Find \(\frac{dy}{dx}, \) if \(y = t^3+1\) and \(x = t^2+3\):
1. \(\frac{t^2}{3}\)
2. \(\frac{t}{2}\)
3. \(\frac{3t}{2}\)
4. \(t^2\)

The velocity of a body moving along the \(x\)-axis varies with \(x\) as \(v   =   \left(x^{3} -x^{2}\right)\) m/s. Find the acceleration of the body at \(x= 2~\text{m}\), if the acceleration is defined as \(a = v\frac{dv}{dx}\).
1. \(132~\text{m/s}^2\)
2. \(32~\text{m/s}^2\)
3. \(8~\text{m/s}^2\)
4. \(4~\text{m/s}^2\)

The volume flow rate of water flowing out of a tubewell is given by \(Q = \left( 3 t^{2}- 4 t +1\right)~\text{m}^3/\text{sec}  \). What volume of water will flow out of the tubewell in the third second if the volume flow rate is defined as \(Q=\frac{dV}{dt}\)?
1. \(10\) ${\mathrm{m}}^3$
2. \(17\) ${\mathrm{m}}^3$
3. \(36\) ${\mathrm{m}}^3$
4. \(34\) ${\mathrm{m}}^3$

The unit vector perpendicular to vectors \(\overrightarrow a= \left(3 \hat{i}+\hat{j}\right)  \) and \(\overrightarrow B = \left(2\hat i - \hat j -5\hat k\right)\) is:
1. \(\pm \frac{\left(\right. \hat{i} - 3 \hat{j} + \hat{k} \left.\right)}{\sqrt{11}}\)
2. \(\pm \frac{\left(3 \hat{i} + \hat{j}\right)}{\sqrt{11}}\)
3. \(\pm \frac{\left(\right. 2 \hat{i} - \hat{j} - 5 \hat{k} \left.\right)}{\sqrt{30}}\)
4. None of these

The maximum or the minimum value of the function \(y= 25x^{2}-10x +5\) is:
1. \(y_{\text{min}}= 4\)
2. \(y_{\text{max}}= 8\)
3. \(y_{\text{min}}= 8\)
4. \(y_{\text{max}}= 4\)$\mathrm{}$

If \(\overrightarrow {A} = 2\hat{i} + \hat{j} - \hat{k},\) \(\overrightarrow {B} = \hat{i} + 2\hat{j} + 3\hat{k},\) and \(\overrightarrow {C} = 6 \hat{i} - 2\hat{j} - 6\hat{k},\) then the angle between \(\left(\overrightarrow {A} + \overrightarrow{B}\right)\) and \(\overrightarrow{C}\) will be:
1. \(30^{\circ}\)
2. \(45^{\circ}\)
3. \(60^{\circ}\)
4. \(90^{\circ}\)

The value of $\int \frac{1}{x+1}d\mathrm{x\; is:}$

1. ln (\(x\) + 1) + C

2. ${\left(x+1\right)}^{-2}+C$

3. $\frac{1}{{\left(x-1\right)}^2}+C$

4. ln (\(x\) – 1) + C

The magnitude of the resultant of two vectors of magnitude \(3\) units and \(4\) units is \(1\) unit. What is the value of their dot product?

1. \(-12\) units

2. \(-7\) units

3. \(-1\) unit

4. \(0\)

Identify the unit vector in the following:
1. \(\hat i + \hat j\)
2. \(\frac{\hat i}{\sqrt{2}}\)
3. \(\hat k - \frac{\hat i}{\sqrt{2}}\)
4. \(\frac{\hat i +\hat j}{\sqrt{2}}\)

If \(\overrightarrow A= 3\hat i + 4\hat j\) and \(\overrightarrow B = 7\hat i + 24\hat k\), then the vector having the same magnitude as that of \(\overrightarrow {B}\) and parallel to \(\overrightarrow {A}\) is:
1. \(15\hat i + 20\hat j\)
2. \(\frac{7}{5}\hat i + \frac{24}{5}\hat j\)
3. \(20\hat i + 15\hat j\)
4. \(15\hat i + 20\hat k\)