Motion in plane practice · Mixed difficulty

Statement [A]: If a body is in equilibrium, then the resultant of the forces acting on the body is zero

Statement [B]: Action and reaction cancel each other as they act on the same body

Statement [C]: Centrifugal force is a pseudo force

Options:

If vectors $\bar A = 2i + 3j + Pk$ and $\bar B = 3\bar i - 8\bar j + 2\bar k$ are perpendicular to each other. Then value of $P$ is

If $\mathrm{\vec A}\mathrm{.\vec B} = AB,$, then the angle between $\mathrm{\vec A}\,\mathrm{and}\,\mathrm{\vec B}$ is

In the case of an oblique projectile, the velocity is perpendicular to acceleration

If the magnitude of resultant of two vectors is equal to magnitude of either vectors, then angle between them is

A man is swimming with speed 10 km/h in still water crosses a river of width 1 km along the shortest path in 10 minutes. The velocity of river flow is

Given that A and B are greater than 1. The magnitude of $\left( {\mathrm{\vec A} \times \mathrm{\vec B}} \right)$ can not be

With what minimum speed a particle be projected from the origin so that it is able to pass through a given point (30 m, 40 m)?

The component of a vector along any other direction is

The value of $\left( {\bar A + \bar B} \right) \times \left( {\bar A - \bar B} \right)$ is

The area of a parallelogram whose adjacent sides are given by vectors $\vec A = \hat i - 2\hat j + 3\hat k$ and $\vec B = 4\hat i + 5\hat j$ is

Statement-I: If $\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}$, then $\vec{a}$ may not always be equal to $\vec{{c}}$.
Statement-II: The dot product of two vectors involves cosine of the angle between the two vectors.

Angle (in rad) made by the vector $\sqrt 3 i + j$ with the x-axis is

Vector $\bar A = 4i + 2j - 3k,\,\,\bar B = 5\bar i + \bar j - 2\bar k$ and $\bar C = 3i - j + 2k$. Then $\bar A.\left( {\bar B \times \bar C} \right)$ is

A vector $\mathrm{\vec A}$ points vertically upwards and $\mathrm{\vec B}$ points upwards North. The vector product $\mathrm{\vec A} \times \mathrm{\;\vec B}$ is

A vector $\mathrm{\vec A}$ when added to the vector $\mathrm{\vec B = }3\hat i + 4\hat j$ yields a resultant vector that is in the positive y direction and has a magnitude equal to that of $\mathrm{\vec B}$ Find the magnitude of $\mathrm{\vec A}$

Following forces start acting on a particle at rest at the origin of the co-ordinate system simultaneously ${\mathrm{\vec F}_1} = 5\hat i - 5\hat j + 5\hat k,\mathrm{\;\;\;}{\mathrm{F}_2} = 2\hat i + 8\hat j + 6\hat k,{\mathrm{F}_3} = - 6\hat i + 4\hat j - 7\hat k,{\mathrm{F}_4} = - \hat i - 3\hat j - 2\hat k.$ The particle will move

Assertion If a particle is projected vertically upwards with velocity $u$, the maximum height attained by the particle is $h_1$. The same particle is projected at angle $30^{\circ}$ from horizontal with the same speed $u$. Now the maximum height is $h_2$. Thus, $h_1=4 h_2$.

Reason In first case, $v=0$ at highest point and in second case, $v \neq 0$ at highest point.

If $a^2+b^2+c^2=1$ where $a, b, c \in R$, then the maximum value of $(4 a-3 b)^2+(5 b-4 c)^2+(3 c-5 a)^2$ is

Two vectors $\vec{A}$ and $\vec{B}$ are such that $\vec{A}+\vec{B}=\vec{C}$ and $A^2+B^2=C^2$. Which of the following statements is correct?

Component of the vector $\mathbf{A}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}$ along the vector $\mathbf{B}=(\hat{\mathbf{i}}+\hat{\mathbf{j}})$ is

A: Two row boats moving parallel to each other and nearby, are attracted towards each other
R: Increase of velocity of fluid flow decrease the pressure

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

Assertion: The angle between the two vectors $(\hat{i}+\hat{j})$ and $(\hat{i}+\hat{k})$ is $\frac{\pi}{3}$ radian.
Reason: Angle between two vectors $\vec{A}$ and $\vec{B}$ is given by $\theta=\cos ^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{A B}\right)$.

Two bodies are projected with the same velocity. If one is projected at an angle of $30^\circ$ and the other at angle of $60^\circ$ to the horizontal, the ratio of the maximum heights reached is

A boat takes two hours to travel 8 km and back in still water. If the velocity of water 4 $\mathrm{km}{\mathrm{h}^{ - 1}}$, the time taken for going ups tream 8km and coming back is

A man can swim with a speed of $4\,\,km{h^{ - 1}}$ in still water. He crosses a river 1 km wider that flows steadily at $3\,\,km{h^{ - 1}}$ If he makes his strokes normal to the river current how far down the river does he go when he reaches the other bank?

A ball is projected with velocity ${v_0}$ at an angle of elevation ${30^0}$. Mark the correct statement

Consider a vector $\overrightarrow F = 4\widehat i - 3\widehat j$. Another vector that is perpendicular to $\overrightarrow F$ is

The resultant of $A\,\, \times \,\,0$ will be equal to