Motion in plane practice · Mixed difficulty

The unit vectors along the three co-ordinate axes are related as

Statement I Two forces (P+Q) and (P-Q) where $P \bot Q$, when act at an angle ${\theta _1}$ to each other, the magnitude of their resultant is $\sqrt {3({P^2} + {Q^2})} ,$ when they act at an angle ${\theta _2},$ the magnitude of their resultant becomes $\sqrt {2({P^2} + {Q^2})}$. This is possible only when ${\theta _1} < {\theta _2}.$
Statement II In the situation given above. ${\theta _1} = {60^ \circ }\,\,\mathrm{and}\,\,{\theta _2} = {90^ \circ }$
In the light of the above statements, choose the most appropriate answer from the options given below.

$\left( {\mathrm{\vec P} + \mathrm{\vec Q}} \right)$ is a unit vector along X-axis. If $\mathrm{\vec P} = \hat i - \hat j + \hat k,$ then what value is $\mathrm{\vec Q}?$

The magnitudes of a set of 3 vectors are given below. The set of vectors for which the resultant cannot be zero is

A bomb is dropped from an aeroplane moving horizontally at constant speed. When air resistance is taken into consideration, the bomb

Consider the following statements and select the correct statements from the following.
I. Addition and subtraction of scalars make sense only for quantities with same units
II. Multiplication and division of scalars with different units is possible
III. Addition, subtraction, multiplication and division of scalars with same unit is possible

A ship $A$ sailing due east with a velocity of $10 km / h$ happens to appear sailing due north with a velocity of $5 km / h$, to a person, sitting in a moving ship $B.$ Determine the velocity (absolute) of ship $B.$

Assertion : A shell fired from a gun is moving along the parabolic path. If it explodes at the top of the trajectory, then no part of the shell can fly vertically.
Reason : The vertical momentum of the shell at the top of the trajectory is zero.

A mosquito is moving with a velocity $V\, = \,(0.5\,{t^2}\hat i\, + \,3\,t\hat j\, + \,9\hat k)\,m/s$ and accelerating in uniform conditions. What will be the directions of mosquito after 2s?

If $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, then $\mathbf{a} \times \mathbf{b}$ is equal to

The ranges and heights for two projectiles projected with the same initial velocity at angles ${42^\mathrm{o}}\,\,\mathrm{and }\,\mathrm{4}{\mathrm{8}^\mathrm{o}}$ with the horizontal are ${R_1},{R_2}\mathrm{ }\,\,\mathrm{and }\,{H_1},{H_2},$ respectively. Choose the correct option.

Assertion: Three projectiles are moving in different paths in the air. Vertical component of relative velocity between any of the pair does not change with time as long as they are in air. (Neglecting the effect of air friction)
Reason: Relative acceleration between any of the pair of projection is zero.

Out of the following quantities, which is scalar?

Assertion At highest point of a projectile, dot product of velocity and acceleration is zero.

Reason At highest point, velocity and acceleration are mutually perpendicular.

If the angle between two forces increases, the magnitude of their resultant

The sum of two unit vectors is also a vector of unit magnitude, then the magnitude of the difference of the two unit vectors is

A projectile is fired with a velocity v at right angle to the slope which is inclined at an angle $\theta$ with the horizontal. What is the time of flight?

At the point of a projectile motion, acceleration and velocity are perpendicular to each other?

Given that $\mathrm{\vec A} + \mathrm{\vec B} + \mathrm{\vec C} = 0$. Out of three vectors, two are equal in magnitude and the magnitude of third vector is&nbsp;$\sqrt 2$ times that of either of the two having equal magnitude. Then the angles between vectors are given by

Given $\theta$ is the angle between $\mathrm{\vec A}\,and\,\mathrm{\vec B}$. Then $\left| {\hat A \times \hat B\;} \right|$ is equal to

A river flows with a speed more than the maximum speed with which a person can swim in still water. He intends to cross the river by the shortest possible path (i.e., he wants to reach the point on the opposite bank which directly opposite to the starting point). Which of the following is correct?

Given $\mathrm{\vec c} = \mathrm{\vec a} \times \mathrm{\vec b}$. The angle which $\mathrm{\vec a}$ makes with $\mathrm{\vec c}$ is

If $\vec A = \hat i + \hat j + \hat k$ and $\vec B = 2\hat i - \hat j + 4\hat k$ then the unit vector along $\vec A + \vec B$ is

Statement-I: If $\vec{a}$ is perpendicular to $\vec{b}$ and $\overrightarrow{{c}}$, then $\overrightarrow{{a}} \times(\overrightarrow{{b}} \times \overrightarrow{{c}})=0$
Statement-II: If $\overrightarrow{{b}}$ is perpendicular to $\overrightarrow{{c}}$, then $\overrightarrow{{b}} \times \overrightarrow{{c}}=0$

Assertion: If $|\vec{A}+\vec{B}|=| \vec{A}-\vec{B} \mid$, then the angle between $\vec{A}$ and $\vec{B}$ is $90^{\circ}$.
Reason: $\vec{A}+\vec{B}=\vec{B}+\vec{A}$

A force $2\hat i + \hat j - \hat k$ newton acts on a body which is initially at rest. If the velocity of the body at the end of 20 seconds is $4\hat i + 2\hat j + 2\hat k\,\,m{s^{ - 1}}$, the mass of the body.

The angle between the diagonal of a cube with edges of unit length is

Two bullets are fired horizontally with different velocities from the same height. Which will reach the ground first?

For a particle projected with speed of 20 m/s from ground, the kinetic energy at top most point of trajectory is $\left( {\frac{1}{4}} \right)th$ of the initial kinetic energy. The angle of projection from horizontal is

The unit vector parallel to the resultant of vectors $\overrightarrow A = 4\hat i + 3\hat j + 6\hat k$ and $\overrightarrow B = - \hat i + 3\hat j - 8\hat k$ is