AP PGECET 2026 Nano Technology Question Paper: Download Answer Key with Solutions

Major challenge in nanotechnology is

• (A) Quantum effects
• (B) scalability and control
• (C) high density
• (D) low reactivity

The value of bulk modulus of a fluid is required to determine

• (A) Reynold's number
• (B) Froude's number
• (C) Mach number
• (D) Euler's number

A car starts from rest and accelerates uniformly at 2 m/s\(^2\). Find its velocity after 10 sec?

• (A) 20 m/s
• (B) 25 m/s
• (C) 15 m/s
• (D) 30 m/s

What is the primary reason why materials exhibit vastly different properties at the Nano-scale compared to the Bulk scale?

• (A) A significant increase in the Surface-to-Volume ratio and the emergence of Quantum Confinement effects
• (B) The disappearance of atoms from the lattice
• (C) The loss of all chemical reactivity
• (D) Increased gravitational forces at the small scale

The kinematic viscosity is the

• (A) ratio of absolute viscosity to the density of the liquid
• (B) ratio of density of the liquid to the absolute viscosity
• (C) product of absolute viscosity and density of the liquid
• (D) product of absolute viscosity and mass of the liquid

Which of the following supports can resist both vertical and horizontal forces but not moments?

• (A) Roller support
• (B) Pin support
• (C) Fixed support
• (D) Hinge support

In a Fibre-reinforced composite, the "Critical Fibre Length" (\(l_c\)) is defined as the length required for

• (A) the fibre to break before the matrix fails
• (B) the maximum tensile stress to be transferred from the matrix to the centre of the fibre
• (C) the composite to become transparent
• (D) the matrix to undergo recrystallization

Coefficient of resistance is the ratio of

• (A) actual velocity of jet at vena contracta to the theoretical velocity
• (B) area of jet at vena contracta to the area of orifice
• (C) loss of head in the orifice to the head of water available at the exit of the orifice
• (D) actual discharge through an orifice to the theoretical discharge

The efficiency of a furnace is defined as:

• (A) (Heat supplied / Heat utilized) \(\times\) 100
• (B) (Heat utilized / Heat supplied) \(\times\) 100
• (C) (Heat losses / Heat supplied) \(\times\) 100
• (D) (Heat utilized / Total heat supplied) \(\times\) 100

Why are ceramics typically much stronger in compression than in tension?

• (A) Because they contain inherent micro-cracks that act as stress concentrators under tensile loads
• (B) Because they have a high density of mobile dislocations
• (C) Because their ionic bonds are only stable under compressive forces
• (D) Because they undergo significant plastic deformation before fracture

The pressure measured above or below the atmospheric pressure is known as

• (A) suction pressure
• (B) vacuum pressure
• (C) negative pressure
• (D) gauge pressure

Which of the following has bcc crystal structure?

• (A) Ferrite
• (B) Austenite
• (C) Martensite
• (D) Cementite

The Glass Transition Temperature (Tg) of a polymer is a critical parameter. Which of the following statements correctly describes the behaviour of a polymer below Tg?

• (A) The polymer chains are in a liquid-like, flowable state
• (B) The polymer is in a "rubbery" state with high molecular mobility
• (C) The long-range segmental motion of polymer chains is "frozen," making the material brittle
• (D) The polymer undergoes a first-order phase transition from liquid to solid

The coefficient of viscosity may be determined by using an instrument is called_______

• (A) accelerometer
• (B) viscometer
• (C) vibrometer
• (D) pressure gauge

Which among the following is NOT a factor in Hume-Rothery rules of formation of substitutional solid solutions

• (A) crystal structure
• (B) relative size
• (C) chemical affinity
• (D) chemical Reactivity

In the band theory of solids, the 'Effective Mass' of an electron is determined by

• (A) the number of valence electrons
• (B) the velocity of light in the crystal
• (C) the atomic weight of the lattice ions
• (D) the curvature of the E - k diagram

Which of the following statement is wrong?

• (A) The heat transfer in liquid and gases takes place according to convection
• (B) The amount of heat flow through a body is dependent upon the material of the body
• (C) The thermal conductivity of solid metals increases with rise in temperature
• (D) Logarithmic mean temperature difference is not equal to the arithmetic mean temperature difference.

Curie temperature of ferrite is

• (A) 768 \(^\circ\)C
• (B) 910 \(^\circ\)C
• (C) 1147 \(^\circ\)C
• (D) 727 \(^\circ\)C

If a material has a band gap (\(E_g\)) of 4.0 eV, it will most likely be

• (A) transparent to visible light
• (B) opaque and metallic
• (C) a p-type semiconductor at room temperature
• (D) highly absorbent in the infrared region

Thermal diffusivity is a

• (A) function of temperature
• (B) physical property of a substance
• (C) dimensionless parameter
• (D) function of pressure

Which one of the following elements is ferrite stabilizer?

• (A) Nickel
• (B) Copper
• (C) Chromium
• (D) Manganese

According to the Free Electron Theory, if the temperature of a pure metal is increased, the collision frequency of electrons increases. How does this specifically affect the Fermi energy (\(E_F\)) of the metal?

• (A) It decreases linearly with temperature
• (B) It drops to zero at the melting point
• (C) It increases significantly due to higher thermal energy
• (D) It remains essentially constant

In a loaded beam, the point of contra flexure occurs at a section where

• (A) bending moment is minimum
• (B) bending moment is zero or changes sign
• (C) bending moment is maximum
• (D) shearing force is maximum

The property that cannot be obtained from a tensile test is

• (A) Yield Strength
• (B) Ultimate Tensile Strength
• (C) Ductility
• (D) Endurance Limit

In a combustion process, the "Theoretical Adiabatic Flame Temperature" is the maximum temperature the products can reach. This temperature is lower in practice because

• (A) real reactions are always endothermic
• (B) part of the energy is consumed by the dissociation of product molecules
• (C) the heat capacity of the products decreases at high temperatures
• (D) the nitrogen in the air acts as a catalyst, absorbing energy

If the shear force along a section of a beam is zero, the bending moment at the section is

• (A) zero
• (B) maximum
• (C) minimum
• (D) average of maximum-minimum

Yield point phenomenon observed in mild steel is due to the presence of

• (A) Silicon
• (B) Manganese
• (C) Carbon
• (D) Chromium

Adiabatic flame temperature is maximum when

• (A) the reaction incomplete
• (B) no heat loss and complete combustion
• (C) the heat loss is maximum
• (D) excess air is high

The stress necessary to initiate yielding, is considerably

• (A) more than that necessary to continue it
• (B) less than that necessary to continue it
• (C) more than that necessary to stop it
• (D) less than that necessary to stop it

Cold working of metal is carried out at

• (A) below recrystallization temperature
• (B) above recrystallization temperature
• (C) below room temperature
• (D) just above the grain growth temperature

Mixing identical gases results in

• (A) no change in entropy
• (B) decrease in entropy
• (C) increase in entropy
• (D) infinite entropy

Strain energy of any member may be defined as work done to

• (A) deformation
• (B) pressure
• (C) moment
• (D) Force

Which of the following metal working operation is a direct compression process?

• (A) Extrusion
• (B) Forging
• (C) Wire drawing
• (D) Stretch forming

According to the Gibbs Phase Rule (F = C - P + 2), at the triple point of a pure metal (e.g., where solid, liquid, and vapour coexist), the degrees of freedom F is

• (A) 1 (P can be varied)
• (B) 2 (Both T and P can be varied)
• (C) 0 (neither T nor P can be varied)
• (D) 3

If \(\Delta G\) is positive, the reaction is:

• (A) Spontaneous
• (B) Non-spontaneous
• (C) At equilibrium
• (D) Reversible

Seamless tubes can be produced by

• (A) rolling
• (B) extrusion
• (C) deep drawing
• (D) forging

In statistical thermodynamics, Stirling's Approximation (\(\ln N! \approx N \ln N - N\)) is valid only when

• (A) N is a small integer
• (B) N is very large
• (C) The system is at 0 K
• (D) The gas behaves non-ideally

For ideal solution, activity (a) equals:

• (A) Density
• (B) Pressure
• (C) Volume
• (D) Mole fraction

In an adiabatic process:

• (A) Heat transfer occurs
• (B) No heat transfer occurs
• (C) Temperature is constant
• (D) Pressure is constant

The Zeroth Law of Thermodynamics provides the basis for temperature measurement. If Body A is in thermal equilibrium with Body B, and Body B is in thermal equilibrium with Body C, which property must be identical for all three?

• (A) Internal Energy (U)
• (B) Total Enthalpy (H)
• (C) Temperature (T)
• (D) The product of Pressure and Volume (PV)

Chemical potential is equal to:

• (A) Enthalpy
• (B) Gibbs free energy per mole
• (C) Entropy
• (D) Internal energy

Which of the following processes produces maximum work?

• (A) Rapid expansion of gas
• (B) Free expansion
• (C) Reversible expansion
• (D) Irreversible expansion

Two parallel edge dislocations of the same sign (same Burger's vector) are located on the same slip plane. As they approach each other, what happens?

• (A) Energy decreases; Force is attractive
• (B) Energy increases; Force is repulsive
• (C) Energy remains constant; Force is zero
• (D) Energy increases; Force is attractive

At eutectic point, degree of freedom is:

• (A) 0
• (B) 1
• (C) 2
• (D) 3

Stirling's approximation is used to simplify:

• (A) Integrals
• (B) Factorials
• (C) Logarithms
• (D) Derivatives

Which of the following statements regarding mechanical twinning is FALSE?

• (A) Twinning occurs more frequently at high strain rates and low temperatures.
• (B) The atomic displacement in twinning is a fraction of the interatomic distance.
• (C) Twinning changes the orientation of the crystal lattice in the deformed region.
• (D) Twinning produces the same total strain as slip for the same amount of atomic movement.

Second law states:

• (A) Energy conserved
• (B) Entropy increases
• (C) Heat = work
• (D) \(\Delta H = 0\)

The efficiency of a Carnot engine depends on:

• (A) Pressure
• (B) Volume
• (C) Temperatures of reservoirs
• (D) Working substance

The volume \(V\) of a cubic unit cell is related to the lattice parameter \(a\). Which of the following expressions is dimensionally and physically correct?

• (A) \(V = a + a^2\)
• (B) \(V = 2a\)
• (C) \(V = a^2\)
• (D) \(V = a^3\)

Which material is diamagnetic?

• (A) Iron
• (B) Copper
• (C) Nickel
• (D) Cobalt

At equilibrium, Gibbs free energy change is:

• (A) Positive
• (B) Negative
• (C) Zero
• (D) Infinite

Drag force \(F\) depends on density \(\rho\), velocity \(V\), and diameter \(D\): \(F = k \rho^a V^b D^c\). Find exponents \(a, b, c\).

• (A) \((1, 2, 1)\)
• (B) \((2, 2, 1)\)
• (C) \((1, 1, 2)\)
• (D) \((1, 2, 2)\)

Paramagnetic materials have:

• (A) Large negative susceptibility
• (B) Infinite susceptibility
• (C) Zero susceptibility
• (D) Small positive susceptibility

Magnetic susceptibility (\(\chi\)) of paramagnetic materials follows:

• (A) Curie's Law (\(\chi \propto 1/T\))
• (B) Ohm's law
• (C) Hooke's law
• (D) Wiedemann–Franz law

For a laminar boundary layer developing over a flat plate, the displacement thickness is defined as: \(\delta^* = \int_0^\delta \left(1 - \frac{u}{U_\infty}\right) dy\). Physically, \(\delta^*\) represents

• (A) Momentum loss in the boundary layer
• (B) Reduction in mass flow due to the presence of the boundary layer
• (C) Energy lost in the boundary layer
• (D) Vorticity in the flow

Curie temperature is the temperature at which:

• (A) Ferromagnetic \(\rightarrow\) Paramagnetic
• (B) Paramagnetic \(\rightarrow\) Diamagnetic
• (C) Solid \(\rightarrow\) Liquid
• (D) Magnetic \(\rightarrow\) Non-magnetic

The core of optical Fiber has refractive index:

• (A) Less than cladding
• (B) Equal to cladding
• (C) Greater than cladding
• (D) Zero

For Inlet diameter \(>\) Outlet diameter the Velocity and pressure along the flow will be as

• (A) Velocity decreases, pressure increases
• (B) Velocity increases, pressure decreases
• (C) Velocity constant, pressure decreases
• (D) Velocity increases, pressure constant

Retentivity means:

• (A) Resistance to magnetization
• (B) Ability to retain magnetism
• (C) Loss of magnetism
• (D) Increase in magnetization

A major thermodynamic reason for nanoparticle agglomeration is:

• (A) Low density
• (B) High melting point
• (C) High surface energy
• (D) Low band gap

In a horizontal pipe, velocity doubles. Pressure gets

• (A) Unchanged
• (B) Decreased
• (C) Increased
• (D) Doubled

Which has highest magnetic susceptibility?

• (A) Diamagnetic
• (B) Paramagnetic
• (C) Ferromagnetic
• (D) Antiferromagnetic

The Third Law of thermodynamics states that entropy at absolute zero is:

• (A) Maximum
• (B) Minimum
• (C) Zero for perfect crystal
• (D) Infinite

A solid circular shaft is subjected to bending moment and torsional moment. What is the nature of maximum principal stress at the outer surface?

• (A) Pure bending stress
• (B) Pure shear stress
• (C) Combination of normal and shear stress
• (D) Zero stress

Ceramic materials are generally:

• (A) Ductile and soft
• (B) Brittle and hard
• (C) Flexible and elastic
• (D) Conductive

The packing sequence of atoms in FCC is

• (A) AB AB AB...
• (B) BC BC BC...
• (C) AC AC AC...
• (D) ABC ABC ABC....

A particle under force \(F = -kx\). Then Motion is

• (A) uniform
• (B) simple harmonic
• (C) circular
• (D) random

Zirconia ceramics exhibit:

• (A) High electrical conductivity
• (B) High hardness
• (C) Low hardness
• (D) Transformation toughening

Hall-Petch equation predicts the relation between grain size and

• (A) Yield Strength
• (B) Ductility
• (C) %Elongation
• (D) Tensile Strength

Three forces 3 N, 4 N, and 5 N act at a point. For equilibrium, angle between 3 N and 4 N is

• (A) \(60^\circ\)
• (B) \(90^\circ\)
• (C) \(120^\circ\)
• (D) \(180^\circ\)

A composite material is composed of:

• (A) Single phase material
• (B) Only metals
• (C) Only polymers
• (D) Two or more distinct phases

Given the variables: velocity (V), density (\(\rho\)), viscosity (\(\mu\)), and characteristic length (L), how many dimensionless groups can be formed?

• (A) 1
• (B) 2
• (C) 3
• (D) 4

A shaft is subjected to pure torsion. The state of stress at any point on the surface of the shaft, relative to the axis of the shaft, consists of

• (A) pure normal stress
• (B) pure shear stress
• (C) equal normal and shear stresses
• (D) zero stress

Cross slip is observing

• (A) screw dislocation
• (B) edge dislocation
• (C) twinning
• (D) bending

A particle starts from rest and moves with a constant acceleration of \(2 m/s^2\). What is the distance traveled by the particle in the \(5^{th}\) second of its motion?

• (A) \(25 m\)
• (B) \(9 m\)
• (C) \(10 m\)
• (D) \(12.5 m\)

A rigid body rotates about a fixed axis under a constant moment (M). If the Mass Moment of Inertia (I) of the body is doubled while the moment remains the same, the angular acceleration (\(\alpha\)) will

• (A) double
• (B) remain the same
• (C) be halved
• (D) quadruple

One of the following is not a strengthening mechanism:

• (A) Cold working
• (B) Precipitation hardening
• (C) Strain aging
• (D) Annealing

Water flows from pipe diameter \(200 mm\) to \(100 mm\). If velocity in larger pipe is \(2 m/s\), find velocity in smaller pipe.

• (A) \(2 m/s\)
• (B) \(4 m/s\)
• (C) \(6 m/s\)
• (D) \(8 m/s\)

Removing a hole from a plate shifts centroid:

• (A) Depends on material
• (B) Toward hole
• (C) Away from hole
• (D) No change

Which of the following crystal structure is ductile?

• (A) Face centered cubic (FCC)
• (B) Body centered cubic (BCC)
• (C) Hexagonal close packed (HCP)
• (D) Body centered tetragonal (BCT)

Ideal c/a ratio of HCP is

• (A) 1.633
• (B) 1.88
• (C) 1.58
• (D) 1.48

A surface with emissivity 0.8 and area \(1 m^2\) is maintained at \(500 K\) and is surrounded by air at \(300 K\). Given Stefan–Boltzmann constant \(\sigma = 5.67 \times 10^{-8} W/m^2K^4\). What is the rate of radiative heat transfer?

• (A) \(3.5 kW\)
• (B) \(2.47 kW\)
• (C) \(2.0 kW\)
• (D) \(1.5 kW\)

Which of the following is a natural polymer?

• (A) Polyethylene
• (B) Nylon
• (C) Cellulose
• (D) PVC

The forces of \(60 N\) and \(80 N\) are in equilibrium at a point, acting at an angle of \(90^\circ\) to each other. Find the magnitude of the third force?

• (A) \(100 N\)
• (B) \(120 N\)
• (C) \(140 N\)
• (D) \(160 N\)

Two metals, A and B, have Stacking Fault Energies of \(20 mJ/m^2\) and \(160 mJ/m^2\) respectively. Which of the following is true regarding their deformation behaviour?

• (A) Metal A will cross-slip more easily than Metal B.
• (B) Metal B will exhibit a higher rate of work hardening than Metal A.
• (C) Metal B will cross-slip more easily than Metal A.
• (D) Metal A will not exhibit any plastic deformation.

The hot working temperature of lead

• (A) \(300°C\)
• (B) \(100°C\)
• (C) \(200°C\)
• (D) Room temperature

An activity represents:

• (A) Effective concentration
• (B) Pressure
• (C) Volume
• (D) Temperature

Consider an adiabatic process in which a gas is compressed. Which of the following statements is strictly CORRECT regarding the energy transfer?

• (A) The internal energy of the system remains constant because Q = 0.
• (B) Heat is converted into work during the compression.
• (C) Work is a mode of energy transfer that is stored as "Work Energy" in the system.
• (D) Work is done on the system, increasing the internal energy, but the system contains no "Heat" or "Work."

Total recoverable strain energy stored within the elastic limit in the stress-strain curve is called-

• (A) Ductility
• (B) Hardness
• (C) Resilience
• (D) Tensile strength

Efficiency of Carnot engine is always:

• (A) Greater than 1
• (B) Less than 1
• (C) Equal to 1
• (D) Infinite

Which of the following materials possesses anti-parallel spins of unequal magnitude, resulting in a spontaneous magnetic moment below a critical temperature?

• (A) Ferromagnetic
• (B) Paramagnetic
• (C) Antiferromagnetic
• (D) Ferrimagnetic

Which of the following has the highest thermal diffusivity:

• (A) Iron
• (B) Lead
• (C) Concrete
• (D) Wood

At 0K, an intrinsic semiconductor behaves like a:

• (A) Conductor
• (B) Insulator
• (C) Superconductor
• (D) Metal

Thermosetting polymers differ from Thermoplastics primarily because

• (A) Thermosets have a linear molecular structure and can be reshaped by heating
• (B) Thermosets possess a three-dimensional cross-linked network that prevents melting
• (C) Thermoplastics have higher thermal stability than thermosets
• (D) Thermosets are formed exclusively by addition polymerization

If two forces acting at a joint are not along the straight line, then for the equilibrium of the joint

• (A) one of the forces must be zero
• (B) each force must be zero
• (C) forces must be equal and of the same sign
• (D) forces must be equal in magnitude but opposite in sign

Vacancy diffusion coefficient: \(D = D_0 \exp(-Q/RT)\)

• (A) Dimensionally correct
• (B) Requires \(D_0/L\) to be correct
• (C) Requires Q dimensionless
• (D) All formulas fail dimensionally

The "Glassy" state in ceramics is characterized as

• (A) A perfectly ordered crystalline lattice
• (B) A material that lacks any atomic bonding
• (C) A liquid that has a very low viscosity
• (D) A metastable, non-crystalline solid with short-range order but no long-range order

Centroid of triangle lies at:

• (A) Midpoint
• (B) Intersection of medians
• (C) Circumcenter
• (D) Vertex

Which of the following statements is FALSE?

• (A) Entropy is a state function
• (B) Heat is a state function
• (C) Internal energy is a state function
• (D) Gibbs free energy is a state function

Transparency of glass is due to

• (A) absence of grain boundaries
• (B) presence of free electrons
• (C) metallic bonding
• (D) lack of scattering centres

If \(\Delta U = 100 J\) and work done = \(40 J\), heat supplied is:

• (A) \(60 J\)
• (B) \(140 J\)
• (C) \(100 J\)
• (D) \(40 J\)

According to the Boltzmann Equation \(S = k_B \ln \Omega\), if a system is in a state of "Perfect Order" (a perfect crystal at \(0 K\)), the number of microstates \(\Omega\) and the entropy \(S\) are

• (A) \(\Omega = 0, S = 0\)
• (B) \(\Omega = 1, S = 0\)
• (C) \(\Omega = 1, S = 1\)
• (D) \(\Omega \to \infty, S \to \infty\)

Which combination gives highest strength-to-weight ratio?

• (A) Polymer + fibres
• (B) Metal + fibres
• (C) Polymer + particles
• (D) Ceramic + voids

Ferromagnetic materials show:

• (A) Weak repulsion
• (B) Strong attraction
• (C) No interaction
• (D) Random behaviour

A material loses its strong magnetization suddenly at a critical temperature and becomes weakly magnetic. This transition is from

• (A) Diamagnetic \(\to\) Paramagnetic
• (B) Paramagnetic \(\to\) Ferromagnetic
• (C) Ferromagnetic \(\to\) Paramagnetic
• (D) Superconducting \(\to\) Diamagnetic

The 'Fugacity Coefficient (\(\phi = f/P\))' of a real gas approaches unity when

• (A) the pressure approaches infinity
• (B) the temperature approaches \(0 K\)
• (C) the pressure approaches zero
• (D) the gas is compressed into a liquid

According to equation of continuity,

• (A) \(w_1 a_1 = w_2 a_2\)
• (B) \(w_1 v_1 = w_2 v_2\)
• (C) \(a_1 v_1 = a_2 v_2\)
• (D) \(a_1/v_1 = a_2/v_2\)

Assertion (A): Gibbs free energy determines spontaneity at constant T and P.

Reason (R): A process is spontaneous when Gibbs free energy is minimum.

• (A) Both A and R are true, and R is the correct explanation of A
• (B) Both are true, but R is not the correct explanation of A
• (C) A is true, R is false
• (D) A is false, R is true

A plane wall of thickness \(0.1 m\) and thermal conductivity 50 W/m·K has a surface area of \(2 m^2\). The temperature difference across the wall is \(100 K\). What is the rate of heat transfer through the wall?

• (A) \(200 kW\)
• (B) \(100 kW\)
• (C) \(50 kW\)
• (D) \(25 kW\)

Diamagnetic materials are:

• (A) Attracted by magnetic field
• (B) Repelled by magnetic field
• (C) Neutral
• (D) Strongly attracted

Let \(A\) be an \(n \times n\) matrix such that \(A^3 = A\). Then which one of the following is true?

• (A) \(A\) must be an identity matrix.
• (B) \(A^2\) must be an identity matrix.
• (C) \(A\) is invertible.
• (D) \(A\) is diagonalisable.

The function \(f(x) = \begin{cases} x \sin\frac{1}{x}, & x \neq 0
0, & x = 0 \end{cases}\) is

• (A) continuous everywhere
• (B) continuous only at \(x = 0\)
• (C) not continuous at \(x = 0\)
• (D) nowhere continuous

Let \(\vec{f} = x^2 y z \hat{i} + x y^2 z \hat{j} + x y z^2 \hat{k}\) be a vector field. If \(\vec{F}(x,y,z) = curl \vec{f}\), then the vector \(\vec{F}(1,-2,1)\) is equal to _______

• (A) \(-3\hat{i} + 4\hat{j} + 3\hat{k}\)
• (B) \(-3\hat{i} + 3\hat{k}\)
• (C) \(3\hat{i} - 3\hat{k}\)
• (D) \(-3\hat{i} - 4\hat{j} + 3\hat{k}\)

The integrating factor of the differential equation \((1 + x^2)\frac{dy}{dx} + 2xy - 4x^2 = 0\) is

• (A) \(\log(1 - x)\)
• (B) \(\log(1 + x^2)\)
• (C) \(1 + x^2\)
• (D) \(1 + 2x\)

The number of linearly independent solutions of the differential equation \(\frac{d^4y}{dx^4} - \frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 5\frac{dy}{dx} - 2y = 0\) of the form \(e^{ax}\) (\(a\) is a real number) is

• (A) 1
• (B) 2
• (C) 3
• (D) 4

If the probability distribution of a random variable X is as follows.



Then, the value of k is _______

• (A) 0.1
• (B) 0.2
• (C) 0.3
• (D) 0.4

The inverse Laplace transform of \(F(s) = \frac{s^2 - 3s + 4}{s^3}\) is _______

• (A) \(1 - 3t + 2t^2\)
• (B) \(1 - 3t + 4t^2\)
• (C) \(1 + 3t + 2t^2\)
• (D) \(1 + 3t + 4t^2\)

The general solution of \((x^2 D^2 - xD)y = 0\) is

• (A) \(y = c_1 + c_2 e^x\)
• (B) \(y = c_1 + c_2 x\)
• (C) \(y = c_1 + c_2 x^2\)
• (D) \(y = c_1 x + c_2 x^2\)

Two independent random variables X and Y have variances 0.2 and 0.5 respectively. Let Z = 5X - 2Y. The variance of Z is

• (A) 3
• (B) 7
• (C) 4
• (D) 5

Consider the function \(f(x) = x^3 - x - 1\) for finding the root the equation \(f(x) = 0\). If the initial approximation is \(x_0 = 1\), then the next approximation \(x_1\) using the Newton-Raphson method is

• (A) 1.25
• (B) 1.75
• (C) 1.5
• (D) 2