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

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

1. $12$

2. $17$

3. $7$

4. $13$

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: 

If $\overrightarrow {A}$ $\mathrm{and}$ $\overrightarrow{B}$ are two vectors inclined to each other at an angle $\theta,$ then the component of $\overrightarrow {A}$ perpendicular to $\overrightarrow {B}$ and lying in the plane containing $\overrightarrow {A}$ and $\overrightarrow {B}$ will be:
1. $\frac{\overrightarrow {A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}$
2. $\overrightarrow{A}   -   \frac{\overrightarrow{A} \overrightarrow{.B}}{B^{2}} \overrightarrow{B}$
3. $\overrightarrow{A} -\overrightarrow{B}$
4. $\overrightarrow{A} + \overrightarrow{B}$

A force of $20$ N acts on a particle along a direction, making an angle of $60^\circ$ with the vertical. The component of the force along the vertical direction will be: