p/q q∈ 𝑅, is undefined

For undefined denominators equals to zero i.e q=0,p≠ 0

Hence, the correct answer is option a. p ≠ 𝑜 , 𝑞 = 0

[𝑎, 𝑏] means a & b both belongs to the sets i.e {x: a≤ 𝑥 ≤ 𝑏}

Hence, the correct answer is option a. {x: a≤ 𝑥 ≤ 𝑏}

Given,

f(x)=√(4 − 𝑥^2) ,  f(-3)=?

then,

f(-3)=√{(4 − (-3)^2 }

=√(4-9)

=√-5

=√(5 x i^2)    (∴ i^2= -1 )

=i√5

Hence, the correct answer is option c. i√5

Given,

|x|=1

Here modulus means +ve & -ve sign

so {-1,1}

Hence, the correct answer is option c. {-1,1}

Given,

𝑘 + 2 ,4𝑘 − 6 ,3𝑘 − 2 are in AP

we know,

AP;  b=(a+c)/2

then,

4k-6 = {(k+2)+(3k-2)}/2

or, 8k-12=4k

or, 4k =12

k=3

Hence, the correct answer is option c. 3

given, roots are -3 & 2

from option

a. 𝑥^2 + 𝑥 − 6 = 0

put x= -3

(-3)^2 + (-3) − 6 = 0

9-3-6=0

9-9=0

0=0

Again put x=2

(2)^2 + 2-6 =0

4+2-6=0

6-6=0

0=0

Both value satisfy in option a.

Hence, the correct answer is option a. 𝑥^2 + 𝑥 − 6 = 0

kx+3y+5=0

slope(m1)= -k/3

and 5x-2y-6=0

slope(m2)= -5/-2 = 5/2

For Perpendicular,

m1 x m2 = -1

(-k/3) x (5/2) = -1

5k = 6

k= 6/5

Hence, the correct answer is option a. 6/5

Given,

5𝑥^2 + 3𝑥𝑦 + 𝑘𝑦^2 = 0 comparing with a𝑥^2 + 2h𝑥𝑦 + b𝑦^2 = 0

a=5, 2h=3 =>h = 3/2,    b=k

For  perpendicular

a+b=0

5+k=0

k= -5

Hence, the correct answer is option d. -5

For shortcut trick;

put the given point (x,y)=(1,1) is given option

a. 𝑥 + 𝑦 = 2

1+1=2

2=2

Hence, the correct answer is option a. 𝑥 + 𝑦 = 2

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 c. 4

The word “commemorate” has the stress on second syllable.

Hence, the correct answer is option b. second

The word "go" and "sew" share the same vowel sound /oʊ/. The other options ("nor," "four") have different vowel sounds.

Hence, the correct answer is option c. sew

The correct answer is "is opened" because the sentence implies that the school will be opened by someone (passive voice). It reflects a wish or hope for something to be done.

Hence, the correct answer is option b. is opened

"Bellicose" describes someone who is eager to fight or inclined towards aggression.

Hence, the correct answer is option a. bellicose

The word "ambiguous" means unclear or having more than one possible interpretation. The antonym is "clear," meaning easily understood or obvious. Although "obvious" is also a correct option, "clear" is typically the most straightforward antonym.

Hence, the correct answer is option b. clear

The word "ominous" refers to something that is threatening or foreboding, indicating a bad outcome. The opposite would be "auspicious," which suggests a favorable or promising outcome.

Hence, the correct answer is option b. auspicious

The correct prepositions are "of" and "from." We say someone dies "of" a disease (e.g., cancer) and "from" a cause such as loss of blood. Therefore, "He died of cancer but she died from the loss of blood."

Hence, the correct answer is option a. of, from

The correct preposition here is "to." The question is asking to whom something was given, so "Whom did you give it to?"

Hence, the correct answer is option d. to

The phrase "sprang upon" is correct, as it indicates the dog suddenly jumped towards or attacked the cat. "Upon" is often used to convey sudden movement or action directed at something.

Hence, the correct answer is option a. upon

The missing vowels are "i," "o," and "e," which complete the words as "decease" and "suitors."

Hence, the correct answer is option b. i-o-e

"Rice and curry" is considered a single dish, so it is treated as a singular subject. Therefore, the correct verb form is "is."

Hence, the correct answer is option c. is

The word "snag" refers to a hidden problem or drawback.

Hence, the correct answer is option a. drawback

Polygamy refers to the practice of having more than one wife simultaneously

Hence, the correct answer is option b. Polygamy

The correct term is "gibber," which describes the chattering sounds that monkeys make.

Hence, the correct answer is option b. gibber

The phrase "to his sister" is the indirect object of the sentence. It tells to whom the book (direct object) was handed.

Hence, the correct answer is option d. indirect object

"Its" is the correct possessive pronoun to indicate that the hood belongs to the car. "It’s" is a contraction of "it is," which wouldn't make sense in this context.

Hence, the correct answer is option a. its

The correct phrase is "the same." The article "the" is used before "same" to indicate that two things are identical.

Hence, the correct answer is option c. the

"Versatile" refers to someone who is capable of doing many different things, often indicating that they are talented or gifted in multiple areas.

Hence, the correct answer is option b. Gifted

The correct prefix is "non," forming the word "non-aggression," which refers to the absence of aggression. The other prefixes do not correctly modify "aggression."

Hence, the correct answer is option a. non

"Pyrophobia" is the correct term for the fear of fire.

Hence, the correct answer is option b. pyrophobia

"Reality" is a noun, referring to the state of things as they actually exist.

Hence, the correct answer is option d. reality

"Until" is the correct conjunction to use here, as it indicates that the waiting continued up to the point when the bus arrived.

Hence, the correct answer is option b. until

"Itinerant" refers to a person who travels from place to place, typically for work or a nomadic lifestyle.

Hence, the correct answer is option a. Itinerant

"Aren’t you" is the correct tag question to follow the statement "You're coming to the party," because the statement is positive, and the tag question must be negative.

Hence, the correct answer is option  a. aren’t you

This question typically has a rising tone on "friend" and a falling tone on "enemy," indicating the speaker is seeking confirmation or clarity, which is common in questions that contrast two options.

Hence, the correct answer is option b. rising and falling

The radius of an atom is typically on the order of 10^−10 meters, which is also known as 1 Angstrom (Å). This scale is characteristic of atomic structures.

Hence, the correct answer is option b. 10^-10

For a given initial speed, the range of a projectile is maximized at an angle of 45° when projected with respect to the horizontal plane, assuming no air resistance.

Hence, the correct answer is option b. 45°

The pendulum comes to rest because of air resistance, which dissipates the mechanical energy of the pendulum over time.

Hence, the correct answer is option a. friction of air

Adhesion is the force that causes different materials to stick together. Powder clings to the skin due to the adhesive forces between the powder particles and the skin.

Hence, the correct answer is option c. adhesion

Viscosity is a measure of a fluid's resistance to deformation and flow, which is fundamentally related to the transfer of momentum between fluid layers.

Hence, the correct answer is option c. transfer of momentum

Given,

length(l)= 700m = 700 x 100 cm = 70000cm

coefficient of linear expansion(α)=11𝑋10^−6/K

change in temperature(ΔT)=(243K−131K)= 112K

The change in length due to temperature variation can be calculated using the formula:

ΔL=L⋅α⋅ΔT

where $L$ is the original length, α is the coefficient of linear expansion, and $\Delta T$ is the change in temperature.

Then,

=70000cm x 11 x 10^−6/K x 112K

=86.24 cm

Hence, the correct answer is option d. 86.24cm

During an isothermal process, the temperature of the system remains constant. For an ideal gas, internal energy is a function of temperature, so if the temperature doesn’t change, the internal energy remains unchanged.

Hence, the correct answer is option b. isothermal process

Efficiency of steam engine lies in the order of (20-30)%

Hence, the correct answer is option b. 20%

In an isobaric process, the pressure remains constant, resulting in the largest area under the pressure-volume curve. This maximizes the work done by the gas during expansion.

Hence, the correct answer is option b. isobarically

Constructive interference occurs when two waves overlap in such a way that their amplitudes add together, resulting in a larger disturbance or wave amplitude.

Hence, the correct answer is option a. The interference is constructive.

Given,

original frequency(f)=524 Hz

speed of sound(v)=340 m/s

speed of the observer(v0)=5 m/s

The formula for the observed frequency f′ when the source is stationary and the observer is moving towards it is:

f′=f x {(v+v0)/v}

f' = 524 x {(340 +5)/340} ≈532Hz

Hence, the correct answer is option d. 532Hz

In a longitudinal wave, such as a sound wave, the particles of the medium oscillate back and forth in the same direction (parallel) as the wave is traveling.

Hence, the correct answer is option a. parallel to the wave velocity

luminous intensity(I)=100 cd

distance(d)=2m

Illuminance $E$ can be calculated using the inverse square law:

E=I/d^2

= 100​/2^2

=100/4

=25lux

Hence, the correct answer is option c. 25lux​

When light passes through a prism, it both deviates (bends) from its original path and disperses (splits into its component colors) due to the varying refractive indices for different wavelengths of light.

Hence, the correct answer is option d. dispersion and deviation both

The charge of one electron is approximately 1.6×10^−19 C. The number of excess electrons $n$ is calculated using the formula:

n=(Total Charge)/Charge of one electron)= (1C)/(1.6×10^−19C/electron) ≈6.25×10^18electrons

Hence, the correct answer is option a. 6.25×10^18​​

When a liquid is heated, its density typically decreases, leading to a decrease in the refractive index. This is because the refractive index depends on the density of the medium.

Hence, the correct answer is option b. decreases

The torque τ on an electric dipole in a uniform electric field is given by:

τ=p​×E

𝜏 = 𝑃𝐸𝑠𝑖𝑛𝜃 𝑜𝑟 𝜏 𝛼PE

Hence, the correct answer is option c. P.E

A Wheatstone bridge is an electrical circuit used to precisely measure an unknown resistance by balancing two legs of a bridge circuit.

Hence, the correct answer is option d. resistance

The Curie temperature is the temperature above which a ferromagnetic material loses its permanent magnetic properties and becomes paramagnetic. At this point, the magnetism effectively becomes zero.

Hence, the correct answer is option d. becomes practically zero at Curie temperature

Given,

Self-inductance(L)= 10 mH = 10×10^−3 H

Current(I)=2A

The magnetic flux Φ through the coil can be calculated using the formula:

Φ=L×I

=10×10^−3×2

=0.02Wb

Hence, the correct answer is option a. 0.02Wb

Positive rays, also known as canal rays, consist of positively charged particles (ions) that are produced in a gas discharge tube when atoms lose electrons.

Hence, the correct answer is option c. ions

Given,

Planck's constant(h)=6.626×10^−34 J·s

speed of light(c)=3×10^8 m/s

wavelength(λ)=4000 Å = 4000×10^−10 m

Using Formula,

E=hc/λ

={(​6.626×10^−34  x 3×10^8 )} / (4000×10^−10) 

≈3.1eV

Hence, the correct answer is option d. 3.1eV 

In Bohr's model, the radius of the $n$-th orbit is given by:

rn​∝n^2

This means the radius increases with the square of the principal quantum number.

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

Semiconductors have a conductivity that decreases at low temperatures because the number of charge carriers (electrons and holes) decreases as the temperature drops. At higher temperatures, thermal energy is sufficient to excite electrons across the band gap, allowing charge to flow.

Hence, the correct answer is option a. semiconductors

We know,

Molar mass of CaCO3=40 (Ca)+12(C)+3×16.00 (O)=100g/mol 

Number of moles of CaCO3 in 25.0 g

Moles of CaCO3​= (25.0g​)/(100g/mol) ≈0.25mol

Since each molecule of CaCO3 contains 1 atom of carbon, the number of carbon atoms in 0.25 mol of CaCO3​ is:

Number of carbon atoms=0.25×6.022×10^23≈1.505×10^23atoms

Hence, the correct answer is option a. 1.505×10^23

Chromium is an exception to the typical electron configuration rules. Instead of [Ar]3d^4 4s^2, it adopts [Ar] 3d^5 4s^1 to achieve a more stable half-filled 3d subshell.

Hence, the correct answer is option a. [Ar] 3d^5 4s^1

The CO2​ molecule has a linear structure due to its O=C=O bond angles being 180°, with no lone pairs on the carbon atom to distort the shape.

Hence, the correct answer is option a. linear

Chlorine has the highest electron affinity among the halogens listed. Although fluorine has a high electron affinity, chlorine's is slightly higher due to its larger atomic size, which reduces electron repulsion when gaining an electron.

Hence, the correct answer is option a. chlorine

Let $x$ be the average oxidation state of iron. In Fe3O4 the total charge is:

3x+4(−2)=0(since oxygen is −2)

3x−8=0

x= +8/3

Hence, the correct answer is option d. +8/3

Magnesium oxide (MgO) is a basic oxide. When it reacts with water, it forms magnesium hydroxide (Mg(OH)₂), which is a strong base. Basic oxides are those that react with acids to form salt and water.

Hence, the correct answer is option a. basic oxide

The Downs' process is an electrolytic method used to extract sodium metal from molten sodium chloride (NaCl). The process involves the electrolysis of molten sodium chloride, producing sodium at the cathode and chlorine gas at the anode.

Hence, the correct answer is option a. sodium

The given structure represents a straight-chain alkane with five carbon atoms, which is named "pentane" in IUPAC nomenclature. The structure can be written as CH3−CH2−CH2−CH2−CH3​, which is simply pentane.

Hence, the correct answer is option d. Pentane

When aluminum carbide (Al4C3​) reacts with water, it produces methane gas (CH4) and aluminum hydroxide (Al(OH)3) according to the reaction:

Al4​C3​+12H2​O→3CH4​+4Al(OH)3​

Hence, the correct answer is option a. CH4

Proteins are polymers made up of long chains of α-amino acids linked together by peptide bonds. These α-amino acids are the building blocks of proteins, and their sequence determines the structure and function of the protein.

Hence, the correct answer is option c. α-amino acid

Tollen’s test is used to detect the presence of an aldehyde group. Glucose, being an aldose (containing an aldehyde group), gives a positive Tollen’s test, forming a silver mirror on the inner walls of the test tube. Fructose, although a ketose, can also give a positive Tollen’s test because it can isomerize to glucose in basic conditions, but the primary carbohydrate tested is glucose.

Hence, the correct answer is option a. glucose

DDT (Dichlorodiphenyltrichloroethane) is a well-known synthetic insecticide. It was widely used to control mosquito-borne diseases like malaria, as well as agricultural pests.

Hence, the correct answer is option a. insecticide

Tranquilizers are a class of drugs used to treat mental health conditions such as anxiety and tension. They help in calming the mind and reducing symptoms of anxiety.

Hence, the correct answer is option b. mental anxiety

In benzene, each carbon atom is sp^2 hybridized. This hybridization involves the mixing of one s orbital and two p orbitals to form three sp^2 hybrid orbitals, which participate in the formation of the sigma bonds. The remaining unhybridized p orbital overlaps to form the delocalized π-bond system.

Hence, the correct answer is option a. sp^2 hybridized

Victor Meyer’s test is a chemical test used to distinguish between primary, secondary, and tertiary alcohols based on the color reaction with specific reagents. The test involves converting the alcohol into an alkyl iodide, which is then treated with nitrous acid and potassium hydroxide to yield different colors for primary, secondary, and tertiary alcohols.

Hence, the correct answer is option c. Victor Meyer’s test

The Clemensen reduction is a chemical reaction used to reduce aldehydes and ketones to alkanes using zinc amalgam and concentrated hydrochloric acid. This method is particularly useful for reducing carbonyl groups in the presence of other functional groups that might be sensitive to other reducing agents.

Hence, the correct answer is option a. Clemenssion reduction

The basicity of amines is influenced by the availability of the lone pair of electrons on the nitrogen atom for protonation. In trimethylamine ((CH3)3N, the nitrogen is bonded to three methyl groups. This environment makes the lone pair of electrons on the nitrogen more available for protonation compared to ammonia (NH3) and other amines. The steric hindrance caused by the three methyl groups does not significantly affect the availability of the lone pair for protonation, making trimethylamine the most basic among the options listed.

Hence, the correct answer is option d. (CH3)3N

White vitriol refers to zinc sulphate in its heptahydrate form. The molecular formula is ZnSO4⋅7H2O, which includes zinc sulphate combined with seven molecules of water

Hence, the correct answer is option a. ZnSO4⋅7H2O

Cinnabar is a mineral consisting primarily of mercury sulfide (HgS) and is the most important ore of mercury.

Hence, the correct answer is option c. mercury

HNO₃ (nitric acid) is a strong acid and dissociates completely in solution. The pH of a solution is given by:

pH=−log[H^+]

For a 0.04 M solution of HNO₃:

pH=−log[0.04]≈2

Hence, the correct answer is option b. 2

The unit of cell constant is 𝑚^−1 or  c𝑚^−1

Hence, the correct answer is option c. c𝑚^−1

We know,

Rate=k[𝐴]^1 𝑜r, 𝑘 = (𝑚𝑜𝑙 𝐿^−1 𝑠^−1)/(𝑚𝑜𝑙 𝐿^−1) = 𝑠^−1

Hence, the correct answer is option c. 𝑠^−1

Intensive properties are those that do not depend on the amount of substance present. Concentration is an intensive property because it remains constant regardless of the quantity of the substance.

Hence, the correct answer is option d. concentration

The unit of radioactivity is the curie (Ci), which measures the rate of radioactive decay.

Hence, the correct answer is option a. curie

Sodium nitroprusside is a compound used in medicine as a vasodilator and is characterized by the formula $]$. It is used to manage acute hypertension and is known for its nitroso group.

Hence, the correct answer is option a. Sodium nitroprusside