### Direction Reasoning Problem 1: A certain city has a circular wall around it, and this wall has four gates pointing north, south, east and west. A house stands outside the city, three km north of the north gate, and it can just be seen from a point nine km east of the south gate. What is the diameter of the wall that surrounds the city?

A. 6 km

B. 9 km

C. 12 km

D. None of these

We can represent the information provided in the form of the following diagram: In this diagram, ‘N’ is the north gate, ‘S’ is the south gate, ‘H’ is the house 3 km north of the north gate, ‘A’ is 9 km to the east of the south gate, ‘O’ is the center of the circular well.

Let $r$ be the radius of the circular wall and $d$ be the diameter of the circular wall.

If the house can just be seen from point ‘A’, the line passing through ‘A’ and ‘H’ must be tangent to the circle. We have represented the point of contact of this tangent with the circle by ‘T’.

Since AS and AT are tangents from an external point A, AT = AS = 9

Using Pythagoras theorem for Δ OTH,

OH2 = OT2 + HT2

⇒ HT = $\sqrt{ 9 + 3d}$

In Δ OTH, $\tan(\theta) = \frac{\sqrt{9 + 3d}}{r}​$

In Δ HSA, $\tan(\theta) = \frac{3 + d}{9}​$

Therefore, $\frac{\sqrt{9 + 3d}}{r} = \frac{3 + d}{9}$

Checking the options, only $d$ = 9 satisfies the above equation. So, B is the correct option.

### Direction Reasoning Problem 2: The figure below shows the network connecting cities A, B, C, D, E and F. The arrows indicate the permissible direction of travel. What is the number of distinct paths from A to F? A. 9

B. 10

C. 11

D. None of these

The number of the distinct paths from A to F is 10.

1. A → D → E → F

2. A → D → C → F

3. A → D → C → E → F

4. A → B → F

5. A → B → E → F

6. A → B → C → F

7. A → B → C → E → F

8. A → B → D → E → F

9. A → B → D → C → F

10. A → B → D → C → E → F

Therefore, B is the correct answer.

### Direction Reasoning Problem 3: Consider the following statements:

> There are six villages A, B, C, D, E, and F.

> F is 1 km to the west of D.

> B is 1 km to the east of E.

> A is 2 km to the north of E.

> C is 1 km to the east of A.

> D is 1 km to the south of A.

Which three villages are in a line?

A. A, C, B

B. A, D, E

C. C, B, F

D. E, B, D

The following diagram shows the arrangement of the villages: Clearly, A, D and E are in a straight line. So, B is the correct option.

### Direction Reasoning Problem 4: Location of B is north of A and the location of C is east of A. The distances AB and AC are 5 km and 12 km respectively. The shortest distance (in km) between the locations B and C is

A. 60

B. 13

C. 17

D. 7

Clearly, ABC forms a right-angle triangle with BC as the hypotenuse. The shortest distance between B and C is the length of the hypotenuse which is 13 km. So, B is the correct option.

### Direction Reasoning Problem 5: Shahid and Rohit start from the same point in opposite directions. After each 1 km, Shahid always turns left and Rohit always turns right. Which of the following statement is correct?

A. After both have traveled 2 km, the distance between them is 4 km.

B. They meet after each has traveled 3 km.

C. They meet for the first time after each has traveled 4 km.

D. They go on without ever meeting again.

The paths followed by Shahid and Rohit would look like this: Clearly, they would meet after traveling 3 km each. Therefore, B is the correct option.