Motion in plane practice · Mixed difficulty

Assertion: The magnitude of the resultant of two vectors $\vec{P}$ and $\vec{Q}$ is maximum $(P+Q)$, when the two vectors act in the same direction and minimum $(P-Q)$, when they act in opposite directions.

Reason: The resultant of two vectors can be found by using the relation

$R=\sqrt{P^2+Q^2+2 P Q \cos \theta}$

A boat can travel at a speed of 8 kmph in still water on a lake. In the flowing water of a stream, it can move at 8kmph relative to the water in the stream. If the stream speed is 3kmph, how fast can the boat move past a tree on the shore in travelling (i) upstream (ii) downstream?

Assertion: The time taken by a boat to cover a certain distance upstream is more than the time taken to cover the same distance downstream.

Reason: The speed of the boat is lesser while moving upstream as compared to downstream due to the speed of the stream.

What is the angle between $\left( {i + j} \right)\,$ and $\left( {\bar k + \bar j} \right)$?

Of the vectors given below, the parallel vectors are,

$\begin{aligned} & \vec{A}=6 \hat{i}+8 \hat{j} \\ & \vec{B}=210 \hat{i}+280 \hat{k} \\ & \vec{C}=5.1 \hat{i}+6.8 \hat{j} \\ & \vec{D}=3.6 \hat{i}+6 \hat{j}+48 \hat{k} \end{aligned}$

Given$\mathrm{\vec P}\mathrm{.\vec Q} = 0\,then\,\left| {\mathrm{\vec P} \times \mathrm{\vec Q}} \right|\,is$

The resultant of two vector of equal magnitude is twice of magnitude of either of the vectors the angle between them is

The adjacent sides of a parallelogram are represented by co-initial vectors $2\hat i + 3\hat j\,and\,\hat i + 4\hat j$. The area of the parallelogram is

The $x$ and $y$-coordinates of a particle moving in a plane are given by $x(t)=a \cos (p t)$ and $y(t)=b \sin (p t)$, where $a, b( < a)$ and $p$ are positive constants of appropriate dimensions and $t$ is time. Then, which of the following is not true?

If the vector $\mathrm{\vec A} = 2\hat i + 4\hat j\,\,and\,\,\mathrm{\vec B} = 5\hat i + \mathrm{p\hat j}$ are parallel to each other, the magnitude of $\mathrm{\vec B}$ is

The retardation experienced by a moving motor boat, after its engine is cut off, is given by $\frac{{du}}{{dt}} = - k{v^3},$, where k is a constant. If ${v_0}$ is the magnitude of the velocity at cut-off, the magnitude of the velocity at time t after the cut-off is

Magnitude of component of vector $2 \hat{i}+6 \hat{k}$ along $y$-axis is

If $\overrightarrow A = 2\hat i + 3\hat j + 4\hat k\,\,and\,\,\mathrm{\vec B} = 4\hat i + 3\hat j + 2\hat k$, then angle between $\overrightarrow A$ and $\mathrm{\vec B}$ is

Two stones are projected so as to reach the same distance from the point of projection on a horizontal surface. The maximum height reached by one exceeds the other by an amount equal to half the sum of the height attained by them. Then, angle of projection of the stone which attains smaller height is

A ball is projected with kinetic energy $E$ at an angle of $45^\circ$ to the horizontal. At the highest point during its flight, its kinetic energy will be

If $\vec A.\vec B = \sqrt 3 \left| {\vec A \times \vec B} \right|$ then angle between $\vec A$ and $\vec B$ will be

Two vectors $\mathrm{\vec A}\,and\,\mathrm{\vec B}$ are inclined at an angle $\theta$. Now if the vectors are interchanged then the resultant turns through an angle $\beta$. Which of the following relation is true

Let $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{0},|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{b}}|=4,|\overrightarrow{\mathrm{c}}|=5$
Statement-I: Angle between $\vec{\mathrm{a}}$ and $\vec{\mathrm{b}}$ is $\frac{\pi}{2}$
Statement-II: $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}|^2=|\overrightarrow{\mathrm{a}}|^2+|\overrightarrow{\mathrm{b}}|^2+2|\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}|$

At what angle must the two forces $(x+y)$ and $(x-y)$ act, so that the resultant may be $\sqrt{x^2+y^2}$ ?

Which of the following statement is correct?

Statement-I: Angle between $\hat{i}+\hat{j}$ and $\hat{i}$ is $45^{\circ}$.
Statement-II: $\hat{i}+\hat{j}$ is equally inclined to both $\hat{i}$ and $\hat{j}$ and the angle between $\hat{i}$ and $\hat{j}$ is $90^{\circ}$

Which of the following statement is true?

A body throws a ball with a velocity of 15 m/s at an angle of $15^\circ$ with the horizontal. The distance at which the ball strikes the ground is

If $0.5\hat i + 0.8\hat j + \mathrm{c\hat k}$ is a unit vector, then the value of c is

The angle made by the vector $\vec A = \hat i + \hat j\,$ with y-axis is

A vector is added to an equal and opposite vector of similar nature, forms a

Statement-I: A null vector is a vector whose magnitude is zero and direction is arbitrary.
Statement-II: A null vector does not exist.

A vector perpendicular to the vector $\left( {i + 2j} \right)$ and having magnitude $3\sqrt 5$ units is

At the top of the trajectory of a projectile, the acceleration is

If $\bar A = 5i + 7j - 3k$ and $\bar B = 2i + 2j - ak$ are perpendicular vectors. The value of $a$ is