Motion in plane practice · Mixed difficulty

The area of triangle whose adjacent sides are represented by the vectors $(4i + 3j + 4k)$ and $5\bar i$ in sq. unit is

Three vectors $A={a} \hat{{i}}+\hat{{j}}+\hat{{k}} ; {B}=\hat{{i}}+{b} \hat{{j}}+\hat{{k}}$ and $\mathrm{C}=\hat{{i}}+\hat{{j}}+\mathrm{c} \hat{{k}}$ are mutually perpendicular $(\hat{{i}}, \hat{{j}}$ and $\hat{{k}}$ are unit vectors along $X, Y$ and $Z$ - axis respectively). The respective values of $a, b$ and $c$ are

A projectile is thrown upward with a velocity $v_0$ at an angle $\alpha$ to the horizontal. The change in velocity of the projectile when it strikes the same horizontal plane is

Let the cosine of angle between the vectors $\mathbf{p}$ and $\mathbf{q}$ be $\lambda$ such that $2 \mathbf{p}+\mathbf{q}=\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\mathbf{p}+2 \mathbf{q}=\hat{\mathbf{i}}-\hat{\mathbf{j}}$, then $\lambda$ is equal to

The range of a projectile is $R$ when the angle of projection is $40^{\circ}.$ For the same speed of projection and same range, the other possible angle of projection is:

When two vectors $\vec{A}$ and $\vec{B}$ of magnitude $a$ and $b$ are added, the magnitude of the resultant vector is always

Vector $\mathbf{P}=6 \hat{\mathbf{i}}+4 \sqrt{2} \hat{\mathbf{j}}+4 \sqrt{2} \hat{\mathbf{k}}$ makes an angle with $Z$-axis which is equal to

The resultant of two forces $P$ and $Q$ is of magnitude $P$. If $P$ be doubled, the resultant will be inclined to $Q$ at an angle.

Let two non-collinear vectors $\mathbf{a}$ and $\mathbf{b}$ inclined at an angle $\frac{2 \pi}{3}$ be such that $|\mathbf{a}|=3$ and $|\mathbf{b}|=2$. If a point $P$ moves so that at any time $t$ its position vector $\mathbf{O P}$ (where $O$ is the origin) is given as $\mathrm{OP}=\left(t+\frac{1}{t}\right) \mathbf{a}+\left(t-\frac{1}{t}\right) \mathbf{b}$, then least distance of $P$ from the origin is

A body of mass $10 \mathrm{~kg}$ is projected at an angle of $45^{\circ}$ with the horizontal. The trajectory of the body is observed to pass through a point $(20,10)$. If $T$ is the time of flight, then its momentum vector, at time $t=\frac{T}{\sqrt{2}}$, is ______. [Take $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

Four projectiles are fixed with the same velocities at angles $25^{\circ}, {4 0}^{\circ}, 55^{\circ}$ and $70^{\circ}$ with the horizontal. The range of projectile will be largest for the one projected at angle

From a point on the ground, a particle is projected slantwise with some initial velocity making an acute angle with the horizontal. During its flight, the dot product of its velocity and acceleration vectors

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

Surface area is

If $O A B C$ is a tetrahedron such that $\mathrm{OA}^2+\mathrm{BC}^2$ $=\mathrm{OB}^2+\mathrm{CA}^2=\mathrm{OC}^2+\mathrm{AB}^2$, then which of the following is not true?

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.

Two vectors are given by $\vec{A}=(3 \hat{i}+\hat{j}+3 \hat{k})$ and $\vec{B}=(3 \hat{ i }+5 \hat{ j }-2 \hat{ k })$. Find the third vector $\vec{ C }$ if $\vec{ A }+3 \vec{ B }-\vec{ C }=0$

An aeroplane is flying at a height of 1960 m in horizontal direction with a velocity of 360 km / hr. When it is vertically above the point. A on the ground, it drops a bomb. The bomb strikes a point B on the ground, then the time taken by the bomb to reach the ground 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

If the range of oblique projectile is ${R}$, then its K.E. is maximum after (from start) a displacement equal to

There are two values of time for which a projectile is at the same height, The sum of these two times is equal to ($T =$ time of fight of the projectile)

What is the angle between $\mathbf{P}$ and the cross product of $(\mathbf{P}+\mathbf{Q})$ and $(\mathbf{P}-\mathbf{Q})$ ?

Assertion: The sum of squares of cosines of angles made by a vector with $\mathrm{X}, \mathrm{Y}$ and $\mathrm{Z}$ axes is equal to unity.
Reason: A vector making $45^{\circ}$ with $\mathrm{X}$-axis must have equal components along $\mathrm{X}$ and $\mathrm{Y}$-axes.

If a,b and $\mathbf{c}$ are three non-coplanar uni-modular vectors, each inclined with other at an angle $30^{\circ}$, then volume of tetrahedron whose edges are $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ is

A particle reaches its highest point when it has covered exactly one half of its horizontal range. The corresponding point on the displacement-time graph is characterised by

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

Some persons hire a boat for 4 hours. The river flows at a speed of $5 \mathrm{~km} / \mathrm{hr}$ and the boat moves with speed of $15 \mathrm{~km} / \mathrm{hr}$ relative to the water. How far along the flow direction, they can go if they have to return in 4 hrs?

Which of the following conditions are sufficient and essential for a quantity to be a vector?

If $u$ is the initial velocity of a projectile and $v$ is the velocity at any instant, then the maximum horizontal range $\mathrm{R}_{\mathrm{m}}$ is equal to

Position of a particle at any instant $t$ is given $\vec{r}=3 \hat{i}+2 t^2 \hat{j}+5 \hat{k}$, Its velocity at same instant will be: