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

1. \(12\)

2. \(17\)

3. \(7\)

4. \(13\)

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 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. \(34\) ${\mathrm{m}}^3$

If \(\overrightarrow{A} \times \overrightarrow{B} = \overrightarrow{C} + \overrightarrow{D}\), then which of the following statement is correct?

Given below are two statements: 

Given below are two statements: 

Given are two vectors, \(\overrightarrow{A} =   \left(\right. 2 \hat{i}   -   5 \hat{j}   +   2 \hat{k} \left.\right)\) and \(\overrightarrow{B} =   \left(4 \hat{i}   -   10 \hat{j}   +   c \hat{k} \right).\) What should be the value of \(c\) so that vector \(\overrightarrow A \) and \(\overrightarrow B\) would becomes parallel to each other?
1. \(1\)
2. \(2\)
3. \(3\)
4. \(4\)

Two forces of the same magnitude are acting on a body in the East and North directions, respectively. If the body remains in equilibrium, then the third force should be applied in the direction of:

1. North-East

2. North-West

3. South-West

4. South-East

If $\overrightarrow{\mathrm{R}}$ is the resultant of two vectors $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ and $\overrightarrow{\mathrm{R}}$' is the difference in them, and $\left|\overrightarrow{\mathrm{R}}\right| = \left|\overrightarrow{\mathrm{R}}\text{'}\right|$, then:

(1) $\overrightarrow{\mathrm{A}} \parallel \overrightarrow{\mathrm{B}}$

(2) $\overrightarrow{\mathrm{A}} \perp \overrightarrow{\mathrm{B}}$

(3) $\overrightarrow{\mathrm{A}}$ is antiparallel to $\overrightarrow{\mathrm{B}}$

(4) $\overrightarrow{\mathrm{A}}$ makes an angle of 120° with $\overrightarrow{\mathrm{B}}$

If \(\left|\overrightarrow A\right|\ne \left|\overrightarrow B\right|\) and \(\left|\overrightarrow A \times \overrightarrow B\right|= \left|\overrightarrow A\cdot \overrightarrow B\right|\), then: