Units & Measurement - 3 - Free MCQ Practice Test with solutions, JEE Chapter

Fundamental quantities are those which cannot be defined in terms of other quantities.

• Length and time are independent and can be considered fundamental.
• Velocity depends on both length and time, so it cannot be fundamental.
• Thus, length, time, and mass are suitable as fundamental quantities.

Velocity cannot be a fundamental quantity because it is derived from length and time.

Parallactic second is the unit of distance because parallactic second is an abbreviation of parsec. Parsec = Parsec is the Unit for larger distances.It is the distance at which a star would make parallax of one Second of arc.

The newton second (also newton-second, symbol N s or N.s) is the derived SI unit of impulse. It is dimensionally equivalent to the momentum unit kilogram metre per second (kg.m/s). One newton second corresponds to a one-newton force applied for one second.

F = ma
[F] = kg m/s²

Unit of velocity = m/s
1m = 10⁶  Micro meter
And 1 sec = 10⁶  Micro sec
Thus 1 m/s = 10⁶ 10⁶  micro meter /micro second

Dimension of KE is ML²T^-2
That of pressure is ML^-2T^-2
That of momentum is ML/T 
That of power is ML²T^-3

A dimensionless quantity can have unit. for example angle (radian). But oppisite is not true. A unitless quantity can never have dimensions.it is the unit that give dimensions.

Quantities with different dimensions cant be added or subtracted. Also the dimension of the product of the quantities in power is always one. But when two quantities are multiplied the dimension of the product is the product of the dimensions of the initial quantities.

The two physical quantities which do not have the same dimensions can’t be added or subtracted in the same expression because doing so would lead to equating two quantities of different dimensions, which is non possible. Ex [F]  [P].

Correct Answer :- d

Explanation : a) A topological sort of a directed acyclic graph (DAG) is any ordering m1, m2, …, mn of the nodes

of the graph, such that if mimj is an edge then mi appears before mj . Any topological sort of a

the dependency graph gives a valid evaluation order for the semantic rules. 

b) The parse tree can be annotated with synthesized or inherited attributes. The parse tree can also be indicated with an arrow mark to indicate the manner in which the value gets propagated between the nodes of the parse tree. This graph is called as dependency graph as it indicates the dependency between nodes for deriving the values. This graph is an acyclic graph which doesn’t have a cycle. The presence of a cycle indicates that the graph is incorrect as the dependence of nodes for deriving values cannot be predicted. Edges in the dependence graph show the evaluation order for attribute values and thus the graph is a directed one.

We know that E = hv , where E is energy and v is frequency.
Thus we get h = E/v
And [h] = [E/v] = ML²T^-2 / T^-1
= ML²T^-1
[P] = MLT^-1
[F] = MLT^-2
[Angular momentum] = [P x r] = ML²T^-1

Electron volt (eV) is a unit of measurement used primarily in the fields of physics and electronics. It quantifies the energy gained by an electron when it is accelerated through a potential difference of one volt.

• One electron volt is equivalent to approximately 1.6 x 10^-19 joules.
• It is commonly used to express the energy of particles at the atomic and subatomic levels.
• In practical applications, eV is often used in semiconductors and in the study of atomic interactions.
• Understanding eV is essential for grasping concepts in quantum mechanics and nuclear physics.

The vander Waals gas equation is 

where P is the pressure, V is molar volume and T is the temperature of the given sample of gas. R is called mola gas constant, a and b are called vander Waals constants.

Moment of inertia =1/2​×mass×(Radius of gyration)²=[M¹L²T⁰]
Moment of force = torque =N.m=[M¹L²T⁻²]
Both are different from each other.

Volume=4πR³/3
ΔV/V×100=3∆R/R×100
=3×1/100​×100
=3
∴ Percentage error in volume is 3%

Time period of oscillation of pendulam 

where L is length of pendulam and g is acceleration due to gravity.
hence we have  ...........................(1)
hence 

We know that any quantity in power as a dimension of 1, thus [at²] = 1
Thus we get [a] = T^-2

Dimension of [A]=[MLT^-2]
Dimension of [B]=[MLT^-2]
 
 [A]/[B]= [ M⁰L⁰T⁰]
CT=1
Dimension of  [C]=1/[T]=[T^-1]
DL=1
Dimension of  [D]=1/[L]=[L^-1]
[C][D] =[T^-1]/[L^-1]
=LT^-1
 
A/B=Force/Force=[M⁰L⁰T⁰]
Ct=∠⇒C=Angle/Time=1/T=T^-1
Dx=∠⇒D=Angle/Distance=1/L=L^-1
∴C/D=T^-1L^-1 =[M0 LT^-1 ]

The image provides a solution for finding the dimensions of a in the Van der Waals equation for 1 mole of a real gas:
The Van der Waals equation is:

Step-by-step solution:
To find the dimensions of a, we analyze the term a/V², which has the same dimensions as pressure P.
The dimensional formula of pressure P is ML⁻¹T⁻².
The dimensional formula of V² is L⁶, so:

Now, we calculate the dimensions of a:

Check the options:
The dimensions of a match those of PV²:

Thus, the correct answer is B.

The dimensional formula of the coefficient of viscosity is derived from the relationship between force, area, and velocity. It represents how a fluid resists flow and deformation.

• The coefficient of viscosity is a measure of a fluid's internal friction.
• It can be expressed in terms of dimensions:
• Mass (M): Represents the quantity of matter.
• Length (L): Indicates the distance or space.
• Time (T): Refers to the duration of an event.

The dimensional formula can be derived as follows:

• Viscosity relates shear stress to shear rate.
• Shear stress has dimensions of force per unit area, which is ML^-1T^-2.
• Shear rate has dimensions of velocity, which is LT^-1.
• Thus, the dimensional formula for viscosity is:

[ML^-1T^-1]

According to the principle of homogeneity of dimension, an equation is dimensionally correct when each term has same dimension on both sides of the equation. 

Since left side has dimension of force so the term Pt⁻¹ will have also dimension of force. 

Thus, [P][T⁻¹]=[F]=[MLT⁻²]

or [P]=[MLT⁻¹]

We know that the dimension of momentum is [p]=[MLT⁻¹]

Both are different from each other.
Energy × Time=(M¹L²T⁻²)×(T¹)=M¹L²T⁻¹
Force × Velocity=(M¹L¹T⁻²)×(L¹T⁻¹)=M¹L²T⁻³
Impulse × distance=(M¹L¹T⁻¹)×(L¹)=M¹L²T⁻¹
Power=M¹L²T⁻³
Angular Energy=M¹L¹T⁻²

Argument of sin is dimensionless and using principle of dimensional homogeneity, we get
∣bt∣=∣cx∣=[M₀L₀T₀]
Thus, ∣b∣=T−1 and ∣c∣=[L⁻¹]
Thus, dimensions of b/c are [LT⁻¹], which is the same as the velocity of wave.

• Convert masses to the same unit:

  • The mass of the bag = 1.6 kg.
  • The masses of the two objects are given in grams: 20 g and 15 g.
  • Convert grams to kilograms:
    20 g = 0.020 kg,  15 g = 0.015 kg

• Add the masses:

  Total mass = 1.6 kg +0 .020 kg + 0.015 kg = 1.635 kg

• Apply the rule for significant figures: The limiting factor in this calculation is the least precise measurement, which is the mass of the bag (1.6 kg) with 2 significant figures. Hence, the final answer should also be rounded to 2 significant figures.

• Round the result:

  • 1.635 kg rounded to 2 significant figures is 1.6 kg.