For the given figure, which of the following is true?

                          

1. $\overrightarrow{\mathrm{B}}$ = $\overrightarrow{\mathrm{A}}$ + $\overrightarrow{\mathrm{C}}$

2. $\overrightarrow{\mathrm{A}}$ = $\overrightarrow{\mathrm{B}}$ + $\overrightarrow{\mathrm{C}}$

3. $\overrightarrow{\mathrm{C}}$ = $\overrightarrow{\mathrm{A}}$ + $\overrightarrow{\mathrm{B}}$

4. All of these

The value of the unit vector, which is perpendicular to both A = \(\hat{i} + 2\hat{j} + 3 \hat{k}\) and B = \(\hat{i} - 2\hat{j} - 3 \hat{k}\) $\mathrm{is\; equal\; to}$:

1.  $\frac{\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}}{6}$

2.  $\frac{6\hat{\mathrm{j}} - 4\hat{\mathrm{k}}}{\sqrt{52}}$

3.  $\frac{6\hat{\mathrm{j}} + 4\hat{\mathrm{k}}}{\sqrt{52}}$

4.  $\frac{2\hat{\mathrm{i}} - \hat{\mathrm{j}}}{\sqrt{5}}$

The area of a blot of ink, A, is growing such that after t seconds, \(A=3t^2+\frac{t}{5}+7~m^2\). Then the rate of increase in the area at t= 5s will be :

1. 30.1 m²/s

2. 30.2 m²/s

3. 30.3 m²/s

4. 30.4 m²/s

The current in a circuit is defined as $I=\frac{dq}{dt}$. The charge (q) flowing through a circuit, as a function of time (t), is given by $q=5t^2-20t+3$. The minimum charge flows through the circuit at:

1. t=4 sec

2. t=2 sec

3. t= 6 sec

4. t=3 sec

If a vector is inclined at angles, $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{and} \mathrm{\gamma }$ with x, y, and z-axis respectively, then the value of ${\sin }^2 \mathrm{\alpha } + {\sin }^2 \mathrm{\beta }+ {\sin }^2 \mathrm{\gamma }$ is equal to

1. 0

2. 1

3. 2

4. $\frac{1}{2}$

If $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors inclined to each other at an angle $\theta$, then the component of $\overrightarrow{\mathrm{A}}$ perpendicular to $\overrightarrow{\mathrm{B}}$ and lying in the plane containing $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ will be:

1.  $\frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

2.  $\overrightarrow{\mathrm{A}} - \frac{\overrightarrow{\mathrm{A}}\overrightarrow{\mathrm{.B}}}{{\mathrm{B}}^2}\overrightarrow{\mathrm{B}}$

3.  $\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}$

4.  $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$

If $\left|\overrightarrow{\mathrm{A}}\right| \ne \left|\overrightarrow{\mathrm{B}}\right|$ and $\left|\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}\right|$ = $\left|\overrightarrow{\mathrm{A}} · \overrightarrow{\mathrm{B}}\right|$, then 

1.  $\overrightarrow{\mathrm{A}} \perp \overrightarrow{\mathrm{B}}$

2.  $\overrightarrow{\mathrm{A}} \parallel \overrightarrow{\mathrm{B}}$

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

4.  $\overrightarrow{\mathrm{A}}$ is inclined to $\overrightarrow{\mathrm{B}}$ at an angle of 45$°$

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

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

1.  1

2.  2

3.  3

4.  4

A block of weight W is supported by two strings inclined at 60° and 30° to the vertical. The tensions in the strings are ${\mathrm{T}}_1 \mathrm{and} {\mathrm{T}}_2$ 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.