All 65 questions of GATE 2024 CE (Set 2)

If ‘→’ denotes increasing order of intensity, then the meaning of the words [drizzle → rain → downpour] is analogous to [ ______ → quarrel → feud].

Which one of the given options is appropriate to fill the blank?

Statements:

1. All heroes are winners.
2. All winners are lucky people.

Inferences:

I. All lucky people are heroes. II. Some lucky people are heroes. III. Some winners are heroes.

Which of the above inferences can be logically deduced from statements 1 and 2?

A student was supposed to multiply a positive real number $p$ with another positive real number $q$. Instead, the student divided $p$ by $q$. If the percentage error in the student’s answer is 80%, the value of $q$ is

The sum of the first 20 consecutive positive odd integers is divided by $20^2$. What is the result?

The girls-to-boys ratio in class VIII equals the boys-to-girls ratio in class IX. The total numbers of students in classes VIII and IX are 450 and 360, respectively. The two classes contain the same number of girls.

How many girls are there in each class?

In the given text, the blanks are numbered (i)−(iv). Select the best match for all the blanks.

Yoko Roi stands (i) as an author for standing (ii) as an honorary fellow, after she stood (iii) her writings that stand (iv) the freedom of speech.

Seven identical cylindrical chalk-sticks are fitted tightly in a cylindrical container. The figure below shows the arrangement of the chalk-sticks inside the cylinder.

The length of the container is equal to the length of the chalk-sticks. The ratio of the occupied space to the empty space of the container is

The plot below shows the relationship between the mortality risk of cardiovascular disease and the number of steps a person walks per day. Based on the data, which one of the following options is true?

Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction X, the planar image of the assembly appears as Figure B.

If viewed from direction Y, the planar image of the assembly (Figure A) will appear as

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is

A partial differential equation

$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$

is defined for the two-dimensional field $T: T(x, y)$, inside a planar square domain of size 2 m × 2 m. Three boundary edges of the square domain are maintained at value $T = 50$, whereas the fourth boundary edge is maintained at $T = 100$.

The value of $T$ at the center of the domain is

The statements P and Q are related to matrices $\mathbf{A}$ and $\mathbf{B}$, which are conformable for both addition and multiplication.

P: $(\mathbf{A} + \mathbf{B})^T = \mathbf{A}^T + \mathbf{B}^T$

Q: $(\mathbf{A}\mathbf{B})^T = \mathbf{A}^T\mathbf{B}^T$

Which one of the following options is CORRECT?

The second derivative of a function $f$ is computed using the fourth-order Central Divided Difference method with a step length $h$. The CORRECT expression for the second derivative is

The function $f(x) = x^3 - 27x + 4$, $1 \le x \le 6$ has

Consider two Ordinary Differential Equations (ODEs):

P: $\dfrac{dy}{dx} = \dfrac{x^4 + 3x^2y^2 + 2y^4}{x^3y}$

Q: $\dfrac{dy}{dx} = \dfrac{-y^2}{x^2}$

Which one of the following options is CORRECT?

A 3 m long, horizontal, rigid, uniform beam PQ has negligible mass. The beam is subjected to a 3 kN concentrated vertically downward force at 1 m from P, as shown in the figure. The beam is resting on vertical linear springs at the ends P and Q. For the spring at the end P, the spring constant $K_P$ = 100 kN/m.

(Figure NOT to scale)

If the beam DOES NOT rotate under the application of the force and displaces only vertically, the value of the spring constant $K_Q$ (in kN/m) for the spring at the end Q is

Consider:

P. In a pure project organization, the project manager has complete authority and maximum control over the project.

Q. A matrix organization facilitates relatively quick responses to changes, conflicts and project needs.

Choose the correct truth combination.

For a thin-walled section shown in the figure, points P, Q, and R are located on the major bending axis X−X of the section. Point Q is located on the web whereas point S is located at the intersection of the web and the top flange of the section.

(Figure NOT to scale)

Qualitatively, the shear center of the section lies at

Consider the following data for a project of 300 days duration.

Budgeted Cost of Work Scheduled (BCWS) = Rs. 200

Budgeted Cost of Work Performed (BCWP) = Rs. 150

Actual Cost of Work Performed (ACWP) = Rs. 190

The ‘schedule variance’ for the project is

A simply supported, uniformly loaded, two-way slab panel is torsionally unrestrained. The effective span lengths along the short span (x) and long span (y) directions of the panel are $l_x$ and $l_y$, respectively. The design moments for the reinforcements along the x and y directions are $M_{ux}$ and $M_{uy}$, respectively. By using Rankine-Grashoff method, the ratio $M_{ux}/M_{uy}$ is proportional to

Which structural-design method does not apply explicit safety factors to the design loads?

The figure shows a characteristic contact-pressure distribution beneath a shallow footing.

Which footing/soil idealization corresponds to the shown distribution?

Which one of the following saturated fine-grained soils can attain a negative Skempton’s pore pressure coefficient ($A$)?

The source figure plots shear stress against velocity gradient for several idealized rheological behaviours labelled P, Q, R, S and T.

Choose the correct identification.

What is the CORRECT match between the air pollutants and treatment techniques given in the table?

Which one of the following products is NOT obtained in anaerobic decomposition of glucose?

The longitudinal sections of a runway have gradients as shown in the table.

Consider the reduced level (RL) at the starting point of the runway as 100 m. The effective gradient of the runway is

On horizontal curves, superelevation/cant is produced by raising the outer edge relative to the inner edge in which transportation systems?

Various stresses in jointed plain concrete pavement with slab size of 3.5 m × 4.5 m are denoted as follows:

Wheel load stress at interior = $S_{wl}^{i}$

Wheel load stress at edge = $S_{wl}^{e}$

Wheel load stress at corner = $S_{wl}^{c}$

Warping stress at interior = $S_t^{i}$

Warping stress at edge = $S_t^{e}$

Warping stress at corner = $S_t^{c}$

Frictional stress between slab and supporting layer = $S_f$

The critical stress combination in the concrete slab during a summer midnight is

For a reconnaissance survey, it is necessary to obtain vertical aerial photographs of a terrain at an average scale of 1 : 13000 using a camera. If the permissible flying height is assumed as 3000 m above a datum and the average terrain elevation is 1050 m above the datum, the required focal length (in mm) of the camera is

What is the CORRECT match between the survey instruments/parts of instruments shown in the table and the operations carried out with them?

To finalize the direction of a survey, four surveyors set up a theodolite at a station P and performed all the temporary adjustments. From the station P, each of the surveyors observed the bearing to a tower located at station Q with the same instrument without shifting it. The bearings observed by the surveyors are 30°30′00″, 30°29′40″, 30°30′20″ and 30°31′20″. Assuming that each measurement is taken with equal precision, the most probable value of the bearing is

The steel angle section shown in the figure has elastic section modulus of 150.92 cm³ about the horizontal X−X axis, which passes through the centroid of the section.

(Figure NOT to scale)

The shape factor of the section is ________ (rounded off to 2 decimal places).

A reinforced concrete pile of 10 m length and 0.7 m diameter is embedded in a saturated pure clay with unit cohesion of 50 kPa. If the adhesion factor is 0.5, the net ultimate uplift pullout capacity (in kN) of the pile is _________________ (rounded off to the nearest integer).

A 2 m wide rectangular channel is carrying a discharge of 30 m³/s at a bed slope of 1 in 300. Assuming the energy correction factor as 1.1 and acceleration due to gravity as 10 m/s², the critical depth of flow (in meters) is _____________ (rounded off to 2 decimal places).

In a sample of 100 heart patients, each patient has 80% chance of having a heart attack without medicine X. It is clinically known that medicine X reduces the probability of having a heart attack by 50%. Medicine X is taken by 50 of these 100 patients.

The probability that a randomly selected patient, out of the 100 patients, takes medicine X and has a heart attack is

A linearly elastic beam of length $2l$ with flexural rigidity $EI$ has negligible mass. A massless spring with a spring constant $k$ and a rigid block of mass $m$ are attached to the beam as shown in the figure.

The natural frequency of this system is

A critical activity in a project is estimated to take 15 days to complete at a cost of ₹30,000. The activity can be expedited to complete in 12 days by spending a total amount of ₹54,000.

Consider:

P: It is economically advisable to complete the activity early by crashing, if the indirect cost of the project is ₹8,500 per day.

Q: It is economically advisable to complete the activity early by crashing, if the indirect cost of the project is ₹10,000 per day.

Which option is CORRECT?

A homogeneous, prismatic, linearly elastic steel bar fixed at both ends has a slenderness ratio $(l/r)$ of 105, where $l$ is the bar length and $r$ is the radius of gyration. The coefficient of thermal expansion of steel is $12\times10^{-6}/^\circ$C.

Consider the effective length of the steel bar as $0.5l$ and neglect the self-weight of the bar.

The differential increase in temperature, rounded off to the nearest integer, at which the bar buckles is

Consider the statements P and Q related to the analysis/design of retaining walls.

P: When a rough retaining wall moves toward the backfill, the wall friction force/resistance mobilizes in upward direction along the wall.

Q: Most of the earth pressure theories calculate the earth pressure due to surcharge by neglecting the actual distribution of stresses due to surcharge.

Which one of the following options is CORRECT?

A round-bottom triangular lined canal is to be laid at a slope of 1 in 1500, to carry a discharge of 25 m$^3$/s. The side slopes of the canal cross-section are to be kept at 1.25H : 1V. If Manning's roughness coefficient is 0.013, the flow depth (in meters) will be in the range of

A hypothetical multimedia filter consists of anthracite particles (specific gravity 1.50), silica sand (specific gravity 2.60), and ilmenite sand (specific gravity 4.20). After backwashing, the particles should settle to form three layers: coarse anthracite at the top, silica sand in the middle, and small ilmenite particles at the bottom.

Assume:

(i) slow discrete settling, so Stokes' law applies;
(ii) all particles are spherical; and
(iii) the silica-sand particle diameter is 0.20 mm.

Which option correctly satisfies the diameter requirements?

The consolidated data of a spot speed study for a certain stretch of a highway is given below.

The "upper speed limit" (in kmph) for the traffic sign is

Three vectors $\vec p$, $\vec q$, and $\vec r$ are

Which of the following is/are CORRECT?

Consider the statements P, Q, and R.

P: Compacted fine-grained soils with flocculated structure have isotropic permeability.

Q: Phreatic surface/line is the line along which the pore water pressure is always maximum.

R: The piping phenomenon occurring below the dam foundation is typically known as blowout piping.

Which of the following option(s) is/are CORRECT?

In the context of pavement material characterization, the CORRECT statement(s) is/are

The expression for computing the effective interest rate $(i_{\mathrm{eff}})$ using continuous compounding for a nominal interest rate of 5% is

The effective interest rate (in percentage) is ______ (rounded off to 2 decimal places).

Consider

The determinant of the matrix $AB$ is ______ (in integer).

For the 6 m long horizontal cantilever beam PQR shown in the figure, Q is the mid-point. Segment PQ has flexural rigidity $EI=2\times10^5$ kN·m$^2$, whereas segment QR has infinite flexural rigidity. Segment QR is subjected to a uniformly distributed, vertically downward load of 5 kN/m.

The magnitude of the vertical displacement (in mm) at point Q is ______ (rounded off to 3 decimal places).

The horizontal beam PQRS shown in the figure has a fixed support at P, an internal hinge at Q, and a pin support at R. A concentrated vertically downward load $V=10$ kN can act at any point over the entire length of the beam.

The maximum magnitude of the moment reaction (in kN·m) that can act at support P due to $V$ is ______ (in integer).

A concrete column section of size 300 mm × 500 mm is subjected to both axial compression and bending along the major axis. The neutral-axis depth is

as shown in the figure.

The maximum compressive strain $(\epsilon_c)$ at the highly compressed extreme fibre in concrete, where there is no tension in the section, is ______ × $10^{-3}$ (rounded off to 2 decimal places).

The table shows the activities, their durations and dependencies in a project.

The total duration (in days) of the project is ______ (in integer).

A homogeneous earth dam has a maximum water-head difference of 15 m between the upstream and downstream sides. A flownet has 10 potential drops and the average length of the element is 3 m. The specific gravity of soil is 2.65.

For a factor of safety of 2.0 against piping failure, the void ratio of the soil is ______ (rounded off to 2 decimal places).

The in-situ percentage of voids of a sand deposit is 50%. The maximum and minimum densities of sand determined from laboratory tests are 1.8 g/cm$^3$ and 1.3 g/cm$^3$, respectively. The specific gravity of sand is 2.7.

The relative density index of the in-situ sand is ______ (rounded off to 2 decimal places).

A drained triaxial test was conducted on a saturated sand specimen using a stress-path triaxial testing system. The specimen failed when the axial stress reached 100 kN/m$^2$ from an initial confining pressure of 300 kN/m$^2$.

The angle of the shearing plane (in degrees) with respect to horizontal is ______ (rounded off to the nearest integer).

A storm with recorded cumulative precipitation of 11.0 cm produced a direct runoff of 6.0 cm.

The $\phi$-index of the storm is ______ cm/hr (rounded off to 2 decimal places).

A 500 m long water-distribution pipeline P of diameter 1.0 m conveys 0.1 m$^3$/s. A new pipeline Q, with the same length and flow rate, is to replace P. The Darcy friction factors for P and Q are 0.04 and 0.01, respectively.

The diameter of pipeline Q (in m) is ______ (rounded off to 2 decimal places).

A 2 m × 1.5 m tank of 6 m height is provided with a 100 mm diameter orifice at the center of its base. The orifice is plugged and the tank is filled up to 5 m height. Consider the average value of discharge coefficient as 0.6 and acceleration due to gravity ($g$) as 10 m/s². After unplugging the orifice, the time (in seconds) taken for the water level to drop from 5 m to 3.5 m under free discharge condition is ____________ (rounded off to 2 decimal places).

A rectangular channel is 4.0 m wide and carries a discharge of 2.0 m$^3$/s at a depth of 0.4 m. The channel transitions to a maximum width contraction at a downstream location without influencing the upstream flow conditions.

The width (in m) at the maximum contraction is ______ (rounded off to 2 decimal places).

A circular settling tank is to be designed for primary treatment of sewage at a flow rate of 10 million litres/day. Assume a detention period of 2.0 hours and a surface loading rate of 40000 litres/m$^2$/day.

The height (in m) of the water column in the tank is ______ (rounded off to 2 decimal places).

An organic waste is represented as

Atomic weights are S=32, H=1, C=12, O=16, N=14. Assume complete conversion of S to SO$_2$ while burning.

SO$_2$ generated (in grams) per kg of this waste is ______ (rounded off to 1 decimal place).

A horizontal curve of radius 1080 m, with transition curves on either side, in a Broad Gauge railway track was designed and constructed for an equilibrium speed of 70 kmph. Later, express trains are to run on this track. The maximum allowable cant deficiency is 10 cm.

The maximum restricted speed (in kmph) of the express trains is ______ (rounded off to the nearest integer).

A vertical summit curve on a freight corridor is formed at the intersection of two gradients, +3.0% and −5.0%. Assume the following:

• Only large-sized trucks are allowed on this corridor
• Design speed = 80 kmph
• Eye height of truck drivers above the road surface = 2.30 m
• Height of object above the road surface for which trucks need to stop = 0.35 m
• Total reaction time of the truck drivers = 2.0 s
• Coefficient of longitudinal friction of the road = 0.36
• Stopping sight distance gets compensated on the gradient

The design length of the summit curve (in meters) to accommodate the stopping sight distance is __________ (rounded off to 2 decimal places).

A child walks on a level surface from point P to point Q at a bearing of 30°, from point Q to point R at a bearing of 90° and then directly returns to the starting point P at a bearing of 240°. The straight-line paths PQ and QR are 4 m each. Assuming that all bearings are measured from the magnetic north in degrees, the straight-line path length RP (in meters) is _________________ (rounded off to the nearest integer).

Differential levelling is carried out from point P (BM: +200.000 m) to point R. The readings taken are given in the table.

Reduced Level (in meters) of the point R is _____ (rounded off to 3 decimal places).