What is the maximum value of \(5\sin\theta-12\cos\theta\)?

1. \(12\)

2. \(17\)

3. \(7\)

4. \(13\)

A block of weight \(W\) is supported by two strings inclined at \(60^{\circ}\) and \(30^{\circ}\) to the vertical. The tensions in the strings are \(T_1\) and \(T_2\) as shown. If these tensions are to be determined in terms of \(W\) using the triangle law of forces, which of these triangles should you draw? (block is in equilibrium):

1.		2.
3.		4.

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. \(0\)

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

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 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. \(y_{\text{max}}= 4\)$\mathrm{}$

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

If \(y = t^3+1\) and \(x = t^2+3,\) what is the value of \(\dfrac{dy}{dx}?\)
1. \(\dfrac{t^2}{3}\)
2. \(\dfrac{t}{2}\)
3. \(\dfrac{3t}{2}\)
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. \(34\) ${\mathrm{m}}^3$