All 65 questions of GATE 2021 CE (Set 1)

Getting to the top is ________________ than staying on top.

The mirror image of the above text about the x-axis is

In a company, 35% of the employees drink coffee, 40% of the employees drink tea and 10% of the employees drink both tea and coffee. What % of employees drink neither tea nor coffee?

$\oplus$ and $\odot$ are two operators on numbers $p$ and $q$ such that

$p \oplus q = \frac{p^2 + q^2}{pq} \quad \text{and} \quad p \odot q = \frac{p^2}{q};$

If $x \oplus y = 2 \odot 2$, then $x =$

Four persons P, Q, R and S are to be seated in a row, all facing the same direction, but not necessarily in the same order. P and R cannot sit adjacent to each other. S should be seated to the right of Q. The number of distinct seating arrangements possible is:

Statement: Either P marries Q or X marries Y

Among the options below, the logical NEGATION of the above statement is:

Consider two rectangular sheets, Sheet M and Sheet N of dimensions 6 cm x 4 cm each.

Folding operation 1: The sheet is folded into half by joining the short edges of the current shape.

Folding operation 2: The sheet is folded into half by joining the long edges of the current shape.

Folding operation 1 is carried out on Sheet M three times.

Folding operation 2 is carried out on Sheet N three times.

The ratio of perimeters of the final folded shape of Sheet N to the final folded shape of Sheet M is ______.

Five line segments of equal lengths, PR, PS, QS, QT and RT are used to form a star as shown in the figure above.

The value of $\theta$, in degrees, is ______

A function, $\lambda$, is defined by

$\lambda(p, q) = \begin{cases} (p - q)^2, & \text{if } p \ge q, \\ p + q, & \text{if } p < q. \end{cases}$

The value of the expression $\dfrac{\lambda\big(-(-3 + 2), (-2 + 3)\big)}{\big(-(-2 + 1)\big)}$ is:

Humans have the ability to construct worlds entirely in their minds, which don’t exist in the physical world. So far as we know, no other species possesses this ability. This skill is so important that we have different words to refer to its different flavors, such as imagination, invention and innovation.

Based on the above passage, which one of the following is TRUE?

The rank of matrix $\begin{bmatrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 2 & 5 \\ 5 & 6 & 2 & 7 \\ 7 & 8 & 2 & 9 \end{bmatrix}$ is

If $\boldsymbol{P} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $\boldsymbol{Q} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ then $\boldsymbol{Q}^T\boldsymbol{P}^T$ is

The shape of the cumulative distribution function of Gaussian distribution is

A propped cantilever beam EF is subjected to a unit moving load as shown in the figure (not to scale). The sign convention for positive shear force at the left and right sides of any section is also shown.

The CORRECT qualitative nature of the influence line diagram for shear force at G is

Gypsum is typically added in cement to

The direct and indirect costs estimated by a contractor for bidding a project is ₹160000 and ₹20000 respectively. If the mark up applied is 10% of the bid price, the quoted price (in ₹) of the contractor is

In an Oedometer apparatus, a specimen of fully saturated clay has been consolidated under a vertical pressure of 50 kN/m² and is presently at equilibrium. The effective stress and pore water pressure immediately on increasing the vertical stress to 150 kN/m², respectively are

A partially-saturated soil sample has natural moisture content of 25% and bulk unit weight of 18.5 kN/m³. The specific gravity of soil solids is 2.65 and unit weight of water is 9.81 kN/m³. The unit weight of the soil sample on full saturation is

If water is flowing at the same depth in most hydraulically efficient triangular and rectangular channel sections then the ratio of hydraulic radius of triangular section to that of rectangular section is

‘Kinematic viscosity’ is dimensionally represented as

Which one of the following statements is correct?

Which one of the following is correct?

The liquid forms of particulate air pollutants are

The shape of the most commonly designed highway vertical curve is

A highway designed for 80 km/h speed has a horizontal curve section with radius 250 m. If the design lateral friction is assumed to develop fully, the required super elevation is

Which of the following is NOT a correct statement?

Which of the following is/are correct statement(s)?

Consider the limit:

$\lim_{x \to 1}\left(\frac{1}{\ln x} - \frac{1}{x - 1}\right)$

The limit (correct up to one decimal place) is _____________

The volume determined from $\iiint_V 8\,xyz\,dV$ for $V = [2, 3] \times [1, 2] \times [0, 1]$ will be (in integer) __________

The state of stress in a deformable body is shown in the figure. Consider transformation of the stress from the x-y coordinate system to the X-Y coordinate system. The angle θ, locating the X-axis, is assumed to be positive when measured from the x-axis in counter-clockwise direction.

The absolute magnitude of the shear stress component $\sigma_{xy}$ (in MPa, round off to one decimal place) in x-y coordinate system is _____________________

The equation of deformation is derived to be $y = x^2 - xL$ for a beam shown in the figure.

The curvature of the beam at the mid-span (in units, in integer) will be _______

A truss EFGH is shown in the figure, in which all the members have the same axial rigidity R. In the figure, P is the magnitude of external horizontal forces acting at joints F and G.

If R = 500×10³ kN, P = 150 kN and L = 3 m, the magnitude of the horizontal displacement of joint G (in mm, round off to one decimal place) is _______________

The cohesion (c), angle of internal friction (ϕ) and unit weight (γ) of a soil are 15 kPa, 20° and 17.5 kN/m³, respectively. The maximum depth of unsupported excavation in the soil (in m, round off to two decimal places) is ______________

Two reservoirs are connected through a homogeneous and isotropic aquifer having hydraulic conductivity (K) of 25 m/day and effective porosity (η) of 0.3 as shown in the figure (not to scale). Ground water is flowing in the aquifer at the steady state.

If water in Reservoir 1 is contaminated then the time (in days, round off to one decimal place) taken by the contaminated water to reach to Reservoir 2 will be __________

A signalized intersection operates in two phases. The lost time is 3 seconds per phase. The maximum ratios of approach flow to saturation flow for the two phases are 0.37 and 0.40. The optimum cycle length using the Webster’s method (in seconds, round off to one decimal place) is _________

The solution of the second-order differential equation $\dfrac{d^2y}{dx^2} + 2\dfrac{dy}{dx} + y = 0$ with boundary conditions $y(0) = 1$ and $y(1) = 3$ is

The value of $\displaystyle\int_0^1 e^x\,dx$ using the trapezoidal rule with four equal subintervals is

A 50 mL sample of industrial wastewater is taken into a silica crucible. The empty weight of the crucible is 54.352 g. The crucible with the sample is dried in a hot air oven at 104 °C till a constant weight of 55.129 g. Thereafter, the crucible with the dried sample is fired at 600 °C for 1 h in a muffle furnace, and the weight of the crucible along with residue is determined as 54.783 g. The concentration of total volatile solids is ______________ .

A wedge M and a block N are subjected to forces P and Q as shown in the figure. If force P is sufficiently large, then the block N can be raised. The weights of the wedge and the block are negligible compared to the forces P and Q. The coefficient of friction (μ) along the inclined surface between the wedge and the block is 0.2. All other surfaces are frictionless. The wedge angle is 30°.

The limiting force P, in terms of Q, required for impending motion of block N to just move it in the upward direction is given as $P = \alpha Q$. The value of the coefficient ‘$\alpha$’ (round off to one decimal place) is

Contractor X is developing his bidding strategy against Contractor Y. The ratio of Y’s bid price to X’s cost for the 30 previous bids in which Contractor X has competed against Contractor Y is given in the Table.

Based on the bidding behaviour of the Contractor Y, the probability of winning against Contractor Y at a mark up of 8% for the next project is

Based on drained triaxial shear tests on sands and clays, the representative variations of volumetric strain (∆V/V) with the shear strain (γ) is shown in the figure.

Choose the CORRECT option regarding the representative behaviour exhibited by Curve P and Curve Q.

A fluid flowing steadily in a circular pipe of radius R has a velocity that is everywhere parallel to the axis (centerline) of the pipe. The velocity distribution along the radial direction is

$V_r = U\left(1 - \frac{r^2}{R^2}\right),$

where r is the radial distance as measured from the pipe axis and U is the maximum velocity at $r = 0$. The average velocity of the fluid in the pipe is

A water sample is analyzed for coliform organisms by the multiple-tube fermentation method. The results of confirmed test are as follows:

The most probable number (MPN) of coliform organisms for the above results is to be obtained using the following MPN Index.

MPN Index for Various Combinations of Positive Results when Five Tubes used per Dilution of 10.0 mL, 1.0 mL and 0.1 mL

The MPN of coliform organisms per 100 mL is

Ammonia nitrogen is present in a given wastewater sample as the ammonium ion (NH$_4^+$) and ammonia (NH$_3$). If pH is the only deciding factor for the proportion of these two constituents, which of the following is a correct statement?

On a road, the speed – density relationship of a traffic stream is given by $u = 70 - 0.7k$ (where speed, u, is in km/h and density, k, is in veh/km). At the capacity condition, the average time headway will be

The values of abscissa (x) and ordinate (y) of a curve are as follows:

By Simpson’s 1/3rd rule, the area under the curve (round off to two decimal places) is ______________

Vehicular arrival at an isolated intersection follows the Poisson distribution. The mean vehicular arrival rate is 2 vehicle per minute. The probability (round off to two decimal places) that at least 2 vehicles will arrive in any given 1-minute interval is ___________

Refer the truss as shown in the figure (not to scale).

If load, F = $10\sqrt{3}$ kN, moment of inertia, I = 8.33 × 10⁶ mm⁴, area of cross-section, A = 10⁴ mm², and length, L = 2 m for all the members of the truss, the compressive stress (in kN/m², in integer) carried by the member Q-R is _______

A prismatic cantilever prestressed concrete beam of span length, L = 1.5 m has one straight tendon placed in the cross-section as shown in the following figure (not to scale). The total prestressing force of 50 kN in the tendon is applied at $d_c$ = 50 mm from the top in the cross-section of width, b = 200 mm and depth, d = 300 mm.

If the concentrated load, P = 5 kN, the resultant stress (in MPa, in integer) experienced at point ‘Q’ will be ________

A column is subjected to a total load (P) of 60 kN supported through a bracket connection, as shown in the figure (not to scale).

The resultant force in bolt R (in kN, round off to one decimal place) is __________________

Employ stiffness matrix approach for the simply supported beam as shown in the figure to calculate unknown displacements/rotations. Take length, L = 8 m; modulus of elasticity, E = 3×10⁴ N/mm²; moment of inertia, I = 225×10⁶ mm⁴.

The mid-span deflection of the beam (in mm, round off to integer) under P = 100 kN in downward direction will be ________

A square plate O-P-Q-R of a linear elastic material with sides 1.0 m is loaded in a state of plane stress. Under a given stress condition, the plate deforms to a new configuration O-P'-Q'-R' as shown in the figure (not to scale). Under the given deformation, the edges of the plate remain straight.

The horizontal displacement of the point (0.5 m, 0.5 m) in the plate O-P-Q-R (in mm, round off to one decimal place) is ______________

A small project has 12 activities – N, P, Q, R, S, T, U, V, W, X, Y, and Z. The relationship among these activities and the duration of these activities are given in the Table.

The total float of the activity “V” (in weeks, in integer) is _______________

The soil profile at a construction site is shown in the figure (not to scale). Ground water table (GWT) is at 5 m below the ground level at present. An old well data shows that the ground water table was as low as 10 m below the ground level in the past. Take unit weight of water, $\gamma_w$ = 9.81 kN/m³.

The overconsolidation ratio (OCR) (round off to two decimal places) at the mid-point of the clay layer is _____________

A retaining wall of height 10 m with clay backfill is shown in the figure (not to scale). Weight of the retaining wall is 5000 kN per m acting at 3.3 m from the toe of the retaining wall. The interface friction angle between base of the retaining wall and the base soil is 20°. The depth of clay in front of the retaining wall is 2.0 m. The properties of the clay backfill and the clay placed in front of the retaining wall are the same. Assume that the tension crack is filled with water. Use Rankine’s earth pressure theory. Take unit weight of water, $\gamma_w$ = 9.81 kN/m³.

The factor of safety (round off to two decimal places) against sliding failure of the retaining wall after ignoring the passive earth pressure will be _________

A combined trapezoidal footing of length L supports two identical square columns (P1 and P2) of size 0.5 m × 0.5 m, as shown in the figure. The columns P1 and P2 carry loads of 2000 kN and 1500 kN, respectively.

If the stress beneath the footing is uniform, the length of the combined footing L (in m, round off to two decimal places) is ______________

An unsupported slope of height 15 m is shown in the figure (not to scale), in which the slope face makes an angle 50° with the horizontal. The slope material comprises purely cohesive soil having undrained cohesion 75 kPa. A trial slip circle KLM, with a radius 25 m, passes through the crest and toe of the slope and it subtends an angle 60° at its center O. The weight of the active soil mass (W, bounded by KLMN) is 2500 kN/m, which is acting at a horizontal distance of 10 m from the toe of the slope. Consider the water table to be present at a very large depth from the ground surface.

Considering the trial slip circle KLM, the factor of safety against the failure of slope under undrained condition (round off to two decimal places) is ______________

An unlined canal under regime conditions along with a silt factor of 1 has a width of flow 71.25 m. Assuming the unlined canal as a wide channel, the corresponding average depth of flow (in m, round off to two decimal places) in the canal will be __________

A cylinder (2.0 m diameter, 3.0 m long and 25 kN weight) is acted upon by water on one side and oil (specific gravity = 0.8) on other side as shown in the figure.

The absolute ratio of the net magnitude of vertical forces to the net magnitude of horizontal forces (round off to two decimal places) is ___________

A tube-well of 20 cm diameter fully penetrates a horizontal, homogeneous and isotropic confined aquifer of infinite horizontal extent. The aquifer is of 30 m uniform thickness. A steady pumping at the rate of 40 litres/s from the well for a long time results in a steady drawdown of 4 m at the well face. The subsurface flow to the well due to pumping is steady, horizontal and Darcian and the radius of influence of the well is 245 m. The hydraulic conductivity of the aquifer (in m/day, round off to integer) is ____________

A baghouse filter has to treat 12 m³/s of waste gas continuously. The baghouse is to be divided into 5 sections of equal cloth area such that one section can be shut down for cleaning and/or repairing, while the other 4 sections continue to operate. An air-to-cloth ratio of 6.0 m³/min-m² cloth will provide sufficient treatment to the gas. The individual bags are of 32 cm in diameter and 5 m in length. The total number of bags (in integer) required in the baghouse is ___________

A secondary clarifier handles a total flow of 9600 m³/d from the aeration tank of a conventional activated-sludge treatment system. The concentration of solids in the flow from the aeration tank is 3000 mg/L. The clarifier is required to thicken the solids to 12000 mg/L, and hence it is to be designed for a solid flux of 3.2 $\dfrac{\text{kg}}{\text{m}^2\cdot\text{h}}$. The surface area of the designed clarifier for thickening (in m², in integer) is _____________

Spot speeds of vehicles observed at a point on a highway are 40, 55, 60, 65 and 80 km/h. The space-mean speed (in km/h, round off to two decimal places) of the observed vehicles is ____________

The longitudinal section of a runway provides the following data:

The effective gradient of the runway (in %, round off to two decimal places) is ________________

Traversing is carried out for a closed traverse PQRS. The internal angles at vertices P, Q, R and S are measured as 92°, 68°, 123°, and 77°, respectively. If fore bearing of line PQ is 27°, fore bearing of line RS (in degrees, in integer) is ______________