JEE Main Mock Test - 3 - JEE MCQ

 Q = 40μC

Length , l = 4,234 m
Breadth, b = 1.005 m
Thickness , t = 2.01 x 10^-2 m
Volume = l x b x t
⇒ V = 4.234 x 1.005 x 0.0201 = 0.0855289 = 0.0855 m³ (significant figure = 3)

157900  implies that the population is believed to be within the range of about 157850 to about 157950.

In other words, the population is 157900 ± 50. The “plus-or-minus 50” appended to this number means that we consider the absolute uncertainty of the population measurement to be 50 – (–50) = 100.

We can also say that the relative uncertainty is 100/157900, which we can also express as 1 part in 1579, or 1/1579 = 0.000633, or about 0.06 percent.

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

The answer is PV²
Solution,
As we know the Vander Waals equation is {P+(a/V²)} (V−b) =RT
Then,
The dimension of a is,
(a/V²) = (P)
Or, a=PV²

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 ]

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]

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.

We define length and time separately as it is not possible to define velocity without using these quantities. This means that one fundamental quantity depends on the other. So, these quantities cannot be listed as fundamental quantities in any system of units.

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.

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

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.

The Measurement of a given quantity is essentially an act or result of comparison between a quantity whose magnitude (amount) is unknown, with a similar quantity whose magnitude (amount) is known, the latter quantity being called a Standard.

Resistance after temperature increases by 500°C = 220/2 = 110 Ω
The variation of the resistance with temperature is given by:
R₂ = R₁(1 + α(T₂ - T₁))
110 = 100(1 + α × 500)

Doppler effect in light (speed of observer is not very small compared to the speed of light)

Due to thermal expansion,
Change in the volume is:

Due to external pressure, volumetric strain,

From 1 and 2, we have

Hence, change in temperature is given by:

Let's consider the equation for radioactive decay.

Based on the data given in the figure,

Energy of photon = 12500/980 = 12.75 eV

∴ Electron will excite to n = 4

Since 'R' ∝ n²

∴ Radius of atom will be 16a₀.

For adiabatic process : TV^γ-1 = constant

For diatomic process : γ-1 = 7/5 - 1

∴ x = 2/5

Here, the temperature inside the refrigerator can be provided as, T₁ = 9^oC = 282 K
Room temperature is given as, T₂ = 36^oC = 309 K
Coefficient of performance can be given by the relation, COP = T₁/(T₂ - T₁)

Given, points P and Q are on the ellipse defined by 9x² + 4y² = 36 which simplifies to . This is the standard form of the equation of an ellipse centered at the origin, with semi-major axis a = 3 along the y-axis and semi-minor axis b = 2 along the x-axis.
OP is the distance from origin O to point P, which is given by :

1. Let's represent the given point  in polar coordinates. We can write . 

Simplifying this, we obtain :

2. Similarly, if we represent the point R as ,   (the negative sign is due to the fact that line RS is perpendicular to line PQ. Since they are perpendicular, the angle between them is 90 degrees or π/2  radians. In terms of sin and . 
it too should satisfy the equation of the ellipse. We have :

Simplifying this, we obtain :

3. From equations (1) and (2), we have :

4. Now, note that lines PQ and RS are perpendicular and pass through the origin, so PQ =2OP and RS = 2OR. Thus,

5. Substituting the values of r₁ and r₂, we obtain :

6. Hence, p= 13 and q = 144.

7. So, the final answer p + q = 13 + 144=157.

Given that S be the set of values of λ for which given system of equations has no solution.

Therefore for the given set of equations

Also for each values of  we have

which implies that, for each values of λ, the given system of equations has no solution.