If a vector \(2\hat{i}+3\hat{j}+8\hat{k}\) is perpendicular to the vector \(-4\hat{i}+4\hat{j}+\alpha \hat{k},\) then the value of \(\alpha\) will be:$\mathrm{}$
1. $-1$
2. $-\frac{1}{2}$
3. $\frac{1}{2}$
4. \(1\)

Six vectors $\overrightarrow{a} through \overrightarrow{f}$ have the directions as indicated in the figure. Which of the following statements may be true?

1. $\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{-f}$                       
2. $\overrightarrow{d}+\overrightarrow{c}=\overrightarrow{f}$
3. $\overrightarrow{d}+\overrightarrow{e}=\overrightarrow{f}$                       
4. $\overrightarrow{b}+\overrightarrow{e}=\overrightarrow{f}$

If the magnitude of the sum of two vectors is equal to the magnitude of the difference between the two vectors, the angle between these vectors is:

1. 90°

2. 45° 

3. 180° 

4. 0°

In the given figure

1.  Angle between $\overrightarrow{\mathrm{A}}$ and $\overrightarrow{\mathrm{B}}$ is $110°$

2.  Angle between $\overrightarrow{\mathrm{C}}$ and $\overrightarrow{\mathrm{D}}$ is $60°$

3.  Angle between $\overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{C}}$ is $110°$

4.  Angle between $\overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{C}}$ is $70°$

A child pulls a box with a force of $200 \mathrm{N}$ at an angle of $60°$above the horizontal. Then the horizontal and vertical components of the force will be:
              

1. 100 N, 175 N

2.  86.6 N, 100 N  

3.  100 N, 86.6 N

4. 100 N, 0 N

A man rows a boat at a speed of $18 \mathrm{km}/\mathrm{hr}$ in the northwest direction. The shoreline makes an angle of $15°$ south of west. The component of the velocity of the boat along the shoreline is:

1.  $9 \mathrm{km}/\mathrm{hr}$

2.  $18\frac{\sqrt{3}}{2} \mathrm{km}/\mathrm{hr}$

3.  $18 \cos 15° \mathrm{km}/\mathrm{hr}$

4.  $18 \cos 75° \mathrm{km}/\mathrm{hr}$

The linear velocity of a rotating body is given by $\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{\omega }}\times \overrightarrow{\mathrm{r}}$, where $\mathrm{\omega }$ is the angular velocity and r is the radius vector. The angular velocity of a body, $\overrightarrow{\mathrm{\omega }}=\hat{\mathrm{i}}-2\hat{\mathrm{j}}+2\hat{\mathrm{k}}$ and their radius vector is  $\overrightarrow{\mathrm{r}}=4\hat{\mathrm{j}}-3\hat{\mathrm{k}}, \mathrm{then\; value\; of} \left|\overrightarrow{\mathrm{v}}\right|$ will be:

1.  $\sqrt{29} \mathrm{units}$

2.  $31 \mathrm{units}$

3.  $\sqrt{37} \mathrm{units}$

4.  $\sqrt{41} \mathrm{units}$

The displacement of a particle is given by $\mathrm{y}=\mathrm{a}+\mathrm{bt}+{\mathrm{ct}}^2-{\mathrm{dt}}^4$. The initial velocity and initial acceleration, respectively, are: (\(Given: v=\frac{dx}{dt}~and~a=\frac{d^2x}{dt^2}\))

1.  b, -4d

2.  -d, 2c

3.  b, 2c

4.  2c, -4d

The position x of the particle varies with time t as $\mathrm{x}={\mathrm{at}}^2-{\mathrm{bt}}^3$.  The acceleration of the particle will be zero at a time equal to: (\(Given: a=\frac{d^2x}{dt^2}\))

1.  $\frac{\mathrm{a}}{\mathrm{b}}$

2.  $\frac{2\mathrm{a}}{\mathrm{b}}$

3.  $\frac{\mathrm{a}}{3\mathrm{b}}$

4.  Zero

A body is moving according to the equation $\mathrm{x}=\mathrm{at}+{\mathrm{bt}}^2-{\mathrm{ct}}^3$ where x represents displacement and a, b and c are constants. The acceleration of the body is: (\(Given: a=\frac{d^2x}{dt^2}\))

1.  $\mathrm{a}+2\mathrm{bt}$

2.  $2\mathrm{b}+6\mathrm{ct}$

3.  $2\mathrm{b}-6\mathrm{ct}$

4.  $3\mathrm{b}-6{\mathrm{ct}}^2$