All 65 questions of GATE 2022 CE (Set 1)

You should _________ when to say _________.

Two straight lines pass through the origin $(x_0,y_0)=(0,0)$. One passes through $(x_1,y_1)=(1,3)$ and the other through $(x_2,y_2)=(1,2)$. What is the area enclosed between them in the interval $[0,1]$ on the $x$-axis?

If $p:q=1:2$, $q:r=4:3$, $r:s=4:5$, and $u$ is 50% more than $s$, what is the ratio $p:u$?

Given the statements:

• P is the sister of Q.
• Q is the husband of R.
• R is the mother of S.
• T is the husband of P.

Based on the above information, T is ______ of S.

In the diagram, R is the center of the circle. PQ and ZV are tangential to the circle. The relation among the areas of the squares PXWR, RUVZ and SPQT is

Healthy eating is a critical component of healthy aging. When should one start eating healthy? It turns out that it is never too early. For example, babies who start eating healthy in the first year are more likely to have better overall health as they get older.

Which one of the following is the CORRECT logical inference based on the information in the above passage?

P invested ₹5000 per month for 6 months of a year and Q invested ₹$x$ per month for 8 months of the year in a partnership business. The profit is shared in proportion to the total investment made in that year. If Q receives $4/9$ of the total profit, what is $x$ (in ₹)?

The above frequency chart shows the frequency distribution of marks obtained by a set of students in an exam.

From the data presented above, which one of the following is CORRECT?

In the square grid shown on the left, a person standing at P2 position is required to move to P5 position. The only movement allowed for a step involves, “two moves along one direction followed by one move in a perpendicular direction”. The permissible directions for movement are shown as dotted arrows in the right. For example, a person at a given position Y can move only to the positions marked X on the right.

Without occupying any of the shaded squares at the end of each step, the minimum number of steps required to go from P2 to P5 is

Consider a cube made by folding a single sheet of paper of appropriate shape. The interior faces of the cube are all blank. However, the exterior faces that are not visible in the above view may not be blank.

Which one of the following represents a possible unfolding of the cube?

Consider the following expression:

$z = \sin(y + it) + \cos(y - it)$

where $z$, $y$, and $t$ are variables, and $i = \sqrt{-1}$ is a complex number. The partial differential equation derived from the above expression is

For $\dfrac{d^3y}{dx^3}+x\left(\dfrac{dy}{dx}\right)^{3/2}+x^2y=0$, the correct description is

The hoop stress at a point on the surface of a thin cylindrical pressure vessel is 30.0 MPa. The maximum shear stress at this point is

In the elastic theory of reinforced concrete, the modular ratio is the ratio of

Which equation is correct for the Pozzolanic reaction?

Consider the cross-section of a beam made up of thin uniform elements having thickness $t$ ($t \ll a$) shown in the figure. The (x, y) coordinates of the points along the center-line of the cross-section are given in the figure.

The coordinates of the shear center of this cross-section are:

Four different soils are classified as CH, ML, SP, and SW, as per the Unified Soil Classification System. Which one of the following options correctly represents their arrangement in the decreasing order of hydraulic conductivity?

Let $\sigma_v'$ and $\sigma_h'$ denote the effective vertical stress and effective horizontal stress, respectively. Which one of the following conditions must be satisfied for a soil element to reach the failure state under Rankine’s passive earth pressure condition?

With respect to fluid flow, match the following in Column X with Column Y:

Which one of the following combinations is correct?

Let $\psi$ denote soil suction head and $K$ hydraulic conductivity. If soil moisture content $\theta$ increases, which statement is TRUE?

A rectangular channel with Gradually Varied Flow has a bed-slope change from a steeper slope to a steep slope. The resulting GVF profile is

The total hardness in raw water is 500 milligram per liter as CaCO$_3$. The total hardness of this raw water, expressed in milligram equivalent per liter, is

(Consider the atomic weights of Ca, C, and O as 40 g/mol, 12 g/mol, and 16 g/mol, respectively.)

An aerial photograph is taken from a flight at a height of 3.5 km above mean sea level, using a camera of focal length 152 mm. If the average ground elevation is 460 m above mean sea level, then the scale of the photograph is

A line between stations P and Q laid on a slope of 1 in 5 was measured as 350 m using a 50 m tape. The tape is known to be short by 0.1 m. The corrected horizontal length (in m) of the line PQ will be

The matrix $\mathbf{M}$ is defined as

$\mathbf{M} = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}$

and has eigenvalues $\mathbf{5}$ and $-\mathbf{2}$. The matrix $\mathbf{Q}$ is formed as

$\mathbf{Q} = \mathbf{M}^3 - 4\mathbf{M}^2 - 2\mathbf{M}$

Which of the following is/are the eigenvalue(s) of matrix $\mathbf{Q}$?

For wastewater coming from a wood pulping industry, Chemical Oxygen Demand (COD) and 5-day Biochemical Oxygen Demand (BOD$_5$) were determined. For this wastewater, which of the following statement(s) is/are correct?

Which of the following process(es) can be used for conversion of salt water into fresh water?

A horizontal curve must be designed in a region with limited space. Which measures can decrease its radius?

Consider the following recursive iteration scheme for different values of variable $P$ with the initial guess $x_1 = \mathbf{1}$:

$x_{n+1} = \frac{1}{2}\left(x_n + \frac{P}{x_n}\right), \qquad n = 1, 2, 3, 4, 5$

For $P = \mathbf{2}$, $x_5$ is obtained to be 1.414, rounded-off to three decimal places.

For $P = \mathbf{3}$, $x_5$ is obtained to be 1.732, rounded-off to three decimal places.

If $P = \mathbf{10}$, the numerical value of $x_5$ is _________ . (round off to three decimal places)

The Fourier cosine series of a function is given by:

$f(x) = \sum_{n=0}^{\infty} f_n \cos nx$

For $f(x) = \cos^4 x$, the numerical value of $(f_4 + f_5)$ is ______ . (round off to three decimal places)

An uncompacted soil heap has volume $10000\ \mathrm{m^3}$ and void ratio 1. If compacted to $7500\ \mathrm{m^3}$, find its void ratio (one decimal place).

A concentrated vertical load of 3000 kN is applied on a horizontal ground surface. Points P and Q are at depths 1 m and 2 m below the ground, respectively, along the line of application of the load. Considering the ground to be a linearly elastic, isotropic, semi-infinite medium, the ratio of the increase in vertical stress at P to the increase in vertical stress at Q is ________. (in integer)

At a site, Static Cone Penetration Test was carried out. The measured point (tip) resistance $q_c$ was 1000 kPa at a certain depth. The friction ratio ($f_r$) was estimated as 1 % at the same depth. The value of sleeve (side) friction (in kPa) at that depth was ___________________ . (in integer)

During a stage of crop growth, consumptive use is 2.8 mm/day. Water available in the soil is 50% of the maximum depth of available water in the root zone. The maximum root zone depth is 80 mm and irrigation efficiency is 70%. Find the irrigation interval in days (nearest integer).

A survey line has bearing $\mathrm{N}31^\circ17'\mathrm{W}$. Find its azimuth observed from north in degrees (two decimal places).

P has Cartesian coordinates (1,1,1) in a right-handed coordinate system. What are its coordinates after a 45° clockwise rotation of the coordinate system about the positive x-axis?

A semi-circular bar of radius R m, in a vertical plane, is fixed at the end G, as shown in the figure. A horizontal load of magnitude P kN is applied at the end H. The magnitude of the axial force, shear force, and bending moment at point Q for θ = 45°, respectively, are

A weld is used for joining an angle section ISA 100 mm × 100 mm × 10 mm to a gusset plate of thickness 15 mm to transmit a tensile load. The permissible stress in the angle is 150 MPa and the permissible shear stress on the section through the throat of the fillet weld is 108 MPa. The location of the centroid of the angle is represented by $C_{yy}$ in the figure, where $C_{yy}$ = 28.4 mm. The area of cross-section of the angle is 1903 mm².

Assuming the effective throat thickness of the weld to be 0.7 times the given weld size, the lengths $L_1$ and $L_2$ (rounded-off to the nearest integer) of the weld required to transmit a load equal to the full strength of the tension member are, respectively

The project activities are given in the following table along with the duration and dependency.

Which one of the following combinations is correct?

The correct match between the physical states of the soils given in Group I and the governing conditions given in Group II is

As per Rankine’s theory of earth pressure, the inclination of failure planes is $\left(45 + \dfrac{\phi}{2}\right)^\circ$ with respect to the direction of the minor principal stress.

The above statement is correct for which one of the following options?

Henry’s law constant for transferring O$_2$ from air into water, at room temperature, is $1.3\ \dfrac{\text{mmol}}{\text{liter-atm}}$. Given that the partial pressure of O$_2$ in the atmosphere is 0.21 atm, the concentration of dissolved oxygen (mg/liter) in water in equilibrium with the atmosphere at room temperature is

(Consider the molecular weight of O$_2$ as 32 g/mol)

A water sample contains $\mathrm{Ca^{2+}}=100$ mg/L, $\mathrm{Mg^{2+}}=36$ mg/L and $\mathrm{HCO_3^-}=122$ mg/L. Atomic masses: Ca 40, Mg 24, H 1, C 12, O 16. Find total hardness and temporary hardness, in mg/L as $\mathrm{CaCO_3}$.

Consider the four points P, Q, R, and S shown in the Greenshields fundamental speed-flow diagram. Denote their corresponding traffic densities by $k_P$, $k_Q$, $k_R$, and $k_S$, respectively.

The correct order of these densities is

Let $\max\{a, b\}$ denote the maximum of two real numbers $a$ and $b$. Which of the following statement(s) is/are TRUE about the function $f(x) = \max\{3 - x, x - 1\}$?

A horizontal force of P kN is applied to a homogeneous body of weight 25 kN, as shown in the figure. The coefficient of friction between the body and the floor is 0.3.

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

For cross-drainage structures, which statements about the relative positions of a natural drain and irrigation canal are correct?

Consider the differential equation

$\frac{dy}{dx} = 4(x + 2) - y$

For the initial condition y = 3 at x = 1, the value of $y$ at x = 1.4 obtained using Euler’s method with a step-size of 0.2 is _________. (round off to one decimal place)

A set of observations of independent variable (x) and the corresponding dependent variable (y) is given below.

Based on the data, the coefficient $a$ of the linear regression model $y = a + bx$ is estimated as 6.1. The coefficient $b$ is ______________ . (round off to one decimal place)

The plane truss shown in the figure is subjected to an external force P. It is given that P = 70 kN, a = 2 m, and b = 3 m.

The magnitude (absolute value) of force (in kN) in member EF is _______. (round off to the nearest integer)

Consider the linearly elastic plane frame shown in the figure. Members HF, FK and FG are welded together at joint F. Joints K, G and H are fixed supports. A counter-clockwise moment M is applied at joint F. Consider flexural rigidity EI = 10⁵ kN-m² for each member and neglect axial deformations.

If the magnitude (absolute value) of the support moment at H is 10 kN-m, the magnitude (absolute value) of the applied moment M (in kN-m) to maintain static equilibrium is ___________. (round off to the nearest integer)

Consider a simply supported beam PQ as shown in the figure. A truck having 100 kN on the front axle and 200 kN on the rear axle, moves from left to right. The spacing between the axles is 3 m. The maximum bending moment at point R is _________ kNm. (in integer)

A reinforced concrete beam with rectangular cross section (width = 300 mm, effective depth = 580 mm) is made of M30 grade concrete. It has 1% longitudinal tension reinforcement of Fe 415 grade steel. The design shear strength for this beam is 0.66 N/mm². The beam has to resist a factored shear force of 440 kN. The spacing of two-legged, 10 mm diameter vertical stirrups of Fe 415 grade steel is __________mm. (round off to the nearest integer)

A square concrete pile of 10 m length is driven into a deep layer of uniform homogeneous clay. Average unconfined compressive strength of the clay, determined through laboratory tests on undisturbed samples extracted from the clay layer, is 100 kPa. If the ultimate compressive load capacity of the driven pile is 632 kN, the required width of the pile is _______ mm. (in integer)

(Bearing capacity factor $N_c$ = 9; adhesion factor $\alpha$ = 0.7)

A raft foundation of 30 m × 25 m is proposed to be constructed at a depth of 8 m in a sand layer. A 25 m thick saturated clay layer exists 2 m below the base of the raft foundation. Below the clay layer, a dense sand layer exists at the site. A 25 mm thick undisturbed sample was collected from the mid-depth of the clay layer and tested in a laboratory oedometer under double drainage condition. It was found that the soil sample had undergone 50 % consolidation settlement in 10 minutes. The time (in days) required for 25 % consolidation settlement of the raft foundation will be ____________________. (round off to the nearest integer)

A two-hour duration storm event with uniform excess rainfall of 3 cm occurred on a watershed. The ordinates of streamflow hydrograph resulting from this event are given in the table.

Considering a constant baseflow of 10 m³/s, the peak flow ordinate (in m³/s) of one-hour unit hydrograph for the watershed is ________ . (in integer)

Two reservoirs are connected by two parallel pipes of equal length and of diameters 20 cm and 10 cm, as shown in the figure (not drawn to scale).

When the difference in the water levels of the reservoirs is 5 m, the ratio of discharge in the larger diameter pipe to the discharge in the smaller diameter pipe is ____________. (round off to two decimal places)

(Consider only loss due to friction and neglect all other losses. Assume the friction factor to be the same for both the pipes)

Depth of water flowing in a 3 m wide rectangular channel is 2 m. The channel carries a discharge of 12 m³/s. Take g = 9.8 m/s². The bed width (in m) at contraction, which just causes the critical flow, is _________ without changing the upstream water level. (round off to two decimal places)

A wastewater sample contains two nitrogen species, namely ammonia and nitrate. Consider the atomic weight of N, H, and O as 14 g/mol, 1 g/mol, and 16 g/mol, respectively. In this wastewater, the concentration of ammonia is 34 mg NH$_3$/liter and that of nitrate is 6.2 mg NO$_3^-$/liter. The total nitrogen concentration in this wastewater is _______ milligram nitrogen per liter. (round off to one decimal place)

A 2 % sewage sample (in distilled water) was incubated for 3 days at 27 °C temperature. After incubation, a dissolved oxygen depletion of 10 mg/L was recorded. The biochemical oxygen demand (BOD) rate constant at 27 °C was found to be 0.23 day⁻¹ (at base e). The ultimate BOD (in mg/L) of the sewage will be_____________. (round off to the nearest integer)

A water treatment plant has a sedimentation basin of depth 3 m, width 5 m, and length 40 m. The water inflow rate is 500 m³/h. The removal fraction of particles having a settling velocity of 1.0 m/h is_______. (round off to one decimal place)

(Consider the particle density as 2650 kg/m³ and liquid density as 991 kg/m³)

A two-phase signalized intersection is designed with a cycle time of 100 s. The amber and red times for each phase are 4 s and 50 s, respectively. If the total lost time per phase due to start-up and clearance is 2 s, the effective green time of each phase is ______s. (in integer)

At a traffic intersection, cars and buses arrive randomly according to independent Poisson processes at an average rate of 4 vehicles per hour and 2 vehicles per hour, respectively. The probability of observing at least 2 vehicles in 30 minutes is ______. (round off to two decimal places)

The vehicle count obtained in every 10 minute interval of a traffic volume survey done in peak one hour is given below.

The peak hour factor (PHF) for 10 minute sub-interval is __________. (round off to one decimal place)

For the dual-wheel carrying assembly shown in the figure, P is the load on each wheel, a is the radius of the contact area of the wheel, s is the spacing between the wheels, and d is the clear distance between the wheels. Assuming that the ground is an elastic, homogeneous, and isotropic half space, the ratio of Equivalent Single Wheel Load (ESWL) at depth z = d/2 to the ESWL at depth z = 2s is ___________. (round off to one decimal place)

(Consider the influence angle to be 45° for the linear dispersion of stress with depth)