Units Dimensions And Measurement — DPP 2

20 NEET questions from Physics and Measurement, with answers and explanations. Attempt it as a timed test to get scoring, accuracy breakdown and mistake tracking.

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1
If x = at + bt$^{2}$, where x is in metre and t is in hour(hr), then unit of b will be :-
  1. A $\mathrm { m } ^ { 2 } / \mathrm { hr }$
  2. B m
  3. C $\mathrm { m } / \mathrm { hr }$
  4. D $\mathrm { m } / \mathrm { hr } ^ { 2 }$
Show answer & explanation

Correct answer: D

x = at + bt$^{2}$
[x] = [at] = [bt$^{2}$]
b = $\frac{x}{t^{2}}$
m/hr$^{2}$
2
Given the force $\mathrm { F } = \mathrm { Pt } ^ { - 1 } + \mathrm { Qt }$. Here t is time. The unit of P is same as that of :-
  1. A displacement
  2. B velocity
  3. C acceleration
  4. D momentum
Show answer & explanation

Correct answer: D

$\begin{array}{l}\mathrm{F}=\mathrm{Pt}^{-1}+\mathrm{Q} \Rightarrow[\mathrm{F}]=\left[\mathrm{PT}^{-1}\right] \\ \mathrm{P}=\frac{\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\right]}{\left[\mathrm{T}^{-1}\right]}=\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-1}\right]=\text { Momentum }\end{array}$
3
The time dependence of a physical quantity P is given by $\mathrm { P } = \mathrm { P } _ { 0 } \exp \left( - \alpha \mathrm { t } ^ { 2 } \right)$, where α is a constant and t is time. The constant α
  1. A is dimensionless
  2. B has dimensions $\left[ \mathrm { T } ^ { - 2 } \right]$
  3. C has dimensions of P
  4. D has dimensions $\left[ \mathrm { T } ^ { 2 } \right]$
Show answer & explanation

Correct answer: B

$\mathrm{P}=\mathrm{P}_{0} \mathrm{e}^{-\alpha t^{2}}$
αt$^{2}$ must be dimensionless
Thus dimensions of α = $\frac{1}{\mathrm{t}^{2}}=\left[\mathrm{T}^{-2}\right]$
4
A wave is represented by y = a sin (At - Bx + C) where A, B, C are constants. The dimensions of A, B, C are
  1. A $\mathrm { T } ^ { - 1 } , \mathrm {~L} , \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { 0 }$
  2. B $\mathrm { T } ^ { - 1 } , \mathrm {~L} ^ { - 1 } , \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { 0 }$
  3. C T, L, M
  4. D $\mathrm { T } ^ { - 1 } , \mathrm {~L} ^ { - 1 } , \mathrm { M } ^ { - 1 }$
Show answer & explanation

Correct answer: B

$\begin{array}{l}\mathrm{y}=\mathrm{a} \sin [\mathrm{At}-\mathrm{Bx}+\mathrm{C}] \\ {[\mathrm{At}-\mathrm{Bx}+\mathrm{C}]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]} \\ {[\mathrm{At}]=[\mathrm{Bx}]=[\mathrm{C}]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]} \\ {[\mathrm{A}]=\left[\mathrm{T}^{-1}\right] ;[\mathrm{B}]=\left[\mathrm{L}^{-1}\right] ;[\mathrm{C}]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]}\end{array}$
5
Which pair has the same dimensions ?
  1. A Work and power
  2. B Density and relative density
  3. C Momentum and impulse
  4. D Latent heat and specific heat
Show answer & explanation

Correct answer: C

Impulse = Force × Time
= $\left(\frac{\text { Linear momentum }}{\text { time }}\right) \times$ time
= linear momentum
6
The quantity having dimension -2 in the time is :-
  1. A force
  2. B pressure
  3. C gravitational constant
  4. D all of these
Show answer & explanation

Correct answer: D

Force → [M$^{1}$L$^{1}$T$^{–2}$]
Pressure → [M1L$^{–1}$T$^{–2}$]
Gravitational constant → [M$^{–1}$L$^{3}$T$^{–2}$]
7
Dimensions of relative density is
  1. A $\mathrm { kg } \mathrm { m } ^ { - 3 }$
  2. B $\left[ \mathrm { ML } ^ { - 3 } \right]$
  3. C dimensionless
  4. D $\left[ \mathrm { M } ^ { 2 } \mathrm {~L} ^ { - 6 } \right]$
Show answer & explanation

Correct answer: C

$\begin{aligned} \text { Relative density } & =\frac{\text { Density of substance }}{\text { Density of water }} \\ & =\frac{\mathrm{kgm}^{-3}}{\mathrm{kgm}^{-3}}=\text { dimensionless }\end{aligned}$
8
Which of the following does not have dimensions of length ?
  1. A fermi
  2. B micron
  3. C angstrom
  4. D radian
Show answer & explanation

Correct answer: D

1 Fermi = 10$^{–15}$ m; 1 Micron = 10$^{–6}$ m
1 angstrom = 10$^{–10}$ m; radian → Angle
9
Which of the following is incorrect statement
  1. A A dimensionally correct equation may be correct
  2. B A dimensionally correct equation may be incorrect
  3. C A dimensionally incorrect equation may be correct
  4. D A dimensionally incorrect equation is incorrect
Show answer & explanation

Correct answer: C

Dimensionally correct equation may be correct or incorrect but dimensionally incorrect equation is always incorrect e.g. p = $\frac{mv}{t}$ is dimensionally as well as experimentally wrong.
10
For a body moving along x-axis, the distance travelled by body from a reference point is given as function of time t as x = at$^{2}$ + b, where a and b are constants, then the dimension of $\sqrt { \mathrm { ab } }$ is same as :-
  1. A speed
  2. B distance travelled
  3. C acceleration
  4. D none of these
Show answer & explanation

Correct answer: A

$\begin{array}{l}\mathrm{x}=\mathrm{at}^{2}+\mathrm{b} \\ {[\mathrm{x}]=\left[\mathrm{at}^{2}\right]=[\mathrm{b}]} \\ \mathrm{a}=\left[\mathrm{L}^{1} \mathrm{~T}^{-2}\right] ; \mathrm{b}=\left[\mathrm{L}^{1}\right]\end{array}$
11
Which of the following system of units is not based on units of mass, length and time alone?
  1. A SI
  2. B MKS
  3. C FPS
  4. D CGS
Show answer & explanation

Correct answer: A

The SI system has seven fundamental units.
12
Which of the following is not the name of a physical quantity :-
  1. A kilogram
  2. B impulse
  3. C energy
  4. D density
Show answer & explanation

Correct answer: A

Kilogram is a unit
13
If $\mathrm { S } = \frac { 1 } { 3 } \mathrm { ft } ^ { 3 }$, ' f ' has the dimensions of ( $\mathrm { S } =$ distance, $\mathrm { t } =$ time)
  1. A $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { - 1 } \mathrm {~T} ^ { 3 } \right]$
  2. B $\left[ \mathrm { M } ^ { 1 } \mathrm {~L} ^ { 1 } \mathrm {~T} ^ { - 3 } \right]$
  3. C $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 1 } \mathrm {~T} ^ { - 3 } \right]$
  4. D $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { - 1 } \mathrm {~T} ^ { - 3 } \right]$
Show answer & explanation

Correct answer: C

$\begin{array}{l}\mathrm{S}=1 / 3 \mathrm{ft}^{3} \\ {[\mathrm{~S}]=\left[\mathrm{ft}^{3}\right] \Rightarrow[\mathrm{f}]=\frac{[\mathrm{S}]}{\left[\mathrm{t}^{3}\right]}=\left[\mathrm{L}^{1} \mathrm{~T}^{-3}\right]}\end{array}$
14
The equation $\frac { d v } { d t } = A T - B v$ is describing the rate of change of velocity of a body falling from rest in a resisting medium. The dimensions of A and B are :-
  1. A $\mathrm { LT } ^ { - 3 } , \mathrm {~T}$
  2. B $\mathrm { LT } ^ { - 3 } , \mathrm {~T} ^ { - 1 }$
  3. C LT, T
  4. D LT, $\mathrm { T } ^ { - 1 }$
Show answer & explanation

Correct answer: B

$\begin{array}{l}\frac{d v}{d t}=A T-B v \\ {\left[\frac{d v}{d t}\right]=[A T]=[B v]} \\ {\left[L^{1} T^{-2}\right]=[A T] ;\left[L^{1} T^{-2}\right]=[B]\left[L^{1} T^{-1}\right]} \\ {[A]=\left[L^{1} T^{-3}\right] \quad[B]=\left[T^{-1}\right]}\end{array}$
15
For 10$^{(at + 3)}$, the dimensions of a is :-
  1. A $\mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { 0 }$
  2. B $\mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { 1 }$
  3. C $\mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { - 1 }$
  4. D None of these
Show answer & explanation

Correct answer: C

at + 3 = dimensionless
⇒ at = dimensionless
⇒ aT$^{1}$ = M$^{0}$L$^{0}$T$^{0}$
a = M$^{0}$L$^{0}$T$^{–1}$
16
In the relation $\mathrm { y } = \mathrm { r } \sin ( \omega \mathrm { t } - \mathrm { kx } )$ the dimensions of $\frac { \omega } { \mathrm { k } }$ are :-
  1. A $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 1 } \mathrm {~T} ^ { 0 } \right]$
  2. B $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 1 } \mathrm {~T} ^ { - 1 } \right]$
  3. C $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 0 } \mathrm {~T} ^ { 3 } \right]$
  4. D $\left[ \mathrm { M } ^ { 0 } \mathrm {~L} ^ { 1 } \mathrm {~T} ^ { 0 } \right]$
Show answer & explanation

Correct answer: B

$\begin{array}{l}\mathrm{y}=\mathrm{r} \sin (\omega \mathrm{t}-\mathrm{kx}) \\ {[\omega \mathrm{t}-\mathrm{kx}]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]} \\ {\left[\omega \mathrm{t}=[\mathrm{kx}]=\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\right.} \\ {\left[\frac{\omega}{\mathrm{k}}\right]=\left[\frac{\mathrm{x}}{\mathrm{t}}\right]=\left[\mathrm{L}^{1} \mathrm{~T}^{-1}\right]}\end{array}$
17
Dimensions of one pair is same. Identify the pair:-
  1. A Torque and energy
  2. B Angular momentum and work
  3. C Energy and pressure
  4. D Light year and solar day
Show answer & explanation

Correct answer: A

Torque= (Force) × (distance)
Energy = Work = (Force) × (displacement)
18
The method of dimensional analysis can be used to derive which of the following relations?
  1. A $\mathrm { N } _ { 0 } \mathrm { e } ^ { - \lambda _ { \mathrm { t } } }$
  2. B $A \sin ( \omega t + k x )$
  3. C $\frac { 1 } { 2 } m v ^ { 2 } + \frac { 1 } { 2 } I \omega ^ { 2 }$
  4. D None of the above
Show answer & explanation

Correct answer: D

Dimensional analysis cannot be used to derive relations involving trigonometric, exponential functions and relations involving (+) & (–) sign
19
A unitless quantity
  1. A Does not exist
  2. B Always has a nonzero dimension
  3. C Never has a nonzero dimension
  4. D May have a nonzero dimension
Show answer & explanation

Correct answer: C

For unitless quantity dimensions are always zero i.e. they never have non zero dimension.
20
Which of the following is incorrect?
  1. A All derived quantities may be represented dimensionally in terms of the base quantities
  2. B A base quantity cannot be represented dimensionally in terms of other base quantities
  3. C The dimension of a derived quantity is never zero in any base quantity
  4. D The dimension of a base quantity in other base quantities is always zero.
Show answer & explanation

Correct answer: C

Dimensionally correct equation may be correct or incorrect but dimensionally incorrect equation is always incorrect e.g. p = $\frac{mv}{t}$ is dimensionally as well as experimentally wrong.

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