If \(\overrightarrow {A}= \overrightarrow{B}+ \overrightarrow{C}\) and the magnitudes of \(\overrightarrow {A}, \overrightarrow {B},\overrightarrow {C}\) are \(5,4\) and \(3\) units, respectively. Then the angle between \(\overrightarrow {A}~\text{and}~\overrightarrow{C}\)is:
1. \(\cos^{-1}\left(\frac{3}{5}\right)\)
2. \(\cos^{-1}\left(\frac{4}{5}\right)\)
3. \(\sin^{-1}\left(\frac{3}{4}\right)\)
4. \(\frac{\pi}{2}\)
For the given figure, which of the following is true?
1. \(\overrightarrow{B}= \overrightarrow{A}+\overrightarrow{C}\)
2. \(\overrightarrow{A}= \overrightarrow{B}+\overrightarrow{C}\)
3. \(\overrightarrow{C}= \overrightarrow{A}+\overrightarrow{B}\)
4. All of these
The value of the unit vector, which is perpendicular to both \(A=\hat{i} + 2\hat{j} + 3 \hat{k}\) and \(B= \hat{i} - 2\hat{j} - 3 \hat{k}\) :
1. \(\frac{\hat{i} + 2 \hat{j} + 3 \hat{k}}{6}\)
2. \(\frac{6\hat{j} -4 \hat{k}}{\sqrt{52}}\)
3. \(\frac{6\hat{j} +4 \hat{k}}{\sqrt{52}}\)
4. \(\frac{2\hat{i} - \hat{j}}{\sqrt{5}}\)
The instantaneous velocity (defined as ) at time of a particle, whose position equation is given as s(t)=12 tanm, is:
1. 12 m/s
2. 12 m/s
3. 6 m/s
4. \(6\sqrt2\) m/s
If the acceleration \(a(t)= 4t+6\), the velocity of a particle starting from rest is: \(\left(\text{here} , a = \frac{d v}{d t}\right)\)
1. \(2t+6\)
2. \(4\)
3. \(0\)
4. \(2t^2+6t\)
Given velocity v(t) = . Assume s(t) is measured in meters and t is measured in seconds. If s(0) = 0, the position s(4) at t = 4s is:
1. | \(30\) | 2. | \(31\) |
3. | \(32\) | 4. | \(33\) |
The current through a wire depends on time as \(i = (2+3t)~\text{A}\).
The charge that crosses through the wire in \(10\) seconds is: \(\left(\text{Instantaneous current,}~i= \frac{dq}{dt} \right)\)
1. \(150~\text{C}\)
2. \(160~\text{C}\)
3. \(170~\text{C}\)
4. None of there
The area of a blot of ink, \(A\), is growing such that after \(t\) seconds, \(A=\left(3t^2+\frac{t}{5}+7\right)\text{m}^2\). Then the rate of increase in the area at \(t = 5~\text{s}\) will be:
1. \(30.1~\text{m}^2/\text{s}\)
2. \(30.2~\text{m}^2/\text{s}\)
3. \(30.3~\text{m}^2/\text{s}\)
4. \(30.4~\text{m}^2/\text{s}\)
A particle starts rotating from rest and its angular displacement is given by \(\theta = \frac{t^2}{40}+\frac{t}{5}\). Then, the angular velocity \(\omega = \frac{d\theta}{dt}\) at the end of \(10~\text{s}\) will be:
1. \(0.7\)
2. \(0.6\)
3. \(0.5\)
4. \(0\)