AP PGECET Question Paper 2025 (Available) : Download Geo Engineering and Geo Informatics Question Paper

Where is Earth’s core located relative to the surface?

• (A) 2,900 kilometers below the surface
• (B) 3,900 kilometers below the surface
• (C) At the crust–mantle boundary
• (D) 1,900 kilometers below the surface

What is the term for the process that led to the formation of Earth's core?

• (A) Plate tectonics
• (B) Planetary differentiation
• (C) Seafloor spreading
• (D) Mantle convection

What are polymorphs in mineralogy?

• (A) Rocks formed under identical pressure and temperature conditions
• (B) Two or more minerals with the same chemical formula but different crystal structures
• (C) Crystals that form only in metamorphic environments
• (D) Minerals with different chemical formulas but similar crystal structures

What characterizes a porphyritic texture in igneous rocks?

• (A) Layers of alternating light and dark minerals
• (B) A texture in which some crystals are distinctly larger than the rest
• (C) Crystals that form only from volcanic eruptions
• (D) Uniformly sized crystals throughout the rock

Which of the following correctly describes the composition of Earth’s core?

• (A) Silicate rocks with low-density elements
• (B) Solid iron-nickel outer core and liquid silicate inner core
• (C) Liquid iron-nickel outer core and solid iron-nickel inner core
• (D) Primarily aluminum silicate with traces of magnesium

What is the process called when a parent rock transforms due to heat and pressure without melting?

• (A) Diagenesis
• (B) Metamorphism
• (C) Crystallization
• (D) Erosion

Which of the following best explains the term "continental shield"?

• (A) Submarine ridges with high heat flow
• (B) Oceanic crust covered by continental sediments
• (C) Large regions of exposed Precambrian crystalline igneous and metamorphic rocks
• (D) Young mountain ranges formed by recent tectonic activity

Which of the following is a framework silicate mineral?

• (A) Olivine
• (B) Biotite
• (C) Quartz
• (D) Pyroxene

The cleavage in mica minerals is described as:
 

• (A) Basal
• (B) Prismatic
• (C) Cubic
• (D) Irregular

The primary instrument used for measuring horizontal and vertical angles in surveying is:
 

• (A) Theodolite
• (B) Total station
• (C) Plane table
• (D) Dumpy level

High oblique photographs differ from low oblique photographs in that they:
 

• (A) Are taken from a higher altitude
• (B) Include the horizon in the image
• (C) Are taken with a vertical camera axis
• (D) Show less ground area

Which type of map projection assumes a globe resting on a flat surface, producing a circular map with a central point of tangency?
 

• (A) Conic Projection
• (B) Mercator Projection
• (C) Azimuthal Projection
• (D) Cylindrical Projection

Which map projection is widely used for nautical navigation due to its ability to represent lines of constant true direction?
 

• (A) Azimuthal Projection
• (B) Robinson Projection
• (C) Polyconic Projection
• (D) Mercator Projection

The point on the ground vertically beneath the camera at the time of exposure is called the:
 

• (A) Perceptive point
• (B) Nadir Point
• (C) Principal point
• (D) Isoline

Which of the following characteristics most clearly distinguishes an oblique photograph from a vertical photograph in aerial photography?
 

• (A) Uniform scale throughout the photograph
• (B) Ability to see the horizon in the photograph
• (C) Taken with the camera axis vertical to the ground
• (D) Presence of overlapping areas between photos

In remote sensing, super wide angle cameras are cameras whose angular field of view is:
 

• (A) Less than 60\(^\circ\)
• (B) Between 60\(^\circ\) and 90\(^\circ\)
• (C) Greater than 90\(^\circ\)
• (D) Exactly 180\(^\circ\)

What does the term 'radiance' refer to in radiometry?
 

• (A) Energy per unit wavelength
• (B) Energy per unit area
• (C) Energy per unit area per unit solid angle
• (D) Total energy emitted by a source

Which of the following is not a radiometric quantity?
 

• (A) Radiant flux
• (B) Irradiance
• (C) Reflectance
• (D) Radiance

Which orbit is best suited for continuous weather monitoring over a specific region?
 

• (A) Molniya orbit
• (B) Polar orbit
• (C) Geostationary orbit
• (D) Sun-synchronous orbit

Geostationary satellites orbit at approximately:
 

• (A) 35,786 km
• (B) 786 km
• (C) 1000 km
• (D) 500 km

Across-track stereoscopic imaging is typically obtained by:
 

• (A) Combining radar and optical data
• (B) Not combining radar and optical data
• (C) Using side-looking sensors from different satellite passes
• (D) Capturing nadir images over different regions

Which of the following satellite systems is capable of along-track stereo imaging using fore and aft sensors?
 

• (A) Cartosat-1
• (B) MODIS
• (C) SPOT
• (D) Landsat

Across-track stereoscopic imaging is typically obtained by:
 

• (A) Combining radar and optical data
• (B) Not combining radar and optical data
• (C) Using side-looking sensors from different satellite passes
• (D) Capturing nadir images over different regions

Which of the following satellite systems is capable of along-track stereo imaging using fore and aft sensors?
 

• (A) Cartosat-1
• (B) MODIS
• (C) SPOT
• (D) Landsat

Which among the following is not related to GIS software’s?
 

• (A) STAAD Pro
• (B) QGIS
• (C) ARCVIEW
• (D) CAD

Which of the following best describes topology in GIS?
 

• (A) Raster resolution accuracy
• (B) Geometric measurement of features
• (C) Spatial relationships between vector features
• (D) Storage of attribute information

Which GIS file format is most commonly used for storing vector data, including points, lines, and polygons?
 

• (A) .img
• (B) .txt
• (C) .pps
• (D) .shp

Which of the following represents the correct set of coordinate classification in GIS?
 

• (A) Geographic, spherical systems
• (B) Geographic, projected systems
• (C) Linear, geometric systems
• (D) Spherical, projected systems

In GIS, what is the purpose of indexing spatial data in a database?
 

• (A) To speed up data visualization
• (B) To improve spatial query performance
• (C) To store backup copies of data
• (D) To manage spatial relationships

What is the key advantage of Object-Oriented Database Models in GIS?
 

• (A) They only work with vector data
• (B) They are easier to use than relational databases
• (C) They store data in a flat, three Dimensional table
• (D) They allow for more complex data relationships and inheritance

What is the primary advantage of using a hierarchical database structure in GIS?
 

• (A) Improved spatial data analysis
• (B) Faster query response times
• (C) Handling of data
• (D) Clear parent-child relationships between data sets

Which of the following would be considered a non-spatial data analysis technique?
 

• (A) Raster data interpolation
• (B) Database queries using SQL
• (C) Buffer zone creation
• (D) Network analysis

In GIS, what is the primary purpose of using symbology?
 

• (A) To perform spatial queries
• (B) To represent spatial features visually with symbols
• (C) To convert raster to vector
• (D) To store spatial data efficiently

Which of the following types of visualizations would be most appropriate for displaying elevation data?
 

• (A) Heat map
• (B) Bar chart
• (C) 3D surface model
• (D) Line graph

......... map projection is best suited for regional or hemispheric maps, but unsuitable for entire world map.
 

• (A) Cylindrical
• (B) Equivalent
• (C) Conical
• (D) Azimuthal

Which attribute is not associated with digital maps?
 

• (A) Colour
• (B) Symbology
• (C) South arrow
• (D) Legends

Across track scanning systems are also called as _________ scanner systems.

• (A) Push broom
• (B) Lateral broom
• (C) Pull broom
• (D) Whisk broom

RISAT stands for _________.

• (A) Radio Satellite
• (B) Radio Navigation and Ranging Satellite
• (C) Radar Imaging Satellite
• (D) Reference system

Joint earth-observing mission between ISRO and NASA using advanced radar imaging system is known as ________.

• (A) NASA-ISRO Systematic Radar Mission (NISAR)
• (B) NASA-ISRO Synthetic Aperture Radar (NISAR)
• (C) NASA-ISRO Side-looking Aperture Radar (NISAR)
• (D) NASA-ISRO Side-looking Aperture Radio (NISAR)

Instrument used for monitoring water levels in rivers/reservoirs from space is known as ________.

• (A) Stadiometer or Satellite Stadiometer
• (B) Altimeter or Satellite Altimeter
• (C) Space Theodolite
• (D) LISS

________ resolution defines the smallest difference of radiant energy detected by a sensor.

• (1) Spectral resolution
• (2) Radiometric resolution
• (3) Temporal resolution
• (4) Spatial resolution

Diffused reflection of solar radiation from earth’s surface is known as __________.

• (1) Radiance
• (2) Irradiance
• (3) Albedo
• (4) Reflectance

The energy quantum for radiation is proportional to its __________.

• (1) Frequency
• (2) Reciprocal of energy
• (3) Square of frequency
• (4) Square root of frequency

Output from a thermal sensor is a measurement of ________ temperature of an object.

• (1) Kinetic
• (2) Radiant
• (3) Blackbody
• (4) Infrared

Who is known as father of Indian Space Program?

• (1) Prof. P. R. Pisharoty
• (2) Dr. Vikram Sarabhai
• (3) Dr. Homi J. Bhabha
• (4) Prof. U. R. Rao

National Remote Sensing Day is celebrated on ________ every year.

• (1) August 12
• (2) June 5
• (3) November 14
• (4) August 15

Which colour of EM spectrum is used for bathymetry studies?

• (1) Red
• (2) UV
• (3) SWIR
• (4) Green

Rayleigh scattering occurs when particle size is _____ than wavelength of incoming radiation.

• (1) very small
• (2) equal
• (3) not equal
• (4) very big

The reflectance ________ with presence of moisture, surface roughness or organic content in soils.

• (1) increases
• (2) decreases
• (3) remains same
• (4) zero

In digital image processing, the ________ process adjusts translational and rotational alignment between two images.

• (1) Registration
• (2) Filtering
• (3) Enhancement
• (4) Processing

________ helps to transform a correlated dataset into uncorrelated dataset.

• (1) Rationing
• (2) Principal Component Analysis
• (3) Fusion
• (4) Classification

Rectification accuracy is expressed in terms of ________ error.

• (1) Commission
• (2) Root Mean Square
• (3) Kappa
• (4) Omission

Spectral region of electromagnetic radiation which passes through atmosphere without much attenuation is known as _________.

• (1) Ozone hole
• (2) Atmospheric window
• (3) Ozone window
• (4) Black hole

GIS ideally deals with _________.

• (1) Numeric data
• (2) Binary data
• (3) Spatial data
• (4) Complex data

SDI stands for

• (1) Spatial Data Interface
• (2) Spatial Data Infrastructure
• (3) Spatial Data Intention
• (4) Spatial Data International

________ defines the property of connectivity in GIS.

• (1) Proximity
• (2) Neighbourhood
• (3) Topology
• (4) Boolean Identity

Which of the following are true of AM/FM systems?

• (1) The terms GIS and AM/FM are synonymous
• (2) AM/FM are always vector based
• (3) AM/FM is a specialized GIS
• (4) AM/FM systems are always confined to the utility industry

India-WRIS is a portal for ________.

• (1) Agricultural resources
• (2) Water resources
• (3) Forest resources
• (4) Geological resources

________ is a type of vector data model.

• (1) Spaghetti
• (2) Spatial
• (3) Raster
• (4) Quadtree

The purpose of GIS data models is to ________.

• (1) Generate random data structures
• (2) Analyse and describe fictional features in GIS
• (3) Represent real-world features in various forms like points, lines, arcs, and polygons
• (4) Create abstract sculptures of real world

________ function is performed by data input/capture subsystem of GIS.

• (1) Retrieval of data elements
• (2) Storage of data elements
• (3) Acquiring data
• (4) Transformation of data

A network is a series of interconnecting lines along which there is a flow of ________.

• (1) Entities
• (2) Points
• (3) Nodes
• (4) Data

Which interpolation technique is not used for continuous data?

• (1) Kriging
• (2) Trend surface fitting
• (3) Digitization
• (4) Triangulation

What is the approximate radius of the Earth?

• (1) 5,000 km
• (2) 6,371 km
• (3) 7,500 km
• (4) 8,000 km

The Earth’s core is primarily made up of which two elements?

• (1) Silicon and Oxygen
• (2) Iron and Nickel
• (3) Iron and Magnesium
• (4) Nickel and Oxygen

Which of the following statements accurately describes the formation of an intrusive igneous rock?

• (1) It forms from the rapid cooling of lava on the Earth's surface, resulting in fine-grained crystals.
• (2) It forms from the compaction and cementation of sediments over millions of years.
• (3) It forms from the slow cooling of magma beneath the Earth's surface, allowing for the development of large crystals.
• (4) It forms from the transformation of existing rocks due to high temperatures and pressures.

Which one of the following are the two main constituents of granite?

• (1) Iron and nickel
• (2) Iron and silver
• (3) Silica and aluminium
• (4) Iron Oxide and potassium

Which of the following rocks is an intrusive igneous rock?

• (1) Basalt
• (2) Rhyolite
• (3) Granite
• (4) Pumice

Which rock type primarily forms from cooling of lava on Earth's surface?

• (1) Plutonic igneous
• (2) Volcanic igneous
• (3) Metamorphic
• (4) Clastic sedimentary

Which of the following is true about rocks?

• (1) Rocks are composed of only one mineral.
• (2) Rocks do not contain any non-mineral matter.
• (3) Coal is not considered a true rock.
• (4) Most rocks are a mixture of minerals.

A rock that forms from cooling lava is classified as an:

• (1) Intrusive igneous rock
• (2) Extrusive igneous rock
• (3) Extrusive metamorphic rock
• (4) Intrusive volcanic rock

Which environment is most favourable for the formation of metamorphic rocks?

• (1) High pressure and high temperature
• (2) Low pressure and low temperature
• (3) High pressure and low temperature
• (4) Surface weathering

Igneous rocks that crystallize from magma and are composed almost entirely of quartz and feldspars have a _________.

• (1) Granitic composition
• (2) Ultramafic composition
• (3) Andesitic composition
• (4) Basaltic composition

The mineral group that constitutes most of Earth's crust is _____.

• (1) Carbonates
• (2) Oxides
• (3) Silicates
• (4) Sulfides

Which of the following statements accurately distinguishes between a silicate and a non-silicate mineral?

• (1) Silicate minerals are always harder than non-silicate minerals.
• (2) Non-silicate minerals never contain oxygen in their chemical formula.
• (3) Silicate minerals contain silicon and oxygen as major elements, while non-silicate minerals do not always contain both.
• (4) Non-silicate minerals are always formed through biological processes.

Which instrument is primarily used to measure angles between lines of sight in surveying?

• (1) A GPS receiver.
• (2) A theodolite.
• (3) A level.
• (4) A clinometer.

Which type of aerial photograph is taken with the camera's optical axis directed vertically downward?

• (1) An oblique photograph.
• (2) A tilted photograph.
• (3) A stereoscopic photograph.
• (4) A vertical photograph.

Which of the following is NOT typically found as marginal information on a topographic sheet?

• (1) The sheet name and number.
• (2) The contour interval.
• (3) The magnetic declination.
• (4) The coordinates of specific landmarks within the mapped area.

Parallax in aerial photographs is an error due to \hspace{2cm}.

• (1) Movement of camera and ground relief
• (2) Overlap in photographs
• (3) Distortion caused by camera lens
• (4) Distortion due to lack of focus

In map projections, the Mercator projection is best suited for \hspace{2cm}.

• (1) Polar regions
• (2) Small-scale world maps
• (3) Navigation purposes
• (4) Land use planning

Which of the following is a key characteristic of a map projection?

• (1) It accurately portrays the true shape of the Earth's surface.
• (2) It preserves the scale and area of the map.
• (3) It provides a 3D representation of the terrain.
• (4) It distorts the Earth's shape to fit on a flat surface.

Which aerial photo format typically provides the largest area of coverage on a single photograph?

• (1) Small format (e.g., 70mm)
• (2) Medium format (e.g., 9 inches × 9 inches)
• (3) Large format (e.g., 12 inches × 12 inches)
• (4) Digital imagery with a small sensor size.

Which type of map projection is most likely to distort the shape of large landmasses?

• (1) Cylindrical map projection.
• (2) Conic map projection.
• (3) Planar map projection.
• (4) Gnomonic map projection.

The process of transforming and representing features from the Earth's curved surface onto a flat map is known as \hspace{1.5cm}.

• (1) Geo-referencing.
• (2) Cartography.
• (3) Photogrammetry.
• (4) Remote sensing.

In cartography, map generalization refers to the process of:

• (1) Accurately representing every single feature on the ground at the map scale.
• (2) Simplifying or omitting features on a map to maintain clarity and legibility at a reduced scale.
• (3) Projecting geographic coordinates onto a flat map surface.
• (4) Assigning colors and symbols to different features on a map.

The propagation of electromagnetic waves involves the interplay of oscillating:

• (1) Electric and magnetic fields perpendicular to each other and the direction of propagation.
• (2) Electric and gravitational fields parallel to each other and perpendicular to the direction of propagation.
• (3) Magnetic and gravitational fields perpendicular to each other and parallel to the direction of propagation.
• (4) Electric and magnetic fields parallel to each other and the direction of propagation.

Radiance is best defined as:

• (1) Power per unit area
• (2) Power per unit solid angle
• (3) Power per unit area per unit solid angle
• (4) Power per unit wavelength

Which of the following defines the spatial resolution of a sensor?

• (1) Number of spectral bands
• (2) Smallest object detectable on the ground
• (3) Energy detection capacity
• (4) Frequency of satellite revisits

Higher radiometric resolution means:

• (1) Ability to distinguish finer wavelength differences
• (2) Ability to detect small differences in energy
• (3) Ability to capture more images per day
• (4) Ability to detect larger objects

Along-track stereoscopic imaging involves __________.

• (1) Two separate satellites imaging the same area
• (2) Imaging the same area from different angles during the same orbit
• (3) Successive orbits imaging the same area
• (4) Using a microwave radar for 3D imaging

Which systematic error in digital image processing is caused by the Earth's rotation during image acquisition?

• (1) Scan Skew
• (2) Mirror-Scan Velocity
• (3) Panoramic Distortion
• (4) Earth Rotation

The ALOS satellite belongs to ________.

• (1) Europe
• (2) Japan
• (3) Canada
• (4) India

The Copernicus program for Earth Observation is run by ________.

• (1) NASA
• (2) JAXA
• (3) ESA - European Space Agency
• (4) ISRO

Which image enhancement technique stretches the range of pixel intensity values to cover the full dynamic range of display devices?

• (1) Level slicing
• (2) Contrast stretching
• (3) Gray level thresholding
• (4) Histogram equalization

The European Space Agency's primary Earth observation program is known as _______.

• (1) ENVISAT/Sentinel
• (2) JERS
• (3) ALOS
• (4) CBERS

For urban mapping requiring very detailed building footprints, the ideal satellite would provide _______.

• (1) 30 m spatial resolution multispectral data
• (2) 5 m spatial resolution panchromatic data
• (3) 1 m or finer spatial resolution imagery
• (4) 250 m spatial resolution MODIS data

What does spectral resolution refer to in remote sensing?

• (1) Ability to detect different wavelengths of electromagnetic radiation
• (2) Ability to distinguish between different features on the ground
• (3) Ability to capture images at various times
• (4) Ability to discern details within an image

Which of the following best describes a relational database structure commonly used in GIS?

• (1) Data is organized in a tree-like hierarchy.
• (2) Data elements are linked through predefined relationships, forming a network.
• (3) Data is organized into tables with rows representing entities and columns representing attributes, with relationships established through keys.
• (4) Spatial data and attribute data are stored in separate, unconnected files.

Raster data typically represents geographic phenomena as:

• (1) Points, lines, and polygons.
• (2) A grid of cells, each with an associated value.
• (3) Topologically structured features with nodes and edges.
• (4) A collection of attribute records linked to spatial coordinates.

A topological error where adjacent polygons overlap is called ________.

• (1) Dangle
• (2) Overshoot
• (3) Gap
• (4) Sliver

DEM files are critical in which GIS applications?

• (1) Noise pollution mapping
• (2) Surface water runoff modeling
• (3) Soil pH mapping
• (4) Land use categorization

In raster to vector conversion, errors typically occur because of ______.

• (1) Topological inaccuracies
• (2) Attribute duplication
• (3) File compression
• (4) Metadata loss

In the vector data model, an area is typically stored as ______.

• (1) A single coordinate pair representing its centroid.
• (2) A grid of cells covering its extent.
• (3) A closed sequence of connected coordinate pairs forming a polygon.
• (4) A series of disconnected line segments

TIN (Triangulated Irregular Network) sheeting is a method used to represent ______.

• (1) Categorical spatial data.
• (2) Continuous surface data using non-overlapping triangles.
• (3) Linear features with associated attributes.
• (4) Regularly spaced grid data

A common challenge in raster to vector conversion, especially when dealing with scanned maps, is ______.

• (1) Loss of attribute information.
• (2) Introduction of topological errors.
• (3) Generalization and loss of detail in curved or complex features.
• (4) Difficulty in assigning coordinate values.

Structured Query Language (SQL) is primarily used in GIS to ______.

• (1) Perform spatial analysis operations.
• (2) Manage and query attribute data stored in databases.
• (3) Visualize spatial data in different map projections.
• (4) Convert data between different file formats.

A GPS collected point dataset suffers from high horizontal error.
The best immediate action is:

• (1) Increase the sampling interval
• (2) Correct using differential GPS (DGPS)
• (3) Rasterize the point data
• (4) Smooth using TIN creation

In the context of GIS modeling, a conceptual model represents:

• (1) The physical implementation of the database.
• (2) A high-level, user-oriented understanding of the real-world phenomena and their relationships.
• (3) The logical organization of data elements and their relationships within a database.
• (4) The specific data structures and file formats used to store the data.

A key characteristic of object-oriented database models in GIS is their ability to:

• (1) Only store spatial data efficiently.
• (2) Represent both spatial and non-spatial data as objects with associated attributes and behaviors (methods).
• (3) Organize data in a strictly hierarchical structure.
• (4) Primarily handle raster data.

Recent trends in GIS applications include all EXCEPT:

• (1) Mobile GIS
• (2) Cloud-based GIS
• (3) Real-time spatial analytics
• (4) Paper-based cartography

In GIS, what is the function of a topology rule?

• (1) To enforce spatial relationships between features
• (2) To define the spatial extent of a dataset
• (3) To specify the coordinate system of a map
• (4) To categorize features based on attributes

In the context of hydrological modeling within GIS, a Digital Elevation Model (DEM) is crucial for:

• (1) Determining soil texture and permeability.
• (2) Estimating evapotranspiration rates.
• (3) Delineating watersheds and analyzing flow direction.
• (4) Measuring rainfall intensity directly.

If \( A \) and \( B \) are two non-zero square matrices of same size such that the product matrix \( AB \) is a zero matrix, then which of the following must be true?

• (1) Both \( A \) and \( B \) are invertible
• (2) At least one of \( A \) or \( B \) is singular
• (3) \( A + B \) is invertible
• (4) \( A \) and \( B \) must be symmetric matrices

Consider the system of equations: \[ \begin{aligned} x + y + z &= 6
2x + 2y + 3z &= 13
3x + 4y + 5z &= 20 \end{aligned} \]
Which of the following statements is true about the solution to this system?

• (1) The system has a unique solution.
• (2) The system has no solution.
• (3) The system has infinitely many solutions.
• (4) The system has finitely many solutions but not unique.

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function such that \( f(0) = 3 \) and \( |f'(x)| \leq 2 \), \( \forall x \in \mathbb{R} \). Then \( f(2) \) lies in which of the following intervals?

• (1) \([-2, 5]\)
• (2) \([0, 8]\)
• (3) \([3, 8]\)
• (4) \([-1, 7]\)

The directional derivative of the function \( f(x, y) = 2x^2 + y^2 \)
along a line directed from \( (0, 0) \) to \( (1, 1) \), evaluated at the point \( x = 1, y = 1 \), is \underline{\hspace{2cm

• (1) \( 6\sqrt{2} \)
• (2) \( 2\sqrt{2} \)
• (3) \( 3\sqrt{2} \)
• (4) \( 6 \)

Given \( f(z) = \dfrac{-z}{(z - 2)^2} \), which of the following is the Laurent series expansion of \( f(z) \) around \( z = 2 \)?

• (1) \( f(z) = \dfrac{1}{(z - 2)} + \dfrac{1}{(z - 2)^2} \)
• (2) \( f(z) = \dfrac{2}{(z - 2)} + \dfrac{z}{(z - 2)^2} \)
• (3) \( f(z) = \dfrac{z}{(z - 2)^2} \)
• (4) \( f(z) = \dfrac{1}{(z - 2)} + \dfrac{2}{(z - 2)^2} \)

A complex function \( f(z) = u + iv \) with \( u = \operatorname{Re}(z) + \operatorname{Im}(z) \) is analytic if \( v = \) ________

• (1) \( v = \operatorname{Re}(z) + \operatorname{Im}(z) + c \), where \( c \) is an arbitrary constant
• (2) \( v = -\operatorname{Re}(z) + \operatorname{Im}(z) + c \), where \( c \) is an arbitrary constant
• (3) \( v = \operatorname{Re}(z) - \operatorname{Im}(z) + c \), where \( c \) is an arbitrary constant
• (4) \( v = -\operatorname{Re}(z) - \operatorname{Im}(z) + c \), where \( c \) is an arbitrary constant

Let \( f(x) = x^3 - \dfrac{9}{2}x^2 + 6x - 2 \) be a function defined on the closed interval \( [0, 3] \). Then, the global maximum value of \( f(x) \) is

• (1) 4.5
• (2) 0.5
• (3) 2.5
• (4) 3.0

In a town, the probability that a person attends a gym on weekdays is 0.7, the probability that a person attends the gym on weekends is 0.4, and the probability that a person attends the gym on weekdays or weekends, or both is 0.3. What is the probability that a person attends the gym on both weekdays and weekends?

• (1) 0.6
• (2) 0.7
• (3) 0.8
• (4) 0.9

In a workshop of 100 machines, 25 machines are defective. Assuming the Poisson law for the number of defective machines, the probability that a random sample of 5 machines will have no defective machine is

• (1) \( e^{-0.25} \)
• (2) \( 0.25\, e^{-0.25} \)
• (3) \( 0.25\, e^{-1.25} \)
• (4) \( e^{-1.25} \)

Let \( A, B \) be two events of a sample space such that \( P(A) = \frac{1}{4} \), \( P(B|A) = \frac{1}{2} \), \( P(A|B) = \frac{1}{4} \). If \( \overline{B} \) is the complement of \( B \), then \( P(A | \overline{B}) \) is _____

• (1) \( \dfrac{3}{4} \)
• (2) \( \dfrac{1}{4} \)
• (3) \( \dfrac{1}{3} \)
• (4) \( \dfrac{2}{3} \)