We know,

u cosθ = u/2

θ=cos^-1(1/2) = 60°

R=(u^2 sin 2θ)/g = (u^2 sin 2x60)/g= (u^2 sin 120)/g = (u^2 x √3) / 2g

Hence, the correct answer is option d. (u^2 x √3) / 2g

Rifle & Bullet have same momentum

i.e KE ∝ 1/m

Hence, the correct answer is option a. Less

The graph between length and time period is A parabola

Hence, the correct answer is option c. A parabola

The surface tension of a liquid decreases with increase of temperature. At the boiling point, every molecule of a liquid is in motion from bottom to surface, and due to rise in temperature, molecules lose the adhesion and hence, surface tension becomes zero. The surface tension of a liquid becomes zero at its boiling point and vanishes at critical temperature

Hence, the correct answer is option a. Boiling point

An ideal liquid is defined as a liquid with zero viscosity and zero compressibility.

Hence, the correct answer is option c. Zero viscosity & Zero compressibility

A gas thermometer measures temperature by the change in pressure of a gas. To ensure accurate and sensitive measurements, the gas should be filled under conditions that avoid liquefaction and maintain ideal gas behavior. Ideal gas behavior is more closely approximated at low pressure to minimize interactions between gas molecules, and high temperature to ensure the gas remains in the gaseous state.

Hence, the correct answer is option d. High temperature & low-pressure

For an ideal gas, the internal energy is purely a function of temperature. This is derived from the kinetic theory of gases, which states that the kinetic energy of gas molecules is proportional to the temperature of the gas. The internal energy of an ideal gas does not depend on pressure or volume.

Hence, the correct answer is option b. Temperature

Beats occur when two sound waves of slightly different frequencies interfere with each other. The result is a variation in amplitude at a frequency equal to the difference between the frequencies of the two waves. This phenomenon is known as the beat frequency.

Hence, the correct answer is option d. Superposition of 2 waves of nearly same frequencies

In an open organ pipe vibrating in its fundamental mode, the pressure variations are maximum at the middle of the pipe. This is because the middle of the pipe is where the pressure antinode is located, while the ends of the pipe are pressure nodes

Hence, the correct answer is option c. middle of pipe

The two are related by the equation c=λf, where $c$ is the speed of light, λ is the wavelength, and f is the frequency. In practical terms, the color we perceive depends on the frequency of the light, but not on wavelength. The inverse relationship between wavelength and frequency.

Hence, the correct answer is option b. Frequency and not on wavelength

The number of images formed by two plane mirrors inclined at an angle θ can be calculated using the formula:

n=(360°/θ) − 1

According to the question θ=30°

n=(360°/30°) − 1 = 12-1 = 11

Hence, the correct answer is option b. 11

This phenomenon occurs due to atmospheric refraction. The Earth's atmosphere bends the light from the sun, allowing us to see the sun slightly before it has actually risen and after it has actually set.

Hence, the correct answer is option a. Refraction

According to electrostatic principles, the electric field inside a conductor in electrostatic equilibrium is zero because the charges reside on the surface of the conductor and not within its interior.

Hence, the correct answer is option b. Zero

The work done in moving a point charge in an electric field is independent of the path taken; it depends only on the initial and final positions. This is because the electric field is a conservative field.

Hence, the correct answer is option d. No work is done in some cases and independent of path in other cases

When an object is given a positive charge, it loses electrons. Since electrons have mass (although very small), the loss of electrons will result in a decrease in the mass of the object.

Hence, the correct answer is option b. Decreases

Let's see from the option

Joule effect (Joule heating): The process of converting electrical energy into heat energy in a conductor. This process is not reversible because once energy is dissipated as heat, it cannot be fully recovered.

Peltier effect: This is a reversible process where heating or cooling occurs at the junction of two different conductors when an electric current flows through them.

Seebeck effect: This is also a reversible process where a temperature difference between two different conductors produces a voltage difference.

Thomson effect: This is a reversible process where heat is absorbed or evolved along the length of a conductor with a temperature gradient when an electric current flows through it.

Hence, the correct answer is option a. Joule effect

The time period $T$ of a freely suspended magnet in the Earth's magnetic field is given by:

T=2π√(I/mH)

where $I$ is the moment of inertia of the magnet, m is the magnetic moment, and H is the horizontal component of the Earth's magnetic field. The time period depends on the magnetic moment and the horizontal component of the Earth's magnetic field, but not on the length of the magnet or the length of suspension.

Hence, the correct answer is option d. Length of suspension

According to Ampere's Law, the magnetic field inside a hollow conductor (cylindrical shell) carrying current is zero because there is no enclosed current within the hollow region.

Hence, the correct answer is option b. Zero

The capacitive reactance XC is given by:

XC=1/2πfC

where $f$ is the frequency and $C$ is the capacitance. As the frequency increases, the capacitive reactance decreases.

Hence, the correct answer is option b. Decreases

When a current passes through a spring, a magnetic field is created around the wire of the spring. The current in the spring generates a magnetic field that will induce a force in each coil of the spring. This force tends to bring the coils closer together, causing the spring to compress.

Hence, the correct answer is option b. Compress

We know from de Broglie wavelength λ

λ=h/p

where h is the Planck constant and $p$ is the momentum.

For a photon, the energy $E$ is related to its momentum by:

E=pc

where c is the speed of light.

For an electron, the total energy $E$ is the sum of its kinetic energy and rest mass energy:

E=√{(pc)^2+(me​c^2)^2 }

where me ​ is the rest mass of the electron and c is the speed of light.

Given that both the photon and the electron have the same de Broglie wavelength, they have the same momentum. However, the photon’s energy is purely kinetic (since it has no rest mass), while the electron's total energy includes its rest mass energy, which is significant compared to its kinetic energy.

​Hence, the correct answer is option b. Electron

Hence, the correct answer is option b. 27:5

The maximum frequency of X-rays produced in an X-ray tube is determined by the potential difference between the cathode and the anode (anti-cathode). This potential difference accelerates the electrons, and when they strike the target material, they decelerate rapidly, emitting X-rays. The higher the potential difference, the higher the energy (and frequency) of the emitted X-rays.

Hence, the correct answer is option d. Potential difference between cathode and anti-cathode

In the given notation

$Z$ is the atomic number, representing the number of protons.

$A$ is the mass number, representing the total number of protons and neutrons.

the number of neutrons is A−Z.

Hence, the correct answer is option a. Z protons and A - Z neutrons

The average binding energy per nucleon for a typical heavy nucleus (such as Uranium or Lead) is approximately 7.6 MeV. This value represents the energy required to disassemble the nucleus into its constituent protons and neutrons.

Hence, the correct answer is option c. 7.6Mev

Let's see from the option

a. MgSO₄·H₂O:

Mg: 1 atom

S: 1 atom

O: 4 (from SO₄) + 1 (from H₂O) = 5 atoms

H: 2 atoms (from H₂O)

Total atoms = 1 (Mg) + 1 (S) + 5 (O) + 2 (H) = 9 atoms

b. MgSO₄·7H₂O:

Mg: 1 atom

S: 1 atom

O: 4 (from SO₄) + 7 × 1 (from 7H₂O) = 11 atoms

H: 7 × 2 = 14 atoms (from 7H₂O)

Total atoms = 1 (Mg) + 1 (S) + 11 (O) + 14 (H) = 27 atoms

c.SiO₂:

Si: 1 atom

O: 2 atoms

Total atoms = 1 (Si) + 2 (O) = 3 atoms

d.Al₂O₃:

Al: 2 atoms

O: 3 atoms

Total atoms = 2 (Al) + 3 (O) = 5 atoms

Hence, the correct answer is option b. MgSO₄·7H₂O with 27 atoms.

1 joule is equal to 10^7 ergs. This can be derived from the conversion factor between joules and ergs where 1 erg=10^−7 joules.

Hence, the correct answer is option c. c. 10^7 erg

Chemical equations are balanced to obey the Law of Conservation of Mass, which states that mass is neither created nor destroyed in a chemical reaction. This ensures that the same number of atoms of each element is present on both the reactant and product sides of the equation.

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

Robert Millikan's oil-drop experiment determined the charge of the electron. Through his experiment, he measured the charge of individual oil droplets, leading to the quantification of the elementary charge.

Hence, the correct answer is option c. Electron

The sp^3 hybridization involves the mixing of one s orbital and three p orbitals, resulting in four equivalent sp^3 hybrid orbitals arranged in a tetrahedral geometry.

Hence, the correct answer is option d. Tetrahedral

Glucose is a simple sugar with the molecular formula C6H12O6.

Hence, the correct answer is option a. C6H12O6.

Let's see from the option

Mg²⁺: Magnesium ion with a 2+ charge, smaller due to greater nuclear charge.

Na⁺: Sodium ion with a 1+ charge, relatively larger than Mg²⁺ but smaller than neutral Na.

O²⁻: Oxide ion with a 2- charge, larger due to the addition of electrons.

F⁻: Fluoride ion with a 1- charge, larger than neutral fluorine but smaller than O²⁻.

The ion with the largest radius is typically the one with the greatest negative charge because the added electrons increase the size of the electron cloud relative to the nuclear charge.

Hence, the correct answer is option c. O²⁻

Hence, the correct answer is option c. 0.4N

Let's see from the option

Acids: Free hydrogen is present as H⁺ ions in acids.

Water: Hydrogen is bonded to oxygen in water (H₂O).

Marsh gas: Common name for methane (CH₄), which contains hydrogen in the form of CH₄, not as free hydrogen.

Water gas: A mixture of hydrogen and carbon monoxide (CO). Free hydrogen is present in water gas.

Hence, the correct answer is option d. Water gas

When dilute nitric acid (HNO₃) reacts with non-metals, the typical reaction involves the formation of nitric oxide (NO) and water (H₂O). The specific product can depend on the non-metal involved and the reaction conditions, but generally, nitric acid reacts with non-metals to form NO and water.

Here’s a general reaction:

4 HNO3​+C→4 NO2​+2 H2​O+CO2​

For other non-metals, a similar pattern of nitric oxide formation can be seen.

Hence, the correct answer is option a. No reaction occur

Sulphur trioxide (SO₃) has a structure where sulphur is doubly bonded to three oxygen atoms in a trigonal planar arrangement. There are no S-S bonds in SO₃.

Hence, the correct answer is option c. Zero

HCl (Hydrochloric acid) is considered non-combustible. It is a strong acid that does not burn or support combustion. Instead, it reacts with bases and metals to produce salts and water, and its fumes can be corrosive and harmful.

Hence, the correct answer is option a. HCL

The term "king of metals" is often used to refer to gold, due to its historical value, rarity, and significance in various cultures.

Hence, the correct answer is option b. Gold

Heating borax (sodium tetraborate) with a platinum wire in a flame produces a characteristic colored bead known as the borax bead. This test is used in qualitative analysis to identify metal ions.

Hence, the correct answer is option c. Borax bead

Let's see from the option

PCl₃: Phosphorus trichloride is not typically associated with war gases.

COCl₃: Carbonyl chloride (phosgene) is used as a war gas.

COCl₂: Also known as phosgene, which is used in chemical warfare.

PCl₅: Phosphorus pentachloride is not used as a war gas.

Hence, the correct answer is option c. COCl₂ (phosgene)

chemical formula of Nessler's reagent is: K2[Hgl4

Hence, the correct answer is option b. . K2[Hgl4

Wurtz Reaction: This reaction involves the coupling of two alkyl halides using sodium metal in dry ether to form a higher alkane.

The general reaction is: 2R-X+2Na→R-R+2NaX 

Kolbe's Electrolysis: This reaction involves the electrolysis of a carboxylate salt to produce alkanes.

Bayer's Reagent: This is used for detecting phenolic compounds, not for forming alkanes.

Cannizzaro Reaction: This is a reaction of aldehydes without alpha-hydrogens, leading to disproportionation.

Hence, the correct answer is option a. Wurtz reaction

This describes the reaction of acetylene (ethyne, HC≡CH) when heated in the presence of a copper tube at 300°C. Under these conditions, acetylene undergoes a cyclization reaction to form benzene.

Benzene (C₆H₆) is formed from the dehydrocyclization of acetylene when treated with copper at high temperatures. This is a classic reaction known as the Benzene Synthesis or Benzene Formation Reaction.

Hence, the correct answer is option a. Benzene

The chemical formula of picric acid is 2,4,6-trinitropheno

Hence, the correct answer is option b. 2,4,6-trinitropheno

𝐶𝐻𝐶𝑙3 + 𝑁𝑎𝑜𝐻 + Phenol → Salicydehyde is obtained. This reaction is called as Rimer Hermann reaction

Hence, the correct answer is option a. Rimer Hermann reaction

The reaction between phenol and benzenediazonium chloride to form p hydroxyazobenzene is a condensation reaction.

Condensation reactions involve the joining of two molecules with the elimination of a small molecule, such as water   

In this specific case, a water molecule is eliminated during the coupling of the diazonium ion with the phenol.

The resulting product, p-hydroxyazobenzene, is an azo dye with a characteristic color.

Hence, the correct answer is option c. Condensation reaction

This reaction is known as the Rosenmund reduction. It involves the reduction of an acid chloride to an aldehyde using palladium on barium sulfate (Pd/BaSO₄) as a catalyst, often in the presence of hydrogen gas.

Hence, the correct answer is option d. Rosenmund reaction

When glycerol and oxalic acid are heated together, they undergo a reaction to produce formic acid and carbon dioxide.

Hence, the correct answer is option a. Formic acid

When methyl chloride (CH₃Cl) reacts with ammonia (NH₃) at high temperatures (1000°C), it undergoes a reaction known as the ammonolysis of an alkyl halide, leading to the formation of a primary amine.

Hence, the correct answer is option a. 1° amine

Nylon is not a homopolymer. It is a condensation polymer, a polyamide, and a copolymer

Hence, the correct answer is option d. Homopolymer

The symmetric difference of two sets $A$ and $B$, denoted AΔB, is defined as the set of elements that are in either of the sets but not in their intersection. Mathematically, it's given by: AΔB=(A−B)∪(B−A)

Hence, the correct answer is option a. AΔB=(A−B)∪(B−A)

The property described in the question is that if for all (x,y)∈R and (y,z)∈R, it implies (x,z)∈R. This is the definition of a transitive relation.

Hence, the correct answer is option c. Transitive

The inverse of an exponential function f(x)= a^x is the logarithmic function. If f(x)=a^x, then its inverse is f^{-1}(x) =loga​(x).

Hence, the correct answer is option b. Logarithmic function

Given,

P=m x n

Q= n x p

For Multiplication of any two matrices 1st matrix of Column must equal to 2nd matrix of row so

the column of P matrix must be qual to row of Q matrix  and the result is row of P matrix and Column of Q matrix

so, PQ=m x p

Hence, the correct answer is option c. m x p

From the relationship between A.M, G.M and H.M is A.M>G.M>H.M

Hence, the correct answer is option b. A.M>G.M>H.M

Given,

C(n,4)=C(n,6)

so n=4+6=10

and we have to find the value of

C(n,2)

=C(10,2)

=10! / 2! (10-2)!

= (10 x 9 x 8!)/ (2! x 8!)

= 45

Hence, the correct answer is option a. 45

From option

Ө= sec^−1 (1/2) = cos^-1 (2)

which is not defined

Hence, the correct answer is option b. Ө= sec^−1 (1/2)

Given,

2sin2x = sin x

or, 2 x 2sinx cosx = sinx       (using sin2x=2sinx cosx)

or, 4sinx cosx = sinx

or, 4sinxcosx-sinx=0

sinx(4cosx-1)=0

either,                                       or,

sinx=0                                      cosx=1/4

x=n∏                                            x= 2n∏ ± cos^-1 (1/4)

Hence, the correct answer is option c. n∏

Given,

a=5, b=6, c=7

then,

CosC= (a^2 + b^2 -c^2) / (2ab)

= (5^2 + 6^2 - 7^2)/ (2 x 5 x 6)

=(25+36-49)/(60)

=1/5

Hence, the correct answer is option a. 1/5

Given,

line is 2x+3y+5=0  -----------(i)

If equation of a straight line parallel to the line A1x+B1y+C1=0 is A1x+B1y+K=0

then equation(i) becomes

2x+3y+k=0

Hence, the correct answer is option b. 2x+3y+k=0

Given,

Line is ax^2+2hxy+by^2+2gx+2fy+c=0

since, it intersect on y-axis, putting x=0

we get,

by^2 + 2fy +c=0

two line intersect at unique point on y-axis

i.e b^2 - 4ac=0

(2f)^2 - 4bc=0

f^2-bc=0

Hence, the correct answer is option a. f^2-bc=0

Given,

2x^2+2y^2+8x-2y-3=0

2( x^2+y^2+4x-y-3/2 )=0

x^2+y^2+4x-y-3/2 =0 comparing with x^2+y^2+2gx+2fy+c =0

then, 2gx=4x           2f= -1           c= -3/2

g=2                 f= -1/2

Using formula

Center(-g, -f)= (-2, 1/2)

Hence, the correct answer is option c. (-2, 1/2)

Given,

y=2x+λ comparing with y=mx+c

and parabola y^2=2x does not meet the parabola when

c>a/m

λ> 1/2x2

λ>1/4

Hence, the correct answer is option b. λ>1/4

Given,

eccentricity(e)=√2

if e>1 then it is hyperbola

so √2>1 it is hyperbola

Hence, the correct answer is option b. Hyperbola

x^3 x y^4 = (x+y)^4

if this type of questions asked then you have to search y/x in the option

Hence, the correct answer is option a. y/x

Given,

f(x)=x^4 – 6x^2 + 8x -1 ---------------(i)

for point of inflection f''(x)=0

then differentiate equation (i) with respect to x

f'(x)=4x^3 - 12x+8

f''(x)=12x^2 - 12

then,

f''(x)=0

or, 12x^2 - 12

or, 12(x^2 - 1)=0

x^2 - 1 =0

x = ±1

Hence, the correct answer is option a. x = ±1

Given,

x^2 + y^2 = 16 is comparing with  x^2 + y^2 = r^2

r^2 = 16

r=4

Using area of semi-circle= (∏ r^2)/2

= (∏ x 16)/2

=8∏

Hence, the correct answer is option c. 8∏

Given,

b⃗=(5,7)= 5i + 7j ,

a⃗−b⃗=(12,4)=12i + 4j

now adding both

b⃗ + (a⃗−b⃗)

= 5i + 7j  + 12i + 4j

=17i + 11j

(17,11)

Hence, the correct answer is option c. (17,11)

The cross product a⃗×b⃗    of two vectors a⃗   and b⃗   results in a vector that is perpendicular to both a⃗  and b⃗. This is a fundamental property of the cross product in vector algebra.

So, the vector a⃗×b⃗   is perpendicular to both a⃗ and b⃗.

Hence, the correct answer is option a. Perpendicular to both a⃗×b⃗

Given,

Total Bulbs= 1000 bulbs

Defective Bulbs P(A) = 250 / 1000 = 1/4

Non-defective Bulbs P(A¯ ) = 1- 1/ 4 = 3/4

Hence, the correct answer is option c. 3/4

The sentence "………cow give milk" is incorrect because "give" should be "gives" when referring to a singular noun. Therefore, none of the options (a, b, c) correctly complete the sentence. The correct sentence would be "A cow gives milk" using option a ("a"), but since the verb is not correctly conjugated in the question,

Hence, the correct answer is option d. none

From the option

a. Will be attending: This is the correct option. It uses the future continuous tense, which is appropriate for expressing an action that will be ongoing at a specific time in the future.

b. Will be having: While "having a party" is correct, the future continuous is not necessary here. It suggests the party will be in progress at the exact moment of 4 o'clock, which might not be the case.

c. Will attend: This uses the simple future tense, which is correct for general future actions but doesn't specify that the action will be in progress at a particular time.

d. Will attended: This is incorrect grammar. "Will" is used for future tense, not "will attended."

Hence, the correct answer is option a. Will be attending

"Must be" is used here to express a strong assumption or certainty about the identity of the person at the door. The speaker believes strongly that the person at the door is Pravin. "Might be" and "could be" suggest less certainty, and "would be" is not appropriate in this context.

Hence, the correct answer is option c. must be

"Among" is used to indicate that he is the best in English in the context of a group of people (us). "Between" is typically used for two entities, "over" and "with" do not fit the context.

Hence, the correct answer is option d. Among

"On" is the correct preposition to use with specific dates. Therefore, "World Environment Day is observed on the 5th of June" is the correct sentence.

Hence, the correct answer is option b. on

When using "neither...nor" with subjects that differ in number, the verb agrees with the subject closest to it. Here, "they" is closest to the verb, and "they" requires "are." So, "Neither he nor they are going on the picnic" is correct.

Hence, the correct answer is option c. are

This sentence should be in the present perfect tense to indicate an action that started in the past and continues to the present: "He has known me for a long time." The correct passive form is "I am known to him for a long time," though it is not common to use this structure in English. The passive construction typically used would be "I have been known to him for a long time," but since it is not an option, we choose the closest one.

Hence, the correct answer is option a. I am known to him for a long time.

The statement "Water freezes at 0° degree Celsius" is a general scientific fact, and the simple present tense is used to express general truths or laws of nature.

Hence, the correct answer is option d. Water freezes at 0° degree Celsius.

The sentence "It is I am to be invited" is grammatically incorrect because it lacks proper structure and does not follow standard English grammar rules. The other sentences are grammatically correct, although (b) is less formal and less preferred compared to (c) and (d).

Hence, the correct answer is option a. It is I am to be invited.

The correct answer is "misfire." Prefix "miss" forms a word that means to fail to fire properly or to fail to achieve the intended effect. The other prefixes (mono, micro, mind) do not combine with "fire" to form standard English words.

Hence, the correct answer is option a. miss

The correct conjunction to use with "both" is "and." The phrase "Both Mary and Ellen are going on tour" is grammatically correct. "Both... and..." is a correlative conjunction pair used to link two elements equally.

Hence, the correct answer is option b. and

The correct term for someone who kills oneself is "suicide," but since the closest option given is "selfcide," it would be the best fit, although it's not a standard term in English. The other options are incorrect:

"Murder" is the killing of another person.

"Senicide" is the killing of elderly people.

"Matricide" is the killing of one's mother.

Hence, the correct answer is option a. Selfcide

Anemia is a condition characterized by a lack of healthy red blood cells or hemoglobin, which leads to insufficient oxygen being carried to the body's tissues. Thus, "Lack of blood" is the correct cause of anemia.

Hence, the correct answer is option b. Lack of blood

The antonym of "granted permission" is "refused." If someone is not granted permission, they are refused. The other options are not antonyms:

"Confined" means restricted.

"Allowed" is synonymous with granted permission.

"Prolonged" means extended.

Hence, the correct answer is option a. Refused

"Bawdy" refers to something that is humorously indecent or vulgar. Therefore, the synonym is "vulgar." The other options are not synonyms:

"Humble" means modest.

"Swerve" means to turn aside sharply.

"Soft" means gentle or not hard.

Hence, the correct answer is option a. Vulgar

HTML (HyperText Markup Language) is used to structure and design the content of web pages. It defines elements like headings, paragraphs, links, and other content.

Hence, the correct answer is option a. Designing the Web Page

All the options listed (Worms, Trojans, and I Love You) are types of malicious software. Worms and Trojans are different kinds of malware, and "I Love You" refers to a specific email worm that spread rapidly in 2000.

Hence, the correct answer is option d. all

Non-volatile memory retains its data even when the power is turned off. ROM (Read-Only Memory) is a type of non-volatile memory used to store firmware or software that is rarely changed. RAM (Random Access Memory) is volatile, meaning it loses its data when power is lost. Cache memory and video memory are also types of volatile memory.

Hence, the correct answer is option b. ROM

Analog computers process continuous data and work with physical quantities that vary smoothly. They are used to measure and manipulate continuous variables, like voltage or mechanical movements.

Hence, the correct answer is option c. Analog Computer

Optical Mark Readers (OMR) are used to check and read answers on standardized test answer sheets. They detect marks made on paper, such as those in multiple-choice tests.

Hence, the correct answer is option b. Optical Mark Readers (OMR)

Windows, Linux, and Unix are all examples of operating systems. An operating system manages hardware and software resources on a computer.

Hence, the correct answer is option d. All

Pressing Ctrl+A selects all the content in the current document or field, allowing users to quickly highlight everything.

Hence, the correct answer is option c. Select all

The wheel on a mouse is typically used for scrolling through documents or web pages, allowing users to move up or down easily.

Hence, the correct answer is option a. Scrolling

Microsoft PowerPoint is a software application used to create and deliver presentations, including slideshows and multimedia content.

Hence, the correct answer is option d. Presentation

Worldlink and CG net are Internet Service Providers (ISPs) that offer internet connectivity services. The World Wide Web (WWW) is not an ISP; it is a system of interlinked hypertext documents accessed via the internet.

Hence, the correct answer is option d. both a and b