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. \(\cos\theta = \frac{-3P}{4Q}\)

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?

The vector \(\overrightarrow b\) which is collinear with the vector \(\overrightarrow a = \left(2, 1, -1\right)\) and satisfies the condition \(\overrightarrow a. \overrightarrow b=3\) is:
1. \(\left(1, \frac{1}{2}, \frac{-1}{2}\right)\)
2. \(\left(\frac{2}{3}, \frac{1}{3}, \frac{-1}{3}\right)\)
3. \(\left(\frac{1}{2}, \frac{1}{4}, \frac{-1}{4}\right)\)
4. \(\left(1, 1, 0\right)\)