AP PGECET Question Paper 2025 (Available): Download Metallurgy Question Paper with Solution PDF

The Gibbs free energy of a reaction is minimized when the system is at equilibrium. Which of the following conditions must hold at equilibrium?

• (1) The reaction quotient is greater than the equilibrium constant.
• (2) The reaction quotient is equal to the equilibrium constant.
• (3) The chemical potentials of all components are equal.
• (4) The Gibbs free energy is positive.

What is the use of the Ellingham diagram in metallurgy?

• (1) To calculate the boiling point of different metals.
• (2) To predict the solubility of gases in metals.
• (3) To determine the temperature at which a metal oxide can be reduced.
• (4) To visualize phase changes in metal alloys.

In a second-order reaction, the rate of the reaction is proportional to:

• (1) The square of the concentration of the reactant
• (2) The concentration of the reactant
• (3) The concentration of the reactant raised to the third power
• (4) The inverse of the concentration of the reactant

Which of the following statements is true regarding phase equilibrium in a binary alloy system?

• (1) The phase diagram of a binary alloy system typically includes solid, liquid and gas phases.
• (2) At the eutectic composition in a binary alloy system, the system consists of a single phase.
• (3) The liquid phase is always more stable than the solid phase in a binary system.
• (4) The eutectic temperature is the lowest temperature at which the solid phase forms.

Which of the following conditions is necessary for a metallurgical reaction to be spontaneous at a given temperature?

• (1) The reaction must have a negative entropy change.
• (2) The enthalpy change must be negative.
• (3) The Gibbs free energy change must be negative.
• (4) The reaction must involve an increase in temperature.

In a diffusion process, if the concentration of a species is higher at the surface of a material, the diffusion flux will:

• (1) Increase with time
• (2) Remain constant
• (3) Decrease with time
• (4) Be zero

The first law of thermodynamics is related to:

• (1) Entropy
• (2) Enthalpy
• (3) Conservation of energy
• (4) Free energy

Which of the following diagrams is used to predict oxide stability?

• (1) Phase diagram
• (2) TTT diagram
• (3) Ellingham diagram
• (4) Lever rule diagram

Activity of a pure element in its standard state is:

• (1) 0
• (2) 1
• (3) \(\infty\)
• (4) -1

The rate-limiting step is the:

• (1) Fastest step
• (2) Slowest step
• (3) Initial step
• (4) Final step

For an ideal solution, volume of mixing is ...........

• (1) Negative
• (2) Infinity
• (3) Zero
• (4) Fractional

All adiabatic processes are known as ............

• (1) Isoentropic
• (2) Isothermal
• (3) Isochore
• (4) Isobar

Henry’s law is applicable to ............ solutions.

• (1) Ideal
• (2) Regular
• (3) Very dilute
• (4) Solid

\(C_p\), the heat capacity at constant pressure is given by:

• (1) \(\left(\frac{\partial H}{\partial T}\right)_P\)
• (2) \(\left(\frac{\partial E}{\partial T}\right)_P\)
• (3) \(\left(\frac{\partial A}{\partial T}\right)_P\)
• (4) \(\left(\frac{\partial G}{\partial T}\right)_P\)

For ideal solutions, activity of a component is ............ mole fraction.

• (1) Equal to
• (2) Greater than
• (3) Lesser than
• (4) Not related to

Chemical potential of a system is regarded as ............

• (1) Partial molar property
• (2) Colligative Property
• (3) State function
• (4) Electrochemical property

The order of a reaction can be determined from:

• (1) Heat change
• (2) Stoichiometry
• (3) Experimental data
• (4) Ellingham diagram

For the reaction A + B → Products, the rate law is \( r = k[A]^{1/2}[B]^2 \). What is the order of the reaction?

• (1) 1.5
• (2) 2.5
• (3) 2
• (4) 3

Equilibrium constant of any reaction is ............ at constant temperature.

• (1) Negative
• (2) Positive
• (3) Fraction
• (4) Constant

Which of the following is true for a reaction to be zero-order?

• (1) The rate of reaction is independent of the concentration of reactants.
• (2) The rate constant has units of mol/L
• (3) The rate increases with the concentration of reactants
• (4) The reaction is independent of temperature

According to Fick's law of diffusion, the diffusion flux is proportional to:

• (1) The concentration gradient
• (2) The temperature gradient
• (3) The pressure gradient
• (4) The velocity of the fluid

In a fluid flowing over a flat plate, the Reynolds number (Re) determines:

• (1) The rate of mass transfer
• (2) Whether the flow is laminar or turbulent
• (3) The pressure drop in the fluid
• (4) The temperature distribution

The diffusion mechanism that involves atoms moving through vacancies in a solid is called:

• (1) Interstitial diffusion
• (2) Vacancy diffusion
• (3) Grain boundary diffusion
• (4) Surface diffusion

In radiation heat transfer, the Stefan–Boltzmann constant (\(\sigma\)) is used in the equation for:

• (1) Thermal diffusivity
• (2) The rate of heat loss by convection
• (3) The total radiation emitted by a black body
• (4) The rate of heat transfer by conduction

What is the primary cause of surface tension in a liquid?

• (1) The cohesive forces between the molecules of the liquid
• (2) The gravitational pull on the liquid molecules
• (3) The adhesive forces between the liquid and the container
• (4) The chemical reactions occurring at the surface of the liquid

Which of the following metals is most susceptible to stress corrosion cracking (SCC) in chloride environments?

• (1) Carbon steel
• (2) Copper
• (3) Titanium
• (4) Austenitic stainless steel

Which of the following is a common method used for the concentration of sulfide ores?

• (1) Magnetic separation
• (2) Leaching
• (3) Froth flotation
• (4) Gravity separation

Comminution refers to:

• (1) Electrolytic refining
• (2) Chemical separation
• (3) Crushing and grinding of ores
• (4) Melting of ores

Which of the following is a pyro-metallurgical process?

• (1) Electrolysis
• (2) Roasting
• (3) Ion exchange
• (4) Leaching

The process of agglomeration is primarily used to:

• (1) Increase ore porosity
• (2) Improve floatability
• (3) Prepare fine particles for furnace use
• (4) Reduce slag volume

In size classification of ore particles, which equipment is commonly used?

• (1) Rotary kiln
• (2) Froth column
• (3) Cyclone separator
• (4) Ball mill

Which method is most suitable for extraction of aluminium from bauxite?

• (1) Roasting
• (2) Electrolytic reduction
• (3) Magnetic separation
• (4) Froth flotation

The main reducing agent in a blast furnace is:

• (1) Carbon monoxide
• (2) Hydrogen
• (3) Limestone
• (4) Coke ash

Zinc is commonly extracted using:

• (1) Electrolysis directly
• (2) Roasting followed by electrolysis
• (3) Roasting and reduction
• (4) Smelting with carbon

In steelmaking, slag primarily functions to:

• (1) Improve thermal conductivity
• (2) Remove impurities
• (3) Add carbon
• (4) Increase viscosity

Ladle metallurgy involves:

• (1) Ore crushing
• (2) Secondary steel refining
• (3) Electrolysis
• (4) Ore concentration

Which element is commonly used for desulphurization in ladle metallurgy?

• (1) Calcium
• (2) Oxygen
• (3) Aluminium
• (4) Carbon

Which of the following processes is used for the extraction of copper from low-grade ores?

• (1) Electro-refining
• (2) Liquation
• (3) Roasting
• (4) Bioleaching

Which of the following conditions favour dephosphorization in steel making?

• (1) Acid slag and oxidizing atmosphere
• (2) Basic slag and oxidizing atmosphere
• (3) Acid slag and reducing atmosphere
• (4) Basic slag and reducing atmosphere

In LD process, silicon removal takes place before carbon removal since:

• (1) % Si in hot metal is less than % C
• (2) \(SiO_2\) is thermodynamically more stable than CO under LD conditions
• (3) LD slag is acidic
• (4) Activity of silicon is higher than activity of carbon in hot metal

High top pressure in the blast furnace:

• (1) Increases the silicon content in hot metal
• (2) Increases the sulphur content in hot metal
• (3) Increases the phosphorous content in hot metal
• (4) Decreases the silicon content in hot metal

If the contact angle between a mineral and water is \(0^\circ\), the mineral will

• (1) Be completely wetted by water
• (2) Be partly wetted by water
• (3) Not be wetted by water
• (4) Float in water

Match the following processes with their physical principles:

• (1) A-3, B-1, C-4, D-2
• (2) A-2, B-4, C-3, D-1
• (3) A-3, B-1, C-4, D-3
• (4) A-2, B-4, C-1, D-3

In refining process, ............ of a system works against the refiner.

• (1) Enthalpy
• (2) Gibbs free energy
• (3) Entropy
• (4) Internal energy

Forsterite is a mineral of:

• (1) Co
• (2) Fe
• (3) Ni
• (4) Mg

The Rhombohedral crystal structure has (where a, b, c are lattice parameters and \(\alpha, \beta, \gamma\) are their angles):

• (1) \( a = b = c, \, \alpha = \beta = \gamma = 90^\circ \)
• (2) \( a \ne b \ne c, \, \alpha \ne \beta \ne \gamma \ne 90^\circ \)
• (3) \( a \ne b = c, \, \alpha \ne \beta = \gamma = 90^\circ \)
• (4) \( a = b = c, \, \alpha = \beta = \gamma \ne 90^\circ \)

The relationship between spread out film (S) and the radius of the powder camera (R) in Debye-Scherrer powder diffraction is (where \(\theta\) is the angle):

• (1) \( R = 4S\theta \)
• (2) \( S = 4R\theta \)
• (3) \( R = S\theta \)
• (4) \( S = R\theta \)

In X-ray diffraction analysis, the condition for BCC crystal where the reflections are allowed is (where h, k, l are Miller indices):

• (1) \( h + k + l is even \)
• (2) \( h, k, l are all odd \)
• (3) \( (h + k + l)/4 is odd \)
• (4) \( h^2 + k^2 + l^2 = 0 \)

An example of Secondary bond is:

• (1) Covalent bond
• (2) Metallic bond
• (3) Vander Waals bond
• (4) Ionic bond

If the mean bond length increases with temperature in material, then the material exhibits:

• (1) Thermal stability
• (2) Thermal expansion
• (3) Thermal contact
• (4) Thermal conductivity

Which of the following elements/compound has high bonding energy?

• (1) Silicon
• (2) Diamond
• (3) Germanium
• (4) Silicon Carbide

An example of long chain polymer which exhibit unique rubbery behaviour is:

• (1) Plastics
• (2) Elastomers
• (3) Fibres
• (4) Thermosets

The polymer used as sound proofing in refrigerators and buildings is:

• (1) Polypropylene
• (2) Polyvinylchloride
• (3) Polyethylene
• (4) Polystyrene

Which of the following is a peritectic reaction?

• (1) Solid\(_1\) + Solid\(_2\) gives Solid\(_3\)
• (2) Liquid gives Solid\(_1\) + Solid\(_2\)
• (3) Solid\(_1\) gives Solid\(_2\) + Solid\(_3\)
• (4) Liquid + Solid\(_1\) gives Solid\(_2\)

Razor blades are made of:

• (1) Medium carbon steel
• (2) Mild steel
• (3) High carbon steel
• (4) Pure Iron

To produce malleable cast iron, the silicon content must be:

• (1) \textbf{1%}
• (2) \textbf{2.5%}
• (3) \textbf{5%}
• (4) \textbf{10%}

Which of the following is in the decreasing order of hardness in Rockwell C-scale?

• (1) \textbf{Bainite, Martensite, Fine Pearlite, Coarse Pearlite}
• (2) \textbf{Martensite, Bainite, Fine Pearlite, Coarse Pearlite}
• (3) \textbf{Coarse Pearlite, Fine Pearlite, Bainite, Martensite}
• (4) \textbf{Martensite, Bainite, Coarse Pearlite, Fine Pearlite}

For an ASTM grain size number of 8, approximately how many grains would be visible per square inch at a magnification of 100\(\times\)?

• (1) \textbf{64 grains/in\(^2\)}
• (2) \textbf{100 grains/in\(^2\)}
• (3) \textbf{128 grains/in\(^2\)}
• (4) \textbf{256 grains/in\(^2\)}

The structure of a polymer depends on:

• (1) \textbf{Crystal size}
• (2) \textbf{Chain structure and crosslinking}
• (3) \textbf{Grain size}
• (4) \textbf{Surface finish}

The incubation period at nose temperature region of TTT diagram of eutectoid steel is shorter than below and above nose regions. This is due to:

• (1) \textbf{Slow cooling rate}
• (2) \textbf{More diffusion}
• (3) \textbf{Optimum nucleation and diffusion}
• (4) \textbf{Less nucleation}

Pearlitic content of normalized steel is slightly higher than slowly cooled steel, because:

• (1) \textbf{Cooling rate shifts the eutectic point towards left side}
• (2) \textbf{Cooling rate increases the carbon concentration}
• (3) \textbf{More diffusion due to faster cooling}
• (4) \textbf{Cooling rate reduces the nucleation sites}

Spheroidal grey cast iron (SG Iron) will be produced by:

• (1) \textbf{Controlling carbon content}
• (2) \textbf{Inoculation with Mg}
• (3) \textbf{Malleablisation treatment}
• (4) \textbf{Controlling Carbon Equivalent}

The (\(\alpha + \beta\)) Ti alloys are solution treated in the:

• (1) \textbf{\(\alpha\) region}
• (2) \textbf{\(\beta\) region}
• (3) \textbf{(\(\alpha + \beta\)) region}
• (4) \textbf{\(\gamma\) region}

Addition of Pb in Brass improves:

• (1) \textbf{Machinability}
• (2) \textbf{Strength}
• (3) \textbf{Toughness}
• (4) \textbf{Ductility}

The strength of grain boundary and grains are equal:

• (1) \textbf{At equi cohesive temperature}
• (2) \textbf{Above equi cohesive temperature}
• (3) \textbf{Below equi cohesive temperature}
• (4) \textbf{At recrystallisation temperature}

The Larson-Miller parameter is used to:

• (1) \textbf{Estimate the fatigue life of materials under variable loading}
• (2) \textbf{Predict the creep rupture life of materials at high temperatures}
• (3) \textbf{Determine the fracture toughness of materials}
• (4) \textbf{Assess the impact resistance of materials}

According to Griffith's theory, the critical condition for crack propagation in a brittle material is when the energy release rate \( G \) equals:

• (1) \textbf{The strain energy density at the crack tip.}
• (2) \textbf{The total strain energy in the material.}
• (3) \textbf{The surface energy per unit area of the crack.}
• (4) \textbf{The elastic modulus of the material.}

In which crystal structure is dislocation motion most difficult due to limited slip systems?

• (1) \textbf{FCC}
• (2) \textbf{BCC}
• (3) \textbf{HCP}
• (4) \textbf{SC}

Strain hardening in metals leads to:

• (1) \textbf{A decrease in dislocation density}
• (2) \textbf{An increase in porosity}
• (3) \textbf{An increase in hardness and strength}
• (4) \textbf{A reduction in grain size}

In a compression test, a ductile material specimen typically exhibits:

• (1) \textbf{Necking and fracture is observed}
• (2) \textbf{Brittle fracture occur}
• (3) \textbf{Significant elongation takes place}
• (4) \textbf{Significant shortening and bulging take place}

Stage II creep is characterized by:

• (1) \textbf{Accelerating strain rate}
• (2) \textbf{Constant strain rate}
• (3) \textbf{Decelerating strain rate}
• (4) \textbf{No strain}

Which of the following is a key characteristic of hot working processes?

• (1) \textbf{Material is worked below its recrystallization temperature}
• (2) \textbf{Material is worked above its recrystallization temperature}
• (3) \textbf{Material does not undergo significant plastic deformation}
• (4) \textbf{Material retains its shape and structure after deformation}

In rolling, the term "frictional shear stress" is used to describe which of the following?

• (1) \textbf{The resistance of the workpiece to deformation during rolling}
• (2) \textbf{The pressure exerted by the rolls on the material}
• (3) \textbf{The tangential force at the work-roll interface causing plastic deformation}
• (4) \textbf{The normal stress experienced by the workpiece as it enters the roll gap}

Fiber strengthening is effective in:

• (1) \textbf{Isotropic materials}
• (2) \textbf{Anisotropic materials with aligned fibers}
• (3) \textbf{Materials with high dislocation density}
• (4) \textbf{Materials with fine grain size}

Critical resolved shear stress is zero when slip plane is ........... degree to loading axis.

• (1) \textbf{0}
• (2) \textbf{45}
• (3) \textbf{90}
  s
• (4) \textbf{60}

The coefficient of friction increases by 5 times when a slab is hot rolled in a specific mill. The maximum possible reduction:

• (1) \textbf{increases by 5 times}
• (2) \textbf{increases by 25 times}
• (3) \textbf{decreases by 5 times}
• (4) \textbf{decreases by 25 times}

Which type of dislocation has the Burgers vector parallel to the dislocation line?

• (1) \textbf{Edge}
• (2) \textbf{Screw}
• (3) \textbf{Mixed}
• (4) \textbf{Partial}

Stress fields around dislocations cause:

• (1) \textbf{Ductility increase}
• (2) \textbf{Lattice distortion}
• (3) \textbf{Phase transformation}
• (4) \textbf{Grain refinement}

The multiplication of dislocations is explained by:

• (1) \textbf{Twinning mechanism}
• (2) \textbf{Griffith theory}
• (3) \textbf{Frank–Read source}
• (4) \textbf{Edge slip}

Partial dislocations are associated with:

• (1) \textbf{FCC metals}
• (2) \textbf{BCC metals}
• (3) \textbf{HCP metals}
• (4) \textbf{Ceramics only}

Fracture toughness is a measure of:

• (1) \textbf{Strength}
• (2) \textbf{Hardness}
• (3) \textbf{Resistance to crack propagation}
• (4) \textbf{Ductility}

Slip occurs on planes with:

• (1) \textbf{Maximum bond strength}
• (2) \textbf{Highest energy}
• (3) \textbf{Highest atomic density}
• (4) \textbf{Random orientation}

Irwin modified the Griffith's theory and introduced the concept of:

• (1) \textbf{Stress intensity factor.}
• (2) \textbf{Plastic zone size.}
• (3) \textbf{Energy release rate.}
• (4) \textbf{Crack tip opening displacement.}

The strain rate sensitivity of flow stress for occurrence of superplasticity is in the range of:

• (1) \textbf{0.3--0.6}
• (2) \textbf{0.01--0.1}
• (3) \textbf{0.1--0.2}
• (4) \textbf{0}

The stress ratio for completely reversed fatigue cycle is:

• (1) \textbf{1}
• (2) \textbf{0.5}
• (3) \textbf{-1}
• (4) \textbf{-0.5}

Which of the following gating ratio is normally followed for Al alloys:

• (1) \textbf{1 : 0.75 : 0.5}
• (2) \textbf{2 : 1 : 0.5}
• (3) \textbf{4 : 3 : 1}
• (4) \textbf{1 : 2 : 2}

Usual casting method for making dental crowns:

• (1) \textbf{Sand casting}
• (2) \textbf{Die casting}
• (3) \textbf{Continuous casting}
• (4) \textbf{Investment casting}

Compression ratio is defined as:

• (1) \textbf{Ratio of green density to apparent density}
• (2) \textbf{Ratio of height to diameter}
• (3) \textbf{Ratio of sintered strength to green strength}
• (4) \textbf{Ratio of apparent density to tap density}

Sinterability of aluminium is poor, because:

• (1) \textbf{Soft nature of powder}
• (2) \textbf{Poor compressibility}
• (3) \textbf{Presence of inherent oxide}
• (4) \textbf{Higher thermal conductivity}

In cored structure:

• (1) \textbf{There are no composition fluctuations from core to the tip}
• (2) \textbf{There are composition fluctuations from core to the tip}
• (3) \textbf{Equiaxed structure}
• (4) \textbf{Dendritic segregation}

Catastrophic oxidation occurs in metals which exhibit:

• (1) \textbf{Parabolic kinetics}
• (2) \textbf{Cubic kinetics}
• (3) \textbf{Linear kinetics}
• (4) \textbf{Logarithmic kinetics}

Meehanite is the proprietary name for a patented series of high duty cast irons inoculated with:

• (1) \textbf{Magnesium}
• (2) \textbf{Manganese}
• (3) \textbf{Ferro silicon}
• (4) \textbf{Calcium silicate}

Earing is common defect in:

• (1) \textbf{Deep drawing}
• (2) \textbf{Rolling}
• (3) \textbf{Extrusion}
• (4) \textbf{Forging}

Fine and spherical powder particles offer poor compressibility and good sinterability. This is due to:

• (1) \textbf{Interparticle friction and more contact points}
• (2) \textbf{Narrow distribution and high compression}
• (3) \textbf{More strength and less porosity}
• (4) \textbf{Good flow rate and less friction}

Densification of green compacts during sintering is predominantly by:

• (1) \textbf{Adhesion mechanism}
• (2) \textbf{Recrystallization}
• (3) \textbf{Volume diffusion}
• (4) \textbf{Surface diffusion}

The following technique is most probably recommended for consolidation of nanostructured and amorphous powders:

• (1) \textbf{Hot pressing}
• (2) \textbf{Metal injection moulding}
• (3) \textbf{Cold isostatic pressing}
• (4) \textbf{Spark Plasma sintering}

Pourbaix diagrams are graphical plots of ...........

• (1) \textbf{current Vs voltage}
• (2) \textbf{Potential Vs pH}
• (3) \textbf{pH Vs current}
• (4) \textbf{potential Vs time}

Which of the following is the major drawback of using X-rays compared to gamma rays for inspecting thick materials in Radiography Test?

• (1) \textbf{X-rays require a higher energy source than gamma rays}
• (2) \textbf{X-rays produce more secondary radiation, which can be hazardous}
• (3) \textbf{X-rays can only be used for surface inspections}
• (4) \textbf{X-rays have a higher attenuation rate in dense materials}

Which of the following is a defect in the rolling process?

• (1) \textbf{Flash}
• (2) \textbf{Alligatoring}
• (3) \textbf{Surface porosity}
• (4) \textbf{Wrinkling}

Which of the following is NOT a part of a typical gating system in sand casting?

• (1) \textbf{Sprue}
• (2) \textbf{Runner}
• (3) \textbf{Ingate}
• (4) \textbf{Chaplets}

What is the main problem when welding stainless steels, particularly in the HAZ (Heat-Affected Zone)?

• (1) \textbf{Formation of carbides at grain boundaries}
• (2) \textbf{Decrease in hardness}
• (3) \textbf{Increased toughness}
• (4) \textbf{Formation of excessive martensite}

What is the effect of increasing the riser-to-casting volume ratio?

• (1) \textbf{Improved casting yield}
• (2) \textbf{Increased casting yield}
• (3) \textbf{Decreased casting yield}
• (4) \textbf{No effect on casting yield}

The wax pattern in investment casting is removed by:

• (1) \textbf{Burning it out in a furnace}
• (2) \textbf{Immersing it in a hot acid bath}
• (3) \textbf{Mechanical breaking}
• (4) \textbf{Dissolving it in a solvent}

Which of the following types of penetrant is most commonly used for detecting very fine surface cracks in non-porous materials?

• (1) \textbf{Fluorescent penetrant}
• (2) \textbf{Visible penetrant}
• (3) \textbf{Water-soluble penetrant}
• (4) \textbf{Oil-based penetrant}

Permanent mould casting is best suited for which of the following production areas?

• (1) \textbf{Low-volume production of large parts with intricate features}
• (2) \textbf{Mass production of large parts with simple geometries}
• (3) \textbf{High-precision casting of metals with high melting points}
• (4) \textbf{High-volume production of small, complex parts with high dimensional accuracy}

In shell moulding, the binder used to hold the sand grains together is:

• (1) Clay
• (2) Sodium silicate
• (3) Thermoplastic resin
• (4) Silica gel

What is the relationship between the frequency of the sound waves and the ability to detect smaller defects in Ultrasonic testing?

• (1) Higher frequency waves detect larger defects
• (2) Lower frequency waves detect smaller defects
• (3) Higher frequency waves detect smaller defects
• (4) There is no relationship between frequency and defect size

How brazing is differ from welding?

• (1) The base metal is heated to a temperature below its melting point
• (2) The filler metal and base metal are both heated above their melting points
• (3) The filler metal is melted at a higher temperature than the base metal
• (4) Both the base metal and filler metal remain below their melting points

Which of the following is a common defect in wire drawing processes, particularly when excessive friction is present at the die-workpiece interface?

• (1) Die wear
• (2) Surface cracks on the wire
• (3) Material elongation
• (4) Necking of the wire

Back Scattered electrons are used in:

• (1) Scanning electron microscope
• (2) Optical microscope
• (3) Transmission electron microscope
• (4) X-ray diffraction analysis

If \( \begin{bmatrix} 2k+7 & 2k^2 - 5k + 3
3k - 3 & 15 \end{bmatrix} \) is a \(2 \times 2\) symmetric matrix, then the value of \(k\) is .........

• (1) 4
• (2) -1
• (3) 3
• (4) 0

The eigenvalues of the matrix \( \begin{bmatrix} 2 & 1 - 2i
1 + 2i & -2 \end{bmatrix} \) are ..........

• (1) \(-3i,\ 3i\)
• (2) \(-3,\ 3\)
• (3) \(-2,\ 2\)
• (4) \(-2i,\ 2i\)

Consider the improper integral \( I = \int_{2025}^{2030} \frac{1}{(x - 2025)^k} \, dx \), where \( k > 0 \). Which of the following is true for \( I \)?

• (1) \( I \) is convergent if \( k < 1 \) and is divergent if \( k \geq 1 \)
• (2) \( I \) is convergent if \( k \leq 1 \) and is divergent if \( k > 1 \)
• (3) \( I \) is convergent if \( k \geq 1 \) and is divergent if \( k < 1 \)
• (4) \( I \) is convergent if \( k > 1 \) and is divergent if \( k \leq 1 \)

The Laplace transform of \( e^{-2t} \cos(\sqrt{2} t) \) is .......

• (1) \( \frac{s - 2}{(s - 2)^2 + 2} ,\ s > 2 \)
• (2) \( \frac{2}{(s + 2)^2 + 2} ,\ s > -2 \)
• (3) \( \frac{\sqrt{2}}{(s + 2)^2 + 2} ,\ s > -2 \)
• (4) \( \frac{s + 2}{(s + 2)^2 + 2} ,\ s > -2 \)

The line integral of the vector field \( \vec{F} = x\hat{i} - 2y\hat{j} + z\hat{k} \) along the straight line path from the point \((-1,2,3)\) to \((2,3,5)\), is ..........

• (1) \( -\frac{1}{2} \)
• (2) \( \frac{9}{2} \)
• (3) \( \frac{13}{2} \)
• (4) \( -\frac{7}{2} \)

Consider the ordinary differential equation \( x^2 \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + 2y = 0 \) with \( y(x) \) as a general solution. Given the values \( y(1) = 1 \), \( y(2) = 5 \), the value of \( y(3) \) is equal to ..........

• (1) 9
• (2) 12
• (3) 15
• (4) -15

If \( X \) is a continuous random variable with the probability density function
\[ f(x) = \begin{cases} cx^3, & 0 \le x \le 2
0, & otherwise \end{cases} \]
then \( P\left( \frac{1}{2} < X < \frac{3}{2} \right) \) is ..........

• (1) \(\frac{5}{16}\)
• (2) \(\frac{1}{8}\)
• (3) \(\frac{5}{8}\)
• (4) \(\frac{1}{16}\)

If \( A \) and \( B \) are two mutually exclusive events with \( P(B) \ne 1 \), then the conditional probability \[ P(A \mid \overline{B}) = \,? \]
where \( \overline{B} \) is the complement of \( B \)

• (1) \(\frac{1}{P(B)}\)
• (2) \(\frac{P(A)}{P(B)}\)
• (3) \(\frac{1}{1 - P(B)}\)
• (4) \(\frac{P(A)}{1 - P(B)}\)

Evaluation of the integral \(\int_{2}^{4} x^2 \, dx\) using the trapezoidal rule (with two equal segments) gives a value of .......

• (1) 14.5
• (2) 22.5
• (3) 19.0
• (4) 18.6

Let \( f(x) = x^3 - x - 3 \). In finding a positive real root of \( f(x) = 0 \) using Newton-Raphson method, if the starting guess \( x_0 = 2 \), then the numerical value of the root \( x_1 \) after the first iteration is .........

• (1) 1.5
• (2) 1.6
• (3) 1.7
• (4) 1.8