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A particle moves in the XY plane and at time t is at the point whose coordinates are $\left({\mathrm{t}}^2, {\mathrm{t}}^3-2\mathrm{t}\right)$. Then at what instant of time, will its velocity and acceleration vectors be perpendicular to each other?
(1)  1/3 sec
(2)  2/3 sec
(3)  3/2 sec
(4)  never

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

A motor boat of mass m moving along a lake with velocity ${\mathrm{V}}_0$. At t=0, the engine of the boat is shut down. Magnitude of resistance force offered to the boat is equal to rV. (V is instantaneous speed). What is the total distance covered till it stops completely? $\left[\mathrm{Hint}: \mathrm{F}\left(\mathrm{x}\right)=\mathrm{mV}\frac{\mathrm{dV}}{\mathrm{dx}}=-\mathrm{rV}\right]$
(1)  ${\mathrm{mV}}_0/\mathrm{r}$
(2)  $3 {\mathrm{mV}}_0/2\mathrm{r}$
(3)  ${\mathrm{mV}}_0/2\mathrm{r}$
(4)  $2{\mathrm{mV}}_0/\mathrm{r}$

If a curve is governed by the equation y = sinx, then the area enclosed by the curve and x-axis between x = 0 and x =$\mathrm{ \pi }$ is (shaded region):
              
1. \(1\) unit
2. \(2\) units
3. \(3\) units
4. \(4\) units

The dimension of ${\int }_u^{v}mvdv$ is
(1) $\left[M^1{L}^{-2}{T}^{-2}\right]$
(2) $\left[M^1{L}^2{T}^{-2}\right]$
(3) $\left[M^1{L}^1{T}^{-1}\right]$
(4) None of these

Component of $3\hat{i}+4\hat{j}$ perpendicular to $\hat{i}+\hat{j}$ and in the same plane as that of $3\hat{i}+4\hat{j}$ is:
1. $\frac{1}{2}(\hat{j}-\hat{i})$
2. $\frac{3}{2}(\hat{j}-\hat{i})$
3. $\frac{5}{2}(\hat{j}-\hat{i})$
4. $\frac{7}{2}(\hat{j}-\hat{i})$

If two force vectors $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ of the same magnitude are arranged in the following manner, the magnitude of resultant is maximum for?
                    
(1) (I)
(2) (II)
(3) Both (II) & (III)
(4) Both (I) & (II)

If vector $\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{R}}$ have magnitude 5, 12 and 13 units and $\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{Q}}=\overrightarrow{\mathrm{R}}$, then the angle between $\overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{R}}$ will be:
1.  ${\cos }^{-1} \frac{5}{12}$
2.  ${\cos }^{-1} \frac{5}{13}$
3.  ${\cos }^{-1} \frac{12}{13}$
4.  ${\cos }^{-1} \frac{2}{13}$

A particle travels with speed \(50\) m/s from the point \((3,7)\) in a direction \(7\hat i -24 \hat j.\) Find its position vector after \(3\) seconds.
1. \(151\hat i+45\hat j\)
2. \(45\hat i-137\hat j\)
3. \(151\hat i-45\hat j\)
4. \(4.5\hat i-151\hat j\)

Two forces A and B have a resultant $R_1$. If B is doubled, the new resultant $R_2$ is perpendicular to A. Then
1. $R_1=A$
2. $R_1=B$
3. $R_2=A$
4. $R_2=B$

Which of the sets given below may represent the magnitude of resultant of three vectors adding to zero?
(1)  2, 4, 8
(2)  4, 8, 16
(3)  1, 2, 1
(4)  0.5, 1, 2

The angle that the vector $\mathrm{A}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}$ makes with the y-axis is:
(1)  ${\tan }^{-1} 3/2$
(2)  ${\tan }^{-1} 2/3$
(3)  ${\sin }^{-1} 2/3$
(4)  ${\cos }^{-1} 3/2$

Two constant forces ${\overrightarrow{\mathrm{F}}}_1=2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+3\hat{\mathrm{k}} \left(\mathrm{N}\right)$ and ${\overrightarrow{\mathrm{F}}}_2=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2\hat{\mathrm{k}} \left(\mathrm{N}\right)$ act on a body and displace it from the position ${\overrightarrow{\mathrm{r}}}_1=\hat{\mathrm{i}}+2\hat{\mathrm{j}}-2\hat{\mathrm{k}} \left(\mathrm{m}\right)$ to the position ${\overrightarrow{\mathrm{r}}}_2=7\hat{\mathrm{i}}+10\hat{\mathrm{j}}+5\hat{\mathrm{k}} \left(\mathrm{m}\right)$. What is the work done W? $[Given : W =\overrightarrow{F}.∆\overrightarrow{r}]$
(A)  9 Joule
(B)  41 Joule
(C)  -3 Joule
(D)  None of these

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. $\sqrt{41} \mathrm{units}$

If $\mathrm{\theta }$ is the angle between vectors $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$, then which of the following is the unit vector perpendicular to $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$?
1. $\frac{\hat{\mathrm{A}} \times \hat{\mathrm{B}}}{\mathrm{AB} \sin \mathrm{\theta }}$
2. $\frac{\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}}{\mathrm{AB} \cos \mathrm{\theta }}$
3. $\frac{\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}}{\mathrm{AB} \sin \mathrm{\theta }}$
4. $\frac{\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}}{\mathrm{AB} }$