The linear velocity of a rotating body is given by , where is the angular velocity and r is the radius vector. The angular velocity of a body, and their radius vector is will be:
1.
2.
3.
4.
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 \(x = at^2-bt^3\).
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 . 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:
1. | \(x=1\). | a minimum at
2. | \(x=1\). | a maximum at
3. | no extreme point. |
4. | no minimum. |
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. \(\cos\theta = \frac{-3P}{4Q}\)
If \(\overrightarrow{A}={2}\hat{i}+\hat{j}\;{\&}\;\overrightarrow{B}=\hat{i}{-}\hat{j},\) then the components of \(\overrightarrow {A}\) along with \(\overrightarrow {B}\) & perpendicular to \(\overrightarrow {B}\) respectively will be:
1. \(\frac{\hat{i} - \hat{j}}{2} ,~\frac{3}{2}\left(\hat i+\hat j\right)\)
2. \(\frac{\hat{i} - \hat{j}}{2} ,~-\frac{2}{3}\left(\hat i+\hat j\right)\)
3. \(\frac{\hat{i} - \hat{j}}{2} ,~-\frac{3}{2}\left(\hat i-\hat j\right)\)
4. \(\frac{\hat{i} - \hat{j}}{2} ,~\frac{2}{3}\left(\hat i-\hat j\right)\)
If \(\overrightarrow{a}\) is a vector and \(x\) is a non-zero scalar, then which of the following is correct?
1. | \(x\overrightarrow{a}\) is a vector in the direction of \(\overrightarrow{a}\). |
2. | \(x\overrightarrow{a}\) is a vector collinear to \(\overrightarrow{a}\). |
3. | \(x\overrightarrow{a}\) and \(\overrightarrow{a}\) have independent directions. |
4. | \(x\overrightarrow{a}\) is a vector perpendicular to \(\overrightarrow{a}\). |