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

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 5sinθ – 12cosθ?

1.  12

2.  17

3.  7

4.  13

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.  

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 \(\vec{A} = 2\hat{i} + \hat{j} - \hat{k}\) ,  \(\vec{B} = \hat{i} + 2\hat{j} + 3\hat{k}\) , and \(\vec{C} = 6 \hat{i} - 2\hat{j} - 6\hat{k}\) , then the angle between \((\vec{A} + \vec{B})\) and \(\vec{C}\) will be

1.  30°

2.  45°

3.  60°

4.  90°

The maximum or the minimum value of the function y = 25$x^2$+ 5 – 10x is:

1. y_min = 4

2. y_max = 8

3. y_min = 8

4. y_max = 4

The unit vector perpendicular to vectors $\overrightarrow{\mathrm{a}}=\left(3\hat{\mathrm{i}}+\hat{\mathrm{j}}\right) \mathrm{and} \overrightarrow{\mathrm{b}}=\left(2\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}}\right)$ is

1. $\pm \frac{(\hat{\mathrm{i}}-3\hat{\mathrm{j}}+\hat{\mathrm{k}})}{\sqrt{11}}$

2. $\pm \frac{\left(3\hat{\mathrm{i}}+\hat{\mathrm{j}}\right)}{\sqrt{11}}$

3. $\pm \frac{(2\hat{\mathrm{i}}-\hat{\mathrm{j}}-5\hat{\mathrm{k}})}{\sqrt{30}}$

4. None of these