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

1.	$\overrightarrow B$ must be perpendicular to $\overrightarrow C$
2.	$\overrightarrow A$ must be perpendicular to $\overrightarrow C$
3.	Component of $\overrightarrow C$ along $\overrightarrow A$ = Component of $\overrightarrow D$ along $\overrightarrow A$
4.	Component of $\overrightarrow C$ along $\overrightarrow A$  = - (Component of $\overrightarrow D$ along $\overrightarrow A$)

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. $34$ ${\mathrm{m}}^3$

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: