Motion in plane practice · Mixed difficulty

If $\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are given vector. A vector $\mathbf{c}$ which is perpendicular to $Z$-axis satisfying $\mathbf{c} \cdot \mathbf{a}=9$ and $\mathbf{c} \cdot \mathbf{b}=-4$. If inclination of $\mathbf{c}$ with $X$-axis and $\gamma$-axis is $\alpha$ and $\beta$ respectively, then which of the following is not true?

A scalar quantity is one that

Three bodies A, B and C projected with the same initial speed with angles of projections $30^{\circ}, 45^{\circ}$ and $60^{\circ}$ respectively. The body/bodies having maximum range

Two balls are projected simultaneously in the same vertical plane from the same point with velocities $V_1$ and $V_2$ with angles $\theta_1$ and $\theta_2$ respectively with the horizontal. If $V_1 \cos \theta_1=$ $V_2 \cos \theta_2$, the path of one ball as seen from the position of other ball is:

Assertion(A): When $\theta=45^{\circ}$ or $135^{\circ}$, the value of $R$ remains the same, only the sign changes.

Reason(R): $R=\frac{u^2 \sin 2 \theta}{g}$

The number of components in which a vector can be resolved into is

Assertion: Time of flight of a projectile is equal to twice the time taken to attain the maximum height.

Reason: At the highest point of the trajectory, the projectile comes to rest and then begins to fall vertically downward under the effect of gravity.

A man can swim with a speed of $4\,\,km{h^{ - 1}}$ in still water. He crosses a river 1 km wider that flows steadily at $3\,\,km{h^{ - 1}}$ If he makes his strokes normal to the river current how far down the river does he go when he reaches the other bank?

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?

What should be the angle between $\Delta \vec{A}$ and $\vec{A}$ so that $|\Delta \vec{A}|=\Delta|\vec{A}|$ ?

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be non-zero vectors such that $\mathbf{a}$ is perpendicular to $\mathbf{b}$ and $\mathbf{c}$ and $|\mathbf{a}|=1,|\mathbf{b}|=2,|\mathbf{c}|=1, \mathbf{b} \cdot \mathbf{c}=1$ and there is a non-zero vector $\mathrm{d}$ coplanar with $\mathbf{a}+\mathbf{b}$ and $2 \mathbf{b}-\mathbf{c}$ and $\mathbf{d} \cdot \mathbf{a}=1$, then minimum value of $|\mathbf{d}|$ is

Two forces $\vec{F}_1=500 \mathrm{~N}$ due east and $\vec{F}_2=250 \mathrm{~N}$ due north have their common initial point. $\vec{F}_2-\vec{F}_1$ 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)

In the projectile motion, if air resistance is ignored, the horizontal motion is at

If $\overrightarrow a = 3\hat i + \hat j - 4\hat k$ and $\overrightarrow b = 6\hat i + 5\hat j - 2\hat k,$ then $\left| {\overrightarrow a \times \overrightarrow b } \right|$ is

For any two vectors $\vec{A}$ and $\vec{B}$, then $(\vec{A} \times \vec{B})^2=$

The linear velocity of a rotating body is given by $\vec v = \vec \omega \times \vec r$ If $\vec \omega = \hat i - 2\hat j + 2\hat k\,\,{\text{and }}\vec r = 4\hat j - 3\hat k,$ then the magnitude of $\vec v$ is

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

Which of the following is true regarding projectile motion?

If $\mathrm{\vec P} = 2\hat i - 3\hat j + \hat k\,and\,\mathrm{\vec Q} = 3\hat i - 2\hat j,\,then\,\mathrm{\vec P}\mathrm{.\vec Q}$ is

A steam boat goes across a lake and comes back (1) On a quite day when the water is still and (2) On a rough day when there is uniform air current so as to help the journey onward and to impede the journey back. If the speed of the launch on both days was same, in which case it will complete the journey in lesser time

Given, ${P}={A}+{B}$ and $P=A+B$. The angle between ${A}$ and ${B}$ is

Select the correct statements about the football thrown in a parabolic path.
I. At the highest point the vertical component of velocity is zero
II. At the highest point, the velocity of the football, acts horizontally
III. At the highest point, the acceleration of the ball acts vertically downwards

The diagonals of a parallelogram are given by the vectors $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}-3 \hat{j}+4 \hat{k}$. Find the area of the parallelogram.

If $A$ is $3 \times 3$ matrix and $\mathbf{u}$ is a vector. If $A \mathbf{u}$ and $\mathbf{u}$ are orthogonal for all real $\mathbf{u}$, then matrix $A$ is $a$

A particle is projected with an initial velocity $\vec{v}=(2 \hat{i}+4 \hat{j}) \mathrm{m} / \mathrm{s}$. The equation of trajectory of the particle will be $\left(g=10 \mathrm{~m} / \mathrm{s}^2\right)$ (Assume horizontal direction as $x$-axis and vertically upward as positive y-axis)

If the magnitudes of two vectors are 3 and 4 and their scalar product is 6, then find the angle between the two vectors.

Assertion: A projectile is thrown with an initial velocity of $(a \hat{i}+b \hat{j}) \mathrm{m} / \mathrm{s}$. If range of projectile is maximum then ${a}={b}$.
Reason: In projectile motion, angle of projection is equal to $45^{\circ}$ for maximum range condition.

Assertion If a particle is projected vertically upwards with velocity $u$, the maximum height attained by the particle is $h_1$. The same particle is projected at angle $30^{\circ}$ from horizontal with the same speed $u$. Now the maximum height is $h_2$. Thus, $h_1=4 h_2$.

Reason In first case, $v=0$ at highest point and in second case, $v \neq 0$ at highest point.

The angle $\theta$ between the vector $\mathbf{p}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and unit vector along $X$-axis is