Given,

𝑓(0) = 1

𝑓(1) = 2

𝑓(2) = 5

𝑓(3) = 10

.

.

.

.

.

.

f(x)=x^2 + 1

then 𝑓(−1) =(-1)^2 +1=2

Hence, the correct answer  is option d. 2

𝐴 ⊆ 𝐵 is complement of B is subset of A so,B' ⊆ A'

Hence, the correct answer is option c. B' ⊆ A'

Let the three number of GP be a/r , a, ar (1st term, 2nd term which is also know as middle term & 3rd term)

then according to the questions

product =1728

so,  a/r x a x ar =1728

a^3 = 1728

∴a=12

Hence, the middle term is a=12

Hence, the correct answer is option a. 12

Fourth roots of unity are ±1, ±𝑖

Hence, the correct answer is option c. ±1, ±𝑖

Given,

f(x)= 𝑘𝑥^2 + 𝑥 − 1,

f(x-a)=x+2k =x-(-2k)

∴a=-2k

then, according to the remainder theory

f(-2k)= k(-2k)^2+(-2k)-1

=k4k^2 - 2k -1

=4k^3 - 2k -1

Hence, the correct answer  is option d. 4k^2 - 2k -1

Given,

𝑥^2 + 𝑘𝑥𝑦 +2𝑦^2 + 3𝑥 + 5𝑦 + 2 = 0 comparing with a𝑥^2 + 2h𝑥𝑦 +b𝑦^2 + 2g𝑥 + 2f𝑦 + c=0

a=1, 2h=k,    b=2,    2g=3,       2f=5,      c=2

h=k/2                 g=3/2        f=5/2

using formula,

abc+2fgh-af^2-bg^2 - ch^2=0

or, 1x2x2 + 2 x 5/2 x 3/2 x k/2  - 1 x (5/2)^2 - 2 x (3/2)^2 - 2 x (k/2)^2=0

or, 4 + 15k/4 -25/4 -19/4 -2k^2/4 =0

∴ 15k-2k^2-27=0

Put the value of k from the given option.Here option a satisfy

i.e 15 x 3 -2 x 3^2 -27 =0

Hence, the correct answer is option a. 3

Given,

𝑥^2 + 𝑦^2 −10𝑥 = 0--------(i) with the chord 𝑦 = 2𝑥--------(ii)

Put the value of (x,y) from the given option

a. (2,4) (0,0)  put (2,4) in equation(i)

2^2 +4^2-10x2=0

4+16-20=0

20-20=0

0=0

again put(2,4) in equation(ii) then,

4=2x2

4=4

if we put (0,0) in both equation it also satisfy

Hence, the correct answer is option a. (2,4) (0,0)

Given,

𝑦 = 𝑥^3 − 12𝑥^2

y' = 3x^2-24x

y''=6x-24

For point of inflection,

y''=0

6x-24=0

6x=24

x=4

Hence, the correct answer is option d. 4

The displacement equation is given as x=a+bt+ct^2. To find the acceleration, we first take the derivative of the displacement with respect to time t to get the velocity:

v=dx/dt​=b+2ct

Then, take the derivative of the velocity to get the acceleration:

a=dv/dt​=2c

Therefore, the acceleration is 2c.

Hence, the correct answer is option d. 2c

Given,

Let the first mass be m1=m

Let the second mass be m2=4m

Kinetic energy of first mass = Kinetic energy of second mass = K.E.

Velocity of first mass = V1

Velocity of second mass = V2

Kinetic energy of a particle with mass m and velocity v is given by:

K.E=(1x mv^2)/2

Linear momentum of a particle of mass m and velocity V is given by:  p=mv

We have two cases as the Kinetic energy is equal for both cases.

K.E=(1x mV1^2)/2--------------(i)

K.E=(1x 4m x V2^2)/2--------------(ii)

Dividing equation (i) with (ii) we get,

V1=2V2

∴Now we have,

p1=m1V1 & p2=m2V2

Dividing P2 by P1 we get

P2/P1= (m2V2)/(m1V1)

P2/P1= (mV2)/(mx 2V2)

P2/P1= 1/2

Hence, the correct answer is option b. 1/2

The force of gravity is weaker at the equator compared to the poles due to the centrifugal force caused by Earth's rotation and the equatorial bulge, which increases the distance from the Earth's center.

Hence, the correct answer is option a. Minimum at the equator

We know,
Surface tension $=\frac{\text{Surface energy/}}{\text{Unit Area}}$

$\therefore$ Surface energy $\propto$ Area $\propto$ ${\mathrm{r^}}^2$

Hence, the correct answer is option b. r^2

The principle of flotation states that the weight of the body is equal to the weight of the displaced water. Let the density of the body be ρb​ and the density of water be ρw​ (which is 1 g/cc):

Submerged Volume/Total Volume = ρb/ρw​ = (3/4)/1 = 0.75

Hence, the correct answer is option b. 0.75​

The energy emitted by a black body is proportional to the fourth power of its absolute temperature, as given by the Stefan-Boltzmann law.

Convert 727°C to Kelvin: T=727+273=1000 K.

Therefore, the energy emitted is proportional to T^4=(1000)^4.

Hence, the correct answer is option d. (1000)^4

According to thermodynamics,

Heat loss by water = heat gain by ice

It is given that,

Mass of water, m=80gm

Specific heat of water, s=1cal/gm/0C

So, heat lost by water during cooling from 30°Cto 0°C

ΔH=msΔT

ΔH=80×1×(30−0)

ΔH=2400cal........(i)

Heat gain by ice, $m_{1}l=m_1\times 80cal/gm.....(ii)$

Where, l is the latent heat of fusion

m1 is the mass of ice melted.

From equation (i) and (ii)

2400=80m1

m1=30g

Hence, the correct answer is option a. 30g

A liquid boils when its vapor pressure equals the external pressure (usually atmospheric pressure) acting on the surface. When this happens, bubbles of vapor can form within the liquid, leading to boiling.

Hence, the correct answer is option d. Equal to atmospheric pressure

The refractive index of a material varies with wavelength, being lower for red light than for violet light. Since the focal length f is inversely proportional to the refractive index $n$, the focal length is maximum for red color.

Hence, the correct answer is option a. Maximum for red color

Hence, the correct answer is option a. 30°

The fringe width $\beta$ in Young’s double-slit experiment is given by:

β=λD/d

where λ is the wavelength of light, $D$ is the distance between the slits and the screen, and d is the distance between the slits. If the slit separation $d$ is increased, the fringe width $\beta$ decreases because β is inversely proportional to $d$

Hence, the correct answer is option b. decreases​

Illumination (E) is given by:

E=(I.A)/(r^2)

where I is the luminous intensity, $A$ is the area of the circular disc, and $r$ is the distance from the source to the surface.

The total luminous flux Φ falling on the surface is:

Φ=I⋅Ω×fraction of light=500×(πr^2)/(r^2) x 0.02=500×0.02×π×0.1^2=10×3.14×0.01=0.314

The area A of the circular disc is:

​A=πr^2=π×0.12=0.0314 m^2

The average illumination $E$ is:

E=Φ/A =0.314/0.0314=10lux=2000lux

Hence, the correct answer is option b. 2000 lux​

For a charged hollow metal sphere, the potential at any point inside the sphere (including the center) is the same as the potential on the surface. Therefore, the potential at the center is 80 V.

Hence, the correct answer is option b. 80 V

When capacitors are connected in parallel, they share the same voltage. The charge $Q$ on each capacitor is given by:

Q=C×V

Since the voltage $V$ is the same, the ratio of the charges Q1 and Q2 on the capacitors is directly proportional to their capacitances:

Q1​​/Q2 = C1​/C2 = 2μF/3μF = 2/3

Hence, the correct answer is option b. 2:3

The angle between the geographic meridian (true north-south direction) and the magnetic meridian (direction indicated by a compass needle) is called the angle of declination. It represents the difference between true north and magnetic north.

Hence, the correct answer is option b. angle of declination

Inside a solenoid, the magnetic field is uniform and parallel to the axis of the solenoid. Since the charged particle is moving along the axis of the solenoid, it is moving parallel to the magnetic field lines. According to the Lorentz force law, the force on a charged particle moving parallel to the magnetic field is zero, so the particle experiences no force.

Hence, the correct answer is option d. no force

Hence, the correct answer is option a

The general form of a traveling wave equation is:

y=Acos(kx−ωt)

Comparing with the given equation y=3cos⁡(π(100t−x)), we see that k=π. The wavelength λ is related to k by:

k=2π/λ ⟹λ=2π/k = 2π/π = 2cm

Hence, the correct answer is option c. 2cm​

The apparent frequency f′ heard by a person due to the Doppler effect is given by:

f′=f{(v+vo​​)/(v−vs​) } -------(i)

f is the original frequency (150 Hz),

v is the speed of sound in air (330 m/s),

vs​ is the speed of the source (110 m/s),

vo​ is the speed of the observer (0 m/s, since the observer is stationary).

then equation(i) becomes

Since the source is moving toward the observer:

​f′=150 x {(330)/(330−110) }  =150× 330/220 =150×1.5=225Hz

Hence, the correct answer is option a. ​225Hz

In a stationary wave, nodes are points where the displacement is always zero, but the strain (which is related to the derivative of displacement) is maximum. Strain is a measure of deformation and is highest where the displacement changes most rapidly, which occurs at the nodes.

Hence, the correct answer is option a. strain is max. at nodes

According to Bohr's theory, the angular momentum of an electron in the nth orbit is quantized and given by mvr=nℏ. The de Broglie wavelength λ is given by λ=h/p=h/mv​. For an electron in a circular orbit, the circumference is equal to an integer multiple of the wavelength: 2πR=nλ

Hence, the correct answer is option c. $n\lambda =2\pi R$.

The frequency of the absorbed photon in a hydrogen atom is given by the energy difference between the two levels, ΔE=E8−E3

where En=−13.6 eV/n^2

The greater the energy difference, the higher the frequency of the absorbed photon. Among the options provided, the transition from n=3 to n=8 corresponds to the largest energy difference, and therefore the highest frequency.

Hence, the correct answer is option b. n=3 to n=8

Hence, the correct answer is option d. 7/8

$\beta$-rays are high-energy, high-speed electrons (or positrons) emitted by certain types of radioactive nuclei during the process of beta decay. These particles are charged, with β^- particles being electrons (negative charge) and β^+ particles being positrons (positive charge).

Hence, the correct answer is option c. Charged particles emitted by the nucleus

The correct pronoun to refer to "the wounded soldiers" is "them" because it is the objective case of the pronoun used to refer to multiple people.

Hence, the correct answer is option a. them

The reflexive pronoun "himself" is used because the subject "he" is performing the action on himself.

Hence, the correct answer is option c. himself

The subject of the sentence is "The leader," which is singular. The phrase "with his helpers" is a prepositional phrase and does not affect the verb. Therefore, the correct verb is "has."

Hence, the correct answer is option b. has

"Her husband and she" forms a compound subject, which is plural, so the correct verb form is "have."

Hence, the correct answer is option b. have

The correct tag question matches the subject "one" and the auxiliary "should," so the correct tag is "shouldn’t one?"

Hence, the correct answer is option a. shouldn’t one

The correct tag question is formed by inverting the subject and verb of the statement. The statement is negative, so the tag question should be positive, making "are they?" the correct option.

Hence, the correct answer is option a. are they

The preposition "for" is used to indicate a period or duration of time. "He lived in this city for ten years" means that he stayed in that city over a span of ten years.

Hence, the correct answer is option a. for

"Beside" means next to someone or something. "Sit beside me" means to sit next to me.

Hence, the correct answer is option b. beside

"Unless" sets a condition, meaning "if not." So the sentence means that if you don't use English, it will not improve. Therefore, "It will not improve" is the correct continuation.

Hence, the correct answer is option a. It will not improve

This sentence is a third conditional sentence, which is used to describe a hypothetical situation in the past and its possible result. "If he had tried" is a condition in the past that didn’t happen, and "he would have won the prize" describes what could have happened as a result.

Hence, the correct answer is option b. He would have won the prize.

When referring to sports or games in general, we don’t use an article. Therefore, "He plays football" is correct.

Hence, the correct answer is option d. no article

"The" is used here to specify a particular theatre that is known to both the speaker and listener. "He went to the theatre" suggests a specific theatre, like one they had plans to visit.

Hence, the correct answer is option c. the

To "sow" means to plant seeds, whereas "reap" means to harvest the results of planting. These two actions are opposites in the context of agriculture.

Hence, the correct answer is option b. reap

"Faulty" means having defects or imperfections. The antonym "flawless" means without any faults or defects.

Hence, the correct answer is option a. flawless

A synonym for "check" in the context of verifying or examining is "test." "Test" means to assess or evaluate something, similar to how "check" is used.

Hence, the correct answer is option a. test

In the context of "present" as a noun, it means a gift. "Suggest," "greet," and "free" do not fit this meaning.

Hence, the correct answer is option b. gift

To convert the question "Do we eat sand?" into passive voice, we change it to "Is sand eaten?" The passive form questions the action being done to the object (sand).

Hence, the correct answer is option a. Is sand eaten?

In the passive voice, "Fish is sold here" reflects that the focus is on the fish being sold rather than the people doing the selling. The singular form "is" is used because "fish" is treated as a singular noun in this context.

Hence, the correct answer is option b. Fish is sold here.

When converting direct speech to reported speech, "will" changes to "would" to reflect the past tense.

Hence, the correct answer is option c. She told me that she would do it herself.

To report an imperative sentence (command), we use "told" and the infinitive form of the verb ("to open").

Hence, the correct answer is option a. Rekha told me to open my mouth.

The verb "goes" is used for habitual actions in the present tense. "Goes" is the correct form to indicate that he regularly travels to Pokhara.

Hence, the correct answer is option a. goes

"Acrophobia" is the term used to describe the fear of heights. The other options are not standard terms for this phobia.

Hence, the correct answer is option a. acrophobia

The past perfect tense "had died" is used to show that the action (dying) occurred before another action in the past (the doctor arriving).

Hence, the correct answer is option b. had died

"Fire" takes the prefix "mis-" to form the word "misfire,"

Hence, the correct answer is option a. mis

The present participle of "die" is "dying."

Hence, the correct answer is option d. dying

The compound in question is butanoic acid. The suffix "oic acid" is used for carboxylic acids in IUPAC nomenclature. The structure given corresponds to a carboxylic acid with the functional group –COOH.

Hence, the correct answer is option c. oic acid

In Markovnikov's addition, the intermediate product is a carbocation. This intermediate is formed during the addition of HX to an alkene, where the more stable carbocation is favored.

Hence, the correct answer is option a. Carbocation

Acetylene (C₂H₂) contains one triple bond between the two carbon atoms, which consists of 2 π bonds and 1 σ bond. Each carbon also forms a single σ bond with hydrogen atoms.

Hence, the correct answer is option a. 2π and 3σ bond

Lewis acids and oxidizing agents are electrophiles because they are capable of accepting electrons. They typically seek to gain electrons or form bonds with nucleophiles.

Hence, the correct answer is option a. Electrophiles

The Wurtz reaction involves the coupling of two alkyl halides in the presence of sodium metal to form a new alkane. Methane cannot be prepared using this reaction because it lacks a suitable halide precursor for the reaction. The Wurtz reaction is typically used to synthesize alkanes with more than one carbon atom.

Hence, the correct answer is option a. Methane

The common ion effect occurs when the addition of an ion common to a weak electrolyte decreases its ionization. In the case of HCl (a strong acid) and NaCl (which dissociates into Na⁺ and Cl⁻), the presence of Cl⁻ from NaCl will affect the ionization of HCl. The common ion effect does not apply to reactions where the species do not share a common ion or where one or both components are strong electrolytes without a common ion.

Hence, the correct answer is option b. HCl + NaCl

Carboxylic acids have the general formula CₙH₂ₙO₂. This formula represents a family of organic compounds with a carboxyl group (-COOH).

Hence, the correct answer is option c. CₙH₂ₙO₂

The molecular formula C₆H₁₂O₆ corresponds to glucose, fructose, and galactose. These are all isomers with the same molecular formula but different structures.

Hence, the correct answer is option d. all of above

The Law of Conservation of Mass states that mass is neither created nor destroyed in a chemical reaction. Therefore, the total mass of reactants equals the total mass of products.

Hence, the correct answer is option c. Law of conservation of mass

The energy levels of orbitals increase as follows: 3d < 4s < 4p. Among the options given, the 4p orbital has the highest energy level.

Hence, the correct answer is option d. 4p

In peroxides (e.g., H₂O₂), the oxidation state of oxygen is -1.

Hence, the correct answer is option b. -1

The standard electrode potential of the hydrogen electrode is defined as zero volts. This value is used as a reference point for measuring other electrode potentials.

Hence, the correct answer is option a. zero

Among Na (sodium), Mg⁺ (magnesium ion), and Al²⁺ (aluminum ion), the neutral Na atom has the largest size. This is because it has the highest number of electron shells and is not positively charged, which generally makes it larger than its positively charged ions.

Hence, the correct answer is option a. Na

The molarity of pure water is approximately 55.55 M. This is calculated from the density of water and the molar mass of water. Pure water has a density of about 1 kg/L, and the molar mass of water (H₂O) is approximately 18 g/mol.

Hence, the correct answer is option d. 55.55M

Tritium (³H) is the radioactive isotope of hydrogen. Protium (¹H) and deuterium (²H) are stable isotopes of hydrogen.

Hence, the correct answer is option c. Tritium

In the contact process for the production of sulfuric acid, vanadium(V) oxide (V₂O₅) is used as a catalyst. It facilitates the oxidation of sulfur dioxide (SO₂) to sulfur trioxide (SO₃).

Hence, the correct answer is option a. V₂O₅

White vitriol is zinc sulfate heptahydrate, and its chemical formula is ZnSO₄·7H₂O. It is a common compound used in various industrial and laboratory processes.

Hence, the correct answer is option b. ZnSO₄.7H₂O

Proteins exhibit several types of bonding

Hydrogen bonds: Stabilize the secondary and tertiary structures of proteins.

Covalent bonds: Include disulfide bonds between cysteine residues.

Peptide bonds: Link amino acids together in the primary structure of proteins.

Hence, the correct answer is option d. all of above

The chromophore is the part of a molecule responsible for its color, typically containing conjugated systems of double bonds or other multiple bonds.

Hence, the correct answer is option a. Chromophore

Molecular formula of calomel is Hg₂Cl₂

Hence, the correct answer is option b. Hg₂Cl₂

The brown ring test is a qualitative test for the presence of nitrate ions (NO₃⁻). When FeSO₄ is added to a solution containing nitrate and concentrated sulfuric acid is carefully added, nitric oxide (NO) forms and reacts with Fe²⁺ to create a brown ring at the interface.

Hence, the correct answer is option c. NO

Chalcopyrite (CuFeS₂) is a copper iron sulfide mineral and is the most important ore of copper. It is not referred to as "matte" (a mixture of metal sulfides), "lunar caustic" (another name for silver nitrate), or "fool’s gold" (which is actually pyrite, FeS₂). Therefore, the correct answer is "none."

Hence, the correct answer is option d. None

Hexane (C₆H₁₄) has five structural isomers: n-hexane, 2-methylpentane (isohexane), 3-methylpentane, 2,2-dimethylbutane, and 2,3-dimethylbutane.

Hence, the correct answer is option c. 5

The rate of a first-order reaction depends on:

Time: The concentration of the reactant decreases exponentially with time.

Temperature: The reaction rate increases with temperature.

Concentration of reactant: The rate is directly proportional to the concentration of the reactant.

Hence, the correct answer is option d. All of above

Neutral oxides do not react with acids or bases to form salts. Water (H₂O) is a neutral oxide. Other examples include carbon monoxide (CO) and nitrous oxide (N₂O). The other options (CO₂, N₂O₃, ZnO) are acidic or amphoteric oxides.

Hence, the correct answer is option b. H₂O