solution (bfs) $v_0 (v_0 \in \mathbb{R}^5)$, the simplex method yields the following form 27. In a supplier-retailer supply chain, the demand of each retailer, the capacity of each supplier, and the unit cost in ru 28. For a ball bearing, the fatigue life in millions of revolutions is given by $L = \left(\frac{C}{P}\right)^n$, where $P$ 29. Which one of the following failure theories is the most conservative design approach against fatigue failure? 30. A solid massless cylindrical member of 50 mm diameter is rigidly attached at one end, and is subjected to an axial force 31. A band brake shown in the figure has a coefficient of friction of 0.3. The band can take a maximum force of 1.5 kN. The 32. The "Earing" phenomenon in metal forming is associated with 33. The grinding wheel used to provide the best surface finish is 34. The allowance provided to a pattern for easy withdrawal from a sand mold is 35. The most suitable electrode material used for joining low alloy steels using Gas Metal Arc Welding (GMAW) process is 36. The preparatory functions in Computer Numerical Controlled (CNC) machine programming are denoted by the alphabet 37. A cutting tool provides a tool life of 60 minutes while machining with the cutting speed of 60 m/min. When the same tool 38. Aluminium is casted in a cube-shaped mold having dimensions as 20 mm × 20 mm × 20 mm. Another mold of the same mold mate 39. A blanking operation is performed on C20 steel sheet to obtain a circular disc having a diameter of 20 mm and a thicknes 40. In an arc welding process, the voltage and current are 30 V and 200 A, respectively. The cross-sectional area of the joi 41. A flat surface of a C60 steel having dimensions of 100 mm (length) × 200 mm (width) is produced by a HSS slab mill cutte 42. The phases present in pearlite are 43. Which of the following beam(s) is/are statically indeterminate? 44. A horizontal beam of length 1200 mm is pinned at the left end and is resting on a roller at the other end as shown in th 45. The figure shows a thin cylindrical pressure vessel constructed by welding plates together along a line that makes an an 46. The change in kinetic energy $\Delta E$ of an engine is $300$ J, and minimum and maximum shaft speeds are $\omega_{min} 47. A linear spring-mass-dashpot system with a mass of 2 kg is set in motion with viscous damping. If the natural frequency 48. The Levi type-A train illustrated in the figure has gears with module $m = 8 \text{ mm/tooth}$. Gears 2 and 3 have 19 an 49. At the instant when OP is vertical and AP is horizontal, the link OD is rotating counter clockwise at a constant rate ω 50. A vibratory system consists of mass $m$, a vertical spring of stiffness $2k$ and a horizontal spring of stiffness $k$. T 51. A furnace can supply heat steadily at 1200 K at a rate of 24000 kJ/min. The maximum amount of power (in kW) that can be 52. Which one of the following statements regarding a Rankine cycle is FALSE? 53. A piston-cylinder arrangement shown in the figure has a stop located 2 m above the base. The cylinder initially contains 54. A heat pump (H.P.) is driven by the work output of a heat engine (H.E.) as shown in the figure. The heat engine extracts 55. Consider an air-standard Brayton cycle with adiabatic compressor and turbine, and a regenerator, as shown in the figure.

General Aptitude

1. If ‘→’ denotes increasing order of intensity, then the meaning of the words [smile → giggle → 2. Find the odd one out in the set: {19, 37, 21, 17, 23, 29, 31, 11} 3. In the following series, identify the number that needs to be changed to form the Fibonacci series.1, 1, 2, 3, 6, 8, 13, 4. The real variables $x, y, z$, and the real constants $p, q, r$ satisfy$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z 5. Take two long dice (rectangular parallelepiped), each having four rectangular faces labelled as 2, 3, 5, and 7. If throw 6. In the given text, the blanks are numbered (i)–(iv). Select the best match for all the blanks.Prof. P __(i)__ merely a m 7. How many combinations of non-null sets A, B, C are possible from the subsets of {2, 3, 5} satisfying the conditions: (i) 8. The bar chart gives the batting averages of VK and RS for 11 calendar years from 2012 to 2022. Considering that 2015 and 9. A planar rectangular paper has two V-shaped pieces attached as shown below.This piece of paper is folded to make the fol 10. Four equilateral triangles are used to form a regular closed three-dimensional object by joining along the edges. The an

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1

GATE ME 2024

MCQ (Single Correct Answer)

+1

-0.33

The value of the surface integral

where *S* is the external surface of the sphere *x ^{2} + y^{2} + z^{2} = R^{2}* is

A

B

$ 4 \pi R^{3} $

C

$ \frac{4\pi}{3} R^{3} $

D

$ \pi R^{3} $

2

GATE ME 2024

MCQ (Single Correct Answer)

+1

-0.33

Let ** f(z) ** be an analytic function, where ** z = x + iy **. If the real part of ** f(z) ** is ** cosh x cos y **, and the imaginary part of ** f(z) ** is zero for ** y = 0 **, then ** f(z) ** is

A

cosh x exp (−iy)

B

cosh z exp z

C

cosh z cos y

D

cosh z

3

GATE ME 2024

MCQ (Single Correct Answer)

+1

-0.33

Consider the system of linear equations

x + 2y + z = 5

2x + ay + 4z = 12

2x + 4y + 6z = b

The values of **a** and **b** such that there exists a non-trivial null space and the system admits infinite solutions are

A

a = 8, b = 14

B

a = 4, b = 12

C

a = 8, b = 12

D

a = 4, b = 14

4

GATE ME 2024

MCQ (Single Correct Answer)

+1

-0.33

Let $f(.)$ be a twice differentiable function from $\mathbb{R}^{2} \rightarrow \mathbb{R}$. If $P, \mathbf{x}_{0}\in \mathbb{R}^{2}$ where $\vert \vert P\vert \vert$ is sufficiently small (here $\vert \vert . \vert \vert$ is the Euclidean norm or distance function), then $f (\mathbf{x}_{0} + p) = f(\mathbf{x}_{0}) + \nabla f(\mathbf{x}_{0})^{T}p + \dfrac{1}{2} p^{T} \nabla^{2}f(\psi)p$ where $\psi \in \mathbb{R}^{2}$ is a point on the line segment joining $\mathbf{x}_{0}$ and $\mathbf{x}_{0} + p$. If $\mathbf{x}_{0}$ is a strict local minimum of $f (\mathbf{x})$, then which one of the following statements is TRUE?

A

$\nabla f(x_{0})^{T}p > 0\ \ and\ \ p^{T} \nabla^{2} f(\psi)p = 0$

B

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f(\psi)p > 0$

C

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f(\psi)p = 0$

D

$\nabla f(x_{0})^{T}p = 0\ and\ p^{T} \nabla^{2} f(\psi)p < 0$

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Paper analysis

Total Questions

Engineering Mathematics

9

Engineering Mechanics

3

Fluid Mechanics

4

Heat Transfer

6

Industrial Engineering

5

Machine Design

4

Production Engineering

10

Strength of Materials

4

Theory of Machines

5

Thermodynamics

4

Turbo Machinery

1

General Aptitude

10

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