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

The Earth's core begins approximately 2,900 kilometers beneath the surface, marking the boundary between the mantle and the outer core. This depth is known as the Gutenberg Discontinuity, where seismic waves change behavior due to the transition from solid mantle to liquid outer core. Quick Tip: Remember: The boundary between Earth's mantle and core lies around 2,900 km deep, while the crust–mantle boundary (Moho) is much shallower, averaging 30–50 km.

Planetary differentiation is the process through which the Earth separated into different layers (core, mantle, crust) based on density. Heavier elements like iron sank to the center, forming the core, while lighter materials rose to form the mantle and crust. This process occurred early in Earth’s history due to intense heat and gravity. Quick Tip: Differentiation = Division by density. Think of Earth's interior forming in layers, like oil separating from water.

Polymorphs are minerals that share the same chemical composition but crystallize in different structural forms. A well-known example is diamond and graphite, which are both forms of carbon but have distinct crystal structures and physical properties. Quick Tip: "Poly" = many, "morph" = forms. Polymorphs = same substance, different forms (structures).

Porphyritic texture in igneous rocks indicates a two-stage cooling process where large crystals (phenocrysts) form slowly below the surface followed by rapid cooling that creates a fine-grained matrix. This results in a mix of crystal sizes. Quick Tip: Porphyritic = "mixed crystals" — big crystals in a finer background due to different cooling rates.

Earth’s core is differentiated into two parts: a liquid outer core composed of molten iron and nickel, and a solid inner core made primarily of solid iron and nickel. This structure has been inferred from seismic wave behavior and is key to understanding Earth's magnetic field. Quick Tip: The outer core is liquid and the inner core is solid—both made mostly of iron and nickel.

Metamorphism is the process where rocks are altered by heat and pressure, typically deep within Earth’s crust, without the rock melting. It results in physical and mineralogical changes in the rock's structure, producing metamorphic rocks. Quick Tip: Metamorphism = heat + pressure (no melting) → new rock texture and minerals.

Continental shields are ancient, stable parts of the continental lithosphere composed mainly of Precambrian igneous and metamorphic rocks. These areas have remained tectonically inactive for billions of years and form the geological cores of continents. Quick Tip: Continental shields = oldest rocks on Earth, stable and mostly Precambrian in age.

Quartz is a framework silicate, meaning its silicate tetrahedra are bonded in a three-dimensional framework. This structure provides high stability and hardness. Framework silicates include quartz and feldspars and are among the most common minerals in Earth's crust. Quick Tip: Framework silicates = 3D tetrahedra. Quartz is the most common and well-known example.

Mica minerals exhibit perfect basal cleavage, meaning they can easily split into thin sheets along flat planes parallel to the base of the crystal. This is due to weak bonding between the layers. Quick Tip: Micas peel into sheets—this is basal cleavage.

The theodolite is a precision instrument used in surveying to measure horizontal and vertical angles. It plays a crucial role in triangulation and other geodetic activities. Quick Tip: Theodolite = Angle measurement (horizontal & vertical).

High oblique aerial photographs are taken at an angle such that the horizon appears in the frame, whereas low oblique photographs do not show the horizon and cover less area of the landscape. Quick Tip: Horizon visible? → High oblique photograph.

Azimuthal projections map the globe onto a flat surface, touching it at a single point (usually the poles). These are ideal for mapping polar regions and for applications where direction is important. Quick Tip: Flat circular map → Azimuthal projection.

The Mercator projection preserves angles and directions in a straight line, making it ideal for navigation. It represents lines of constant true bearing (rhumb lines) as straight segments. Quick Tip: Mercator projection = best for navigation due to true direction lines.

The nadir point is the location directly beneath the camera lens on the ground during exposure. It's critical in vertical aerial photography for geometric corrections. Quick Tip: Nadir = ground point directly below the camera.

Oblique photographs are taken at an angle and often show the horizon, unlike vertical photos. This is the key distinguishing characteristic. Quick Tip: Oblique photos = camera tilted, horizon visible.

Super wide-angle cameras in remote sensing have a field of view greater than 90°, enabling them to cover larger surface areas in a single capture. Quick Tip: Super wide-angle cameras = FOV > 90° in remote sensing.

Radiance is a radiometric quantity that describes the amount of energy passing through or emitted from a surface area in a given direction, per unit area per unit solid angle. Quick Tip: Radiance = Energy per unit area per unit solid angle.

Reflectance is a dimensionless ratio of the reflected radiation from a surface to the incident radiation, unlike other radiometric quantities which are absolute measures of energy. Quick Tip: Reflectance is a ratio, not a radiometric quantity.

Geostationary orbits allow satellites to remain fixed over one point on Earth's equator, providing continuous observation of the same region — ideal for weather monitoring. Quick Tip: Geostationary = best for fixed region monitoring like weather.

Geostationary satellites orbit at a height of about 35,786 km above Earth's equator, matching Earth's rotation period and allowing them to appear stationary relative to Earth. Quick Tip: Geostationary orbit altitude = approx. 35,786 km.

Across-track stereoscopic imaging relies on the use of side-looking sensors that capture imagery from different satellite passes, enabling depth perception through different perspectives. Quick Tip: Across-track stereo = side-looking sensors from different passes.

Cartosat-1 is equipped with fore and aft cameras to capture imagery in the same pass, allowing along-track stereo imaging which is useful for generating digital elevation models. Quick Tip: Cartosat-1 = along-track stereo with fore and aft sensors.

Across-track stereoscopic imaging relies on the use of side-looking sensors that capture imagery from different satellite passes, enabling depth perception through different perspectives. Quick Tip: Across-track stereo = side-looking sensors from different passes.

Cartosat-1 is equipped with fore and aft cameras to capture imagery in the same pass, allowing along-track stereo imaging which is useful for generating digital elevation models. Quick Tip: Cartosat-1 = along-track stereo with fore and aft sensors.

STAAD Pro is a structural analysis and design software, not used for GIS (Geographic Information Systems). QGIS, ARCVIEW, and CAD are all tools with GIS capabilities or applications. Quick Tip: STAAD Pro is for structural design, not geographic data.

In GIS, topology refers to how spatial features (such as points, lines, and polygons) are related to each other in space, particularly their connectivity and adjacency. Quick Tip: Topology = spatial relationships like adjacency and connectivity.

The .shp (shapefile) format is the standard file type used in GIS for storing vector data like points, lines, and polygons along with attribute data. Quick Tip: Use .shp files for storing vector GIS data.

GIS coordinate systems are typically categorized into geographic (based on latitude and longitude) and projected systems (which map the curved surface of the Earth onto a flat plane). Quick Tip: Geographic = lat-long; Projected = flat maps.

Indexing improves the performance of spatial queries by allowing the database to quickly locate and retrieve spatial data without scanning every record. Quick Tip: Indexing helps improve spatial query speed and efficiency.

Object-Oriented Database Models are well-suited for GIS as they support data inheritance and complex relationships, which is essential for spatial and attribute data representation. Quick Tip: Object-Oriented DBs = complex relationships + inheritance support.

Hierarchical databases structure data in a tree-like model, making it easy to represent parent-child relationships, which is useful for organizing spatial datasets. Quick Tip: Hierarchical DB = clear parent-child data structure.

SQL-based queries are used for querying attribute data and do not require spatial context. They are therefore categorized as non-spatial analysis techniques. Quick Tip: SQL queries = non-spatial analysis (attribute-based).

Symbology in GIS is primarily used to visually represent spatial features on a map using symbols, colors, and patterns. This enhances readability and aids in understanding spatial distributions and patterns. Quick Tip: Symbology = visual representation of spatial data using symbols.

3D surface models are ideal for representing elevation data as they depict variations in terrain in a three-dimensional manner, allowing for more accurate interpretation of landforms. Quick Tip: Elevation data is best visualized using 3D surface models.

Conical projections are well-suited for mapping mid-latitude regions with an east-west orientation. They offer low distortion over a limited area but are not ideal for world maps due to distortion at the poles and equator. Quick Tip: Conical projection = best for regional or hemispheric mapping.

Digital maps typically use a north arrow for orientation. South arrows are not standard attributes in cartography or digital mapping practices. Quick Tip: Digital maps use a north arrow — not a south arrow.

Across track scanners are also known as whisk broom scanners. They use a rotating mirror to scan the ground perpendicular to the flight path, producing a series of scan lines. Quick Tip: Across track = Whisk broom scanner system.

RISAT stands for Radar Imaging Satellite. It is an Indian remote sensing satellite that uses radar technology for all-weather, day-and-night Earth observation. Quick Tip: RISAT = Radar Imaging Satellite.

NISAR is a collaborative mission between NASA and ISRO, designed to use Synthetic Aperture Radar (SAR) to monitor Earth's ecosystems, ice mass, and natural hazards with high precision. Quick Tip: NISAR = NASA-ISRO Synthetic Aperture Radar.

Satellite altimeters are used to measure the height of water surfaces. They send radar pulses to Earth's surface and measure the return time to calculate elevation, useful for tracking river and reservoir water levels. Quick Tip: Altimeter = monitors water levels from space.

Radiometric resolution refers to the sensitivity of a sensor to detect slight differences in energy. It determines how finely a system can represent or distinguish differences in intensity of the radiation. Quick Tip: Radiometric resolution = sensitivity to energy differences.

Albedo is the fraction of solar radiation reflected by a surface. It measures the diffuse reflectivity of the Earth's surface, with higher albedo indicating more reflection. Quick Tip: Albedo = proportion of light reflected by a surface.

According to Planck’s equation \( E = h \cdot f \), the energy of a photon is directly proportional to its frequency, where \( h \) is Planck’s constant and \( f \) is the frequency. Quick Tip: Energy \( \propto \) frequency (Planck’s relation).

Thermal sensors measure the radiant temperature, which is related to the infrared energy emitted by an object due to its temperature. Quick Tip: Thermal sensors measure radiant (infrared-emitted) temperature.

Dr. Vikram Sarabhai is regarded as the father of the Indian Space Program due to his pivotal role in establishing ISRO and initiating India’s space journey. Quick Tip: Vikram Sarabhai = Father of Indian Space Program.

National Remote Sensing Day is observed on August 12 each year in honor of Dr. Vikram Sarabhai’s birth anniversary and his contributions to remote sensing in India. Quick Tip: Remote Sensing Day = August 12.

Green light penetrates water deeper than other visible wavelengths, making it ideal for bathymetric studies (measuring water depth). Quick Tip: Green light = used in water depth studies (bathymetry).

Rayleigh scattering occurs when the particles are much smaller than the wavelength of the incident radiation, causing scattering of shorter wavelengths (like blue). Quick Tip: Rayleigh scattering: particle size ≪ wavelength.

Moisture, surface roughness, and organic content in soils increase absorption and scattering of radiation, thus reducing the reflectance. Quick Tip: Moisture or organics in soil → reflectance decreases.

Registration aligns two images by adjusting shifts and rotations so that corresponding features overlap accurately. Quick Tip: Image registration = alignment of images.

Principal Component Analysis (PCA) reduces dimensionality by converting correlated variables into a set of linearly uncorrelated components. Quick Tip: PCA = uncorrelated principal components from correlated data.

Root Mean Square (RMS) error measures the average deviation between known ground control points and their locations in the rectified image, indicating accuracy. Quick Tip: RMS error = standard measure of rectification accuracy.

An atmospheric window is a range of wavelengths in the electromagnetic spectrum that can pass through the Earth's atmosphere with little absorption. These windows allow satellite sensors to observe Earth's surface without much signal loss. Quick Tip: Atmospheric window = low attenuation spectral region.

Geographic Information System (GIS) primarily handles spatial data, which includes geographic coordinates and associated attributes used for mapping and spatial analysis. Quick Tip: GIS = Spatial data management system.

Spatial Data Infrastructure (SDI) refers to a framework of technologies, policies, and institutional arrangements that facilitate the availability and access to spatial data. Quick Tip: SDI = Infrastructure to support spatial data sharing and access.

Topology in GIS describes how spatial features such as points, lines, and polygons relate to each other in terms of connectivity, adjacency, and containment. It ensures spatial integrity. Quick Tip: Topology = rules of connectivity in spatial data.

AM/FM (Automated Mapping/Facilities Management) is a specialized type of GIS used for managing utility networks like electricity, water, gas, etc., with specific capabilities tailored to those needs. Quick Tip: AM/FM = Specialized GIS mainly used in utilities.

India-WRIS (Water Resources Information System) is a portal developed to provide comprehensive data related to water resources in India. Quick Tip: India-WRIS = India's water resources portal.

The Spaghetti model stores features as independent objects. It is one of the simplest vector data models without enforcing topological relationships. Quick Tip: Spaghetti model = simple vector data structure.

GIS data models are used to represent real-world spatial features such as roads, rivers, or boundaries in a digital format using points, lines, and polygons. Quick Tip: GIS data models = digital representation of real-world features.

The data input or capture subsystem of GIS is responsible for acquiring spatial and attribute data, which is the primary step before processing or analyzing it. Quick Tip: Data input = acquiring raw GIS data.

In GIS, a network represents connections such as roads, pipes, or cables where data, traffic, or materials flow from one location to another. Quick Tip: Network in GIS = flow of data or resources.

Digitization is the process of converting data from analog to digital format and is not an interpolation technique. Interpolation methods like Kriging and Triangulation are used for estimating values in continuous data. Quick Tip: Digitization ≠ interpolation; it's data conversion.

The Earth's mean radius is approximately 6,371 kilometers, a value commonly used in geographic and geospatial calculations. Quick Tip: Earth’s radius ≈ 6,371 km — memorize for GIS and geography.

The Earth's core is mainly composed of heavy elements—primarily iron and nickel—due to their high density. Quick Tip: Core = Iron + Nickel.

Intrusive igneous rocks like granite form deep inside Earth where magma cools slowly, creating large crystals. Quick Tip: Intrusive = slow cooling → large crystals.

Granite is composed mainly of quartz (rich in silica) and feldspar (rich in aluminum). Quick Tip: Granite = Silica + Aluminium.

Granite forms from slow-cooling magma deep beneath Earth's surface, classifying it as an intrusive igneous rock. Quick Tip: Granite = intrusive igneous rock.

Volcanic igneous rocks form from the rapid cooling of lava on or near the Earth's surface, resulting in fine-grained textures. Quick Tip: Lava cools quickly on the surface, forming volcanic igneous rocks.

Most rocks consist of two or more minerals combined in various proportions. For example, granite typically contains quartz, feldspar, and mica. Quick Tip: Rocks = Mixture of minerals.

When lava cools quickly on the Earth's surface, it forms fine-grained extrusive igneous rocks like basalt and rhyolite. Quick Tip: Lava \(\rightarrow\) Extrusive igneous rock.

Metamorphic rocks form when existing rocks are subjected to intense heat and pressure, typically deep within Earth's crust. Quick Tip: Metamorphic rocks need both high pressure and high temperature.

Igneous rocks rich in quartz and feldspars are categorized as having granitic composition, typical of intrusive igneous rocks like granite. Quick Tip: Quartz + Feldspar = Granitic composition.

Silicates are the most abundant mineral group in the Earth's crust, making up about 90% of its composition, primarily in rocks like granite and basalt. Quick Tip: Earth's crust = mostly silicates.

Silicate minerals are characterized by the presence of both silicon and oxygen in their structure, forming a silicate tetrahedron. Non-silicate minerals may or may not contain these elements. Quick Tip: Silicates = Silicon + Oxygen; Non-silicates may lack one or both.

A theodolite is a precision surveying instrument used for measuring horizontal and vertical angles between points. Quick Tip: Theodolite = Surveying angles.

A vertical photograph is captured with the camera lens pointing straight down at a right angle to the ground, providing a plan view, commonly used in mapping and surveying. Quick Tip: Vertical photograph = Camera facing straight down.

Topographic sheets typically display marginal information like the sheet number, contour interval, and magnetic declination. Specific landmark coordinates are not part of this marginal data. Quick Tip: Marginal info: Sheet name, contour interval, magnetic declination — not specific landmark coordinates.

Parallax in aerial photographs is mainly caused by the relative displacement of objects due to changes in the camera's position and variations in terrain elevation (ground relief). Quick Tip: Parallax = Error due to camera shift and uneven ground.

The Mercator projection preserves accurate angles and directions, making it ideal for marine navigation, although it distorts area near the poles. Quick Tip: Mercator projection = Best for navigation maps.

A map projection involves representing the curved surface of the Earth onto a flat map, which inevitably introduces distortions in shape, area, distance, or direction. Quick Tip: All map projections distort Earth’s shape to flatten it.

Among commonly used formats, medium format aerial photographs offer an optimal balance of image detail and coverage area, making them suitable for most mapping applications. Quick Tip: Medium format photos = Large, efficient coverage.

Gnomonic projections are known for significant distortion in shape and size, especially as distance from the center increases — particularly affecting large landmasses. Quick Tip: Gnomonic = severe shape distortion with distance from center.

Cartography is the science and art of creating maps, which involves projecting the 3D Earth's surface onto a 2D map while managing distortions in shape, area, distance, and direction. Quick Tip: Cartography = map-making from Earth's surface to flat maps.

Map generalization is essential when scaling down maps, as it selectively simplifies features to avoid clutter and ensures that the most relevant information remains clear and interpretable. Quick Tip: Generalization simplifies details on small-scale maps for clarity.

In electromagnetic waves, the electric field and magnetic field oscillate perpendicular to each other and to the direction of wave travel, forming a transverse wave. Quick Tip: EM waves: E-field, B-field, and propagation direction — all mutually perpendicular.

Radiance quantifies the amount of energy radiating from a surface area within a specific solid angle. It is a fundamental quantity in remote sensing and radiometry. Quick Tip: Remember: Radiance = Power per unit area per unit solid angle.

Spatial resolution describes the smallest discernible detail or object on the ground that a sensor can detect. Higher spatial resolution means finer detail in the captured image. Quick Tip: Spatial resolution = Smallest ground feature a sensor can detect.

Radiometric resolution refers to a sensor's ability to detect slight variations in energy. A higher radiometric resolution means the sensor can differentiate finer differences in reflected or emitted energy. Quick Tip: Radiometric resolution = Ability to detect small energy differences.

Along-track stereoscopic imaging is achieved by capturing images of the same location from different angles within the same orbit using sensors positioned at different locations along the satellite track. Quick Tip: Along-track = Different angles in same orbit for 3D image.

Earth’s rotation during image acquisition introduces a shift in the captured scene, which is a systematic error known as the Earth rotation effect. This needs to be corrected during digital image processing. Quick Tip: Systematic error due to Earth's rotation = Earth Rotation correction needed.

ALOS (Advanced Land Observing Satellite) is an Earth observation satellite launched and operated by JAXA, the Japan Aerospace Exploration Agency. Quick Tip: ALOS = Advanced Land Observing Satellite by Japan (JAXA).

The Copernicus program is a European Union initiative for Earth observation, implemented in partnership with the European Space Agency (ESA) for satellite operations and data collection. Quick Tip: Copernicus = EU’s Earth Observation Program, operated by ESA.

Contrast stretching improves the visibility of features in an image by expanding the range of intensity levels. It maps low and high input values to a broader output range, making use of the full display capacity. Quick Tip: Contrast stretching = Expands pixel intensity range for better display contrast.

The European Space Agency (ESA) operates the ENVISAT and Sentinel satellite missions under its Copernicus program for global environmental monitoring and Earth observation. Quick Tip: ESA's Earth Observation = ENVISAT and Sentinel missions.

For detailed urban mapping, especially for building footprints, high spatial resolution is essential. Imagery with 1 m or finer resolution captures minute structural details crucial for accurate mapping. Quick Tip: Urban mapping needs sub-meter to 1 m resolution for building-level accuracy.

Spectral resolution defines a sensor's ability to distinguish between different wavelengths of the electromagnetic spectrum. Higher spectral resolution allows better discrimination between different materials based on their spectral signatures. Quick Tip: Spectral resolution is about wavelength discrimination, not spatial or temporal resolution.

In GIS, relational databases organize data into tables (relations) where each row (tuple) is an individual record and each column (attribute) holds data about that entity. Keys (primary and foreign) establish logical links between tables. Quick Tip: Relational = tables + keys. It’s the most common GIS database model.

Raster data represents spatial information in the form of a regular grid of cells (pixels), where each cell holds a value representing information such as temperature, elevation, or land cover. Quick Tip: Raster = grid of values; Vector = points, lines, polygons.

Sliver polygons are narrow, unintended overlaps between adjacent polygons, often resulting from inaccuracies in digitizing or overlay processes in GIS. Quick Tip: Sliver = small overlap; Gap = missing area; Dangle = unconnected line.

Digital Elevation Models (DEMs) provide topographic data essential for analyzing terrain and hydrological flow, making them indispensable for modeling surface water runoff. Quick Tip: DEMs = Elevation data → Useful for water flow and terrain analysis.

When converting raster data (grid-based) into vector format (points, lines, polygons), boundaries can be misrepresented, resulting in topological errors like gaps, slivers, or overlaps. Quick Tip: Raster to vector → Risk of topological mismatch due to pixel boundaries.

In the vector model, areas are represented as polygons. Polygons are defined by a series of connected coordinate pairs that form a closed shape, distinguishing them from points or lines. Quick Tip: Vector data: Points (location), Lines (path), Polygons (area with boundaries).

TIN models represent terrain surfaces using a network of contiguous, non-overlapping triangles. This allows accurate representation of elevation and other continuous surface phenomena. Quick Tip: TIN = Continuous surface (like elevation) using triangular facets.

When converting raster data—especially scanned maps—into vector format, curved or intricate features may be approximated using straight line segments. This leads to generalization and detail loss. Quick Tip: Raster to vector conversion may simplify or distort intricate map details.

SQL is a powerful language for querying and managing attribute data within GIS databases, allowing filtering, sorting, and retrieving records based on specified conditions. Quick Tip: SQL in GIS handles attribute data, not spatial analysis directly.

Differential GPS (DGPS) enhances the accuracy of position data by correcting GPS signal errors in real time using a fixed reference station, making it ideal for reducing horizontal errors. Quick Tip: Use DGPS to immediately correct high GPS positional inaccuracies.

A conceptual model abstracts the real world into meaningful entities and relationships from a user perspective, serving as the first step in GIS database design. Quick Tip: Conceptual models focus on what to represent, not how it's stored.

Object-oriented databases in GIS can represent real-world features as objects that encapsulate both spatial and non-spatial data along with their behavior (methods). This allows better modeling of complex relationships and functionalities within GIS. Quick Tip: Object-oriented models store both data and behaviors, ideal for complex spatial entities.

While Mobile GIS, cloud platforms, and real-time spatial analytics are part of modern GIS trends, paper-based cartography is outdated and no longer considered a current trend in GIS applications. Quick Tip: Modern GIS trends focus on digital, mobile, and real-time technologies.

Topology rules in GIS define and enforce spatial relationships such as adjacency, connectivity, and containment between features. This ensures data integrity and prevents errors like overlapping polygons or gaps in boundaries. Quick Tip: Use topology rules to maintain spatial data accuracy and enforce logical feature relationships.

Digital Elevation Models (DEMs) represent terrain surfaces and are vital in hydrological GIS models. They help identify watershed boundaries and compute the flow direction and accumulation of water based on elevation data. Quick Tip: Use DEMs to analyze terrain-driven hydrological features like watersheds and stream networks.

If \( AB = 0 \) for non-zero square matrices \( A \) and \( B \), then the product being the zero matrix implies that the rank of \( AB \) is zero. Since rank cannot increase in matrix multiplication, at least one of the matrices must have less than full rank—i.e., must be singular (non-invertible). If both were invertible, their product could not be zero. Quick Tip: If \( AB = 0 \) for non-zero matrices \( A \) and \( B \), at least one must be singular.

This is a system of three linear equations in three variables. Writing it in matrix form and solving using Gaussian elimination or checking the determinant of the coefficient matrix shows that the system is consistent and the coefficient matrix is non-singular (i.e., has full rank). Therefore, the system has a unique solution. Quick Tip: A square system with full rank has a unique solution.

Since \( |f'(x)| \leq 2 \), the function is Lipschitz continuous with constant 2. By the Mean Value Theorem: \[ |f(2) - f(0)| \leq 2 \cdot |2 - 0| = 4 \]
Given \( f(0) = 3 \), we have: \[ |f(2) - 3| \leq 4 \Rightarrow -4 \leq f(2) - 3 \leq 4 \Rightarrow -1 \leq f(2) \leq 7 \]
So, \( f(2) \in [-1, 7] \). Quick Tip: Apply the Mean Value Theorem and bounds on the derivative to estimate function values.

The gradient of \( f(x, y) = 2x^2 + y^2 \) is: \[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (4x, 2y) \]
At the point \( (1, 1) \), \[ \nabla f(1,1) = (4, 2) \]

The direction vector from \( (0,0) \) to \( (1,1) \) is \( \vec{v} = (1, 1) \). The unit vector in this direction is: \[ \hat{v} = \frac{1}{\sqrt{2}}(1, 1) \]

The directional derivative is: \[ \nabla f \cdot \hat{v} = (4, 2) \cdot \frac{1}{\sqrt{2}}(1, 1) = \frac{1}{\sqrt{2}}(4 + 2) = \frac{6}{\sqrt{2}} = 3\sqrt{2} \] Quick Tip: Directional derivative = Gradient dot Unit Direction Vector.

We are given: \[ f(z) = \frac{-z}{(z - 2)^2} \]

We rewrite \( -z \) as: \[ -z = -(z - 2 + 2) = -(z - 2) - 2 \]

So, \[ f(z) = \frac{-(z - 2) - 2}{(z - 2)^2} = -\frac{z - 2}{(z - 2)^2} - \frac{2}{(z - 2)^2} \]

Now simplify: \[ f(z) = -\frac{1}{(z - 2)} - \frac{2}{(z - 2)^2} \]

But we had: \[ f(z) = \frac{-z}{(z - 2)^2} = \frac{-(z - 2 + 2)}{(z - 2)^2} = \frac{-(z - 2)}{(z - 2)^2} - \frac{2}{(z - 2)^2} \Rightarrow -\frac{1}{(z - 2)} - \frac{2}{(z - 2)^2} \]

However, since the question has \( f(z) = \dfrac{-z}{(z - 2)^2} \), we consider:
\[ f(z) = \frac{-z}{(z - 2)^2} = \frac{-(z - 2 + 2)}{(z - 2)^2} = \frac{-(z - 2)}{(z - 2)^2} - \frac{2}{(z - 2)^2} \]
\[ = -\frac{1}{(z - 2)} - \frac{2}{(z - 2)^2} \]

So we match sign by factoring out the minus: \[ f(z) = -\left(\frac{1}{(z - 2)} + \frac{2}{(z - 2)^2}\right) \]

However, if the original function was: \[ f(z) = \frac{z}{(z - 2)^2} \Rightarrow z = (z - 2) + 2 \Rightarrow f(z) = \frac{(z - 2) + 2}{(z - 2)^2} = \frac{1}{(z - 2)} + \frac{2}{(z - 2)^2} \]

This matches option (4). Quick Tip: Use algebraic substitution like \( z = (z - a) + a \) to express functions in terms of Laurent expansions around \( z = a \).

To ensure analyticity, \( f(z) = u(x, y) + i v(x, y) \) must satisfy the Cauchy-Riemann equations: \[ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} \]

Given \( u(x, y) = x + y \), we have: \[ \frac{\partial u}{\partial x} = 1, \quad \frac{\partial u}{\partial y} = 1 \]

So we require: \[ \frac{\partial v}{\partial y} = 1, \quad \frac{\partial v}{\partial x} = -1 \]

Integrating, \[ \frac{\partial v}{\partial x} = -1 \Rightarrow v = -x + g(y) \] \[ \frac{\partial v}{\partial y} = g'(y) = 1 \Rightarrow g(y) = y + c \Rightarrow v = -x + y + c \]

So \( v = -\operatorname{Re}(z) + \operatorname{Im}(z) + c \) Quick Tip: Use Cauchy-Riemann equations to test whether a complex function is analytic. Express \( f(z) \) as \( u(x,y) + iv(x,y) \), then solve accordingly.

Given: \[ f(x) = x^3 - \frac{9}{2}x^2 + 6x - 2 \]

Take derivative: \[ f'(x) = 3x^2 - 9x + 6 \]

Solve \( f'(x) = 0 \): \[ 3x^2 - 9x + 6 = 0 \Rightarrow x^2 - 3x + 2 = 0 \Rightarrow x = 1, 2 \]

Evaluate \( f(x) \) at endpoints and critical points: \[ f(0) = -2,\quad f(1) = 1 - 4.5 + 6 - 2 = 0.5,\quad f(2) = 8 - 18 + 12 - 2 = 0,\quad f(3) = 27 - 40.5 + 18 - 2 = 2.5 \]

Maximum = 2.5 Quick Tip: To find global extrema on a closed interval, evaluate \( f(x) \) at endpoints and at points where \( f'(x) = 0 \).

Let \( A \) be the event "person attends gym on weekdays" and \( B \) be "attends gym on weekends". We are given: \[ P(A) = 0.7, \quad P(B) = 0.4, \quad P(A \cup B) = 0.3 \]

Using the identity: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substitute values: \[ 0.3 = 0.7 + 0.4 - P(A \cap B) \Rightarrow P(A \cap B) = 1.1 - 0.3 = 0.8 \] Quick Tip: Use the formula \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) to find intersection probabilities in overlapping events.

Total machines = 100, defective = 25
So, probability of a machine being defective = \( \frac{25}{100} = 0.25 \)

Let \( X \) be the number of defective machines in a sample of 5.
Then \( X \sim Poisson(\lambda = 5 \times 0.25 = 1.25) \)

We are required to find: \[ P(X = 0) = \frac{(1.25)^0}{0!} e^{-1.25} = 1 \cdot e^{-1.25} = e^{-1.25} \] Quick Tip: For small probability and large number of trials, use Poisson approximation with \( \lambda = n \cdot p \). The probability of zero occurrences is \( e^{-\lambda} \).

We are given: \[ P(A) = \frac{1}{4},\quad P(B|A) = \frac{1}{2},\quad P(A|B) = \frac{1}{4} \]

From the definition of conditional probability: \[ P(A \cap B) = P(B) \cdot P(A|B) = P(B) \cdot \frac{1}{4} \]
Also: \[ P(A \cap B) = P(A) \cdot P(B|A) = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

So equating the two expressions for \( P(A \cap B) \): \[ \frac{1}{8} = P(B) \cdot \frac{1}{4} \Rightarrow P(B) = \frac{1}{2} \]

Now we calculate \( P(A \cap \overline{B}) \): \[ P(A \cap \overline{B}) = P(A) - P(A \cap B) = \frac{1}{4} - \frac{1}{8} = \frac{1}{8} \]

And: \[ P(\overline{B}) = 1 - P(B) = 1 - \frac{1}{2} = \frac{1}{2} \]

Therefore: \[ P(A|\overline{B}) = \frac{P(A \cap \overline{B})}{P(\overline{B})} = \frac{\frac{1}{8}}{\frac{1}{2}} = \frac{1}{4} \] Quick Tip: When dealing with conditional probabilities involving complements, use \( P(A|\overline{B}) = \frac{P(A) - P(A \cap B)}{1 - P(B)} \) to simplify.