Given,

𝑥 =𝜋/3(𝑎^2 − 𝑏^2)ℎ where $a$, $b$, and $h$ are lengths. Then quantity $x$ is:

The given formula has $a$, $b$, and h as lengths. When calculating a^2 – b^2, you get an area. Multiplying by another length h gives a volume. Thus, $x$ is a volume.

Hence, the correct answer is option d. volume

When a vector is multiplied by -2, its direction reverses (because of the negative sign) and its magnitude is doubled (because of the factor 2).

Hence, the correct answer is option b. direction reverses and magnitude is doubled

Given,

height(h)=45 m

Acceleration due to gravity(g)=$9.8$ m/s²

Use the equations of motion. The total time of fall $t$ can be found using

h=1/2gt^2

or, 45=1/2​×9.8×t^2

t≈3 seconds

The distance travelled in the last second can be calculated using

s=ut+1/2​at^2

For the first 2 seconds:

s1=1/2×9.8×2^2=19.6 m

Therefore, the distance travelled in the last second (from 2 to 3 seconds)

s=45−19.6=25.4 m

Since the closest option is 25 m:

Hence, the correct answer is option c. 25m 

At the maximum height, the vertical component of velocity is zero, and only horizontal velocity remains.

The momentum of the system remains conserved in the absence of external forces. However, breaking the projectile into two pieces does not change the total momentum but can change the kinetic energy distribution.

Hence, the correct answer is option d. kinetic energy decreases

A jet plane operates on the principle of conservation of linear momentum. It expels gas backward, and the forward motion of the plane is a reaction to this action.

Hence, the correct answer is option c. linear momentum

Viscosity is a measure of a fluid’s resistance to flow. Higher viscosity means the fluid flows more slowly.

Let’s see from the option

a. Water: Water has low viscosity, meaning it flows easily.

b. Blood: Blood has higher viscosity than water due to the presence of cells and plasma proteins, but it’s still relatively low compared to thicker substances.

c. Glycerine: Glycerine has a much higher viscosity than water and blood. It is a thick, syrupy liquid that flows slowly.

d. Honey: Honey has a very high viscosity. It is thick and flows very slowly, much more so than water, blood, or even glycerine.

Hence, the correct answer is option d. honey

We know the formula for kinetic energy is:

KE=1/2 m v^2

where $m$ is the mass of the body and $v$ is its velocity. From this equation, we can see that kinetic energy is directly proportional to the square of the velocity.

Let’s see from the option

Graph (a): This graph shows a linear relationship between KE and V, which is incorrect because KE is proportional to v^2, not $v$.

Graph (b): This graph shows a curve that looks like it could represent a relationship where KE increases more rapidly than V, but it is not clearly parabolic and seems more like an exponential curve, which is not correct.

Graph (c): This graph shows a parabolic relationship, which matches the v^2 dependency in the kinetic energy equation.

Graph (d): This graph shows a constant KE irrespective of the velocity, which is incorrect because KE depends on v^2.

Hence, the correct answer is option c.

Initial Configuration:

In the initial configuration, two identical springs with spring constants $k$ are in parallel. The effective spring constant keff is:

keff​=k+k=2k

The frequency $f$ of oscillation for a mass $m$ on a spring with spring constant keff​ is given by:

f=(1/2π) x √(keff/m)

Substituting keff=2k:

​f=(1/2π) x √(2k/m)

New Configuration:

When one of the springs is removed, only one spring with spring constant k remains. The new frequency fnew​ is given by:

​fnew=(1/2π) x √(k/m)

Finding the Relationship:

To find the relationship between the new frequency fnew and the original frequency f, we can compare the two frequencies:

f=(1/2π) x √(2k/m)

fnew=(1/2π) x √(k/m)

Thus, we can write:

fnew=(1/2π) x √(k/m) = (1/2π) x √(k x 2/2m) = 1 x f/√2

fnew​=f÷√2 = f/√2

Since the frequency will be affected by a factor of √2​ when comparing the two configurations, the correct answer should be:f/√2

Hence, the correct answer is option c. f/√2

When a metal sheet is heated, it expands uniformly in all directions. This includes the material around the hole. As the entire sheet expands, the material surrounding the hole also expands, causing the hole itself to increase in size.

Hence, the correct answer is option a. increases

The mean free path (λ) of a gas molecule is the average distance a molecule travels between collisions. The mean free path is inversely proportional to the pressure of the gas. This relationship can be expressed as:

λ∝1/p

Therefore, if the pressure increases, the mean free path decreases, and if the pressure decreases, the mean free path increases.

Hence, the correct answer is option b. p^−1

Hence, the correct answer is option d. 43°C

Given,

wavelength of the light($\lambda )=550$ nm = 550×10^−9m,

slit width(a)=22×10^−7 m,

For the second minimum(n)=2,

The condition for minima in single-slit diffraction is given by:

asin⁡θ=nλ

or, 22×10^−7sinθ=2×550×10^−9

or, sinθ={(2×550×10^−9)/(22×10^−7)

or, sinθ={(1100×10^−9)/(22×10^−7)

or, sinθ=0.5

θ=arcsin(0.5)=30°

Hence, the correct answer is option c. 30°

To determine the relationship between the temperatures T1​, T2​, and T3​ based on the intensity vs. wavelength graph, we use Wien’s Displacement Law. This law states that the wavelength at which the intensity of radiation is maximum (peak wavelength) is inversely proportional to the temperature of the black body. Mathematically:

λmax​T=b

where λmax​ is the peak wavelength, $T$ is the temperature, and $b$ is Wien’s constant.

From the given graph

T1​ has its peak intensity at the shortest wavelength.

T2 has its peak intensity at a longer wavelength than T1​ but shorter than T3​.

T3 has its peak intensity at the longest wavelength.

From Wien’s Displacement Law

Since peak wavelength λmax​ is inversely proportional to the temperature $T$:

A shorter peak wavelength corresponds to a higher temperature.

A longer peak wavelength corresponds to a lower temperature.

From the graph:

λmax,1​<λmax,3​<λmax,2​

Therefore:

T1>T3>T2

Hence, the correct answer is option b. T1>T3>T2​

Refractive Index

The refractive index μ of a material is defined as the ratio of the speed of light in a vacuum (c) to the speed of light in the material ($v$):

μ=c​/v

Visibility in Vacuum

A material will be invisible in vacuum if light passes through it without any refraction or reflection. For this to happen, the speed of light in the material must be the same as in the vacuum. This condition implies that the refractive index of the material must be equal to 1.

Explanation

When μ=1: The speed of light in the material is the same as in the vacuum. There will be no change in the direction of light (no refraction) as it enters or exits the material. Consequently, the material will be invisible because light will not bend or reflect at its boundaries.

When μ>1: The speed of light in the material is less than in the vacuum. Light will bend towards the normal when entering the material, making the material visible due to the change in direction (refraction).

When μ<1: This scenario is not physically realistic for transparent materials because it would imply the speed of light in the material is greater than in a vacuum, which contradicts the principles of relativity.

When μ=∞: This implies the speed of light in the material is zero, which is also not physically realistic for a transparent material.

Hence, the correct answer is option a. unity

To determine the correct graph representing the variation of object distance (u) and image distance ($v$) for a convex mirror, we need to refer to the mirror equation and understand the nature of images formed by a convex mirror.

Mirror Equation

The mirror equation is given by:

1/f​=1/v​+1/u

For a convex mirror:

The focal length ($f$) is positive.

The object distance (u) is taken as negative (since the object is placed in front of the mirror).

The image distance ($v$) is positive (since the image formed by a convex mirror is virtual and appears to be behind the mirror).

​Image Characteristics of a Convex Mirror

The image formed by a convex mirror is always virtual, erect, and diminished.

The image distance ($v$) is always less than the focal length ($f$) and hence always positive but less than $f$.

As the object distance ($u$) increases (i.e., the object moves further away from the mirror), the image distance ($v$) decreases and approaches the focal length asymptotically.

From the given option

Graph (a): This graph shows $v$ increasing asymptotically towards a maximum value as $u$ increases. This is consistent with the behaviour of a convex mirror where $v$ approaches the focal length but never exceeds it.

Graph (b): This is a linear relationship, which is not consistent with the mirror equation for a convex mirror.

Graph (c): This graph shows v decreasing rapidly initially and then levelling off, which is also not consistent with the behaviour of a convex mirror.

Graph (d): This graph shows $v$ decreasing linearly with u, which is not consistent with the mirror equation for a convex mirror.

Hence, the correct answer is option a

The “sweetness” of sound, often referred to as the timbre or quality of sound, depends on the periodicity and regularity of the sound waves.

Let’s see from the option

a. Wavelength: This primarily affects the pitch of the sound.

b. Periodicity: This refers to the regular repeating pattern of the sound waves.

c. Periodicity and regularity: This combination affects the timbre of the sound. A sound that is periodic and regular is often perceived as more pleasant or “sweeter”.

d. Amplitude: This affects the loudness of the sound.

Hence, the correct answer is option c. periodicity and regularity

Consider a parallel plate capacitor that is charged and then disconnected from the battery. When the plates are moved further apart using insulating handles, several properties of the capacitor change:

Let’s see from the option

Charge on the Capacitor: The charge remains constant because the capacitor is isolated once the battery is disconnected.

Voltage across the Plates: As the plates are moved apart, the capacitance decreases (since capacitance C∝1/d​, where $d$ is the distance between the plates). For a given charge, the voltage $V$ is given by V=Q/C​. Since C decreases, V increases.

Capacitance: The capacitance decreases as the distance between the plates increases.

Energy Stored: The energy stored in the capacitor is given by U=1x Q^2/2C​. As $C$ decreases, the energy stored increases.

Hence, the correct answer is option b. The voltage across the plate increases

Identify Parallel and Series Combinations:

The 10Ω and 20Ωresistors in the middle are in parallel.

The equivalent resistance of these two resistors in parallel can be calculated using the formula:

1/Req1​​=(1/10Ω​) + (1/20Ω)

​or, 1/Req1​​=(2/20Ω) ​+ (1/20Ω) = 3/20Ω 

Req1​=20Ω/3​≈6.67Ω

Combine the Resultant Resistance with Series Resistors:

The equivalent resistance Req1≈6.67Ω is in series with the two 10Ω resistors on either side of it.

Thus, the total resistance in the top branch is:

Rtop​=10Ω+6.67Ω+10Ω=26.67Ω

Identify Parallel Combination with the Bottom 10Ω Resistor:

The top branch (26.67Ω) is in parallel with the bottom 10Ω resistor.

The equivalent resistance of these two resistors in parallel can be calculated using the formula:

1/Req2​​=(1/26.67Ω)​ + (1/10Ω)

1/Req2​​=(0.0375/1Ω)​ + (0.1/1Ω​) =0.1375

Req2​=1Ω/0.1375 ​≈ 7.27Ω

The equivalent resistance between points $X$ and Y is approximately:10Ω

Hence, the correct answer is option a.10Ω

A magnetic needle in a non-uniform magnetic field experiences both a force and a torque.

Torque: A magnetic needle experiences torque in any magnetic field, which tends to align it with the field lines.

Force: In a non-uniform magnetic field, there is also a net force on the magnetic needle because different parts of the needle experience different field strengths.

Hence, the correct answer is option c. Force and a torque

Given:

Source voltage (Vs) = 34 V

Frequency (f) = 50 Hz

Inductance of the coil ($L$) = 17 mH = 0.017 H

Potential difference across the resistor (VR) = 10 V

We need to find the potential difference across the coil (VL​).

Impedance of the Resistor (R):Potential difference across the resistor (VR​) = 10 V

Reactance of the Coil (XL​):

Reactance XL=2πfL

XL=2π×50×0.017

XL​=2×3.14×50×0.017 

XL​≈5.34Ω

Calculate Current (I):The total voltage in the circuit is the vector sum of the voltages across the resistor and the inductor.

Using Ohm’s Law for the resistor: I=VR/R

I=10/R

Potential Difference across the Coil (VL​):The voltage across the inductor is VL=I×XL​

Total Voltage: The total voltage (Vs​) is the root of the sum of the squares of the voltages across the resistor and the inductor.

Vs^2​=VR^2​+VL^2

34^2=10^2+VL^2

VL^2​=1056​​

VL​=√1056​≈32.49V

Hence, the correct answer is option d. 32V

The strong argument for the particle nature of cathode rays is that they get deflected by electric and magnetic fields. This behaviour indicates that cathode rays are composed of charged particles, as only particles with charge would experience such deflection

Hence, the correct answer is option c. Get deflected by electric and magnetic

When objects placed in a room are exposed to X-rays, they appear invisible. X-rays can penetrate through many materials and are not visible to the human eye, hence the objects exposed to X-rays do not become visible in any specific color.

Hence, the correct answer is option a. Invisible

The series of the hydrogen atom that lies in the infrared region is the Pfund series. Each series corresponds to electron transitions between specific energy levels. The Pfund series involves transitions where the final energy level is n=5.

Hence, the correct answer is option c. Pfund

Hence, the correct answer is option b. 7

In a P-type semiconductor, the majority charge carriers are holes, and the minority charge carriers are electrons. This is because P-type semiconductors are created by doping the material with an element that has fewer valence electrons than the semiconductor, resulting in the creation of “holes” where an electron is missing.

Hence, the correct answer is option d. Holes and electrons

Molar mass of Na2CO3=2×23+12+3×16=106 g/mol

0.1 N solution means 0.1 equivalents per liter.

To find the amount needed for 100 mL:

Equivalents=Normality × Volume (L)=0.1×0.1=0.01 equivalents

Since 1 mole of Na2CO3 provides 2 equivalents, 0.01 equivalents corresponds to 0.005 moles

Weight of 0.005 moles of Na2CO3​:

Weight=0.005 moles×106 g/mol=0.53 g

Hence, the correct answer is option a. 0.53 g

The concept of pH was introduced by Soren Pedder Lauritz Sorensen in 1909.

Hence, the correct answer is option d. Sorensen

The rate of a chemical reaction increases with an increase in temperature. Higher temperatures provide reactant molecules with more kinetic energy, leading to more frequent and energetic collisions, thus increasing the reaction rate.

Hence, the correct answer is option b. increase in temperature

A spontaneous change is one in which the system has a lowering of free energy. In thermodynamics, spontaneous processes occur naturally and are associated with a decrease in Gibbs free energy (ΔG<0).

Hence, the correct answer is option c. lowering of free energy

Cobalt (Co) is an example of a transition metal. Transition metals are found in the d-block of the periodic table and have partially filled d-orbitals.

Hence, the correct answer is option a. cobalt

Blue vitriol is the common name for copper(II) sulphate pentahydrate, which is CuSO4⋅5H2O. It is known for its bright blue color.

Hence, the correct answer is option b. CuSO4⋅5H2O

Silver is extracted from its sulphide ore using the cyanide process. In this process, silver sulphide is treated with sodium cyanide solution, forming a soluble silver cyanide complex, from which silver is then recovered.

Hence, the correct answer is option c. cyanide process

2,4-Dichlorophenoxyacetic acid (2,4-D) is commonly used as a herbicide. It is a synthetic auxin, which is a type of plant hormone used to kill broadleaf weeds.

Hence, the correct answer is option a. a herbicide

The class of organometallic compounds of magnesium with the general formula $\text{RMgX}$ (where $R$ is an organic group and $X$ is a halogen atom) is known as Grignard reagents. They are widely used in organic synthesis.

Hence, the correct answer is option d. Grignard’s reagent

Aldehydes can be distinguished from ketones using Fehling’s solution. Aldehydes reduce Fehling’s solution, changing its color from blue to a brick-red precipitate, while ketones do not react.

Hence, the correct answer is option a. Fehling’s solution

Picric acid, also known as 2,4,6-trinitrophenol, is obtained by the nitration of phenol. This process involves treating phenol with nitric acid, resulting in the formation of picric acid.

Hence, the correct answer is option b. phenol

Chloroform (CHCl3​) is historically known for its use as an anesthetic. It was widely used in surgeries in the 19th and early 20th centuries until safer alternatives were found.

Hence, the correct answer is option c. chloroform

Cinnamic acid (C9H8O2​) is an aromatic carboxylic acid. It contains a benzene ring and a carboxyl group, fitting the definition of an aromatic carboxylic acid.

Hence, the correct answer is option d. cinnamic acid

Urea is NH2CONH2. It is a compound that contains two amine groups (NH2) connected by a carbonyl functional group (C=O).

Hence, the correct answer is option a. NH2CONH2​

Ethylene on reaction with alkaline $KMn{O}_4$ gives ethylene glycol.

H2C=CH2 $+H_{2}O+\; [O]$$\to {HO-C{H}_2-C{H}_2-OH}^{}$

Hence, the correct answer is option a.. ethylene glycol

Acetaldehyde is the common name, and its IUPAC name is ethanal

Hence, the correct answer is option c. ethanal

Methylmercury (CH3Hg^+) is the most toxic form of mercury, particularly due to its ability to bioaccumulate and bio magnify in food chains.

Hence, the correct answer is option d. CH3Hg^+

32 grams of O₂ corresponds to 1 mole of oxygen molecules. One mole of any substance contains Avogadro’s number of molecules, which is 6.02 × 10^23.

Hence, the correct answer is option a. 6.02 × 10^23

Chromium has an exception to the expected configuration due to stability considerations. Its actual configuration is [Ar] 3d^5 4s^1 rather than [Ar] 3d^4 4s^2.

Hence, the correct answer is option b. 3d^5 4s^1

Among the given options, Al^3+ has the lowest ionic radius because it has the highest positive charge, which causes a greater pull on the electrons, leading to a smaller radius.

Hence, the correct answer is option c. Al^3+

The dipole moment of a molecule depends on both the difference in electronegativity between the atoms and the distance between them. As the size of the halogen increases, the bond length increases and the dipole moment decreases. Iodine is the largest halogen, leading to the lowest dipole moment for HI.

Hence, the correct answer is option d. HI

The universal gas constant R is 8.314 J K⁻¹ mol⁻¹ in SI units, which is used in the ideal gas law and other thermodynamic equations.

Hence, the correct answer is option a. 8.314 J K⁻¹ mol⁻¹

Among the options given, HCl (hydrochloric acid) is an example of an acidic oxide. The other options are halogen acids (HF, HBr, HI) and do not represent acidic oxides.

Hence, the correct answer is option a. HCl

The ionic character of the H-X bond is highest in HF because fluorine has the highest electronegativity among the halogens, which leads to a greater ionic character in the H-F bond.

Hence, the correct answer is option c. HF

Magnesia refers to magnesium hydroxide (Mg(OH)₂). The other options are either hydrated magnesium sulfate (MgSO₄·7H₂O) or magnesium chloride (MgCl₂).

Hence, the correct answer is option d. Mg(OH)₂

The word “headland” is stressed on the first syllable: HEAD-land.

Hence, the correct answer is option a. First syllable

A headland is a narrow piece of land that projects from a coastline into the sea, similar to a promontory.

Hence, the correct answer is option c. promontory

“Remarkable” and “notable” both refer to something worthy of attention or notice. While “astonishing” can also be close, it more specifically refers to something that causes astonishment or surprise.

Hence, the correct answer is option d. notable

“Pompous” refers to someone who is grandiose or self-important, while “humble” is the opposite, meaning modest or unassuming.

Hence, the correct answer is option d. humble

“Reciprocal” refers to something that involves mutual action or give-and-take, whereas “unilateral” refers to something that is one-sided or done by one party without the involvement of others.

Hence, the correct answer is option b. unilateral

Orthography refers to the conventional spelling system of a language.

Hence, the correct answer is option a. orthography

“Abhor” is a stronger term than “loathe,” and “hate” is the least strong among the three. The sequence should progress from the strongest to the least strong emotion, making the correct order “abhor, loathe, hate.”

Hence, the correct answer is option d. abhor, loathe, hate

Pantheism is the belief that the universe and nature are equivalent to God, or that God is inherent in all of nature.

Hence, the correct answer is option c. pantheism

The correct prepositions are “on” and “in.” She lived on banana and milk (subsisting on them) as she believed in the prescription of her doctor.

Hence, the correct answer is option b. on, in

The correct prepositions are “of,” “in,” and “on.” The expression should be “I am scared of loss in my business but keen on gambling.”

Hence, the correct answer is option d. of, in, on

“Reared on” means to be brought up with or to be raised with something, in this case, stories of the sea.

Hence, the correct answer is option a. reared on

“Break up” is the correct phrasal verb used to indicate the end of a term or school session.

Hence, the correct answer is option b. break up

“Bright and breezy” means being cheerful and energetic, ready to tackle tasks with enthusiasm.

Hence, the correct answer is option c. in a cheerful mood to do things quickly

“Go by the board” means that plans or proposals have been discarded or abandoned.

Hence, the correct answer is option a. plans discontinued and abandoned

The correct conditional expression is “whether.” The sentence should read: “You must go tomorrow whether you are ready or not.” This indicates that the action must take place regardless of readiness.

Hence, the correct answer is option d. whether

MS-Word is a word processing application software used to create and edit documents.

Hence, the correct answer is option c. Application software

An operating system is essential for booting a computer as it manages hardware and software resources and provides common services for computer programs.

Hence, the correct answer is option c. Operating system

The default print mode for documents in most word processing software, including MS-Word, is portrait mode, where the document is printed vertically.

Hence, the correct answer is option b. Portrait

Ctrl, Shift, and Alt are known as modifier keys because they modify the function of other keys when used in combination with them.

Hence, the correct answer is option a. modifier

Non-volatile storage retains data even when the power is turned off. Examples include hard drives, SSDs, and flash memory.

Hence, the correct answer is option b. Non-volatile storage

To connect to the internet, you need a telephone line (or another form of connection), a modem to convert digital signals to analog and vice versa, a computer to access the internet, and an Internet Service Provider (ISP) to provide internet access.

Hence, the correct answer is option a. telephone line, modem, computer, and an ISP

Booting is the process of starting up a computer and loading the operating system into the computer’s RAM.

Hence, the correct answer is option a. Booting

The gutter is the space added to the margin to allow for binding. This ensures that text is not obscured by the binding process.

Hence, the correct answer is option d. the space left on one side for stitching and binding

Firmware refers to the permanent software programmed into a read-only memory (ROM). It provides low-level control for a device’s hardware.

Hence, the correct answer is option d. Firmware

Statement A is correct: MS-Excel creates workbooks, and each workbook may contain multiple worksheets.

Statement B is correct: A worksheet in MS-Excel 2000 has a maximum of 256 columns and 65536 rows.

Statement D is correct: Cells in MS-Excel are identified by the column letter and the row number (e.g., A1, B2).

Statement C is incorrect: Columns are identified by letters and rows are identified by numbers.

Hence, the correct answer is option d. A,B,D

It is a statement because, it is either true or false

Hence, the correct answer is option d. a statement

𝑃 ⊂ Q

by the properties of subsets;

A ⊂ B = A¯ ⊂ B¯

so, 𝑃 ⊂ Q= 𝑄¯ ⊂ P¯

Hence, the correct answer is option b. 𝑄¯ ⊂ P¯

Given,

|𝑥 − 3| = 7

As we know, modulus represents the -ve & +ve sign

so,

taking +ve sign                                     taking -ve sign

(x-3)=7                                                    -(x-3)=7

x=7+3                                                    -x+3=7

x=10                                                        -x=7-3

x=-4

x={10,-4}

Hence, the correct answer is option a. {10,-4}

√(−1)

We know that there won’t be negative sign under root

so √i^2  (i^2= -1)

= i which is complex number

Hence the correct answer is option d. a complex number

Range

𝑓(x)=√(9 − 𝑥^2)          For Shortcut trick

=> √9 =3                    Domain [-3,3,] & Range [0,3]

OR,

Let y=𝑓(x)=√(9 − 𝑥^2)

squaring on both sides

y^2 = 9-x^2

y^2 ≥0

x^2=9-y^2

x=√(9-y^2)

x^2 ≥0

As we know,

y≥0

So, Range [0,3]

Hence, the correct answer is option b. [0,3]