List of top Questions asked in GATE CE

Suppose \( \lambda \) is an eigenvalue of matrix \( A \) and \( x \) is the corresponding eigenvector. Let \( x \) also be an eigenvector of the matrix \( B = A - 2I \), where \( I \) is the identity matrix. Then, the eigenvalue of \( B \) corresponding to the eigenvector \( x \) is equal to:

An object is said to have an n-fold rotational symmetry if the object, rotated by an angle of \( \frac{2\pi}{n} \), is identical to the original. 
Which one of the following objects exhibits 4-fold rotational symmetry about an axis perpendicular to the plane of the screen? 
 

A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K). Which one of the following options displays the color codes that are consistent with the color model? 

A circle with center at \( (x, y) = (0.5, 0) \) and radius = 0.5 intersects with another circle with center at \( (x, y) = (1, 1) \) and radius = 1 at two points. One of the points of intersection \( (x, y) \) is:
Rohit goes to a restaurant for lunch at about 1 PM. When he enters the restaurant, he notices that the hour and minute hands on the wall clock are exactly coinciding. After about an hour, when he leaves the restaurant, he notices that the clock hands are again exactly coinciding. How much time (in minutes) did Rohit spend at the restaurant?

“Why do they pull down and do away with crooked streets, I wonder, which are my delight, and hurt no man living? Every day the wealthier nations are pulling down one or another in their capitals and their great towns: they do not know why they do it; neither do I. It ought to be enough, surely, to drive the great broad ways which commerce needs and which are the life-channels of a modern city, without destroying all history and all the humanity in between: the islands of the past.” (From Hilaire Belloc’s “The Crooked Streets”) 
Based only on the information provided in the above passage, which one of the following statements is true? 
 

The figures, I, II, and III are parts of a sequence. Which one of the following options comes next in the sequence as IV? 

A thin wire is used to construct all the edges of a cube of 1 m side by bending, cutting and soldering the wire. If the wire is 12 m long, what is the minimum number of cuts required to construct the wire frame to form the cube?
The sum of the following infinite series is: \[ \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \dots \]
As the police officer was found guilty of embezzlement, he was .......... dismissed from the service in accordance with the Service Rules.
Is there any good show .......... television tonight?
A single pile with 450 mm diameter has been driven into a homogeneous clay layer, which has an undrained cohesion (\(c_u\)) of 20 kPa and unit weight of 18 kN/m³. The ground water table is found to be at the surface of the clay layer. The adhesion factor (\(\alpha\)) of the soil is 0.95 and bearing capacity factor (\(N_c\)) is 9. The pile is supporting a column load of 144 kN with a factor of safety of 3.0 against ultimate axial pile capacity in compression. The required embedment depth of the pile (in m) is ......... (rounded off to the nearest integer).
Lacey’s regime equations, followed in India for making scour calculations while designing hydraulic structures across alluvial channels, are given below. Regarding these equations, which of the following statements is/are true:
A facultative pond system is employed for wastewater treatment. Which of the following statements is/are true:
The Surface Overflow Rate (SOR) in a rectangular sedimentation tank is 45 m\(^3\)/m\(^2\)·day. Minimum diameters of spherical inorganic and organic particles expected to be completely removed in this tank are calculated. Assume that Stoke’s law is applicable. Which of the following options is/are correct:
All the vehicles that come during a particular peak hour come during a 10-minute period within this hour. The 15-minute peak hour factor for this peak hour is
If the Fore Bearing of the lines AB and BC are 60° and 122°, respectively, then the interior angle \( \angle ABC \) (in degrees) is .......... (round off to the nearest integer).

A settling chamber is used for the removal of discrete particulate matter from air with the following conditions. Horizontal velocity of air = 0.2 m/s; Temperature of air stream = 77°C; Specific gravity of particle to be removed = 2.65; Chamber length = 12 m; Chamber height = 2 m; Viscosity of air at 77°C = 2.1 × 10\(^{-5}\) kg/m·s; Acceleration due to gravity (g) = 9.81 m/s²; Density of air at 77°C = 1.0 kg/m³; Assume the density of water as 1000 kg/m³ and Laminar condition exists in the chamber. 
The minimum size of particle that will be removed with 100% efficiency in the settling chamber (in $\mu$m is .......... (round off to one decimal place). 
 

The analyses results of a water sample are given below. The non-carbonate hardness of the water (in mg/L) as CaCO\(_3\) is ......... (in integer).
Ca\(^{2+}\) = 150 mg/L as CaCO\(_3\)
Mg\(^{2+}\) = 40 mg/L as CaCO\(_3\)
Fe\(^{2+}\) = 10 mg/L as CaCO\(_3\)
Na\(^+\) = 50 mg/L as CaCO\(_3\)
K\(^+\) = 10 mg/L as CaCO\(_3\)
CO\(_3^{2-}\) = 120 mg/L as CaCO\(_3\)
HCO\(_3^{-}\) = 30 mg/L as CaCO\(_3\)
Cl\(^{-}\) = 50 mg/L as CaCO\(_3\); Other anions were not analysed.

The shaft of a 6 m wide gate in the figure will fail at a moment of 3924 kN.m about the hinge P. The maximum value of water depth \( h \) (in m) that the gate can hold is ......... (round off to the nearest integer). 

The peak of flood hydrograph due to a 3-hour duration storm in a catchment is 180 m\(^3\)/s. The total rainfall depth is 6.6 cm. It can be assumed that the average infiltration loss is 0.2 cm/h. There are no other losses. The base flow is constant at a value of 30 m\(^3\)/s.
The peak value of the 3-hour unit hydrograph for this catchment (in m\(^3\)/s) is ......... (round off to the nearest integer).

A 4 x 4 group pile, with each pile 20 m long and 500 mm in diameter, is installed in a square pattern in a clayey soil, as shown in the figure. The average unconfined compressive strength of the soil is 100 kN/m\(^2\), and the adhesion factor is 0.8. Neglect the bearing at the tip of the piles. For a group efficiency factor of 1.0, the centre to centre spacing(s) of the piles (in m) would be ......... (round off to two decimal places). 

The bank of a canal has the profile shown in the figure. The material is a homogeneous clay with a bulk unit weight of 20 kN/m\(^3\), undrained cohesion of 30 kPa and is fully saturated (\( \phi = 0 \)). For the trial slip circle shown, the area ABCDEA is 150 m\(^2\) and the centroid is at P. A tension crack (DE) of 2.5 m deep was also observed. Assume unit weight of water is 9.81 kN/m\(^3\) and consider 1 m run of the bank for the analysis. 
Considering the canal is empty and the tension crack is completely filled with water, the factor of safety against slope failure of the bank is ......... (round off to two decimal places). 

A reinforced concrete beam has a support section with width of 300 mm and effective depth of 500 mm. The support section is reinforced with 3 bars of 20 mm diameter at the tension side. Two-legged vertical stirrups of 10 mm diameter and Fe415 steel at a spacing of 100 mm are provided as shear reinforcement. Assume that there is no possibility of diagonal compression failure at the section.
As per IS 456:2000, the maximum shear resisted by the vertical stirrups (in kN), as per limit state design, is ......... (round off to one decimal place).

The figure shows a network diagram for a construction project. The activities A, B, C, D, E, and F are represented by arrows and their durations are in the figure. The total float available for the activity E in day(s) is equal to ......... (round off to the nearest integer).