Motion in plane practice · Mixed difficulty

Minimum number of unequal vectors which can give zero resultant are

If a vector $\hat i + 2\hat j - 8\hat k$ is perpendicular $4\hat j - 4\hat i + \alpha \hat k$ then value of $\alpha$ is

Statement-I: The vector product of a force $\vec{F}$ and displacement $\vec{r}$ is equal to the work done.
Statement-II: Work done is not a vector.

Neglecting the air resistance, the time of flight of a projectile is determined by

Resultant of which of the following may be equal to zero?

Statement-I: The value of $\hat{\mathrm{i}}(\hat{\mathrm{j}} \times \hat{\mathrm{k}})+\hat{\mathrm{j}}(\hat{\mathrm{k}} \times \hat{\mathrm{i}})+\hat{\mathrm{k}} \cdot(\hat{\mathrm{i}} \times \hat{\mathrm{j}})$ is equal to 3
Statement-II: If $\hat{a}, \hat{b}, \hat{c}$ are mutually perpendicular unit vectors, then $[\hat{a}, \hat{b}, \hat{c}]=1$

Two bodies are projected from the same point with equal speeds in such directions that they both strike the same point on a plane whose inclination is $\beta$. If $\alpha$ be the angle of projection of the first body with the horizontal the ratio of their times of flight is

If the angle between two non-zero vectors $\mathbf{A}$ and $\mathbf{B}$ is $120^{\circ}$, its resultant $C$ will be

A body starts from rest from the origin with an acceleration of $6\;m/{s^2}$ along the x-axis and $8\;m/{s^2}$ along the y-axis. Its distance from the origin after 4 seconds will be

Assertion: To cross the river in minimum time, swimmer should swim in a direction perpendicular to the water current.
Reason: In this case river flow helps to cross the river.

Identify the scalar quantity

Two towers $A$ and $B$ each of height $20 \mathrm{~m}$ are separated by a distance $200 \mathrm{~m}$. A body thrown horizontally from the tower $A$ with a velocity $20 \mathrm{~m} / \mathrm{s}$ towards the tower $B$ hits the ground at a point $P$. Similar body projected horizontally from the tower $B$ towards $A$ with same velocity hits the ground at $Q.$ The uniform velocity with which a truck can move from $P$ to $Q$ in 4 seconds is $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$

If $\bar A = \bar i + 2\bar k$ and $\bar B = \bar i + \bar j - \bar k$. Then $\bar A \times \bar B$ is equal to

As a rocket is accelerating vertically upwards at $9.8 \mathrm{~ms}^{-2}$ near the earth's surface, it releases a projectile with zero speed relative to rocket. Immediately after release, the acceleration (in $\mathrm{ms}^{-2}$ ) of the projectile is [take, $g=9.8 \mathrm{~ms}^{-2}$ ]

The vector projection of a vector $3\widehat i + 4\widehat k$ on y-axis is

The angle between the z-axis and the vector $\hat i + \hat j + \sqrt 2 \hat k$ is

Which of the following is correct?

A steam boat goes across a lake and comes back (i) on a quiet day when the water is still and (ii) on a rough day when there is a uniform current so as to help the journey onwards and to impede the journey back. If the speed of the launch on both days was same, the time required for complete journey on the rough day, as compared to the quiet day will be

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

Rain is falling vertically at $20\sqrt 3$ kmph. A man is running at 20 kmph. The angle with vertical at which he should hold his umbrella is

A body is thrown with a velocity of 9.8 m/s making an angle of $30^\circ$ with the horizontal. It will hit the ground after a time

Angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}\,\,is\,\theta$ What is the value of $\mathrm{\vec A}.\left( {\mathrm{\vec B} \times \mathrm{\vec A}} \right)?$

Let $\vec{a}$ and $\vec{b}$ be two vectors inclined at an angle $\theta$, where $\frac{\pi}{2} < \theta < \pi$.
Statement-I: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}| < |\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|$
Statement-II: $|\vec{a}+\vec{b}|^2=a^2+b^2+2\vec{a} \cdot \vec{b}$

The minimum number of vectors having different planes which can be added to give zero resultant is

What is the maximum number of components into which a vector can be split

A projectile can have the same range R for two angles of projection. If ${T_1}$ and ${T_2}$ be the times of flights in the two cases, then the product of the two times of flights is directly proportional to

Two swimmers start swimming perpendicular to river flow with different constant speeds at the same time. After reaching the opposite bank they have to walk back to reach the point exactly opposite to the starting point at the same time. Assume constant walking speed for both the swimmers.

Two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ are inclined to each other at an angle &theta;. Which of the following is the unit vector perpendicular to both $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ ?

Two particles are projected simultaneously in the same vertical plane, from the same point, both with different speeds and at different angles with horizontal. The path followed by one, as seed by the other, is

A projectile is thrown in the upward direction making an angle of ${60^0}$ with the horizontal direction with a velocity of $147\mathrm{m}{\mathrm{s}^{ - 1}}$. Then the time after which its inclination with the horizontal is ${45^0}$, is