Q 6. Direction and Distance Tests
Two friends X and Y start running and they run together for 50 m in the same direction and reach a point. X turns right and runs 60 m, while Y turns left and runs 40 m. Then X turns left and runs 50 m and stops, while Y turns right and runs 50 m and then stops. How far are the two friends from each other now?
a)
100 m
b)
90 m
c)
60 m
d)
50 m
Answer:
a
Option A: 100 m
Let's break down the movements of X and Y step by step:
● Initial Position: Both X and Y start from the same point and run together for 50 m in the same direction.
● Position after 50 m: They reach a common point after running 50 m together.
● X's Movement:
○ X turns right and runs 60 m.
○ Then, X turns left and runs 50 m.
● Y's Movement:
○ Y turns left and runs 40 m.
○ Then, Y turns right and runs 50 m.
Now, let's visualize their final positions:
● X's Final Position:
○ After turning right and running 60 m, X is 60 m perpendicular to the initial direction.
○ After turning left and running 50 m, X is now 50 m parallel to the initial direction from the point where X turned right.
● Y's Final Position:
○ After turning left and running 40 m, Y is 40 m perpendicular to the initial direction.
○ After turning right and running 50 m, Y is now 50 m parallel to the initial direction from the point where Y turned left.
To find the distance between X and Y, we need to calculate the straight-line distance between their final positions:
● Horizontal Distance: Both X and Y have moved 50 m parallel to the initial direction, so the horizontal distance between them is 0 m.
● Vertical Distance: The vertical distance between X and Y is the sum of the perpendicular distances they moved:
○ X moved 60 m perpendicular to the initial direction.
○ Y moved 40 m perpendicular to the initial direction.
○ Total vertical distance = 60 m + 40 m = 100 m.
Therefore, the distance between X and Y is the vertical distance, which is 100 m.
Hence, the correct answer is Option A: 100 m.
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Let's break down the movements of X and Y step by step:
● Initial Position: Both X and Y start from the same point and run together for 50 m in the same direction.
● Position after 50 m: They reach a common point after running 50 m together.
● X's Movement:
○ X turns right and runs 60 m.
○ Then, X turns left and runs 50 m.
● Y's Movement:
○ Y turns left and runs 40 m.
○ Then, Y turns right and runs 50 m.
Now, let's visualize their final positions:
● X's Final Position:
○ After turning right and running 60 m, X is 60 m perpendicular to the initial direction.
○ After turning left and running 50 m, X is now 50 m parallel to the initial direction from the point where X turned right.
● Y's Final Position:
○ After turning left and running 40 m, Y is 40 m perpendicular to the initial direction.
○ After turning right and running 50 m, Y is now 50 m parallel to the initial direction from the point where Y turned left.
To find the distance between X and Y, we need to calculate the straight-line distance between their final positions:
● Horizontal Distance: Both X and Y have moved 50 m parallel to the initial direction, so the horizontal distance between them is 0 m.
● Vertical Distance: The vertical distance between X and Y is the sum of the perpendicular distances they moved:
○ X moved 60 m perpendicular to the initial direction.
○ Y moved 40 m perpendicular to the initial direction.
○ Total vertical distance = 60 m + 40 m = 100 m.
Therefore, the distance between X and Y is the vertical distance, which is 100 m.
Hence, the correct answer is Option A: 100 m.
Q 37. Direction and Distance Tests
Consider the Question and two Statements given below in respect of three cities, P, Q and R in a state:
Question: How far is city P from city Q?
Statement-1: City Q is 18 km from city R.
Statement-2: City P is 43 km from city R.
Which one of the following is correct in respect of the Question and the Statements?
a)
Statement-1 alone is sufficient to answer the Question
b)
Statement-2 alone is sufficient to answer the Question
c)
Both Statement-1 and Statement-2 are sufficient to answer the Question
d)
Both Statement-1 and Statement-2 are not sufficient to answer the Question
Answer:
d
Option D: Both Statement-1 and Statement-2 are not sufficient to answer the Question.
● Statement-1: City Q is 18 km from city R.
○ This statement provides the distance between city Q and city R but gives no information about the distance between city P and city Q. Therefore, it is not sufficient to answer the question.
● Statement-2: City P is 43 km from city R.
○ This statement provides the distance between city P and city R but does not provide any information about the distance between city P and city Q. Therefore, it is not sufficient to answer the question.
● Combining Statement-1 and Statement-2:
○ Even when both statements are combined, we only know the distances of city P and city Q from city R. Without additional information about the relative positions of the cities (e.g., whether they are in a straight line or form a triangle), we cannot determine the exact distance between city P and city Q. Therefore, both statements together are still not sufficient to answer the question.
● Conclusion: Since neither statement alone nor both statements together provide enough information to determine the distance between city P and city Q, the correct answer is Option D.
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● Statement-1: City Q is 18 km from city R.
○ This statement provides the distance between city Q and city R but gives no information about the distance between city P and city Q. Therefore, it is not sufficient to answer the question.
● Statement-2: City P is 43 km from city R.
○ This statement provides the distance between city P and city R but does not provide any information about the distance between city P and city Q. Therefore, it is not sufficient to answer the question.
● Combining Statement-1 and Statement-2:
○ Even when both statements are combined, we only know the distances of city P and city Q from city R. Without additional information about the relative positions of the cities (e.g., whether they are in a straight line or form a triangle), we cannot determine the exact distance between city P and city Q. Therefore, both statements together are still not sufficient to answer the question.
● Conclusion: Since neither statement alone nor both statements together provide enough information to determine the distance between city P and city Q, the correct answer is Option D.
Q 18. Direction and Distance Tests
A bank employee drives 10 km towards South from her house and turns to her left and drives another 20 km. She again turns left and drives 40 km, then she turns to her right and drives for another 5 km. She again turns to her right and drives another 30 km to reach her bank where she works. What is the shortest distance between her bank and her house?
a)
20 km
b)
25 km
c)
30 km
d)
35 km
Answer:
b
Option B: 25 km
Let's break down the movements step by step:
● Step 1: The bank employee drives 10 km towards the South.
○ This means she is now 10 km south of her starting point (house).
● Step 2: She turns to her left and drives 20 km.
○ Turning left from South means she is now heading East. She drives 20 km East.
● Step 3: She again turns left and drives 40 km.
○ Turning left from East means she is now heading North. She drives 40 km North.
● Step 4: She turns to her right and drives 5 km.
○ Turning right from North means she is now heading East. She drives 5 km East.
● Step 5: She again turns to her right and drives 30 km.
○ Turning right from East means she is now heading South. She drives 30 km South to reach her bank.
Now, let's determine her final position relative to her starting point (house):
● East-West Position:
○ She drove 20 km East in Step 2 and an additional 5 km East in Step 4, totaling 25 km East.
● North-South Position:
○ She drove 10 km South in Step 1, then 40 km North in Step 3, and finally 30 km South in Step 5.
○ Net North-South position = 10 km South - 40 km North + 30 km South = 0 km (she is at the same North-South level as her starting point).
Therefore, the shortest distance between her bank and her house is the direct East-West distance, which is 25 km.
Hence, the correct answer is Option B: 25 km.
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Let's break down the movements step by step:
● Step 1: The bank employee drives 10 km towards the South.
○ This means she is now 10 km south of her starting point (house).
● Step 2: She turns to her left and drives 20 km.
○ Turning left from South means she is now heading East. She drives 20 km East.
● Step 3: She again turns left and drives 40 km.
○ Turning left from East means she is now heading North. She drives 40 km North.
● Step 4: She turns to her right and drives 5 km.
○ Turning right from North means she is now heading East. She drives 5 km East.
● Step 5: She again turns to her right and drives 30 km.
○ Turning right from East means she is now heading South. She drives 30 km South to reach her bank.
Now, let's determine her final position relative to her starting point (house):
● East-West Position:
○ She drove 20 km East in Step 2 and an additional 5 km East in Step 4, totaling 25 km East.
● North-South Position:
○ She drove 10 km South in Step 1, then 40 km North in Step 3, and finally 30 km South in Step 5.
○ Net North-South position = 10 km South - 40 km North + 30 km South = 0 km (she is at the same North-South level as her starting point).
Therefore, the shortest distance between her bank and her house is the direct East-West distance, which is 25 km.
Hence, the correct answer is Option B: 25 km.
Q 20. Direction and Distance Tests
A woman runs 12 km towards her North, then 6 km towards her South and then 8 km towards her East. In which direction is she from her starting point?
a)
An angle less than 45° South of East
b)
An angle less than 45° North of East
c)
An angle more than 45° South of East
d)
An angle more than 45° North of East
Answer:
b
1. Correct Answer: b: An angle less than 45° North of East
2. Explanation:
● Step 1: Determine the final position of the woman relative to her starting point.
○ She runs 12 km North, then 6 km South. The net movement in the North-South direction is:
\[
12 \, \text{km North} - 6 \, \text{km South} = 6 \, \text{km North}
\]
○ She then runs 8 km East.
● Step 2: Calculate the resultant displacement from the starting point.
○ The woman is now 6 km North and 8 km East from her starting point. This forms a right triangle where:
○ The North-South leg is 6 km.
○ The East leg is 8 km.
● Step 3: Determine the direction of the resultant vector.
○ The angle \(\theta\) with respect to the East direction can be calculated using the tangent function:
\[
\tan(\theta) = \frac{\text{North-South displacement}}{\text{East displacement}} = \frac{6}{8} = 0.75
\]
○ Calculate \(\theta\) using the arctangent function:
\[
\theta = \tan^{-1}(0.75) \approx 36.87^\circ
\]
● Conclusion: The angle \(\theta\) is approximately \(36.87^\circ\), which is less than \(45^\circ\). Therefore, the woman is at an angle less than \(45^\circ\) North of East from her starting point. This corresponds to Option B.
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2. Explanation:
● Step 1: Determine the final position of the woman relative to her starting point.
○ She runs 12 km North, then 6 km South. The net movement in the North-South direction is:
\[
12 \, \text{km North} - 6 \, \text{km South} = 6 \, \text{km North}
\]
○ She then runs 8 km East.
● Step 2: Calculate the resultant displacement from the starting point.
○ The woman is now 6 km North and 8 km East from her starting point. This forms a right triangle where:
○ The North-South leg is 6 km.
○ The East leg is 8 km.
● Step 3: Determine the direction of the resultant vector.
○ The angle \(\theta\) with respect to the East direction can be calculated using the tangent function:
\[
\tan(\theta) = \frac{\text{North-South displacement}}{\text{East displacement}} = \frac{6}{8} = 0.75
\]
○ Calculate \(\theta\) using the arctangent function:
\[
\theta = \tan^{-1}(0.75) \approx 36.87^\circ
\]
● Conclusion: The angle \(\theta\) is approximately \(36.87^\circ\), which is less than \(45^\circ\). Therefore, the woman is at an angle less than \(45^\circ\) North of East from her starting point. This corresponds to Option B.
Q 31. Direction and Distance Tests
A man walks down the backside of his house straight 25 metres, then turns to the right and walks 50 metres again; then he turns towards left and again walks 25 metres. If his house faces to the East, what is his direction from the starting point?
a)
South-East
b)
South-West
c)
North-East
d)
North-West
Answer:
d
Option D: North-West
Let's break down the man's movements step by step:
● Initial Position: The man starts at the backside of his house, which faces East. Therefore, he is initially facing West.
● Step 1: The man walks straight 25 metres. Since he is at the backside of the house facing West, he walks 25 metres towards the West.
● Step 2: He then turns to the right. Facing West, a right turn means he is now facing North. He walks 50 metres in this direction.
● Step 3: Next, he turns towards the left. Facing North, a left turn means he is now facing West again. He walks another 25 metres in this direction.
Now, let's determine his final position relative to the starting point:
○ From the starting point, he first moved 25 metres West.
○ Then, he moved 50 metres North.
○ Finally, he moved another 25 metres West.
Combining these movements, his final position is 50 metres North and 50 metres West from the starting point.
Therefore, his direction from the starting point is North-West.
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Let's break down the man's movements step by step:
● Initial Position: The man starts at the backside of his house, which faces East. Therefore, he is initially facing West.
● Step 1: The man walks straight 25 metres. Since he is at the backside of the house facing West, he walks 25 metres towards the West.
● Step 2: He then turns to the right. Facing West, a right turn means he is now facing North. He walks 50 metres in this direction.
● Step 3: Next, he turns towards the left. Facing North, a left turn means he is now facing West again. He walks another 25 metres in this direction.
Now, let's determine his final position relative to the starting point:
○ From the starting point, he first moved 25 metres West.
○ Then, he moved 50 metres North.
○ Finally, he moved another 25 metres West.
Combining these movements, his final position is 50 metres North and 50 metres West from the starting point.
Therefore, his direction from the starting point is North-West.
Q 30. Direction and Distance Tests
P, Q and R are three towns. The distance between P and Q is 60 km, whereas the distance between P and R is 80 km. Q is in the West of P and R is in the South of P. What is the distance between Q and R?
a)
140 km
b)
130 km
c)
110 km
d)
100 km
Answer:
d
1. Correct Answer: Option D: 100 km
2. Explanation:
To solve this problem, we can use the Pythagorean theorem. The towns P, Q, and R form a right-angled triangle with P as the vertex of the right angle. Here's how we can visualize and solve it:
● Step 1: Identify the positions of the towns.
○ Town Q is to the West of P, so we can consider the line segment PQ as the horizontal leg of the triangle.
○ Town R is to the South of P, so we can consider the line segment PR as the vertical leg of the triangle.
● Step 2: Assign the given distances to the respective line segments.
○ The distance between P and Q (horizontal leg) is 60 km.
○ The distance between P and R (vertical leg) is 80 km.
● Step 3: Use the Pythagorean theorem to find the distance between Q and R (the hypotenuse of the right triangle).
○ According to the Pythagorean theorem:
\[
\text{(Distance between Q and R)}^2 = \text{(Distance between P and Q)}^2 + \text{(Distance between P and R)}^2
\]
○ Substitute the known values:
\[
\text{(Distance between Q and R)}^2 = 60^2 + 80^2
\]
○ Calculate the squares:
\[
\text{(Distance between Q and R)}^2 = 3600 + 6400 = 10000
\]
○ Take the square root to find the distance:
\[
\text{Distance between Q and R} = \sqrt{10000} = 100 \text{ km}
\]
Therefore, the distance between Q and R is 100 km.
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2. Explanation:
To solve this problem, we can use the Pythagorean theorem. The towns P, Q, and R form a right-angled triangle with P as the vertex of the right angle. Here's how we can visualize and solve it:
● Step 1: Identify the positions of the towns.
○ Town Q is to the West of P, so we can consider the line segment PQ as the horizontal leg of the triangle.
○ Town R is to the South of P, so we can consider the line segment PR as the vertical leg of the triangle.
● Step 2: Assign the given distances to the respective line segments.
○ The distance between P and Q (horizontal leg) is 60 km.
○ The distance between P and R (vertical leg) is 80 km.
● Step 3: Use the Pythagorean theorem to find the distance between Q and R (the hypotenuse of the right triangle).
○ According to the Pythagorean theorem:
\[
\text{(Distance between Q and R)}^2 = \text{(Distance between P and Q)}^2 + \text{(Distance between P and R)}^2
\]
○ Substitute the known values:
\[
\text{(Distance between Q and R)}^2 = 60^2 + 80^2
\]
○ Calculate the squares:
\[
\text{(Distance between Q and R)}^2 = 3600 + 6400 = 10000
\]
○ Take the square root to find the distance:
\[
\text{Distance between Q and R} = \sqrt{10000} = 100 \text{ km}
\]
Therefore, the distance between Q and R is 100 km.
Q 57. Direction and Distance Tests
'A' started from his house and walked 20 m towards East, where his friend 'B' joined him. They together walked 10 m in the same direction. Then 'A' turned left while 'B' turned right and travelled 2 m and 8 m respectively. Again 'B' turned left to travel 4 m followed by 5 m to his right to reach his office. 'A' turned right and travelled 12 m to reach his office. What is the shortest distance between the two offices?
a)
15 m
b)
17 m
c)
19 m
d)
20 m
Answer:
b
• Using Pythagoras Theorem in triangle, PQR: Shortest Distance = Hypotenuse
• So, PQ2 = PR2 + QR2
• Thus, PQ = 17 m.
Q 19. Direction and Distance Tests
A person X was driving in a place where all roads ran either north-south or east-west, forming a grid. Roads are at a distance of 1 km from each other in a parallel. He started at the intersection of two roads, drove 3 km north, 3 km west and 4 km south. Which further route could bring him back to his starting point, if the same route is not repeated?
a)
3 km east, then 2 km south
b)
3 km east, then 1 km north
c)
1 km north, then 2 km west
d)
3 km south, then 1 km north
Answer:
b
Option B: 3 km east, then 1 km north
Let's break down the movements step by step:
● Starting Point: Let's denote the starting point as (0, 0).
● Step 1: Drive 3 km north.
○ New position: (0, 3)
● Step 2: Drive 3 km west.
○ New position: (-3, 3)
● Step 3: Drive 4 km south.
○ New position: (-3, -1)
Now, we need to determine which route will bring him back to the starting point (0, 0) without repeating the same route.
● Option A: 3 km east, then 2 km south
○ New position after 3 km east: (0, -1)
○ New position after 2 km south: (0, -3)
○ This does not bring him back to the starting point.
● Option B: 3 km east, then 1 km north
○ New position after 3 km east: (0, -1)
○ New position after 1 km north: (0, 0)
○ This brings him back to the starting point.
● Option C: 1 km north, then 2 km west
○ New position after 1 km north: (-3, 0)
○ New position after 2 km west: (-5, 0)
○ This does not bring him back to the starting point.
● Option D: 3 km south, then 1 km north
○ New position after 3 km south: (-3, -4)
○ New position after 1 km north: (-3, -3)
○ This does not bring him back to the starting point.
Therefore, the correct route to bring him back to the starting point is Option B: 3 km east, then 1 km north.
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Let's break down the movements step by step:
● Starting Point: Let's denote the starting point as (0, 0).
● Step 1: Drive 3 km north.
○ New position: (0, 3)
● Step 2: Drive 3 km west.
○ New position: (-3, 3)
● Step 3: Drive 4 km south.
○ New position: (-3, -1)
Now, we need to determine which route will bring him back to the starting point (0, 0) without repeating the same route.
● Option A: 3 km east, then 2 km south
○ New position after 3 km east: (0, -1)
○ New position after 2 km south: (0, -3)
○ This does not bring him back to the starting point.
● Option B: 3 km east, then 1 km north
○ New position after 3 km east: (0, -1)
○ New position after 1 km north: (0, 0)
○ This brings him back to the starting point.
● Option C: 1 km north, then 2 km west
○ New position after 1 km north: (-3, 0)
○ New position after 2 km west: (-5, 0)
○ This does not bring him back to the starting point.
● Option D: 3 km south, then 1 km north
○ New position after 3 km south: (-3, -4)
○ New position after 1 km north: (-3, -3)
○ This does not bring him back to the starting point.
Therefore, the correct route to bring him back to the starting point is Option B: 3 km east, then 1 km north.
Q 59. Direction and Distance Tests
A person climbs a hill in a straight path from point 'O' on the ground in the direction of north-east and reaches a point 'A' after travelling a distance of 5 km. Then, from the point 'A' he moves to point 'B' in the direction of north-west. Let the distance AB be 12 km. Now, how far is the person away from the starting point 'O'?
a)
7 km
b)
13 km
c)
17 km
d)
11 km
Answer:
b
Option B: 13 km
To solve this problem, we need to determine the distance between the starting point 'O' and the final point 'B'.
Step-by-step Solution:
● Step 1: Understand the directions and distances.
○ The person starts at point 'O' and travels 5 km in the north-east direction to reach point 'A'.
○ From point 'A', the person travels 12 km in the north-west direction to reach point 'B'.
● Step 2: Break down the movements into components.
○ When moving north-east, the movement can be broken down into equal north and east components. Therefore, the north and east components from 'O' to 'A' are each \( \frac{5}{\sqrt{2}} \) km.
○ When moving north-west, the movement can be broken down into equal north and west components. Therefore, the north and west components from 'A' to 'B' are each \( \frac{12}{\sqrt{2}} \) km.
● Step 3: Calculate the total north and east/west components.
○ Total north component = \( \frac{5}{\sqrt{2}} + \frac{12}{\sqrt{2}} = \frac{17}{\sqrt{2}} \) km.
○ Total east component = \( \frac{5}{\sqrt{2}} \) km.
○ Total west component = \( \frac{12}{\sqrt{2}} \) km.
● Step 4: Determine the net east-west component.
○ Since the person moves east from 'O' to 'A' and then west from 'A' to 'B', the net east-west component is:
\[
\text{Net east-west component} = \frac{5}{\sqrt{2}} - \frac{12}{\sqrt{2}} = -\frac{7}{\sqrt{2}} \text{ km (west)}
\]
● Step 5: Calculate the distance from 'O' to 'B'.
○ The distance from 'O' to 'B' can be found using the Pythagorean theorem:
\[
OB = \sqrt{\left(\frac{17}{\sqrt{2}}\right)^2 + \left(-\frac{7}{\sqrt{2}}\right)^2}
\]
\[
OB = \sqrt{\frac{289}{2} + \frac{49}{2}}
\]
\[
OB = \sqrt{\frac{338}{2}}
\]
\[
OB = \sqrt{169} = 13 \text{ km}
\]
Therefore, the person is 13 km away from the starting point 'O'.
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Explained
To solve this problem, we need to determine the distance between the starting point 'O' and the final point 'B'.
Step-by-step Solution:
● Step 1: Understand the directions and distances.
○ The person starts at point 'O' and travels 5 km in the north-east direction to reach point 'A'.
○ From point 'A', the person travels 12 km in the north-west direction to reach point 'B'.
● Step 2: Break down the movements into components.
○ When moving north-east, the movement can be broken down into equal north and east components. Therefore, the north and east components from 'O' to 'A' are each \( \frac{5}{\sqrt{2}} \) km.
○ When moving north-west, the movement can be broken down into equal north and west components. Therefore, the north and west components from 'A' to 'B' are each \( \frac{12}{\sqrt{2}} \) km.
● Step 3: Calculate the total north and east/west components.
○ Total north component = \( \frac{5}{\sqrt{2}} + \frac{12}{\sqrt{2}} = \frac{17}{\sqrt{2}} \) km.
○ Total east component = \( \frac{5}{\sqrt{2}} \) km.
○ Total west component = \( \frac{12}{\sqrt{2}} \) km.
● Step 4: Determine the net east-west component.
○ Since the person moves east from 'O' to 'A' and then west from 'A' to 'B', the net east-west component is:
\[
\text{Net east-west component} = \frac{5}{\sqrt{2}} - \frac{12}{\sqrt{2}} = -\frac{7}{\sqrt{2}} \text{ km (west)}
\]
● Step 5: Calculate the distance from 'O' to 'B'.
○ The distance from 'O' to 'B' can be found using the Pythagorean theorem:
\[
OB = \sqrt{\left(\frac{17}{\sqrt{2}}\right)^2 + \left(-\frac{7}{\sqrt{2}}\right)^2}
\]
\[
OB = \sqrt{\frac{289}{2} + \frac{49}{2}}
\]
\[
OB = \sqrt{\frac{338}{2}}
\]
\[
OB = \sqrt{169} = 13 \text{ km}
\]
Therefore, the person is 13 km away from the starting point 'O'.
Q 68. Direction and Distance Tests
A person walks 12 km due north, then 15 km due east, after that 19 km due west and then 15 km due south. How far is he from the starting point ?
a)
5 km
b)
9 km
c)
37 km
d)
61 km
Answer:
a
1. Correct Answer: Option A: *5 km*
2. Explanation:
To determine how far the person is from the starting point, we need to calculate the net displacement after all the movements.
Step-by-step Calculation:
● Initial Position: Let's assume the starting point is at the origin of a coordinate system, (0, 0).
● First Movement: The person walks 12 km due north.
○ New position: (0, 12)
● Second Movement: The person walks 15 km due east.
○ New position: (15, 12)
● Third Movement: The person walks 19 km due west.
○ New position: (15 - 19, 12) = (-4, 12)
● Fourth Movement: The person walks 15 km due south.
○ New position: (-4, 12 - 15) = (-4, -3)
Net Displacement:
○ The net displacement is the straight-line distance from the starting point (0, 0) to the final position (-4, -3).
○ Using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (-4, -3)\).
○ Calculate:
\[
\text{Distance} = \sqrt{(-4 - 0)^2 + (-3 - 0)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km}
\]
Therefore, the person is *5 km* away from the starting point.
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2. Explanation:
To determine how far the person is from the starting point, we need to calculate the net displacement after all the movements.
Step-by-step Calculation:
● Initial Position: Let's assume the starting point is at the origin of a coordinate system, (0, 0).
● First Movement: The person walks 12 km due north.
○ New position: (0, 12)
● Second Movement: The person walks 15 km due east.
○ New position: (15, 12)
● Third Movement: The person walks 19 km due west.
○ New position: (15 - 19, 12) = (-4, 12)
● Fourth Movement: The person walks 15 km due south.
○ New position: (-4, 12 - 15) = (-4, -3)
Net Displacement:
○ The net displacement is the straight-line distance from the starting point (0, 0) to the final position (-4, -3).
○ Using the distance formula:
\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (-4, -3)\).
○ Calculate:
\[
\text{Distance} = \sqrt{(-4 - 0)^2 + (-3 - 0)^2} = \sqrt{(-4)^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km}
\]
Therefore, the person is *5 km* away from the starting point.
Q 33. Direction and Distance Tests
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 statements is correct?
a)
After both have travelled 2 km, the distance between them is 4 km.
b)
They meet after each has travelled 3 km.
c)
They meet for the first time after each has travelled 4 km.
d)
They go on without ever meeting again.
Answer:
b
1. Correct Answer: Option B: They meet after each has travelled 3 km.
2. Explanation:
Let's analyze the movements of Shahid and Rohit step by step:
● Initial Position: Both Shahid and Rohit start from the same point.
● After 1 km:
○ Shahid turns left.
○ Rohit turns right.
○ They are now 1 km apart, moving perpendicular to their original direction.
● After 2 km:
○ Shahid turns left again.
○ Rohit turns right again.
○ They are now 2 km apart, moving parallel to their original direction but in opposite directions.
● After 3 km:
○ Shahid turns left once more.
○ Rohit turns right once more.
○ They both turn towards each other and meet at the same point after traveling 3 km each.
Therefore, the correct statement is that they meet after each has traveled 3 km, which corresponds to Option B.
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2. Explanation:
Let's analyze the movements of Shahid and Rohit step by step:
● Initial Position: Both Shahid and Rohit start from the same point.
● After 1 km:
○ Shahid turns left.
○ Rohit turns right.
○ They are now 1 km apart, moving perpendicular to their original direction.
● After 2 km:
○ Shahid turns left again.
○ Rohit turns right again.
○ They are now 2 km apart, moving parallel to their original direction but in opposite directions.
● After 3 km:
○ Shahid turns left once more.
○ Rohit turns right once more.
○ They both turn towards each other and meet at the same point after traveling 3 km each.
Therefore, the correct statement is that they meet after each has traveled 3 km, which corresponds to Option B.
Q 10. Direction and Distance Tests
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
Answer:
b
Option B: A, D, E
Let's analyze the given statements step by step to determine the positions of the villages:
● Statement 1: F is 1 km to the west of D.
○ This means if D is at a certain point, F is 1 km to the left of D.
● Statement 2: B is 1 km to the east of E.
○ This means if E is at a certain point, B is 1 km to the right of E.
● Statement 3: A is 2 km to the north of E.
○ This means if E is at a certain point, A is 2 km directly above E.
● Statement 4: C is 1 km to the east of A.
○ This means if A is at a certain point, C is 1 km to the right of A.
● Statement 5: D is 1 km to the south of A.
○ This means if A is at a certain point, D is 1 km directly below A.
Now, let's place these villages on a coordinate plane for clarity:
○ Place E at the origin (0,0).
○ From Statement 3, A is 2 km north of E, so A is at (0,2).
○ From Statement 5, D is 1 km south of A, so D is at (0,1).
○ From Statement 2, B is 1 km east of E, so B is at (1,0).
○ From Statement 4, C is 1 km east of A, so C is at (1,2).
○ From Statement 1, F is 1 km west of D, so F is at (-1,1).
Now, let's check which villages are in a straight line:
● Option A: A, C, B:
○ A is at (0,2), C is at (1,2), B is at (1,0).
○ These points do not form a straight line.
● Option B: A, D, E:
○ A is at (0,2), D is at (0,1), E is at (0,0).
○ These points are on the same vertical line (x = 0).
● Option C: C, B, F:
○ C is at (1,2), B is at (1,0), F is at (-1,1).
○ These points do not form a straight line.
● Option D: E, B, D:
○ E is at (0,0), B is at (1,0), D is at (0,1).
○ These points do not form a straight line.
Therefore, the correct answer is Option B: A, D, E, as they are in a straight line vertically.
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Read Full Explanation and All the Options
Explained
Let's analyze the given statements step by step to determine the positions of the villages:
● Statement 1: F is 1 km to the west of D.
○ This means if D is at a certain point, F is 1 km to the left of D.
● Statement 2: B is 1 km to the east of E.
○ This means if E is at a certain point, B is 1 km to the right of E.
● Statement 3: A is 2 km to the north of E.
○ This means if E is at a certain point, A is 2 km directly above E.
● Statement 4: C is 1 km to the east of A.
○ This means if A is at a certain point, C is 1 km to the right of A.
● Statement 5: D is 1 km to the south of A.
○ This means if A is at a certain point, D is 1 km directly below A.
Now, let's place these villages on a coordinate plane for clarity:
○ Place E at the origin (0,0).
○ From Statement 3, A is 2 km north of E, so A is at (0,2).
○ From Statement 5, D is 1 km south of A, so D is at (0,1).
○ From Statement 2, B is 1 km east of E, so B is at (1,0).
○ From Statement 4, C is 1 km east of A, so C is at (1,2).
○ From Statement 1, F is 1 km west of D, so F is at (-1,1).
Now, let's check which villages are in a straight line:
● Option A: A, C, B:
○ A is at (0,2), C is at (1,2), B is at (1,0).
○ These points do not form a straight line.
● Option B: A, D, E:
○ A is at (0,2), D is at (0,1), E is at (0,0).
○ These points are on the same vertical line (x = 0).
● Option C: C, B, F:
○ C is at (1,2), B is at (1,0), F is at (-1,1).
○ These points do not form a straight line.
● Option D: E, B, D:
○ E is at (0,0), B is at (1,0), D is at (0,1).
○ These points do not form a straight line.
Therefore, the correct answer is Option B: A, D, E, as they are in a straight line vertically.
Q 51. Direction and Distance Tests
The houses of A and B face each other on a road going north-south, A's being on the western side. A comes out of his house, turns left, travels 5 km, turns right, travels 5 km to the front of D's house. B does exactly the same and reaches the front of C's house. In this context, which one of the following statements is correct?
a)
C and D live on the same street.
b)
C's house faces south.
c)
The houses of C and D are less than 20 km apart.
d)
None of the above
Answer:
c
Option C: The houses of C and D are less than 20 km apart.
● Understanding the Initial Setup:
○ A's house is on the western side of the road, facing east.
○ B's house is on the eastern side of the road, facing west.
○ The road runs north-south.
● A's Movement:
1. A comes out of his house and turns left, which means he heads south (since his house faces east).
2. A travels 5 km south.
3. A then turns right, which means he heads west.
4. A travels 5 km west to reach D's house.
● B's Movement:
1. B comes out of his house and turns left, which means he heads north (since his house faces west).
2. B travels 5 km north.
3. B then turns right, which means he heads east.
4. B travels 5 km east to reach C's house.
● Analyzing the Positions:
○ A and B start facing each other across the road.
○ After their respective movements, A ends up 5 km west and 5 km south of his starting point, at D's house.
○ B ends up 5 km east and 5 km north of his starting point, at C's house.
● Conclusion:
○ The distance between C's house and D's house can be calculated using the Pythagorean theorem. The horizontal distance is 10 km (5 km west + 5 km east), and the vertical distance is 10 km (5 km south + 5 km north).
○ Therefore, the distance between C and D is \(\sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}\) km, which is approximately 14.14 km.
○ Thus, the houses of C and D are indeed less than 20 km apart.
● Verification of Other Options:
● Option A: C and D live on the same street. This is incorrect because C and D are on different streets due to their different directional movements.
● Option B: C's house faces south. This is incorrect because B's house faces west, and after moving, C's house would face east, not south.
● Option D: None of the above. This is incorrect because Option C is correct.
Therefore, the correct answer is Option C: The houses of C and D are less than 20 km apart.
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Read Full Explanation and All the Options
Explained
● Understanding the Initial Setup:
○ A's house is on the western side of the road, facing east.
○ B's house is on the eastern side of the road, facing west.
○ The road runs north-south.
● A's Movement:
1. A comes out of his house and turns left, which means he heads south (since his house faces east).
2. A travels 5 km south.
3. A then turns right, which means he heads west.
4. A travels 5 km west to reach D's house.
● B's Movement:
1. B comes out of his house and turns left, which means he heads north (since his house faces west).
2. B travels 5 km north.
3. B then turns right, which means he heads east.
4. B travels 5 km east to reach C's house.
● Analyzing the Positions:
○ A and B start facing each other across the road.
○ After their respective movements, A ends up 5 km west and 5 km south of his starting point, at D's house.
○ B ends up 5 km east and 5 km north of his starting point, at C's house.
● Conclusion:
○ The distance between C's house and D's house can be calculated using the Pythagorean theorem. The horizontal distance is 10 km (5 km west + 5 km east), and the vertical distance is 10 km (5 km south + 5 km north).
○ Therefore, the distance between C and D is \(\sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}\) km, which is approximately 14.14 km.
○ Thus, the houses of C and D are indeed less than 20 km apart.
● Verification of Other Options:
● Option A: C and D live on the same street. This is incorrect because C and D are on different streets due to their different directional movements.
● Option B: C's house faces south. This is incorrect because B's house faces west, and after moving, C's house would face east, not south.
● Option D: None of the above. This is incorrect because Option C is correct.
Therefore, the correct answer is Option C: The houses of C and D are less than 20 km apart.