Question 1: Palette 1/67

Allen JEE Main-2023 MCQ

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be:

A

$FV^{-2}T^{2}$

B

$FV^{-4}T^{2}$

C

$FV^{4}T^{-6}$

D

$F^{2}V^{-2}T^{6}$

Question 2: Palette 2/67

Allen 2023 MCQ

The speed of a wave produced in water is given by $\upsilon = \lambda^{a} g^{b} \rho^{c}$ . Where $\lambda, g$ and $\rho$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are: (JEE Main-2023)

A

$\frac{1}{2}, \frac{1}{2}, 0$

B

1, 1, 0

C

1, -1, 0

D

$\frac{1}{2}, 0, \frac{1}{2}$

Question 3: Palette 3/67

Allen JEE Main-2023 MCQ

As shown in the figure, a particle is moving with constant speed $\pi m/s$ . Considering its motion from A to B, the magnitude of the average velocity is:

A

$\pi m / s$

B

$\sqrt{3} m / s$

C

$2\sqrt{3} m / s$

D

$1.5\sqrt{3} m / s$

Question 4: Palette 4/67

Allen JEE Main-2023 MCQ

A particle is moving with constant speed in a circular path. When the particle turns by an angle $90^{\circ}$ , the ratio of instantaneous velocity to its average velocity is $\pi: x\sqrt{2}$ . The value of x will be

A

2

B

5

C

1

D

7

Question 5: Palette 5/67

Allen JEE Advanced-2023 MCQ

Young's modulus of elasticity Y is expressed in terms of three derived quantities, namely, the gravitational constant G, Planck's constant h and the speed of light c, as $Y = c^{\alpha} h^{\beta} G^{\gamma}$ . Which of the following is the correct option?

A

$\alpha = 7, \beta = - 1, \gamma = - 2$

B

$\alpha = -7, \beta = -1, \gamma = -2$

C

$\alpha = 7, \beta = - 1, \gamma = 2$

D

$\alpha = - 7, \beta = 1, \gamma = - 2$

Question 6: Palette 6/67

Allen JEE Main-2022 MCQ

Identify the pair of physical quantities which have different dimensions:

A

Wave number and Rydberg's constant

B

Stress and Coefficient of elasticity

C

Coercivity and Magnetisation

D

Specific heat capacity and Latent heat

Question 7: Palette 7/67

Allen JEE Main-2021 MCQ

If time $(t)$ , velocity $(v)$ , and angular momentum $(\ell)$ are taken as the fundamental units. Then the dimension of mass $(m)$ in terms of t, v and $\ell$ is :

A

$[t^{-1} v^{1} \ell^{-2}]$

B

$[t^{1} v^{2} \ell^{-1}]$

C

$[t^{-2} v^{-1} \ell^{1}]$

D

$[t^{-1} v^{-2} \ell^{1}]$

Question 8: Palette 8/67

Allen JEE Main-2021 MCQ

The force is given in terms of time t and displacement x by the equation $F = A \cos Bx + C \sin Dt$ The dimensional formula of $\frac{AD}{B}$ is :

A

$[M^{0} L T^{-1}]$

B

$[M L^{2} T^{-3}]$

C

$[M^{1} L^{1} T^{-2}]$

D

$[M^{2} L^{2} T^{-3}]$

Question 9: Palette 9/67

Allen JEE Main-2021 MCQ

If $E, L, M$ and $G$ denote the quantities as energy, angular momentum, mass and the universal constant of gravitation respectively, then the dimensions of $P$ in the formula $P = EL^2M^{-5}G^{-2}$ are:

A

$[M^0 L^1 T^0]$

B

$[M^{-1}L^{-1}T^2]$

C

$[M^1 L^1 T^{-2}]$

D

$[M^0 L^0 T^0]$

Question 10: Palette 10/67

Allen JEE Main-2021 MCQ

If force (F), length (L) and time (T) are taken as the fundamental quantities, then what will be the dimension of density:

A

$[FL^{-4} T^{2}]$

B

$[FL^{-3} T^{2}]$

C

$[FL^{-5} T^{2}]$

D

$[FL^{-3} T^{3}]$

Question 11: Palette 11/67

Allen JEE Main-2021 Match the Columns

Match List-I with List-II.
Column-I
(A) Torque
(B) Impulse
(C) Tension
(D) Surface Tension
Choose the most appropriate answer from the option given below:

A

(a)-(iii), (b)-(i), (c)-(iv), (d)-(ii)

B

(a)-(ii), (b)-(i), (c)-(iv), (d)-(iii)

C

(a)-(i), (b)-(iii), (c)-(iv), (d)-(ii)

D

(a)-(iii), (b)-(iv), (c)-(i), (d)-(ii)

Question 12: Palette 12/67

Allen JEE Main-2021 MCQ

If velocity [V], time [T] and force [F] are chosen as the base quantities, the dimensions of the mass will be :

A

$[FT^{-1} V^{-1}]$

B

$[FTV^{-1}]$

C

$[FT^{2} V]$

D

$[FVT^{-1}]$

Question 13: Palette 13/67

Allen JEE Main-2021 MCQ

In an octagon ABCDEFGH of equal side, what is the sum of $\overrightarrow{AB} + \overrightarrow{AC} + \overrightarrow{AD} + \overrightarrow{AE} + \overrightarrow{AF} + \overrightarrow{AG} + \overrightarrow{AH}$ if, $\overrightarrow{AO} = 2\hat{i} + 3\hat{j} - 4\hat{k}$

A

$-16\hat{\imath} - 24\hat{j} + 32\hat{k}$

B

$16\hat{\imath} + 24\hat{j} - 32\hat{k}$

C

$16\hat{\imath} + 24\hat{j} + 32\hat{k}$

D

$16\hat{\imath} - 24\hat{j} + 32\hat{k}$

Question 14: Palette 14/67

Allen JEE Main-2021 MCQ

If $\vec{A}$ and $\vec{B}$ are two vectors satisfying the relation $\vec{A}$ . $\vec{B} = |\vec{A} \times \vec{B}|$ . Then the value of $|\vec{A} - \vec{B}|$ will be:

A

$\sqrt{A^{2} + B^{2}}$

B

$\sqrt{A^{2} + B^{2} + \sqrt{2}AB}$

C

$\sqrt{A^{2} + B^{2} + 2AB}$

D

$\sqrt{A^{2} + B^{2} - \sqrt{2}AB}$

Question 15: Palette 15/67

Allen JEE Main-2021 MCQ

What will be the projection of vector $\vec{A} = \hat{i} + \hat{j} + \hat{k}$ on vector $\vec{B} = \hat{i} + \hat{j}$ ?

A

$\sqrt{2}(\hat{i} + \hat{j} + \hat{k})$

B

$2(\hat{i} + \hat{j} + \hat{k})$

C

$\sqrt{2}(\hat{i} + \hat{j})$

D

$(\hat{i} + \hat{j})$

Question 16: Palette 16/67

Allen JEE Main-2021 MCQ

Assertion A: If $A, B, C, D$ are four points on a semi-circular arc with centre at 'O' such that $\left|\overrightarrow{AB}\right| = \left|\overrightarrow{BC}\right| = \left|\overrightarrow{CD}\right|$ , then $$ \overrightarrow {A B} + \overrightarrow {A C} + \overrightarrow {A D} = 4 \overrightarrow {A O} + \overrightarrow {O B} + \overrightarrow {O C} $$ Reason R : Polygon law of vector addition yields $$ \overrightarrow {A B} + \overrightarrow {B C} + \overrightarrow {C D} + \overrightarrow {A D} = 2 \overrightarrow {A O} $$ In the light of the above statements, choose the most appropriate answer from the options given below:

A

$\mathbf{A}$ is correct but $\mathbf{R}$ is not correct.

B

$\mathbf{A}$ is not correct but $\mathbf{R}$ is correct.

C

Both A and R are correct and R is the correct explanation of A.

D

Both A and R are correct but R is not the correct explanation of A.

Question 17: Palette 17/67

Allen JEE Main-2021 MCQ

The magnitude of vectors $\overrightarrow{OA}$ , $\overrightarrow{OB}$ and $\overrightarrow{OC}$ in the given figure are equal. The direction of $\overrightarrow{OA} + \overrightarrow{OB} - \overrightarrow{OC}$ with x-axis will be :

A

$\tan^{-1}\frac{(1 - \sqrt{3} - \sqrt{2})}{(1 + \sqrt{3} + \sqrt{2})}$

B

$\tan^{-1}\frac{(\sqrt{3} - 1 + \sqrt{2})}{(1 + \sqrt{3} - \sqrt{2})}$

C

$\tan^{-1}\frac{(\sqrt{3} - 1 + \sqrt{2})}{(1 - \sqrt{3} + \sqrt{2})}$

D

$\tan^{-1}\frac{(1 + \sqrt{3} - \sqrt{2})}{(1 - \sqrt{3} - \sqrt{2})}$

Question 18: Palette 18/67

Allen JEE Main-2021 MCQ

The angle between vector $(\vec{A})$ and $(\vec{A}-\vec{B})$ is :

A

$\tan^{-1}\left(\frac{-\frac{B}{2}}{A - B\frac{\sqrt{3}}{2}}\right)$

B

$\tan^{-1}\left(\frac{A}{0.7B}\right)$

C

$\tan^{-1}\left(\frac{\sqrt{3}B}{2A - B}\right)$

D

$\tan^{-1}\left(\frac{B\cos\theta}{A - B\sin\theta}\right)$

Question 19: Palette 19/67

Allen JEE Main-2020 MCQ

A quantity $f$ is given by $f = \sqrt{\frac{hc^5}{G}}$ where $c$ is the speed of light, $G$ is the universal gravitational constant and $h$ is the Planck's constant. Dimension of $f$ is that of:

A

Momentum

B

Area

C

Energy

D

Volume

Question 20: Palette 20/67

Allen JEE Main-2020 MCQ

Amount of solar energy received on the Earth's surface per unit area per unit time is defined as solar constant. The dimension of solar constant is:

A

$ML^{2}T^{-2}$

B

$MLT^{-2}$

C

$M^2 L^0 T^{-1}$

D

$ML^0 T^{-3}$

Question 21: Palette 21/67

Allen JEE Advanced-2020 MSQ

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: [position] = $[X^{\alpha}]$ ; [speed] = $[X^{\beta}]$ ; [acceleration] = $[X^{p}]$ ; [linear momentum] = $[X^{q}]$ ; [force] = $[X^{r}]$ . Then

A

$\alpha + p = 2\beta$

B

$p + q - r = \beta$

C

$p - q + r = \alpha$

D

$p + q + r = \beta$

Question 22: Palette 22/67

Allen JEE Main-2019 MCQ

Two vectors $\vec{A}$ and $\vec{B}$ have equal magnitudes. The magnitude of $(\vec{A} + \vec{B})$ is 'n' times the magnitude of $(\vec{A} - \vec{B})$ . The angle between $\vec{A}$ and $\vec{B}$ is:

A

$\sin^{-1}\left[\frac{n^2 - 1}{n^2 + 1}\right]$

B

$\cos^{-1}\left[\frac{n - 1}{n + 1}\right]$

C

$\cos^{-1}\left[\frac{n^2 - 1}{n^2 + 1}\right]$

D

$\sin^{-1}\left[\frac{n - 1}{n + 1}\right]$

Question 23: Palette 23/67

Allen JEE Main-2019 MCQ

In the cube of side 'a' shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be:

A

$\frac{1}{2} a(\hat{i} - \hat{k})$

B

$\frac{1}{2} a(\hat{j} - \hat{i})$

C

$\frac{1}{2} a(\hat{k} - \hat{i})$

D

$\frac{1}{2} a(\hat{j} - \hat{k})$

Question 24: Palette 24/67

Allen JEE Main-2019 MCQ

Let $\left|\vec{A}_{1}\right|=3,\left|\vec{A}_{2}\right|=5$ and $\left|\vec{A}_{1}+\vec{A}_{2}\right|=5$ . The value of $\left(2\vec{A}_{1}+3\vec{A}_{2}\right)\cdot\left(3\vec{A}_{1}-2\vec{A}_{2}\right)$ is:-

A

-112.5

B

-106.5

C

-118.5

D

-99.5

Question 25: Palette 25/67

Allen JEE Advanced-2019 MSQ

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L, which of the following statement (s) is/are correct?

A

The dimension of force is $L^{-3}$

B

The dimension of energy is $L^{-2}$

C

The dimension of power is $L^{-5}$

D

The dimension of linear momentum is $L^{-1}$

Question 26: Palette 26/67

Allen JEE Advanced-2018 MCQ

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $[\in_{0}]$ and $[\mu_{0}]$ stand for dimensions of the permittivity and permeability of free space respectively. $[L]$ and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units. The relation between $[E]$ and $[B]$ is :-

A

$[E] = [B][L][T]$

B

$[E] = [B][L]^{-1}[T]$

C

$[E] = [B][L][T]^{-1}$

D

$[E] = [B][L]^{-1}[T]^{-1}$

Question 27: Palette 27/67

Allen JEE Advanced-2018 MCQ

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $[E]$ and $[B]$ stand for dimensions of electric and magnetic fields respectively, while $[\in_{0}]$ and $[\mu_{0}]$ stand for dimensions of the permittivity and permeability of free space respectively. $[L]$ and $[T]$ are dimensions of length and time respectively. All the quantities are given in SI units. The relation between $[\in_0]$ and $[\mu_0]$ is:-

A

$[\mu_0] = [\in_0][L]^2[T]^{-2}$

B

$[\mu_0] = [\in_0][L]^{-2}[T]^2$

C

$[\mu_0] = [\in_0]^{-1}[L]^2[T]^{-2}$

D

$[\mu_0] = [\in_0]^{-1}[L]^{-2}[T]^2$

Question 28: Palette 28/67

Allen JEE Advanced-2017 MCQ

Three vectors $\vec{P}, \vec{Q}$ and $\vec{R}$ are shown in the figure. Let S be any point on the vector $\vec{R}$ . The distance between the points P and S is $b|\vec{R}|$ . The general relation among vectors $\vec{P}, \vec{Q}$ and $\vec{S}$ is:

A

$\vec{S} = (1 - b)\vec{P} + b^2\vec{Q}$

B

$\vec{S} = (b - 1)\vec{P} + b\vec{Q}$

C

$\vec{S} = (1 - b)\vec{P} + b\vec{Q}$

D

$$<br /> \vec {S} = (1 - b ^ {2}) \vec {P} + b \vec {Q}<br /> $$

Question 29: Palette 29/67

Allen JEE Advanced-2015 MSQ

In terms of potential difference V, electric current I, permittivity $\varepsilon_{0}$ , permeability $\mu_{0}$ and speed of light c, the dimensionally correct equation(s) is(are)-

A

$\mu_{0}I^{2} = \varepsilon_{0}V^{2}$

B

$\varepsilon_{0}I = \mu_{0}V$

C

$I = \varepsilon_{0}cV$

D

$\mu_{0}cI = \varepsilon_{0}V$

Question 30: Palette 30/67

Allen JEE Advanced-2013 MCQ

Match List I with List II and select the correct answer using the codes given below the lists:

A

A-R, B-P, C-Q, D-S

B

A-R, B-Q, C-P, D-S

C

A-S, B-Q, C-P, D-R

D

A-S, B-P, C-Q, D-R

Question 31: Palette 31/67

Allen 2000 MCQ

PARAGRAPH FOR QUESTION NO. 7 TO 9 In a certain system of absolute units, the acceleration produced by gravity in a body falling freely is denoted by 5, the kinetic energy of a 500 kg shot moving with velocity 400 metres per second is denoted by 2000 and its momentum by 100. The unit of length is :-

A

$15m$

B

$50m$

C

$25m$

D

$100m$

Question 32: Palette 32/67

Allen 2000 MCQ

PARAGRAPH FOR QUESTION NO. 7 TO 9 In a certain system of absolute units, the acceleration produced by gravity in a body falling freely is denoted by 5, the kinetic energy of a 500 kg shot moving with velocity 400 metres per second is denoted by 2000 and its momentum by 100. The unit of time is:-

A

10 s

B

20 s

C

5 s

D

15 s

Question 33: Palette 33/67

Allen 2000 MCQ

PARAGRAPH FOR QUESTION NO. 7 TO 9 In a certain system of absolute units, the acceleration produced by gravity in a body falling freely is denoted by 5, the kinetic energy of a 500 kg shot moving with velocity 400 metres per second is denoted by 2000 and its momentum by 100. The unit of mass is :-

A

$200 \mathrm{~kg}$

B

$400 \mathrm{~kg}$

C

$800 \mathrm{~kg}$

D

$1200 \mathrm{~kg}$

Question 34: Palette 34/67

Allen MCQ

If energy $(E)$ , velocity $(V)$ and time $(T)$ are chosen as the fundamental quantities, then the dimensions of surface tension will be. (Surface tension = force / length)

A

$E V^{-2}T^{-1}$

B

$E V^{-1}T^{-2}$

C

$E^{-2} V^{-1}T^{-3}$

D

$E V^{-2}T^{-2}$

Question 35: Palette 35/67

Allen MCQ

Which of the following is incorrect statements?

A

A dimensionally correct equation may be correct

B

A dimensionally correct equation may be incorrect

C

A dimensionally incorrect equation may be incorrect

D

A dimensionally incorrect equation is incorrect

Question 36: Palette 36/67

Allen MCQ

Use the small angle approximation to find approximate value for $\left(\frac{\sin1^{\circ}}{\cos2^{\circ}}\right)$ :

A

1

B

$\frac{\pi}{30}$

C

$\frac{\pi}{60}$

D

$\frac{\pi}{180}$

Question 37: Palette 37/67

Allen MCQ

Expression for time in terms of G (universal gravitational constant), h (Planck constant) and c (speed of light) is proportional to:

A

$\sqrt{\frac{Gh}{c^{3}}}$

B

$\sqrt{\frac{hc^{5}}{G}}$

C

$\sqrt{\frac{c^{3}}{Gh}}$

D

$\sqrt{\frac{Gh}{c^{5}}}$

Question 38: Palette 38/67

Allen MCQ

The density of a material in SI units is $128 \mathrm{~kg} \mathrm{~m}^{-3}$ . In certain units in which the unit of length is $25 \mathrm{~cm}$ and the unit of mass is $50 \mathrm{~g}$ , the numerical value of density of the material is:

A

410

B

640

C

16

D

40

Question 39: Palette 39/67

Allen MCQ

If speed (V), acceleration (a) and force (F) are considered as fundamental units, the dimension of Young's modulus will be :-

A

$V^{-2} a^{2} F^{2}$

B

$V^{-4} a^{2} F$

C

$V^{-4} a^{-2} F$

D

$V^{-2} a^{2} F^{-2}$

Question 40: Palette 40/67

Allen MCQ

The force of interaction between two atoms is given by $F = \alpha\beta \exp\left(-\frac{x^{2}}{\alpha kt}\right)$ ; where x is the distance, k is the Boltzmann constant and T is temperature and $\alpha$ and $\beta$ are two constants. The dimension of $\beta$ is:

A

$M^{2}L^{2}T^{-2}$

B

$M^{2}LT^{-4}$

C

$M^{0}L^{2}T^{-4}$

D

$MLT^{-2}$

Question 41: Palette 41/67

Allen MCQ

If force, acceleration, and time are taken as fundamental quantities, then the dimensions of length will be:

A

$FT^{2}$

B

$F^{-1}A^{2}T^{-1}$

C

$FA^{2}T$

D

$AT^{2}$

Question 42: Palette 42/67

Allen MCQ

In a particular system, the units of length, mass and time are chosen to be 10 cm, 10 g and 0.1 s respectively. The unit of force in this system will be equal to :-

A

0.1 N

B

1 N

C

10 N

D

100 N

Question 43: Palette 43/67

Allen MCQ

The units of three physical quantities x, y and z are $gcm^{2}s^{-5}$ , $gs^{-1}$ and $cms^{-2}$ respectively. The relation among the units of x, y and z is:

A

$z = x^{2}y$

B

$y^{2} = xz$

C

$x = yz^{2}$

D

$x = y^{2}z$

Question 44: Palette 44/67

Allen MCQ

Statement 1 : Method of dimensions cannot tell whether an equation is correct. and Statement 2 : A dimensionally incorrect equation may be correct.

A

Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement 1.

B

Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.

C

Statement-1 is true, statement-2 is false.

D

Statement-1 is false, statement-2 is true.

Question 45: Palette 45/67

Allen MCQ

A man rows a boat with a speed of 18 km/hr in north-west direction. The shoreline makes an angle of $15^{\circ}$ south of west. Obtain the component of the velocity of the boat along the shoreline.

A

9 km/hr

B

$18\frac{\sqrt{3}}{2}$ km/hr

C

$18\cos(15^{\circ})$ km/hr

D

$18\cos(75^{\circ})$ km/hr

Question 46: Palette 46/67

Allen MCQ

Statement 1 : Unit vector has a unit though its magnitude is one and Statement 2 : Unit vector is obtained by dividing a vector by its own magnitude.

A

Statement-1 is true, statement-2 is true, and statement-2 is correct explanation for statement 1.

B

Statement-1 is true, statement-2 is true, and statement-2 is NOT the correct explanation for statement-1.

C

Statement-1 is true, statement-2 is false.

D

Statement-1 is false, statement-2 is true.

Question 47: Palette 47/67

Allen MCQ

A vector of magnitude 10 m in the direction $37^{\circ}$ south of west has its initial point at $(5\ m, 2\ m)$ . If positive x-axis represents the east and positive y-axis the north, the coordinates of its terminal point are

A

$(-3\ m, -4\ m)$

B

$(3\ m, 4\ m)$

C

$(-4\ m, 6\ m)$

D

$(-4\ m, -6\ m)$

Question 48: Palette 48/67

Allen MCQ

After firing, a bullet is found to move at an angle of $37^{\circ}$ to horizontal. Its acceleration is $10 \, m/s^{2}$ downwards. Find the component of acceleration in the direction of the velocity.

A

$-6 \, m/s^{2}$

B

$-4 \, m/s^{2}$

C

$-8 \, m/s^{2}$

D

$-5 \, m/s^{2}$

Question 49: Palette 49/67

Allen MCQ

A string connected with bob is suspended by the point $(O)$ such that it sweeps out conical surface in horizontal plane. Here $\vec{r}$ is the position vector of bob, $\vec{v}$ is its velocity and $\vec{z}$ is the axis of swept cone as shown. Select INCORRECT statement :-

A

$\vec{r}.\vec{z}$ is always zero

B

$\vec{r}.\vec{v}$ is always zero

C

$\vec{z}\cdot\vec{v}$ is always constant

D

$\vec{r}\cdot\vec{z}$ is always non zero constant

Question 50: Palette 50/67

Allen MCQ

Two forces P and Q of magnitude 2f and 3f, respectively, are at an angle $\theta$ with each other. If the force Q is doubled, then their resultant also gets doubled. Then, the angle is:

A

$30^{\circ}$

B

$60^{\circ}$

C

$90^{\circ}$

D

$120^{\circ}$

Question 51: Palette 51/67

Allen MCQ

Consider three vectors $\vec{A} = \hat{i} + \hat{j} - 2\hat{k}, \vec{B} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{C} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ . A vector $\vec{X}$ of the form $\alpha\vec{A} + \beta\vec{B}$ ( $\alpha$ and $\beta$ are numbers) is perpendicular to $\vec{C}$ . The ratio of $\alpha$ and $\beta$ is

A

1:1

B

2:1

C

-1:1

D

3:1

Question 52: Palette 52/67

Allen MCQ

Given the vectors $\vec{A} = 2\hat{i} + 3\hat{j} - \hat{k}$ ; $\vec{B} = 3\hat{i} - 2\hat{j} - 2\hat{k}$ and $\vec{C} = p\hat{i} + p\hat{j} + 2p\hat{k}$ . Find the angle between $(\vec{A} - \vec{B})$ and $\vec{C}$

A

$\theta = \cos^{-1}\left(\frac{2}{\sqrt{3}}\right)$

B

$\theta = \cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$

C

$\theta = \cos^{-1}\left(\frac{\sqrt{2}}{3}\right)$

D

none of these

Question 53: Palette 53/67

Allen MCQ

Two forces $(\hat{i} + \hat{j} + \hat{k})N$ and $(\hat{i} + 2\hat{j} + 3\hat{k})N$ act on a particle and displace it from (2, 3, 4) to point (5, 4, 3). Displacement is in m. Work done is:

A

2 J

B

3 j

C

4 j

D

5 j

Question 54: Palette 54/67

Allen MCQ

If momentum (P), area (A) and time (T) are taken to be the fundamental quantities then the dimensional formula for energy is:

A

$[PA^{-1} T^{-2}]$

B

$[PA^{1/2}T^{-1}]$

C

$[P^{2}AT^{-2}]$

D

$[P^{1/2}AT^{-1}]$

Question 55: Palette 55/67

Allen MCQ

A boy A is standing $20\sqrt{3}$ away in a direction $30^\circ$ north of east from his friend B. Another boy C standing somewhere east of B can reach A, if he walks in a direction $60^\circ$ north of east. In a Cartesian coordinate system with its x-axis towards the east, the position of C with respect to A is

A

$(-20\hat{i}-10\hat{j}),m$

B

$(-10\hat{i}-10\sqrt{3}\hat{j}),m$

C

$(10\hat{i}+10\sqrt{3}\hat{j}),m$

D

It depends on where chose the origin.

Question 56: Palette 56/67

Allen MCQ

A body moves in anticlockwise direction on a circular path in the $x$-$y$ plane. The radius of the circular path is $5,m$ and its centre is at the origin. At an interval of time, displacement of the body is observed to be $6,m$ in the positive $y$-direction. Which of the following is true?

A

Its initial position vector is $5\hat{i}$.

B

Its initial position vector is $(-3\hat{i}+4\hat{j}),m$.

C

Its final position vector is $(4\hat{i}+3\hat{j}),m$.

D

Its final position vector is $6\hat{j},m$.

Question 57: Palette 57/67

Allen MCQ

A particle moves from a position $3\hat{i}+2\hat{j}-6\hat{k}$ to a position $14\hat{i}+13\hat{j}+9\hat{k}$ in $m$ and a uniform force of $4\hat{i}+3\hat{k},N$ acts on it. The work done by the force is :-

A

$200,J$

B

$100,J$

C

$300,J$

D

$500,J$

Question 58: Palette 58/67

Allen MSQ

The vector $\hat{i} + x\hat{j} + 3\hat{k}$ is rotated through an angle $\theta$ and doubled in magnitude, then it becomes $4\hat{i} + (4x - 2)\hat{j} + 2\hat{k}$ . The values of $x$ are

A

1

B

$-\frac{2}{3}$

C

2

D

$\frac{4}{3}$

Question 59: Palette 59/67

Allen MSQ

The value of $\left|\vec{A}+\vec{B}-\vec{C}+\vec{D}\right|$ can be zero if:-

A

$|\vec{A}|=5,|\vec{B}|=3,|\vec{C}|=4;|\vec{D}|=13$

B

$|\vec{A}|=2\sqrt{2},|\vec{B}|=2,|\vec{C}|=2;|\vec{D}|=5$

C

$|\vec{A}|=2\sqrt{2},|\vec{B}|=2,|\vec{C}|=2;|\vec{D}|=10$

D

$|\vec{A}|=5,|\vec{B}|=4,|\vec{C}|=3;|\vec{D}|=8$

Question 60: Palette 60/67

Allen MSQ

Priya says that the sum of two vectors by the parallelogram method is $\vec{R} = 5\hat{i}$ . Subhangi says it is $\vec{R} = \hat{i} + 4\hat{j}$ . Both used the parallelogram method, but one used the wrong diagonal. Which one of the vector pairs below contains the original two vectors?

A

$\vec{A} = +3\hat{i} - 2\hat{j}$ ; $\vec{B} = -2\hat{i} + 2\hat{j}$

B

$\vec{A} = -3\hat{i} - 2\hat{j}$ ; $\vec{B} = +2\hat{i} + 2\hat{j}$

C

$\vec{A} = +3\hat{i} + 2\hat{j}$ ; $\vec{B} = +2\hat{i} - 2\hat{j}$

D

$\vec{A} = +3\hat{i} + 2\hat{j}$ ; $\vec{B} = -2\hat{i} + 2\hat{j}$

Question 61: Palette 61/67

Allen MCQ

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. Which of the following is not a physical quantity

A

Height of a boy

B

Weight of a boy

C

Fever of a boy

D

Speed of a running boy

Question 62: Palette 62/67

Allen MCQ

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. From the concept of directed dimension what is the formula for a range (R) of a cannon ball when it is fired with vertical velocity component $V_{y}$ and a horizontal velocity component $V_{x}$ assuming it is fired on a flat surface. [Range also depends upon acceleration due to gravity, g and k is numerical constant]

A

$R = \frac{k(V_{x}V_{y})}{g}$

B

$R = \frac{k(V_{x})^{2}}{g}$

C

$R = \frac{k(V_{x})^{3}}{V_{y}g}$

D

$R = \frac{k(V_{y})^{3}}{V_{x}g}$

Question 63: Palette 63/67

Allen MCQ

PARAGRAPH FOR QUESTION NO. 10 TO 12 A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement. The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, may represent length in the x direction as $L_{x}$ , and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar or vector) must be dimensionally consistent. As an example, suppose wish to calculate the drift S of a swimmer crossing a river flowing with velocity $V_{x}$ and of width D and he is swimming in direction perpendicular to the river flow with velocity $V_{y}$ relative to river, assuming no use of directed lengths, the quantities of interest are then $V_{x}, V_{y}$ both dimensioned as $\frac{L}{T}$ , S the drift and D width of river both having dimension L. With these four quantities, may conclude that the equation for the drift S may be written: $S = V_{x}^{a} V_{y}^{b} D^{c}$ Or dimensionally $L = \left(\frac{L}{T}\right)^{a + b} \times (L)^{c}$ from which may deduce that $a + b + c = 1$ and $a + b = 0$ , which leaves one of these exponents undetermined. If, however, use directed length dimensions, then $V_{x}$ will be dimensioned as $\frac{L_{x}}{T}$ , $V_{y}$ as $\frac{L_{y}}{T}$ , S as $L_{x}$ and D as $L_{y}$ . The dimensional equation becomes: $L_{x} = \left(\frac{L_{x}}{T}\right)^{a} \left(\frac{L_{y}}{T}\right)^{b} \left(L_{y}\right)^{c}$ and may solve completely as a = 1, b = -1 and c = 1. The increase in deductive power gained by the use of directed length dimensions is apparent. A conveyor belt of width D is moving along x-axis with velocity V. A man moving with velocity U on the belt in the direction perpendicular to the belt's velocity with respect to belt wants to cross the belt. The correct expression for the drift (S) suffered by man is given by (k is numerical constant)

A

$S = k \frac{UD}{V}$

B

$S = k \frac{VD}{U}$

C

$S = k \frac{U^{2}D}{V^{2}}$

D

$S = k \frac{V^{2}D}{U^{2}}$

Question 64: Palette 64/67

Allen Match the Columns

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following combinations unit of force is $10^{6}$ dyne. [F = ma]

A

(IV) (i) (P)

B

(II) (iii) (S)

C

(III) (iv) (P)

D

(I) (ii) (Q)

Question 65: Palette 65/67

Allen Match the Columns

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following system, unit of energy is $10^{9}$ erg? $[E = F \times S]$

A

(III) (i) (S)

B

(IV) (iii) (R)

C

(II) (iv) (Q)

D

(I) (iii) (P)

Question 66: Palette 66/67

Allen Match the Columns

PARAGRAPH FOR QUESTION NO. 13 TO 15
L, M and T are units of length, Mass and Time respectively in a system of units.
Column-I
(I) $M = 1 0 0 \mathrm{gm}$
(II) $M = 10kg$
(III) $M = 10gm$
(IV) M = 1 tonne
Column-III
(P) $T = 0.1\mathrm{sec}.$
(Q) $T = 10ms$
(R) $T = 10$ sec
(S) $T = 0.01\mathrm{sec}.$ In which of the following system, unit for coefficient of viscosity is 100 poiseuille? $[\eta = \frac{F}{6\pi rv}]$

A

(III) (ii) (S)

B

(II) (i) (Q)

C

(III) (iii) (R)

D

(IV) (iv) (P)

Question 67: Palette 67/67

Allen MSQ

A physical quantity $\vec{S}$ is defined as $\vec{S} = (\vec{E} \times \vec{B}) / \mu_{0}$ , where $\vec{E}$ is electric field, $\vec{B}$ is magnetic field and $\mu_{0}$ is the permeability of free space. The dimensions of $\vec{S}$ are the same as the dimensions of which of the following quantity/quantities?

A

$\dfrac{\text{Energy}}{\text{Charge}\times\text{Current}}$

B

$\dfrac{\text{Force}}{\text{Length}\times\text{Time}}$

C

$\dfrac{\text{Energy}}{\text{Volume}}$

D

$\dfrac{\text{Power}}{\text{Area}}$