Motion in plane practice · Mixed difficulty

Two particles are projected with same speed but at angles of projection $\left( {{{45}^0} - \theta } \right)\,and\,\left( {{{45}^0} + \theta } \right)$ Then their horizontal ranges are in the ratio of

Two particles are projected from the same point with the same speed, at different angles $\theta_1$ and $\theta_2$ to the horizontal. They have the same horizontal range. Their times of flight are $t_1$ and $t_2$ respectively incorrect statement is.

The angle made by the vector $A = \hat i + \hat j$ with x – axis is

Statement-I: In $\Delta \mathrm{abc}, \overrightarrow{\mathrm{ab}}+\overrightarrow{\mathrm{bc}}+\overrightarrow{\mathrm{ca}}=0$
Statements-II: If $\overrightarrow{\mathrm{oa}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{ob}}=\overrightarrow{\mathrm{b}}$, then $\overrightarrow{\mathrm{ab}}=\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ (triangle law of addition)

The vector which can give unit vector along x-axis with $\mathrm{\vec A} = 2\hat i - 4\hat j + 7\hat k,\,\mathrm{\;\vec B} = 7\hat i + 2\hat j - 5\hat k$ and $\mathrm{\vec C} = - 4\hat i + 7\hat j + 3\hat k$ is

Unit vector

I. has dimensions and a unit.

II. when multiplied by a scalar quantity, it results a scalar.

Which of the following statement(s) is/are incorrect?

Statement-I: $\vec{a}=\hat{i}+p \hat{j}+2 \hat{k}$ and $\vec{b}=2 \hat{i}+3 \hat{j}+q \hat{k}$ are parallel vectors if $p=\frac{3}{2}, q=4$.
Statement-II: If $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$ are parallel, then $\frac{a_1}{b_1}=\frac{a_2}{b_2}-\frac{a_3}{b_3}$

Identify the vector quantity among the following

Position of a particle in a rectangular coordinate system is $(3,2,5)$. Then, its position vector will be:

Can scalar product be ever negative?

Two bodies are projected from roof with same speed in different directions. If air resistance is not taken into account then

A particle is projected at an angle of ${60^0}$ above the horizontal with a speed of $10\mathrm{\;m}{\mathrm{s}^{ - 1}}$. After some time the direction of its velocity makes an angle of ${30^0}$ above the horizontal. The speed of the particle at this instant is

If $\mathbf{A}$ and $\mathbf{B}$ are two non-zero vectors having equal magnitude, the angle between the vectors $\mathbf{A}$ and $\mathbf{A}-\mathbf{B}$ is

If $\mathbf{A} \cdot \mathbf{B}=0$ and $\mathbf{A} \times \mathbf{B}=1$, then $A$ and $B$ are

A particle moves in XY-plane with $x$ and $y$ varying with time $t$ as $x(t)=5 t, y(t)=5 t(27-$ $\left.t^2\right)$. At what time in seconds, the direction of velocity and acceleration will be perpendicular to each other?

Three particles A, B and C are projected from the same point with the same initial speeds making angles ${30^0},{45^0}\,and\,{60^0}$ respectively with the horizontally. Which of the following statements is correct?

The area of the parallelogram represented by the vectors. $\mathrm{\vec A} = 4\hat i + 3\hat j\,\,and\,\mathrm{\vec B} = 2\hat i + 4\hat j$ is

If $\frac{1}{{\sqrt 3 }}\left| {\vec a \times \vec b} \right| = \overrightarrow a \cdot \overrightarrow b$ then angle between $\overrightarrow a$ and $\overrightarrow b$ is

Two projectiles thrown at $30^{\circ}$ and $45^{\circ}$ with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is:

If angle between two vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ rad then angle between vectors $2 \vec{a}$ and $4 \vec{b}$ is

Two particles are projected from the same point on ground simultaneously with speed $20 \mathrm{~m} / \mathrm{s}$ and $\frac{20}{\sqrt{3}} \mathrm{~m} / \mathrm{s}$ at angles $30^{\circ}$ and $60^{\circ}$ respectively with the horizontal in the same direction. The maximum distance between them till both of them strike the ground is approximately $({g}=10 \mathrm{~m} / \mathrm{s}^2)$

If none of the vectors $\vec{A}, \vec{B}$ and $\vec{C}$ are zero and if $\vec{A} \times \vec{B}=0$ and $\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}}=0$, then the value of $|\vec{A} \times \vec{C}|$ is

Given $\vec{A}+\vec{B}+\vec{C}+\vec{D}=\overrightarrow{0}$, which of the following statements is not correct?

The angle between the two vectors $\vec A\, = \,5\hat i\, + \,5\hat j$ and $\vec B\, = \,5\hat i\, - \,5\hat j$ will be

The unit vector in the direction of resultant of two vectors with components (2,4,-5) and (1,2,3) is

If ${\mathrm{\vec A}_1}\,and\,{\mathrm{\vec A}_2}$ are two non-collinear unit vectors and if $\left| {{{\mathrm{\vec A}}_1} + {{\mathrm{\vec A}}_2}} \right| = \sqrt 3$, then the value of $\left( {{{\mathrm{\vec A}}_1} - {{\mathrm{\vec A}}_2}} \right).\left( {2{{\mathrm{\vec A}}_1} + {{\mathrm{\vec A}}_2}} \right)$

Given two vector $\bar A = 2i + 6j + 4k$ and $\bar B = i - 2j + 8k$. The magnitude of their resultant is

A stone projected with a velocity $u$ at angle $\theta$ with the horizontal reaches a maximum height. When it is projected with velocity $u$ at an angle $\left( {\pi /2 - \theta } \right)$ with the horizontal range of the projectile ${H_1}$ and ${H_2}$ is

The horizontal range is four times the maximum height attained by a projectile. The angle of projection is

If $\frac{1}{{\sqrt 3 }}\left| {\vec a \times \vec b} \right| = \overrightarrow a \cdot \overrightarrow b$ then angle between $\overrightarrow a$ and $\overrightarrow b$ is