All 65 questions of GATE 2023 CE (Set 2)

The line ran _______ the page, right through the centre, and divided the page into two.

Kind : ________ : : Often : Seldom

(By word meaning)

In how many ways can cells in a $3\times3$ grid be shaded, such that each row and each column have exactly one shaded cell? An example of one valid shading is shown.

There are 4 red, 5 green, and 6 blue balls inside a box. If $N$ number of balls are picked simultaneously, what is the smallest value of $N$ that guarantees there will be at least two balls of the same colour?

One cannot see the colour of the balls until they are picked.

Consider a circle with its centre at the origin (O), as shown. Two operations are allowed on the circle.

Operation 1: Scale independently along the x and y axes.
Operation 2: Rotation in any direction about the origin.

Which figure among the options can be achieved through a combination of these two operations on the given circle?

Elvesland is a country that has peculiar beliefs and practices. They express almost all their emotions by gifting flowers. For instance, if anyone gifts a white flower to someone, then it is always taken to be a declaration of one’s love for that person. In a similar manner, the gifting of a yellow flower to someone often means that one is angry with that person.

Based only on the information provided above, which one of the following sets of statement(s) can be logically inferred with certainty?

(i) In Elvesland, one always declares one’s love by gifting a white flower.
(ii) In Elvesland, all emotions are declared by gifting flowers.
(iii) In Elvesland, sometimes one expresses one’s anger by gifting a flower that is not yellow.
(iv) In Elvesland, sometimes one expresses one’s love by gifting a white flower.

Three husband-wife pairs are to be seated at a circular table that has six identical chairs. Seating arrangements are defined only by the relative position of the people.

How many seating arrangements are possible such that every husband sits next to his wife?

Based only on the following passage, which one of the options can be inferred with certainty?

When the congregation sang together, Apenyo would also join, though her little screams were not quite audible because of the group singing. But whenever there was a special number, trouble would begin; Apenyo would try singing along, much to the embarrassment of her mother. After two or three such mortifying Sunday evenings, the mother stopped going to church altogether until Apenyo became older and learnt to behave.

At home too, Apenyo never kept quiet; she hummed or made up silly songs to sing by herself, which annoyed her mother at times but most often made her become pensive. She was by now convinced that her daughter had inherited her love of singing from her father who had died unexpectedly away from home.

[Excerpt from These Hills Called Home by Temsula Ao]

If $x$ satisfies the equation

then $x$ is equal to ______.

Consider a spherical globe rotating about an axis passing through its poles. There are three points $P$, $Q$, and $R$ situated respectively on the equator, the north pole, and midway between the equator and the north pole in the northern hemisphere. Let $P$, $Q$, and $R$ move with speeds $v_P$, $v_Q$, and $v_R$, respectively.

Which one of the following options is CORRECT?

Let $\phi$ be a scalar field, and $\mathbf{u}$ be a vector field. Which of the following identities is true for $\operatorname{div}(\phi\mathbf{u})$?

Which of the following probability distribution functions (PDFs) has the mean greater than the median?

A remote village has exactly 1000 vehicles with sequential registration numbers starting from 1000. Out of the total vehicles, 30% are without pollution clearance certificate. Further, even- and odd-numbered vehicles are operated on even- and odd-numbered dates, respectively.

If 100 vehicles are chosen at random on an even-numbered date, the number of vehicles expected without pollution clearance certificate is ______.

A circular solid shaft of span $L=5$ m is fixed at one end and free at the other end. A torque $T=100$ kN·m is applied at the free end. The torsional rigidity $GJ$ is $50{,}000$ kN·m$^2$/rad.

The following are reported for this shaft:

Statement i) The rotation at the free end is 0.01 rad
Statement ii) The torsional strain energy is 1.0 kN·m

With reference to the above statements, which of the following is true?

M20 concrete as per IS 456:2000 refers to concrete with a design mix having _________________.

When a simply-supported elastic beam of span $L$ and flexural rigidity $EI$ is loaded with a uniformly distributed load $w$ per unit length, the deflection at the mid-span is

If the load on one half of the span is now removed, the mid-span deflection ________.

Muller-Breslau principle is used in analysis of structures for _______________.

A standard penetration test (SPT) was carried out at a location by using a manually operated hammer dropping system with 50% efficiency. The recorded SPT value at a particular depth is 28. If an automatic hammer dropping system with 70% efficiency is used at the same location, the recorded SPT value will be ______.

A vertical sheet pile wall is installed in an anisotropic soil having coefficient of horizontal permeability, $k_H$, and coefficient of vertical permeability, $k_V$. In order to draw the flow net for the isotropic condition, the embedment depth of the wall should be scaled by a factor of ______, without changing the horizontal scale.

Identify the cross-drainage work in the figure.

Which one of the following options provides the correct match of the terms listed in Column-1 and Column-2?

In the context of Municipal Solid Waste Management, ‘Haul’ in ‘Hauled Container System operated in conventional mode’ includes the _________.

Which of the following is equal to the stopping sight distance?

The magnetic bearing of the sun for a location at noon is $183^\circ30'$. If the sun is exactly on the geographic meridian at noon, the magnetic declination of the location is ______.

For the matrix

which of the following statements is/are TRUE?

In the frame shown in the figure (not to scale), all four members (AB, BC, CD, and AD) have the same length and same constant flexural rigidity. All the joints A, B, C, and D are rigid joints. The midpoints of AB, BC, CD, and AD are denoted by E, F, G, and H, respectively. The frame is in unstable equilibrium under the shown forces of magnitude $P$ acting at E and G.

Which of the following statements is/are TRUE?

With regard to the shear design of RCC beams, which of the following statements is/are TRUE?

The reason(s) of the nonuniform elastic settlement profile below a flexible footing, resting on a cohesionless soil while subjected to uniform loading, is/are:

Which of the following is/are NOT active disinfectant(s) in water treatment?

As per the Indian Roads Congress guidelines (IRC 86:2018), extra widening depends on which of the following parameters?

The steady-state temperature distribution in a square plate ABCD is governed by the 2-dimensional Laplace equation. The side AB is kept at a temperature of 100 °C and the other three sides are kept at a temperature of 0 °C. Ignoring the effect of discontinuities in the boundary conditions at the corners, the steady-state temperature at the center of the plate is obtained as $T_0$ °C.

Due to symmetry, the steady-state temperature at the center will be the same ($T_0$ °C) when any one side of the square is kept at 100 °C and the remaining three sides are kept at 0 °C.

Using the principle of superposition, the value of $T_0$ is ______ (rounded off to two decimal places).

An unconfined compression strength test was conducted on a cohesive soil. The test specimen failed at an axial stress of 76 kPa. The undrained cohesion (in kPa, in integer) of the soil is ______.

The pressure in a pipe at X is to be measured by an open manometer as shown in the figure. Fluid A is oil with a specific gravity of 0.8 and Fluid B is mercury with a specific gravity of 13.6.

The absolute pressure at X is ______ kN/m$^2$ (round off to one decimal place).

Assume density of water as 1000 kg/m$^3$, acceleration due to gravity as 9.81 m/s$^2$, and atmospheric pressure as 101.3 kN/m$^2$.

Note: Figure is not to scale.

For the elevation and temperature data given in the table, the existing lapse rate in the environment is ______ °C/100 m (round off to two decimal places).

If the size of the ground area is $6$ km $\times$ $3$ km and the corresponding photo size in the aerial photograph is $30$ cm $\times$ $15$ cm, then the scale of the photograph is $1:$ ______ (in integer).

The solution of the differential equation

is expressed as

where $C_1,C_2,C_3,\alpha,$ and $\beta$ are constants, with $\alpha$ and $\beta$ being distinct and not equal to 2.5.

Which option is correct for $\alpha$ and $\beta$?

Two vectors

belong to the null space of a $4\times4$ matrix of rank 2. Which one of the following vectors also belongs to the null space?

Cholesky decomposition is carried out on the matrix

Let $l_{ij}$ and $a_{ij}$ be the $(i,j)$th elements of matrices $L$ and $A$, respectively. If $l_{22}=1.968$, what is the value (rounded off to the nearest integer) of $a_{22}$?

In a two-dimensional stress analysis, the state of stress at a point is shown in the figure. The values of length of PQ, QR, and RP are 4, 3, and 5 units, respectively.

The principal stresses are ______ (round off to one decimal place).

Two plates are connected by fillet welds of size 10 mm and subjected to tension, as shown in the figure. The thickness of each plate is 12 mm. The yield stress and ultimate stress of steel under tension are 250 MPa and 410 MPa, respectively. The welding is done in the workshop ($\gamma_{mw}=1.25$).

As per the Limit State Method of IS 800:2007, what is the minimum length (in mm, rounded off to the nearest higher multiple of 5 mm) required of each weld to transmit a factored force $P=275$ kN?

In the given figure, Point O indicates the stress point of a soil element at initial non-hydrostatic stress condition. For the stress path (OP), which of the following loading conditions is correct?

The figure shows a vertical retaining wall with backfill consisting of cohesive-frictional soil and a failure plane developed due to passive earth pressure. The forces acting on the failure wedge are: $P$ as the reaction force between the wall and the soil, $R$ as the reaction force on the failure plane, $C$ as the cohesive force along the failure plane and $W$ as the weight of the failure wedge.

Assuming that there is no adhesion between the wall and the wedge, identify the most appropriate force polygon for the wedge.

A compound symmetrical open channel section as shown in the figure has a maximum of ______ critical depth(s).

The critical flow condition in a channel is given by ______.

Note: $\alpha$ = kinetic energy correction factor; $Q$ = discharge; $A_c$ = cross-sectional area of flow at critical flow condition; $T_c$ = top width of flow at critical flow condition; $g$ = acceleration due to gravity.

Match the following air pollutants with the most appropriate adverse health effects.

A delivery agent is at a location R. To deliver the order, she is instructed to travel to location P along straight-line paths RC, CA, AB and BP of 5 km each. The direction of each path is given below as whole circle bearings. Assume that the latitude (L) and departure (D) of R is $(0,0)$ km.

What is the latitude and departure of P (in km, rounded off to one decimal place)?

Which of the following statements is/are TRUE?

In the context of water and wastewater treatments, the correct statements are:

Which of the following statements is/are TRUE for the aerobic composting of sewage sludge?

The figure presents the time-space diagram for when the traffic on a highway is suddenly stopped for a certain time and then released.

Which of the following statements are true?

Consider the Marshall method of mix design for bituminous mix. With the increase in bitumen content, which of the following statements is/are TRUE?

A 5 cm long metal rod AB was initially at a uniform temperature of $T_0$ °C. Thereafter, temperature at both the ends are maintained at 0 °C. Neglecting heat transfer from the lateral surface, the temperature is governed by

where $D=1.0$ cm$^2$/s.

The temperature distribution is written as

where $x$ is in cm and $\beta$ is in s$^{-1}$.

The value of $\beta$ is ______ (rounded off to three decimal places).

A beam is subjected to a system of coplanar forces as shown in the figure.

The magnitude of vertical reaction at Support P is ______ N (round off to one decimal place).

For the frame shown in the figure (not to scale), all members (AB, BC, CD, GB, and CH) have the same length $L$ and flexural rigidity $EI$. The joints at B and C are rigid joints, and the supports A and D are fixed supports. Beams GB and CH carry uniformly distributed loads of $w$ per unit length.

The magnitude of the moment reaction at A is $wL^2/k$. The value of $k$ is ______ (in integer).

Consider the singly reinforced section of a cantilever concrete beam under bending, as shown in the figure (M25 grade concrete, Fe415 grade steel). The stress-block parameters for the section at ultimate limit state, as per IS 456:2000 notations, are given.

The ultimate moment of resistance for the section by the Limit State Method is ______ kN·m (round off to one decimal place).

A 2D thin plate with modulus of elasticity $E=1.0$ N/m$^2$ and Poisson's ratio $\mu=0.5$ is in plane stress condition. The displacement field is

where $C$ is a constant in m$^{-2}$.

The stress in the X direction is

and the shear stress is

What is the value of $\alpha$ (in N/m$^4$, in integer)?

An idealised frame supports a load as shown in the figure.

The horizontal component of the force transferred from the horizontal member PQ to the vertical member RS at P is ______ N (round off to one decimal place).

A square footing is to be designed to carry a column load of 500 kN resting on a soil stratum having the following average properties:

• bulk unit weight = 19 kN/m$^3$
• angle of internal friction = $0^\circ$
• cohesion = 25 kPa

Considering footing depth as 1 m and adopting Meyerhof's bearing capacity theory with a factor of safety of 3, the width of the footing (in m) is ______ (round off to one decimal place).

Assume the applicable shape and depth factor values as unity; ground water level is at greater depth.

A circular pile of diameter 0.6 m and length 8 m was constructed in a cohesive soil stratum having the following properties: bulk unit weight = 19 kN/m$^3$, angle of internal friction = $0^\circ$ and cohesion = 25 kPa.

The allowable load the pile can carry with a factor of safety of 3 is ______ kN (round off to one decimal place).

Adopt adhesion factor $\alpha=1.0$ and bearing capacity factor $N_c=9.0$.

For the flow setup shown in the figure (not to scale), the hydraulic conductivities of Soil 1 and Soil 2 are 10 mm/s and 1 mm/s, respectively. Assume the unit weight of water as 10 kN/m$^3$ and ignore the velocity head.

At steady state, what is the total head (in m, rounded off to two decimal places) at any point located at the junction of the two samples?

A consolidated drained (CD) triaxial test was carried out on a sand sample with known effective shear-strength parameters $c'=0$ and $\phi'=30^\circ$. Prior to failure, while the sample was undergoing axial compression under constant cell pressure, the drainage valve was accidentally closed.

At failure, 360 kPa deviatoric stress was recorded along with 70 kPa pore-water pressure. If the test is repeated without such error, and no back pressure is applied in either test, what is the deviatoric stress (in kPa, in integer) at failure?

A catchment may be idealized as a circle of radius 30 km. There are five rain gauges, one at the center of the catchment and four on the boundary (equi-spaced), as shown in the figure (not to scale).

The annual rainfall recorded at these gauges in a particular year is:

Using the Thiessen polygon method, what is the average rainfall (in mm, rounded off to two decimal places) over the catchment in that year?

The cross-section of a small river is sub-divided into seven segments of width 1.5 m each. The average depth, and velocity at different depths were measured at the middle of each segment width.

The discharge computed by the velocity-area method is ______ m$^3$/s (round off to one decimal place).

The theoretical aerobic oxidation of biomass $(C_5H_7O_2N)$ is

The biochemical oxidation of biomass is assumed as a first-order reaction with a rate constant of 0.23/d at 20 °C (natural logarithm). Neglecting the second-stage oxygen demand, the ratio of BOD$_5$ at 20 °C to total organic carbon (TOC) of biomass is ______ (round off to two decimal places).

Use atomic weights C=12, H=1, O=16, N=14 g/mol.

A system of seven river segments is shown in the schematic diagram. The $R_i$'s, $Q_i$'s, and $C_i$'s $(i=1$ to $7)$ are the river segments, their corresponding flow rates, and concentrations of a conservative pollutant, respectively. Assume complete mixing at intersections, no additional water loss or gain, and steady state.

Given:

What is the steady-state concentration (in kg/m$^3$, rounded off to two decimal places) of the pollutant in river segment 7?