Unit 5: Height, Distance and Analytical Reasoning - Subjective Questions

Angle of elevation: It is the angle between the horizontal line through the observer's eye and the upward line of sight to an object.

Angle of depression: It is the angle between the horizontal line through the observer's eye and the downward line of sight to an object.

If an observer at looks downward at an object at , the horizontal lines through and are parallel. Therefore, by the alternate interior angle property:

• Angle of depression from to = angle of elevation from to .
• Both angles are measured with respect to a horizontal line, not a vertical line.
• These angles allow a height-and-distance situation to be represented by a right-angled triangle.

Let be an acute angle in a right-angled triangle. The three principal trigonometric ratios are:

In height-and-distance problems:

• The vertical height usually represents the perpendicular.
• The horizontal distance represents the base.
• The line of sight represents the hypotenuse.
• If height and horizontal distance are involved, is generally the most direct ratio.
• The observer's eye height must be added or subtracted when the observation is not made from ground level.

Let the height of the tower be m and the horizontal distance be m.

Using the tangent ratio:

Since :

Therefore:

Thus, the height of the tower is m, which is approximately m.

Let the distance from the nearer point to the building be m and the building's height be m. The original distance is therefore m.

From the nearer point:

Hence:

From the original point:

Since :

Equating the two expressions for :

Therefore:

The original distance is:

Thus:

• Height of the building: m
• Original distance: m

Let the horizontal distance between the boat and the foot of the lighthouse be m.

The angle of depression from the lighthouse equals the angle of elevation from the boat. Therefore, the angle at the boat is .

Using the tangent ratio:

Since :

Therefore:

Thus, the boat is m, or approximately m, from the foot of the lighthouse.

Let the height of the pole be m. The shadow is the horizontal base of length m.

Using the tangent ratio:

Since :

Thus, the pole's height is m, approximately m.

For a fixed pole height:

As the angle of elevation increases, increases. Consequently, the shadow becomes shorter.

Let the horizontal distance between the building and tower be m and the tower's height be m.

Using the angle of depression to the tower's foot:

Therefore:

The vertical portion of the tower above the observer's level is m. Using the angle of elevation:

Hence:

Thus, the tower's height is m, approximately m.

A systematic method consists of the following steps:

1. Identify the entities: List all people, objects, positions, days, or attributes involved.
2. Classify the clues: Separate direct clues, negative clues, relative-position clues, and conditional clues.
3. Prepare a diagram or table: Use rows, columns, slots, or circles according to the problem.
4. Place fixed information first: Enter clues that specify exact positions or definite relationships.
5. Form blocks: Treat linked entities such as immediately before as a single block.
6. Apply restrictive clues: Use clues that eliminate the greatest number of possibilities.
7. Use deduction: Combine two or more clues to obtain information not stated directly.
8. Test alternatives: If necessary, consider limited cases and reject any case that violates a clue.
9. Verify the result: Check the final arrangement against every original condition.

This method reduces guesswork and prevents contradictory assignments.

Deductive reasoning moves from general rules or accepted premises to a logically certain conclusion.

• Example: All members of Group are trained. Ravi belongs to Group . Therefore, Ravi is trained.
• If the premises are true and the argument is valid, the conclusion must be true.

Inductive reasoning moves from particular observations to a general or probable conclusion.

• Example: A machine has worked correctly in every test conducted so far. Therefore, it will probably work correctly in the next test.
• The conclusion is probable, but it is not logically guaranteed.

The principal difference is that deduction establishes necessity, whereas induction establishes likelihood based on observed patterns.

Different clues serve different logical functions:

• Direct clue: Gives definite information. Example: sits in the third position.
• Negative clue: Excludes a possibility. Example: does not sit at either end.
• Conditional clue: Makes one fact dependent on another. Example: If is selected, then must also be selected.
• Either-or clue: Restricts the answer to alternatives. Example: sits at either the first or fifth position.

Their importance lies in how they reduce possibilities:

1. Direct clues should generally be placed first.
2. Negative clues eliminate invalid positions.
3. Conditional clues must be checked in both possible cases.
4. Either-or clues often create a small number of cases that can be tested separately.

Correct classification of clues makes complex reasoning problems easier to organize and solve.

Since occupies the left end, place in position .

The linked pairs are:

• , because is immediately to the right of .
• , because is immediately to the left of .

The remaining positions are , , , and . The pair cannot occupy positions and because that would place next to . Therefore, must occupy positions and , and must occupy positions and .

The final arrangement from left to right is:

Verification:

• is at the left end.
• is immediately right of .
• is immediately left of .
• is not adjacent to .

Number the positions to from left to right.

• is third from the left, so is in position .
• is second to the right of , so is in position .
• is immediately left of , so is in position .
• is at the right end, so is in position .

The unoccupied positions are , , and . Since is immediately right of , the pair must occupy positions and . Therefore, occupies position .

The complete arrangement is:

Every clue is satisfied by this arrangement.

The interpretation of left and right depends on the direction in which a person faces:

• A person facing north has the diagram's left on their left and the diagram's right on their right.
• A person facing south has the diagram's right on their left and the diagram's left on their right.

For example, if faces north and sits to 's right, place on the right side of in the diagram. If faces south and sits to 's right, place on the left side of in the diagram.

A useful procedure is:

1. Mark each person's facing direction.
2. Interpret every directional clue from the named person's perspective.
3. Avoid applying a single left-right rule to the entire row when directions differ.
4. Verify all relative positions after completing the arrangement.

Number the positions from to . Taking at the left end and at the right end:

• is in position and is in position .
• is third to the right of , so is in position .
• is immediately right of , so is in position .
• is second to the left of , so is in position .
• is immediately left of , but position is occupied by .

This produces a contradiction. Therefore, the initial assignment of the ends must be reversed: is at position and is at position . Now interpret the fixed-distance clues by position:

• cannot be third to the right of because no position exists beyond position .

Hence, the complete set of clues is inconsistent, and no valid arrangement exists.

This problem demonstrates an essential analytical skill: one must not force an arrangement when the conditions contradict one another. The correct conclusion is that the given information has no solution.

Linear seating arrangement:

• People are placed in a straight row.
• The row usually has identifiable left and right ends.
• Positions such as first, last, third from the left, and second from the right are meaningful.
• Direction depends on whether people face north or south.

Circular seating arrangement:

• People are placed around a circle.
• There are no natural end positions.
• One person is usually fixed at a reference point to remove rotational duplication.
• If everyone faces the centre, left corresponds to clockwise and right corresponds to anticlockwise.
• If everyone faces outward, left corresponds to anticlockwise and right corresponds to clockwise.

In a circle, arrangements obtained by rotating every person together are equivalent. For distinct people, the number of unrestricted circular arrangements is therefore:

Fix at any reference position because rotations of a circular arrangement are equivalent.

Proceed clockwise:

• is immediately clockwise of .
• With six seats, the person opposite is three positions away, so place opposite .
• is immediately anticlockwise of .
• is immediately clockwise of .
• occupies the only remaining seat.

Thus, one valid clockwise order beginning with is:

Therefore:

• sits between and .
• is immediately anticlockwise of .
• Any rotation of this order represents the same circular arrangement.

The direction rule changes with the orientation of the seated persons.

When facing the centre:

• A person's left is in the clockwise direction.
• A person's right is in the anticlockwise direction.

When facing outward:

• A person's left is in the anticlockwise direction.
• A person's right is in the clockwise direction.

A solver should first draw an arrow showing clockwise movement and mark whether the participants face inward or outward. Every left-right clue should then be translated using the appropriate rule. Confusing these rules reverses the arrangement and usually causes several clues to fail.

Fix at a reference seat. Number the seats to clockwise, with at position .

• is opposite , so is at position .
• is immediately clockwise of , so is at position .
• is immediately clockwise of , so is at position .
• is opposite , so is at position .

Positions would therefore have to contain both and , which is impossible.

Thus, the clues are mutually inconsistent, and no complete seating arrangement can be formed. Consequently, the location of cannot be determined.

The correct analytical conclusion is:

• No valid arrangement exists.
• The conflict arises because both and are forced into the same seat.

In a linear arrangement, distinct persons can be arranged in ways. Around a circular table, however, arrangements that differ only by rotation are identical.

To remove this rotational duplication:

1. Fix one person at a reference position.
2. Arrange the remaining persons relative to the fixed person.
3. The remaining persons can be arranged in ways.

Therefore, the number of circular arrangements is:

For seven persons:

Thus, seven distinct persons can be seated around a circular table in different ways.

If mirror-image arrangements were also regarded as identical, the count would be divided by , giving for .

Block technique: When two or more people must sit together, they can initially be treated as one unit. For example, if sits immediately left of , write the fixed block and place it as a single unit.

Gap technique: When certain people must not sit together, first arrange one group and then place the other group in the available gaps. For example, after arranging three persons as , there are four gaps, including the two end gaps, in which other persons may be placed.

Case analysis: When a clue allows alternatives, solve each possible case separately. For example, if sits at either end of a row, consider:

• Case 1: at the left end
• Case 2: at the right end

Reject any case that violates another clue. These techniques reduce a large problem into smaller and more manageable arrangements.