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

1. \(100~\text{N}, ~175~\text{N}\)
2. \(86.6~\text{N}, ~100~\text{N}\)
3. \(100~\text{N}, ~86.6~\text{N}\)
4. \(100~\text{N}, ~0~\text{N}\)

A man rows a boat at a speed of $\mathrm{}$\(18~\text{km/hr}\) in the northwest direction. The shoreline makes an angle of $\mathrm{}$\(15^{\circ}\) south of west. The component of the velocity of the boat along the shoreline is:
1. \(9~\text{km/hr}\)
2. \(18\frac{\sqrt{3}}{2}~\text{km/hr}\)
3. \(18\cos 15^{\circ}~\text{km/hr}\)
4. \(18\cos 75^{\circ}~\text{km/hr}\)

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 \(y = a+bt+ct^2-dt^4\). The initial velocity and initial acceleration, respectively, are: \(\left(\text{Given:}~ v=\frac{dx}{dt}~\text{and}~a=\frac{d^2x}{dt^2}\right)\)
1. \(b, -4d\)
2. \(-d, 2c\)
3. \(b, 2c\)
4. \(2c, -4d\)

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

A body is moving according to the equation \(x = at +bt^2-ct^3\) where \(x\) represents displacement and \(a, b~\text{and}~c\) are constants. The acceleration of the body is: (\(\text{Given:}~ a=\frac{d^2x}{dt^2}\))
1. \(a+ 2bt\)
2. \(2b+ 6ct\)
3. \(2b- 6ct\)
4. \(3b- 6ct^2\)

A particle moves along the X-axis so that its X coordinate varies with time t according to the equation $\mathrm{x}=\left(2-5\mathrm{t}+6{\mathrm{t}}^2\right)\mathrm{m}$. The initial velocity of the particle is: \(\left(\text{Given;}~ v=\frac{dx}{dt}\right)\)
1. -5 m/s
2. 6 m/s
3. 3 m/s
4. 4 m/s

The maximum value of the function \(7 + 6 x - 9 x^{2}\) is:
1. \(8\)
2. \(-8\)
3. \(4\)
4. \(-4\)

If \(f \left(x\right) = x^{2} - 2 x + 4\), then \(f(x)\) has:

The resultant of the forces \(\overrightarrow {P}\) and \(\overrightarrow {Q}\) is \(\overrightarrow {R}\). If \(\overrightarrow {Q}\) is doubled, then the resultant also doubles in magnitude. Find the angle between \(\overrightarrow {P}\) and \(\overrightarrow {Q}\)?
1. \(\cos\theta = \frac{Q}{2P}\)
2. \(\cos\theta = \frac{-4Q}{3P}\)
3. \(\cos\theta = \frac{-2Q}{3P}\)
4. \(\cos\theta = \frac{-3P}{4Q}\)