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

The absolute humidity of air at 101.325 kPa is measured to be 0.02 kg of water per kg of dry air. Then the partial pressure of water vapour in the air is:

• (1) 1.99 kPa
• (2) 2.55 kPa
• (3) 3.16 kPa
• (4) 3.87 kPa

The minimum amount of work required to operate a refrigerator which removes 1000 Cal heat at \(0^\circ\)C and rejects at \(50^\circ\)C will be:

• (1) 170 Cal
• (2) 120 Cal
• (3) 150 Cal
• (4) 183.15 Cal

The entropy of single crystalline Silicon at absolute zero will be:

• (1) 2
• (2) 1
• (3) 0
• (4) 10

Which statement about entropy is incorrect?

• (1) Entropy is zero at 0 K (Third Law)
• (2) Entropy decreases in spontaneous adiabatic processes
• (3) Entropy production is irreversible
• (4) Entropy measures energy dispersal

If the vapour pressure at two temperatures of a solid phase in equilibrium with its liquid phase is known, then the latent heat of fusion can be calculated by the:

• (1) Nernst Heat Theorem
• (2) Maxwell's equation
• (3) Van Laar equation
• (4) Clapeyron–Clausius equation

The theoretical minimum work required to separate one mole of a liquid mixture at 1 atm, containing 50 mole % each of n-heptane and n-octane into pure compounds, each at 1 atm, is:

• (1) \(-2RT \ln 0.5\)
• (2) \(2RT\)
• (3) \(-RT \ln 0.5\)
• (4) \(0.5RT\)

Design an approach to estimate the activity coefficient of a component in a non-ideal liquid mixture:

• (1) Use the ideal gas law
• (2) Apply the Gibbs-Duhem equation with experimental data
• (3) Assume it equals mole fraction
• (4) Use the Second Law only

Which of the following equation must be perfunctorily satisfied while dealing with fluid flow problems?

• (1) Newton’s third law
• (2) Law of conservation of momentum
• (3) Continuity equation
• (4) Newton’s second law

1 mole of Argon gas is heated at constant pressure from 200 K to 600 K. If \(C_p = 4\ Cal.deg^{-1}mol^{-1}\), then the change in entropy will be:

(Given \(\ln 3 = 1.09\))

• (1) 4.36 Cal.deg\(^{-1}\)mol\(^{-1}\)
• (2) 2.76 Cal.deg\(^{-1}\)mol\(^{-1}\)
• (3) 3.34 Cal.deg\(^{-1}\)mol\(^{-1}\)
• (4) 5.39 Cal.deg\(^{-1}\)mol\(^{-1}\)

The change in enthalpy of a system is related to the heat absorbed at a:

• (1) Constant volume
• (2) Constant temperature
• (3) Constant mole fraction
• (4) Constant pressure

If a combustion, 4 kg of organic compound gives 11 kg of CO\(_2\), the percentage of carbon in the compound will be\underline{\hspace{1cm

• (1) 50
• (2) 75
• (3) 25
• (4) 60

Why is the equilibrium constant (K) temperature-dependent?

• (1) Because \(\Delta G^\circ\) depends on T
• (2) Because pressure affects K
• (3) Because entropy is temperature-independent
• (4) Because fugacity coefficients change with T

Filter aids are added to the slurry prior to filtration in order to form \hspace{2cm}.

• (1) Compact cakes of low porosity
• (2) Cakes of increased porosity
• (3) Crystalline cakes
• (4) Baked cake

Pick the best difference between mixing and blending.

• (1) Mixing can only be done for a solid-solid system, while blending is done for solid-liquid mixture
• (2) Mixing is done for liquid-liquid, gas-liquid and solid-liquid systems, while blending is done for solid-solid mixture
• (3) Blending is done for liquid-liquid and gas-liquid systems, while mixing is done for solid-solid mixture
• (4) Mixing is done for liquid-liquid and gas-liquid systems, while blending is done for solid-solid mixture

For transportation of boiler ash, which one of the following conveyors will be useful?

• (1) Belt Conveyor
• (2) Screw Conveyor
• (3) Pipe Conveyor
• (4) Bucket elevator

Centrifugal pumps transfer energy from \textbf{..........}

• (1) rotor to fluid
• (2) fluid to rotor
• (3) draft to rotor
• (4) rotor to draft

Which of the following is true, when operating speed is less than critical speed?

• (1) Bold grinding
• (2) Effective grinding
• (3) No grinding
• (4) Best grinding

What is the term for the ratio of the actual flow rate to the maximum rated flow rate of a flow meter?

• (1) Repeatability
• (2) Linearity
• (3) Turndown ratio
• (4) Accuracy

Which among the following is an assumption of Hagen–Poiseuille equation?

• (1) Fluid is uniform
• (2) Fluid is laminar
• (3) Fluid is turbulent
• (4) Fluid is compressible

While a propeller gives an axial liquid circulation pattern, a turbine gives ..............

• (1) Lateral
• (2) Parallel
• (3) Axial
• (4) Radial

In S.I. system, the unit of filter medium resistance is ..............

• (1) \(\dfrac{kg}{m^2 \cdot s}\)
• (2) \(\dfrac{m^2}{s}\)
• (3) \(\dfrac{kg}{m^3 \cdot s}\)
• (4) \(m^{-1}\)

Surging is a phenomenon associated with:

• (1) Reciprocating pumps
• (2) Centrifugal compressors
• (3) Gear pumps
• (4) Diaphragm pumps

What type of filtration relies on gravity to pull liquid through the filter medium?

• (1) Vacuum filtration
• (2) Centrifugal filtration
• (3) Gravity filtration
• (4) Pressure filtration

Which one of the following is not a Newtonian fluid?

• (1) Water
• (2) Toothpaste
• (3) Glycerol
• (4) Alcohol

The property of the fluid accounts for the major losses in pipes is .............

• (1) density
• (2) specific gravity
• (3) viscosity
• (4) compressibility

If we have very small dp/di < 0.01, then we may use .............

• (1) Ergun Equation
• (2) Kozeny Carman Equation
• (3) Sieder Tate Equation
• (4) Arrhenius Equation

Which of the following is true about the speed of the conveyor belt?

• (1) Fixed conveyors need not be shut down during any speed change
• (2) Adjustable speed belts can be changed only manually
• (3) Fixed speed drives can undergo minor speed changes
• (4) Variations of speed is not possible with conveyors

The number of tube and shell passes of a 3-6 pass exchanger will be:

• (1) 3 and 6
• (2) 3 and 3
• (3) 3 and 5
• (4) 6 and 3

A multiple effect evaporator has a capacity to process 40000 kg of solid caustic soda per day when it is concentrating from 10% to 25% solids. The water evaporated in kilograms per day is:

• (1) 800
• (2) 24000
• (3) 60000
• (4) 48000

The surface temperature of a blackbody is 400 K. What will be the emissive power?

• (1) 1451.26 Wm\(^{-2}\)
• (2) 4521.56 Wm\(^{-2}\)
• (3) 2890 Wm\(^{-2}\)
• (4) 5500 Wm\(^{-2}\)

A cavity with a small hole will always behave as a:

• (1) White body
• (2) Transparent body
• (3) Black body
• (4) Opaque body

The temperature of the liquid is below the saturation temperature and boiling takes place only in vicinity of the heated surface. This type of boiling is known as:

• (1) Subcooled
• (2) Forces
• (3) Saturated
• (4) Pool

Drop wise condensation usually occurs on:

• (1) Oily surface
• (2) Glazed surface
• (3) Smooth surface
• (4) Coated surface

The dimensions of diffusion coefficient is given by:

• (1) \(M\ L\ T^{-2}\)
• (2) \(L^2\ T^{-1}\)
• (3) \(L\ T^{-1}\)
• (4) \(M\ L^{-2}\ T\)

Which of the following materials has the highest thermal conductivity?

• (1) Wood
• (2) Air
• (3) Copper
• (4) Glass

The Wien’s Displacement Law relates the wavelength at which maximum emission occurs to:

• (1) Emissivity of the surface
• (2) Absolute temperature of the blackbody
• (3) Surface area of the emitter
• (4) Stefan-Boltzmann constant

A smaller HTU indicates:

• (1) Poor mass transfer efficiency
• (2) Higher column height required
• (3) Less height needed per transfer unit
• (4) Lower gas flow rates

A hydraulic system operating at \(80^\circ\)C must be cooled to \(45^\circ\)C using a heat transfer oil. A water source of \(30^\circ\)C is available. The effectiveness of heat exchanger will be

• (1) 0.7
• (2) 0
• (3) 1
• (4) 0.5

With respect to incident radiation, transmissivity varies with

• (1) Time
• (2) Temperature
• (3) Surface area
• (4) Wavelength

The evaporative heat transfer coefficient is typically

• (1) Dependent on the latent heat of vaporization
• (2) Independent of air velocity
• (3) Higher for still air compared to moving air
• (4) Not affected by the liquid surface area

Double-pipe heat exchangers are used for cases where the heat transfer area requirement is around

• (1) 10 to 20 m\(^2\)
• (2) 100 to 200 m\(^2\)
• (3) 500 to 1000 m\(^2\)
• (4) 5000 to 10000 m\(^2\)

Consider a uniformly tapered steel rod of circular cross-section of 1 m length. The diameter of the rod at one end is 5 cm, and that at the other end is 2.5 cm. If the heat flux at the end of the larger cross-section is 2500 kcal/m\(^2\cdot\)hr, the heat flux at the other end is equal to

• (1) 2500 kcal/m\(^2\cdot\)hr
• (2) 5000 kcal/m\(^2\cdot\)hr
• (3) 7500 kcal/m\(^2\cdot\)hr
• (4) 10,000 kcal/m\(^2\cdot\)hr

What type of process is used to separate liquid mixtures based on differences in volatility?

• (1) Filtration
• (2) Extraction
• (3) Adsorption
• (4) Distillation

According to penetration theory, the relationship between mass transfer coefficient (k) and diffusivity (D) is given by .............

• (1) \( k \propto D^{1.5} \)
• (2) \( k \propto D^{2} \)
• (3) \( k \propto D^{0.5} \)
• (4) \( k \propto D \)

Pendular state in drying of wet solids generally refers to:

• (1) Constant rate
• (2) First falling rate
• (3) Second falling rate
• (4) Equilibrium

Which of the following equipment is essential in a simple distillation setup?

• (1) Separating funnel
• (2) Condenser
• (3) Chromatography column
• (4) Bunsen burner only

Which of the following is NOT a type of distillation?

• (1) Simple distillation
• (2) Fractional distillation
• (3) Steam distillation
• (4) Magnetic distillation

In a gas absorption process, the gas phase is typically:

• (1) The solvent
• (2) The solute
• (3) The carrier
• (4) The product

The distribution of a solute between two immiscible liquids is governed by:

• (1) Henry's Law
• (2) Le Chatelier's Principle
• (3) Distribution coefficient
• (4) Dalton’s Law

Which of the following increases the efficiency of liquid-liquid extraction?

• (1) Multiple extractions with small volumes of solvent
• (2) Using the same solvent multiple times
• (3) Increasing the temperature excessively
• (4) Using highly miscible solvents

What type of dryer is typically used for drying granular materials like grains?

• (1) Freeze dryer
• (2) Rotary dryer
• (3) Tunnel dryer
• (4) Tray dryer

For temperature sensitive pharmaceuticals which one of the following drying methods is ideal?

• (1) Vacuum Rotary Drying
• (2) Agitated pan Drying
• (3) Vacuum Shelf Drying
• (4) Sublimation Drying

Tannin is extracted from tree barks by leaching with:

• (1) Alkaline solution
• (2) Acidic solution
• (3) Hot water
• (4) Organic solvents

In order to have efficient leaching of the desired material, the process Flaking is done to which of the following raw material:

• (1) Sugar extraction from Sugar beet
• (2) Oil extraction from vegetable seeds
• (3) Iron extraction from hematite
• (4) Aluminium extraction from bauxite

The constant rate period in drying refers to the stage when:

• (1) Surface moisture is completely removed
• (2) Internal moisture starts to migrate
• (3) Drying rate remains constant
• (4) The temperature of the material drops

What happens to the air in a cooling and dehumidification process?

• (1) Temperature and humidity increase
• (2) Temperature increases, humidity decreases
• (3) Temperature and humidity decrease
• (4) Only humidity increases

The rate-determining step in a reaction mechanism is the one with:

• (1) The lowest activation energy
• (2) The highest activation energy
• (3) No energy barrier
• (4) The fastest rate

When the conversion of a liquid phase reaction of first order occurring in a CSTR is 60%, and the molar feed rate is 10 mol/min, and the volume of the reactor is 0.50 litre, then the reaction rate in mol/lit/min will be:

• (1) 12
• (2) 18
• (3) 20
• (4) 6

The reaction type for acid catalysed ester hydrolysis will be:

• (1) Zero order reaction
• (2) First order reaction
• (3) Pseudo first order reaction
• (4) Second order reaction

Mixing of entering and exit fluid does not occur in which one of the following reactors?

• (1) Batch flow
• (2) Plug flow
• (3) Mixed Flow
• (4) Semi batch

The Thiele modulus represents:

• (1) Surface reaction rate to diffusion rate
• (2) Overall reaction order
• (3) Mass transfer rate
• (4) Temperature profile

In a heterogeneous catalytic reaction, the effectiveness factor is:

• (1) Ratio of actual rate to rate if no diffusion limitations
• (2) Ratio of surface area to catalyst weight
• (3) Mass transfer coefficient
• (4) Catalyst deactivation factor

A zero-order reaction has a rate that is:

• (1) Independent of reactant concentration
• (2) Proportional to the reactant concentration
• (3) Proportional to the square of the reactant concentration
• (4) Inversely proportional to reactant concentration

The Damköhler number (Da) relates:

• (1) Reaction rate to diffusion rate
• (2) Heat transfer to mass transfer
• (3) Reactor volume to flow rate
• (4) Pressure to temperature

A catalyst increases the rate of reaction by:

• (1) Increasing activation energy
• (2) Decreasing activation energy
• (3) Changing the equilibrium constant
• (4) Increasing temperature

The reaction A + B → C is elementary. If the concentration of A is doubled, the rate will:

• (1) Remain the same
• (2) Double
• (3) Quadruple
• (4) Decrease by half

For an exothermic reaction, increasing temperature will:

• (1) Shift the equilibrium toward products
• (2) Not affect the equilibrium
• (3) Increase activation energy
• (4) Shift the equilibrium toward reactants

A reversible reaction A⇌B reaches equilibrium when:

• (1) Forward rate = Reverse rate
• (2) All A is converted to B
• (3) Temperature is zero
• (4) Pressure is infinite

For a first order reaction, the starting material reduces to \(\dfrac{1}{4}\)th of its initial value after 20 min. The rate constant of the reaction will be: (Given \(\ln 4 = 1.38\))

• (1) 0.069 min\(^{-1}\)
• (2) 1.3844 min\(^{-1}\)
• (3) 4.3465 min\(^{-1}\)
• (4) 2.1674 min\(^{-1}\)

If the formation of ammonia, shown below, is exothermic at 25°C, what will be the effect of increasing the system's temperature?

3H\(_2\)(g) + N\(_2\)(g) \(\leftrightarrow\) 2NH\(_3\)(g)

• (1) The value of \(K\) will increase
• (2) The value of \(K\) will decrease
• (3) The equilibrium will shift to the right
• (4) The [H\(_2\)] will decrease

Which model is commonly used to describe non-ideal flow with backmixing?

• (1) Ideal PFR model
• (2) Ideal CSTR model
• (3) Axial Dispersion Model
• (4) Batch reactor model

The Arrhenius equation describes the effect of .............. on reaction rate.

• (1) Pressure
• (2) Temperature
• (3) Volume
• (4) Catalyst

Which of the options correctly represents the Laplace inverse of \(\dfrac{2}{s^3}\)?

• (1) \(t^2\)
• (2) \(t^3\)
• (3) \(t^{-2}\)
• (4) \(t^{-3}\)

What type of controller is used for the control of flow of liquid from a pump?

• (1) P controller
• (2) PD controller
• (3) PID controller
• (4) PI controller

For a stable closed loop system, the gain at phase crossover frequency should always be:

• (1) \(< 20\) dB
• (2) \(< 6\) dB
• (3) \(> 0\) dB
• (4) \(> 6\) dB

Gain margin and phase margin are measures of:

• (1) Process efficiency
• (2) Steady-state error
• (3) Controller cost
• (4) Relative stability

Which control strategy is used to anticipate disturbances before they affect the process?

• (1) Feedback control
• (2) Feedforward control
• (3) Cascade control
• (4) On-off control

Dead time in a process refers to:

• (1) Time taken to reach steady state
• (2) Time delay between input change and output response
• (3) Time constant of the system
• (4) Time for the controller to react

In a heat exchanger, the primary controlled variable is often:

• (1) Outlet temperature
• (2) Inlet pressure
• (3) Pump speed
• (4) Pipe diameter

Which one of the following is not a type of actuator?

• (1) Pneumatic
• (2) Hydraulic
• (3) Solenoid
• (4) Ultrasonic

Which one of the following valves are best suitable for corrosive liquids?

• (1) Diaphragm valve
• (2) Rotary plug valve
• (3) Ball valve
• (4) Butterfly valve

For highly accurate incremental position measurement which one of the following systems is used?

• (1) Ultrasonic devices
• (2) Hall effect Sensors
• (3) Microwave devices
• (4) Light interference laser

The device that can change the format of a signal without changing the energy form is known as:

• (1) Converters
• (2) Transducers
• (3) Actuators
• (4) Controllers

Smallest change which a sensor can detect is ...............

• (1) Resolution
• (2) Accuracy
• (3) Precision
• (4) Scale

Which document provides detailed specifications for equipment and materials?

• (1) Process Flow Diagram
• (2) Piping and Instrumentation Diagram
• (3) Material Safety Data Sheet
• (4) Equipment datasheets

Which of the following is the first step in plant design?

• (1) Detailed engineering
• (2) Process synthesis
• (3) Feasibility study
• (4) Piping and instrumentation design

Which cost estimation method uses historical data from similar plants?

• (1) Detailed itemized estimation
• (2) Factored estimation
• (3) Lang factor method
• (4) Turnkey costing

Fixed costs in plant operations include:

• (1) Raw materials
• (2) Labor wages
• (3) Depreciation and plant maintenance
• (4) Utility costs

A pump has an installed cost of Rs. 40,000 and a 10-year estimated life. The salvage value of the pump is zero at the end of 10 years. The pump value (in rupees) after depreciation by the double declining balance method, at the end of 6 years is:

• (1) 4295
• (2) 10486
• (3) 21257
• (4) 37600

Due to a 20% drop in the product selling price, the pay-back period of a new plant increased to 1.5 times that estimated initially, the production cost and the production rate remaining unchanged. If the production cost is \(C_p\) and the new selling price is \(C_s\), then \(C_p/C_s\) is:

• (1) 0.2
• (2) 0.4
• (3) 0.5
• (4) 0.6

Which of the following is a limitation of the payback period method?

• (1) It considers the time value of money
• (2) It ignores cash flows beyond the payback period
• (3) It is difficult to calculate
• (4) It always matches the IRR results

Profitability calculation method that includes time value of money is:

• (1) Payback period
• (2) Net return
• (3) Rate of return on investment
• (4) Net present worth

For a business if the net profit is 10 Lakhs and total capital investment is 5 crore rupees, then the return on investment in percentage will be:

• (1) 5%
• (2) 3%
• (3) 2%
• (4) 50%

A compressor design primarily depends on:

• (1) Suction pressure
• (2) Discharge pressure
• (3) Pressure ratio
• (4) Compressor size

Equipment costs generally follow the relation:

• (1) Linear with size
• (2) Power law with size
• (3) Exponential with size
• (4) Inversely with size

For a commercial fertilizer designated as 5-20-20 the numbers indicate what information about the composition?

• (1) 5% K, 20% N, 20% P
• (2) 5% N, 20% Na, 20% S
• (3) 5% N, 20% P\(_2\)O\(_5\), 20% K\(_2\)O
• (4) 5% K\(_2\)O, 20% N, 20% P\(_2\)O\(_5\)

Polythene is the polymer obtained during polymerization of

• (1) Propylene
• (2) Acetylene
• (3) Methylene
• (4) Ethylene

In the Haber process of ammonia preparation, the catalyst and the promoter are

• (1) Reduced Fe\(_2\)O\(_3\) and Al\(_2\)O\(_3\) respectively
• (2) Reduced Fe\(_2\)O\(_3\) and Mn\(_2\)O\(_3\) respectively
• (3) Reduced Al\(_2\)O\(_3\) and Fe\(_2\)O\(_3\) respectively
• (4) Reduced Mn\(_2\)O\(_3\) and Al\(_2\)O\(_3\) respectively

The Kraft process of chemical pulping involves a mixture of two chemicals in water which are?

• (1) NaOH, and Na\(_2\)S
• (2) NaOH, and NaCl
• (3) NaCl, and Na\(_2\)S
• (4) NaOH, and K\(_2\)S

LPG contains mostly:

• (1) Methane, Ethane, and Propane
• (2) Ethane, Propane, and Butane
• (3) Propane, Butane, and Isobutene
• (4) Methane

For which refractory material the thermal conductivity decreases with increasing temperature?

• (1) Zirconia
• (2) Silica
• (3) Magnesite
• (4) Alumina

Which of the following equipment has the maximum capital cost contribution in a chemical plant?

• (1) Pumps
• (2) Compressors
• (3) Heat exchangers
• (4) Reactors

Hydrogenation is the conversion of unsaturated acid groups into the saturated using ............... catalyst.

• (1) Fe
• (2) Zn
• (3) Ti
• (4) Ni

Which of the following is fully synthetic first produced fibre?

• (1) Jute
• (2) Acrylic
• (3) Rayon
• (4) Nylon

The term "circular economy" in chemical technology refers to:

• (1) Linear production and disposal
• (2) Increasing single-use plastics
• (3) Maximizing waste generation
• (4) Reusing and recycling materials

............... is not an intermediate distillate product in petroleum refining.

• (1) Heavy fuel oils
• (2) Diesel oils
• (3) Lubricating oil
• (4) Gas oil

The Ostwald process is used to produce:

• (1) Nitric acid
• (2) Sulfuric acid
• (3) Hydrochloric acid
• (4) Phosphoric acid

............... is the waste liquor from the Kraft pulping process after pulping is completed.

• (1) White liquor
• (2) Black liquor
• (3) Brown liquor
• (4) Red liquor

............... includes the largest fraction of petroleum crude.

• (1) n-paraffin series
• (2) Naphthene series
• (3) Asphalts
• (4) Isoparaffin series

............... act as a catalyst in anionic polymerisation.

• (1) Grignard reagent
• (2) Lewis acids
• (3) Benzoyl peroxide
• (4) AIBN

If the systems of equations \(3x - 2y + z = 0\), \(5x + ay + 15z = 0\), \(x + 2y - 3z = 0\) have non-zero solution, then \(a =\) ...............

• (1) \(-2\)
• (2) \(2\)
• (3) \(-14\)
• (4) \(14\)

If the matrix \(A = \begin{pmatrix} 3 & -1 & 1
-1 & 5 & -1
1 & -1 & 3 \end{pmatrix}\) has three distinct eigenvalues and one of its eigenvectors is \(\begin{pmatrix} 1
0
-1 \end{pmatrix}\), then which of the following can be another eigenvector of \(A\)?

• (1) \(\begin{pmatrix} -1
  0
  1 \end{pmatrix}\)
• (2) \(\begin{pmatrix} -1
  0
  2 \end{pmatrix}\)
• (3) \(\begin{pmatrix} 1
  1
  -1 \end{pmatrix}\)
• (4) \(\begin{pmatrix} 1
  -2
  1 \end{pmatrix}\)

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

If \(\vec{F}(x, y, z) = 3x^2y\,\hat{i} + 5y^2z\,\hat{j} - 8xyz\,\hat{k}\) is a continuously differentiable vector field, then the curl of \(\vec{F}\) at (1,1,1) is ...............

• (1) \(-13\hat{i} - 8\hat{j} - 3\hat{k}\)
• (2) \(-13\hat{i} + 8\hat{j} + 3\hat{k}\)
• (3) \(-13\hat{i} + 8\hat{j} - 3\hat{k}\)
• (4) \(13\hat{i} + 8\hat{j} - 3\hat{k}\)

If \(\vec{F} = x(x^2 + y^2 + z^2)\hat{i} + 2y(x^2 + y^2 + z^2)\hat{j} + 3z(x^2 + y^2 + z^2)\hat{k}\), then \(div \, \vec{F}\) at \((1,1,1)\) is equal to ______

• (1) 12
• (2) 21
• (3) 30
• (4) 33

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 of \( y(1) = 1 \), \( y(2) = 5 \), the value of \( y(3) \) is equal to ______

• (1) 9
• (2) 12
• (3) 15
• (4) -15

If the Laplace transform of a function \( f(t) \) is given by \[ \frac{2s + 1}{(s + 1)(s + 2)} \]
then \( f(0) \) is equal to ______

• (1) 2
• (2) 4
• (3) -4
• (4) \( 3e - 1 \)

Let \( z \) be a complex variable and \( C: |z| = 3 \) be a circle in the complex plane. Then, \[ \oint_C \frac{z^2}{(z - 1)^2(z + 2)} \, dz = \, ______ \]

• (1) \( \pi i \)
• (2) \( 2\pi i \)
• (3) 0
• (4) \( \pi(2 + i) \)

The probability of a component being defective is 0.01. There are 100 such components in a machine. Then the probability of two or more defective components in the machine is ...........

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

The value of the integral \( \int_1^3 \frac{2}{x} \, dx \), when evaluated by using Simpson’s \( \frac{1}{3} \) rule on two equal subintervals each of length 1, is ...........

• (1) 2.00
• (2) 2.24
• (3) 2.19
• (4) 2.22