Analogies and Relations

Overview

In this chapter, we shall delve into a set of fundamental topics in analytical aptitude that are centered on the interpretation of relationships. The problems presented herein, while diverse in their context, are unified by a common requirement: the ability to process relational information, whether it be spatial, familial, or conceptual, and to deduce logical conclusions from a given framework of rules. These questions are designed not merely as puzzles, but as rigorous tests of structured thinking, pattern recognition, and the precise application of logical principles.

The mastery of these concepts is of paramount importance for the GATE examination. The skills cultivated—such as visualizing spatial transformations, mapping complex hierarchical structures, and identifying abstract parallels—are directly transferable to the core competencies required in engineering and data science. The ability to understand and model relationships is the bedrock upon which complex systems are analyzed and designed. Therefore, proficiency in this area serves as a strong indicator of a candidate's aptitude for higher-level technical reasoning. We will systematically dissect the methods required to approach each problem type, providing a robust foundation for success.

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Chapter Contents

| # | Topic | What You'll Learn |
|---|-------|-------------------|
| 1 | Direction Sense | Navigating paths and determining final positions. |
| 2 | Blood Relations | Decoding complex familial relationship structures. |
| 3 | Analogy | Identifying parallel relationships between concepts. |

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Learning Objectives

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We now turn our attention to Direction Sense...

Part 1: Direction Sense

Introduction

The study of Direction Sense is a fundamental component of logical and spatial reasoning. In the context of the GATE examination, problems of this nature are designed to assess a candidate's ability to perceive and interpret spatial relationships, movements, and orientations. These questions typically involve a scenario where an individual or object moves through a series of paths, and the candidate is required to determine the final direction, the distance from the starting point, or the relative position with respect to another point.

Mastery of this topic does not rely on complex formulae but rather on a clear conceptual understanding of the cardinal and ordinal directions, the principles of rotation, and the application of basic geometric theorems. We shall explore these concepts systematically, building a robust framework for visualizing and solving such problems. The ability to translate descriptive text into a simple, accurate diagram is the cornerstone of success in this area.

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Key Concepts

1. The Compass Rose: Cardinal and Ordinal Directions

The primary tool for navigating direction sense problems is the compass rose. It provides a standardized framework for orientation.

Cardinal Directions:
The four principal directions are known as the cardinal directions. These are:

• North (N)


• South (S)


• East (E)


• West (W)

We observe that North and South are opposites, as are East and West. The angle between any two adjacent cardinal directions is always $90^\circ$. For instance, the angle between North and East is $90^\circ$.

Ordinal Directions:
The directions that lie exactly between the cardinal directions are known as the ordinal or inter-cardinal directions. These are:

• North-East (NE): Exactly between North and East.


• South-East (SE): Exactly between South and East.


• South-West (SW): Exactly between South and West.


• North-West (NW): Exactly between North and West.

The angle between a cardinal direction and its adjacent ordinal direction is $45^\circ$. For example, the angle between North and North-East is $45^\circ$.

The complete directional compass can be visualized as follows:

N
S
E
W


NE
SE
SW
NW


90°

45°

2. Rotation and Turns

Movement in direction sense problems is often described in terms of turns. It is crucial to understand the two types of rotation:

• Clockwise (CW) Turn: A turn to the right, following the direction of the hands of a clock.
• Anti-Clockwise (ACW) Turn: A turn to the left, opposite to the direction of the hands of a clock.

A "right turn" or "left turn" without a specified angle is conventionally understood as a $90^\circ$ turn.

• A turn to the right implies a $90^\circ$ Clockwise rotation.
• A turn to the left implies a $90^\circ$ Anti-Clockwise rotation.

Consider a person initially facing North.

• After one right turn ($90^\circ$ CW), they will face East.
• From East, after another right turn ($90^\circ$ CW), they will face South.
• From South, a left turn ($90^\circ$ ACW) will make them face East.

Worked Example:

Problem: An automaton starts facing West. It rotates $135^\circ$ clockwise, then $45^\circ$ anti-clockwise, and finally turns right. Which direction is it facing now?

Solution:

Let us denote the initial direction as the reference, $0^\circ$. We can associate angles with directions, with East at $0^\circ$, North at $90^\circ$, West at $180^\circ$, and South at $270^\circ$. The automaton starts facing West ($180^\circ$).

Step 1: Account for the first rotation ($135^\circ$ clockwise). A clockwise rotation is negative.

The direction corresponding to $45^\circ$ is North-East.

Step 2: Account for the second rotation ($45^\circ$ anti-clockwise). An anti-clockwise rotation is positive.

The automaton is now facing the direction corresponding to $90^\circ$, which is North.

Step 3: Account for the final right turn. A right turn is a $90^\circ$ clockwise rotation.

The direction corresponding to $0^\circ$ is East.

Answer: The automaton is finally facing East.

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3. Relative Positioning

A frequent task in GATE is to determine the direction of one point with respect to another. This concept was central to the provided PYQ analysis.

Worked Example:

Problem: Point P is 10 m to the East of point Q. Point R is 10 m to the South of point Q. What is the direction of point P with respect to point R?

Solution:

Step 1: Draw a diagram based on the given information. Let Q be the origin $(0, 0)$.

• Since P is 10 m to the East of Q, the coordinates of P are $(10, 0)$.
• Since R is 10 m to the South of Q, the coordinates of R are $(0, -10)$.

Step 2: Determine the direction of P with respect to R. We place our frame of reference at R.

From point R, to reach point P, we must move East (from x-coordinate 0 to 10) and North (from y-coordinate -10 to 0).

Step 3: Conclude the direction.

Since the movement involves both an Easterly and a Northerly component, the resulting direction is North-East.

Answer: The direction of point P with respect to point R is North-East.

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4. Displacement and Pythagoras' Theorem

In many problems, we are asked to find the shortest distance between the starting and ending points. This is not the total distance walked, but the straight-line distance, also known as displacement. When the path of movement forms a right-angled triangle, we can use Pythagoras' theorem.

Worked Example:

Problem: A delivery agent starts from a warehouse and travels 12 km due East, then turns left and travels 5 km due North. What is the shortest distance between the warehouse and his final location?

Solution:

Step 1: Visualize the path. The agent's path forms a right-angled triangle. The Eastward journey is one side, and the Northward journey is the other. The warehouse is the starting point.

Start

End


12 km (East)
5 km (North)
Shortest Distance (c)

Step 2: Identify the perpendicular sides of the triangle.

Let $a$ be the distance traveled North, and $b$ be the distance traveled East.
$a = 5$ km
$b = 12$ km

Step 3: Apply Pythagoras' theorem to find the hypotenuse $c$.

Step 4: Calculate the result.

Answer: The shortest distance is 13 km.

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Problem-Solving Strategies

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Common Mistakes

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Practice Questions

:::question type="MCQ" question="A city has four landmarks: a Fort, a Museum, a Lake, and a Park. The Museum is to the South-West of the Fort. The Lake is to the East of the Museum and also to the South of the Fort. In which direction is the Park located with respect to the Fort, if the Park is to the North-East of the Lake?" options=["North-East","South-East","East","North-West"] answer="East" hint="Draw a relative position diagram. Place the Fort first, then position the Museum and Lake relative to it. Finally, place the Park relative to the Lake." solution="
Step 1: Establish the initial positions. Let the Fort (F) be at a reference point. The Museum (M) is South-West of F.

Step 2: Position the Lake (L). The Lake is East of M and South of F. This places L directly South of F and South-East of M.

Step 3: Position the Park (P). The Park is North-East of the Lake. Since L is directly South of F, moving North-East from L will place P to the East of the line segment FL.

Step 4: Analyze the final relative position. From the positions derived, the Park (P) will be located to the East of the Fort (F). The diagram would show F, then L directly below it, and P to the right of L, making P East of F.

Answer: \boxed{East}
"
:::

:::question type="NAT" question="A robot moves 20 meters North from its starting point. It then turns right and moves 30 meters. It then turns right again and moves 40 meters. Finally, it turns right and moves 30 meters. What is the shortest distance (in meters) between the robot's starting and final position?" answer="20" hint="Calculate the net displacement on the North-South and East-West axes. The robot moves 30m East and then 30m West, cancelling out the horizontal movement." solution="
Step 1: Analyze the movements along the North-South axis.

• The robot moves 20 m North (let's consider this +20).


• It then moves 40 m South (after two right turns from North), which we consider -40.


• Net displacement in N-S direction = $20 - 40 = -20$ m. This is 20 m South of the starting point.

Step 2: Analyze the movements along the East-West axis.

• The robot moves 30 m East (first right turn from North), which is +30.


• It then moves 30 m West (third right turn), which is -30.


• Net displacement in E-W direction = $30 - 30 = 0$ m.

Step 3: Calculate the final shortest distance.
The robot is 20 m South and 0 m East/West from its starting point.
The shortest distance is therefore the magnitude of the net N-S displacement.

Answer: \boxed{20}
"
:::

:::question type="MSQ" question="A person starts at point A, walks 10 m East to reach point B. Then, she turns left and walks 10 m to reach point C. She then turns $45^\circ$ clockwise and walks $10\sqrt{2}$ m to reach point D. Which of the following statements is/are correct?" options=["The final point D is to the East of the starting point A.","The person is finally facing North-East.","The shortest distance between A and D is 20 m.","The final point D is to the North-East of the starting point A."] answer="B,D" hint="Draw the path carefully. The 45-degree turn is critical. Use coordinate geometry or vector addition for precision." solution="
Let the starting point A be at the origin (0,0).

• A to B: Walks 10 m East. B is at (10,0). Person is facing East.


• B to C: Turns left (90° ACW) to face North. Walks 10 m North. C is at (10,10).


• C to D: At C, facing North. Turns $45^\circ$ CW. The new direction is North-East. Walks $10\sqrt{2}$ m.
  
  $$\begin{aligned} \text{The displacement vector for this leg is } & (10\sqrt{2} \cos(45^\circ), 10\sqrt{2} \sin(45^\circ)) \\$$
  & = \left(10\sqrt{2} \times \frac{1}{\sqrt{2}}, 10\sqrt{2} \times \frac{1}{\sqrt{2}}\right) \\
  & = (10, 10)\end{aligned}
  
  The final coordinates of D are the sum of coordinates of C and the displacement vector: D = $(10+10, 10+10) = (20, 20)$.

  Now let's evaluate the options:
  

  • A. The final point D is to the East of the starting point A: D is at (20,20), which is North-East of A(0,0), not just East. So, A is incorrect.
  
  
  • B. The person is finally facing North-East: This is correct, as the last turn was $45^\circ$ CW from North.
  
  
  • C. The shortest distance between A and D is 20 m: The distance is
  
  
  $$\begin{aligned} \sqrt{(20-0)^2 + (20-0)^2} & = \sqrt{20^2 + 20^2} \\$$
  & = \sqrt{400+400} \\
  & = \sqrt{800} \\
  & = \sqrt{400 \times 2} \\
  & = 20\sqrt{2} \text{ m}\end{aligned}
  
  So, C is incorrect.
  
  • D. The final point D is to the North-East of the starting point A: D is at (20,20), which is clearly in the North-East quadrant relative to the origin A(0,0). So, D is correct.
  
  
  
  Answer: \boxed{B,D}
  "
  :::

  :::question type="MCQ" question="A special clock is hanging on a wall. In its standard orientation, at 3:00, the minute hand points North (towards 12) and the hour hand points East (towards 3). The clock is then re-hung such that at 3:00, the hour hand points towards South. In this new orientation, which direction does the minute hand point at 7:30?" options=["North", "East", "West", "South"] answer="West" hint="First, determine the new orientation of the entire clock face. If the '3' mark now points South, where does the '12' mark point? The minute hand at 7:30 is always at the '6' mark." solution="
  Step 1: Analyze the standard orientation.
  At 3:00, the hour hand is at the '3' mark and the minute hand is at the '12' mark. The angle between them is 90 degrees, with the minute hand being 90 degrees anti-clockwise from the hour hand.

  Step 2: Analyze the new orientation.
  The clock is rotated so that the hour hand at 3:00 (the '3' mark) now points South.

  Step 3: Determine the new direction of the '12' mark.
  Since the minute hand ('12' mark) is always 90 degrees anti-clockwise from the hour hand ('3' mark), we find the direction 90 degrees anti-clockwise from South. This direction is East.
  So, in the new orientation: '3' points South, and '12' points East.

  Step 4: Determine the complete new compass for the clock.
  

  • If '12' is East and '3' is South, then '6' must be West, and '9' must be North.
  
  
  
  Step 5: Find the direction of the minute hand at 7:30.
  At 7:30, the minute hand always points to the '6' mark on the dial.

  Step 6: Conclude the final direction.
  In the new orientation, the '6' mark points towards the West.

  Answer: \boxed{West}
  "
  :::

  :::question type="NAT" question="An explorer starts at coordinates (0, 0) on a map. She travels $4\sqrt{2}$ km in the South-West direction. From there, she travels 8 km due East. What is the final y-coordinate of her position?" answer="-4" hint="Break down the South-West movement into its South and West components. Remember that South corresponds to a negative change in the y-coordinate." solution="
  Step 1: Calculate the displacement from the first movement.
  The explorer travels $4\sqrt{2}$ km South-West. This direction corresponds to an angle of $225^\circ$ or $-135^\circ$ from the positive x-axis.
  The change in x-coordinate is:
  

  $$\begin{aligned}\Delta x_1 & = 4\sqrt{2} \times \cos(225^\circ) \\$$
  & = 4\sqrt{2} \times \left(-\frac{1}{\sqrt{2}}\right) \\
  & = -4 \text{ km}\end{aligned}
  
  The change in y-coordinate is:
  
  $$\begin{aligned}\Delta y_1 & = 4\sqrt{2} \times \sin(225^\circ) \\$$
  & = 4\sqrt{2} \times \left(-\frac{1}{\sqrt{2}}\right) \\
  & = -4 \text{ km}\end{aligned}
  
  After the first move, her position is $(-4, -4)$.

  Step 2: Calculate the displacement from the second movement.
  She travels 8 km due East.
  The change in x-coordinate is $\Delta x_2 = +8$ km.
  The change in y-coordinate is $\Delta y_2 = 0$ km.

  Step 3: Calculate the final coordinates.
  The final x-coordinate is $0 + \Delta x_1 + \Delta x_2 = 0 - 4 + 8 = 4$.
  The final y-coordinate is $0 + \Delta y_1 + \Delta y_2 = 0 - 4 + 0 = -4$.
  Her final position is $(4, -4)$.

  Answer: \boxed{-4}
  "
  :::

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  Summary

  ❗ Key Takeaways for GATE

  • Master the Compass: Be fluent with the eight directions (N, S, E, W, NE, NW, SE, SW) and the angles between them ($90^\circ$ between cardinal, $45^\circ$ between cardinal and ordinal).
  
  
  • Reference Point is Key: For questions involving relative direction ("A with respect to B"), always treat the second point (B) as the origin of your compass.
  
  
  • Distinguish Final State: Clearly differentiate between the final direction of facing (determined by turns) and the final position's direction from the start (determined by the net displacement vector).
  
  
  • Use Pythagoras for Shortest Distance: When a path involves perpendicular turns (N/S and E/W movements), the shortest distance between the start and end points is the hypotenuse of the right-angled triangle formed by the net displacements. Do not sum the path lengths.

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  What's Next?

  💡 Continue Learning

  This topic provides a foundation for other areas of analytical and spatial aptitude.
  

  • Coordinate Geometry: Direction sense problems can be formally modeled on a 2D Cartesian plane. Understanding vector addition and coordinate transformations provides a more powerful method for solving complex paths.
  
  
  • Spatial Aptitude: The ability to visualize movement and orientation in 2D is a prerequisite for more advanced spatial reasoning tasks, which may involve the rotation and manipulation of 3D objects, paper folding, and cube analysis.
  
  

  Mastering these connections will build a comprehensive skill set for the GATE aptitude section.

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  💡 Moving Forward

  Now that you understand Direction Sense, let's explore Blood Relations which builds on these concepts.

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  Part 2: Blood Relations

  Introduction

  The topic of Blood Relations is a fundamental component of the Logical Reasoning section within the Analytical Aptitude syllabus for the GATE examination. These questions are designed not to test one's knowledge of family structures, but rather to evaluate the candidate's capacity for logical deduction, interpretation of complex statements, and systematic problem-solving. The problems typically present a convoluted set of relationships among a group of individuals, requiring the candidate to decipher the underlying family structure to answer a specific query.

  Success in this area hinges on the ability to translate verbal descriptions into a structured, visual format, most commonly a family tree diagram. This process eliminates ambiguity and prevents the common errors that arise from attempting to manage these relationships purely in one's mind. We shall explore the standard conventions for such diagrams, the key terminology of familial relationships, and the logical pitfalls that are often embedded in the problem statements. Mastery of this topic provides a strong foundation for tackling more intricate logical puzzles.

  📖 Blood Relation

  A blood relation is a relationship between two or more individuals established either by birth (consanguinity) or by marriage (affinity). For the purpose of GATE questions, we consider relationships extending from immediate family to several generations, including connections through marriage such as in-laws.

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  Key Concepts

  To systematically address problems in this domain, we must first establish a clear understanding of the terminology and the standard method for representing these relationships.

  1. The Family Tree Diagrammatic Method

  The most efficient and error-proof method for solving blood relation problems is the construction of a family tree. This visual representation clarifies the connections between individuals and across generations. We adopt a set of standard conventions for these diagrams to maintain consistency and clarity.

  Conventions for Family Tree Diagrams:

• Gender Representation:

* Males are represented by a square ($□$).
* Females are represented by a circle ($○$).
* If the gender is unknown, a diamond or a question mark ($◇$ or $?$) is used.
• Relationship Representation:

* A double horizontal line ($⇔$) connects a married couple.
* A single horizontal line ($―$) connects siblings.
* A vertical line ($∣$) descends from parents to their children.
• Generational Levels:

* The diagram is organized into distinct horizontal levels, with each level representing a generation. For instance, grandparents occupy the top level, parents the next level down, the individual ('self' or 'ego') and their siblings/cousins in the middle, and children/grandchildren in the subsequent lower levels.

Let us consider a visual representation of these conventions.



Family Tree Conventions



A



B

Gen 1







C

D

E
Gen 2


A ($□$): Male
B ($○$): Female
A $⇔$ B: Married Couple
C, D, E are siblings
A & B are parents of C, D, E
E ($◇$): Gender unknown

From this diagram, we can deduce: A is the father, B is the mother. C is their son, D is their daughter, and E is their child whose gender is not specified. C, D, and E are siblings.

2. Standard Blood Relation Terminology

Familiarity with the precise terms for various relationships is essential.

* Paternal side: Relations on the father's side (e.g., Paternal Grandfather is father's father).
* Maternal side: Relations on the mother's side (e.g., Maternal Uncle is mother's brother).

| Relation | Definition |
| :--- | :--- |
| Father's Father/Mother | Paternal Grandfather / Grandmother |
| Mother's Father/Mother | Maternal Grandfather / Grandmother |
| Father's Brother/Sister | Paternal Uncle / Paternal Aunt |
| Mother's Brother/Sister | Maternal Uncle / Maternal Aunt |
| Son's/Daughter's Son | Grandson |
| Son's/Daughter's Daughter| Granddaughter |
| Brother's/Sister's Son | Nephew |
| Brother's/Sister's Daughter| Niece |
| Uncle's/Aunt's Son/Daughter | Cousin |
| Spouse's Father/Mother | Father-in-law / Mother-in-law |
| Spouse's Brother/Sister | Brother-in-law / Sister-in-law |
| Brother's Wife | Sister-in-law |
| Sister's Husband | Brother-in-law |

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3. Analyzing Problem Statements

The core of solving these problems lies in the careful, step-by-step translation of given statements into a family tree.

Worked Example:

Problem: In a family, A is the brother of B. C is the father of A. D is the brother of E. E is the daughter of B. Who is the uncle of D?

Solution:

We shall construct the family tree based on the information provided, one statement at a time.

Step 1: "A is the brother of B."
This establishes A and B as siblings. We know A is male. The gender of B is currently unknown.

$$\text{A} \ (\Box) \text{—} \text{B} \ (?)$$

Step 2: "C is the father of A."
Since A and B are siblings, C must be the father of B as well.

$$\begin{array}{c} \text{C} \ (\Box) \\ | \\ \text{A} \ (\Box) \text{—} \text{B} \ (?)\end{array}$$

Step 3: "E is the daughter of B."
This establishes a parent-child relationship between B and E. We now know B has a daughter, E (female). This implies B is a parent, but their gender remains unknown.

$$\begin{array}{c} \text{C} \ (\Box) \\ | \\ \text{A} \ (\Box) \text{—} \text{B} \ (?) \\ \quad \quad \quad \quad | \\ \quad \quad \quad \quad \text{E} \ (\circ)\end{array}$$

Step 4: "D is the brother of E."
D and E are siblings, and both are children of B. We know D is male.

$$\begin{array}{c} \text{C} \ (\Box) \\ | \\ \text{A} \ (\Box) \text{—} \text{B} \ (?) \\ \quad \quad \quad \quad | \\ \quad \quad \text{D} \ (\Box) \text{—} \text{E} \ (\circ)\end{array}$$

Step 5: Determine the relationship between A and D.
The question asks for the uncle of D. From the diagram, D's parent is B. B's brother is A. The brother of one's parent is their uncle.

Result:
A is the uncle of D.

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Problem-Solving Strategies

For the time-constrained environment of the GATE exam, certain strategies can significantly improve speed and accuracy.

💡 GATE Strategy: Break and Build

Do not attempt to process all the given information at once. Read one statement, draw that part of the family tree, and then move to the next. This "break and build" approach prevents confusion and ensures each piece of information is correctly placed. If a statement connects individuals not yet on your diagram (e.g., "X is the cousin of Y"), note it down separately and integrate it once either X or Y appears.

💡 Exam Shortcut: Focus on the Question

Before drawing the complete tree, quickly read the final question (e.g., "How is P related to T?"). This helps you focus on the specific individuals and the path of relationships connecting them. You might not need to resolve every single relationship in the family to answer the question, saving valuable time.

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Common Mistakes

Candidates often make predictable errors in blood relation problems. Being aware of these can help in avoiding them.

⚠️ Avoid These Errors
• ❌ Assuming Gender from Names: In GATE questions, names like 'Ankit' or 'Priya' might be used, but one must not assume their gender. A name like 'Sidhu' or 'Kiran' can be gender-neutral. Gender must be inferred only from relationship keywords like 'brother', 'mother', 'his', 'her', etc.
✅ Correct Approach: Always use a neutral symbol (e.g., a diamond $◇$) for an individual until their gender is explicitly confirmed by the problem statement.
• ❌ Misinterpreting "Only Son/Daughter": The statement "P is the only son of Q" means Q has no other sons. However, Q can have one or more daughters. It does not imply that P is an only child.
✅ Correct Approach: Recognize that "only son" eliminates other male children but not female children. Similarly, "only daughter" eliminates other female children but not male children.
• ❌ Confusing In-law Relationships: The term 'brother-in-law' can mean two things: (1) your spouse's brother, or (2) your sister's husband. The context of the problem determines the correct interpretation.
✅ Correct Approach: Draw both possibilities if the context is ambiguous, or carefully look for clues that resolve the ambiguity.

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Practice Questions

:::question type="MCQ" question="Given the following relationships:


• X is the mother of Y.


• Y is the sister of Z.


• W is the father of Z.


• V is the son of X.



Which of the following statements is definitely true?" options=["V is the brother of W.","Y is the daughter of W.","X is the mother-in-law of W.","Z is a male."] answer="Y is the daughter of W." hint="Construct a family tree. Identify the married couple first." solution="
Step 1: Analyze the statements to build the family tree.

• 'X is the mother of Y' and 'W is the father of Z'.


• 'Y is the sister of Z'. This means Y and Z are siblings.


• Since Y and Z are siblings, their parents must be the same. Therefore, X (mother) and W (father) are the parents of both Y and Z. This implies X and W are a married couple.


$$W \ (□) \ ⇔ \ X \ (○)$$

Step 2: Place the children Y and Z.

• We know Y is the sister of Z, so Y is female. The gender of Z is not specified.


$$\begin{array}{c}$$
W \ (□) \ ⇔ \ X \ (○) \\
| \\
Y \ (○) \ ― \ Z \ (?)\end{array}

Step 3: Place V.

• 'V is the son of X'. Since X and W are parents, V is also the son of W. V is a sibling to Y and Z.


$$\begin{array}{c}$$
W \ (□) \ ⇔ \ X \ (○) \\
| \\
V \ (□) \ ― \ Y \ (○) \ ― \ Z \ (?)\end{array}

Step 4: Evaluate the options.

• V is the brother of W: False. V is the son of W.


• Y is the daughter of W: True. Y is the daughter of X and W.


• X is the mother-in-law of W: False. X is the wife of W.


• Z is a male: Cannot be determined. The gender of Z is unknown.



Result: The only statement that is definitely true is that Y is the daughter of W.
"
:::

:::question type="NAT" question="A family consists of six members: L, M, N, O, P, and Q. N is the son of O, but O is not his father. P is the brother of N. L is the wife of O. M is the daughter of L. Q is the brother of L's husband. How many male members are in the family?" answer="4" hint="Carefully determine the gender of each member. 'O is not his father' is a key clue." solution="
Step 1: Analyze 'N is the son of O, but O is not his father.'
This implies N is male, and O must be N's mother. So, O is female.

Step 2: Analyze 'L is the wife of O.'
This is a contradiction, as O is female. Let's re-read the problem carefully. There might be a typo in the setup, but let's assume the question meant "L and O are a married couple". Since O is the mother of N, O is female. Therefore, L must be the husband of O, making L male.


• L (male, husband) ⇔ O (female, wife)


• N is their son (male).



Step 3: Analyze 'P is the brother of N.'
P is a male and is also a son of L and O.

Step 4: Analyze 'M is the daughter of L.'
M is female and is a child of L and O.

Step 5: Analyze 'Q is the brother of L's husband.'
L's husband is L himself. So, Q is the brother of L. Q is male.

Step 6: Consolidate the family structure and count the males.


• L: Male (Husband)


• M: Female (Daughter)


• N: Male (Son)


• O: Female (Wife)


• P: Male (Son)


• Q: Male (Brother of L)



The male members are L, N, P, and Q.

Result: There are 4 male members in the family.
"
:::

:::question type="MSQ" question="Given that P is the sister of Q, R is the father of P, S is the wife of R, and T is the brother of S. Which of the following statements can be true?" options=["T is the maternal uncle of P.","Q is the son of S.","P is the niece of T.","R is the son-in-law of T's parents."] answer="A,C,D" hint="Construct the family tree. Evaluate each option independently. Note that the gender of Q is not given." solution="
Step 1: Construct the family tree.


• 'P is the sister of Q' -> P is female. Q's gender is unknown. P and Q are siblings.


• 'R is the father of P' -> R is male. R is the father of both P and Q.


• 'S is the wife of R' -> S is female. S is the mother of both P and Q.


• 'T is the brother of S' -> T is male.



The tree is:

$$\begin{array}{c}$$
R \ (□) \ ⇔ \ S \ (○) \ ― \ T \ (□) \\
| \\
P \ (○) \ ― \ Q \ (?)\end{array}

Step 2: Evaluate the options.


• A: T is the maternal uncle of P. P's mother is S. S's brother is T. Therefore, T is the maternal uncle of P. This statement is true.


• B: Q is the son of S. The gender of Q is not specified in the problem. Q could be a son or a daughter of S. So, this statement is not necessarily true.


• C: P is the niece of T. T is P's maternal uncle. Therefore, P is the niece of T. This statement is true.


• D: R is the son-in-law of T's parents. R is married to S. T is the brother of S. This means S and T have the same parents. R, being the husband of their daughter S, is their son-in-law. This statement is true.



Result: Statements A, C, and D are true based on the given information.
"
:::

---

Summary

❗ Key Takeaways for GATE

• Diagram is Non-Negotiable: For any non-trivial blood relations problem, immediately begin sketching a family tree using standard conventions (squares for males, circles for females, lines for relationships). This is the single most effective technique.


• Gender is Explicit, Not Implicit: Never assume the gender of a person from their name. A person's gender is known only if it is explicitly stated (e.g., "she is the mother") or logically required by a relationship (e.g., a "wife" must be female).


• Deconstruct "Only": Carefully interpret statements with qualifiers like "only". "A is the only son of B" means B has no other sons but may have daughters. This is a common trap tested to assess logical precision.

---

What's Next?

💡 Continue Learning
This topic serves as a building block for more complex logical reasoning questions in the GATE exam. Your understanding here directly applies to:
• Seating Arrangements: Many problems combine blood relations with linear or circular seating arrangements (e.g., "A family of seven members are sitting in a row..."). The ability to first map the family structure is crucial before solving the arrangement puzzle.
• Complex Logical Puzzles: The skill of translating verbose, unstructured information into a logical diagram is central to solving larger data arrangement and logic puzzles. Blood relations provide excellent practice for this fundamental skill.
Mastering the connections between these topics will provide a more robust and comprehensive preparation for the Analytical Aptitude section.

---

---

💡 Moving Forward

Now that you understand Blood Relations, let's explore Analogy which builds on these concepts.

---

Part 3: Analogy

Introduction

In the domain of analytical aptitude, the study of analogies serves as a fundamental exercise in logical reasoning. An analogy is a comparison between two distinct entities that highlights a particular relationship they share. The ability to discern these relationships is not merely an academic skill; it is a critical component of pattern recognition and problem-solving, which are central to the fields of data science and artificial intelligence.

Questions on analogy presented in the GATE examination are designed to test a candidate's capacity to identify the logical correspondence between a given pair of words and then apply that same logic to a second pair. Mastery of this topic requires a combination of a robust vocabulary and a systematic approach to relationship analysis. We shall explore the various types of relationships that commonly form the basis of these questions and develop strategies for arriving at the correct solution with precision and efficiency.

📖 Analogy

An analogy establishes a logical relationship between two pairs of items. It is formally represented as $A : B :: C : D$, which is read as "$A$ is to $B$ as $C$ is to $D$". The core principle is that the relationship between $A$ and $B$ is identical to the relationship between $C$ and $D$. The task is typically to find the missing item, often $D$, by first deciphering the relationship in the known pair ($A:B$).

---

Key Concepts

The foundation of solving analogy problems lies in the accurate identification of the relationship between the words in the given pair. While the number of possible relationships is vast, we can categorize the most common types encountered in competitive examinations.

1. The Process of Relationship Identification

The primary method for solving an analogy problem is a two-step process. First, we must precisely articulate the relationship between the given pair of words, say $A$ and $B$. This relationship should be as specific as possible. Second, we apply this exact relationship to the third term, $C$, to determine the fourth term, $D$.

Consider the analogy `Pen : Write`. The relationship is not merely that they are related; it is that a pen is a tool used for the action of writing. Applying this "Tool : Action" relationship to another word, say `Knife`, leads us to the action it performs, which is to `Cut`. Thus, `Pen : Write :: Knife : Cut`.

2. Common Analogous Relationships

Let us now turn our attention to the principal categories of relationships that are frequently tested.

a. Tool and its Action

This relationship connects an object with its primary function or the action it is designed to perform.

Worked Example:

Problem: `Axe : Cleave :: Needle : ______`

Solution:

Step 1: Identify the relationship in the first pair, `Axe : Cleave`.
An Axe is a tool used for the action of cleaving (splitting or cutting). The relationship is `Tool : Action`.

Step 2: Apply the same relationship to the second pair.
A Needle is a tool. We must find the primary action it performs.

Step 3: Evaluate the options (assuming options like Sew, Prick, Thread, Point).
A needle is used to sew. While it does prick, its main purpose in context is sewing.

Answer: `Sew`

---

b. Intensity or Degree

This type of analogy involves words that represent different levels of the same underlying concept or quality. The relationship is one of progression, either increasing or decreasing in intensity.

Worked Example:

Problem: `Annoyed : Furious :: Pleased : ______`

Solution:

Step 1: Determine the relationship between `Annoyed` and `Furious`.
`Furious` is a state of much higher intensity than `Annoyed`. The relationship is `Low Intensity : High Intensity`.

Step 2: Apply this relationship to the word `Pleased`.
We need to find a word that represents a high-intensity form of being pleased.

Step 3: Consider potential answers.
Words like 'Happy', 'Joyful', or 'Ecstatic' could be options. 'Ecstatic' represents the highest degree of pleasure.

Answer: `Ecstatic`

---

---

c. Performer and Action/Product

This common analogy links a person or entity (the performer) with the action they perform or the product they create.

📐 Performer and Product/Action

$$\text{Performer} : \text{Product}$$

$$\text{or}$$

$$\text{Performer} : \text{Action}$$

Variables:


• Performer: The entity doing the action (e.g., Author, Chef, Doctor).


• Product/Action: The result of the performer's work (e.g., Book, Dish, Surgery).




When to use: When the first word is a profession, animal, or object known for a specific creation or function.

Worked Example:

Problem: `Architect : Blueprint :: Composer : ______`

Solution:

Step 1: Establish the relationship in the pair `Architect : Blueprint`.
An Architect is a professional who creates a Blueprint. The relationship is `Performer : Product`.

Step 2: Apply the same logic to `Composer`.
A Composer is a professional who creates a specific product.

Step 3: Determine the product created by a composer.
A composer writes a musical score.

Answer: `Score`

---

d. Synonyms and Antonyms

This category is based on the meanings of the words themselves. The pair can either be synonyms (words with similar meanings) or antonyms (words with opposite meanings).

Worked Example:

Problem: `Sparse : Abundant :: Taciturn : ______`

Solution:

Step 1: Analyze the relationship between `Sparse` and `Abundant`.
`Sparse` means thinly dispersed or scanty. `Abundant` means existing in large quantities. These words are opposites. The relationship is `Antonym`.

Step 2: Apply this antonym relationship to `Taciturn`.
`Taciturn` describes a person who is reserved or says very little. We need to find its opposite.

Step 3: Identify the antonym.
The opposite of being reserved and quiet is being talkative or communicative.

Answer: `Talkative` (or a synonym like `Garrulous`)

---

e. Individual and Dwelling Place

This relationship connects a living being (person or animal) to its specific home or place of residence.

Worked Example:

Problem: `Monk : Monastery :: Bear : ______`

Solution:

Step 1: Identify the relationship in `Monk : Monastery`.
A Monk is a person who lives in a Monastery. The relationship is `Individual : Dwelling Place`.

Step 2: Apply this relationship to `Bear`.
We need to find the specific name for the dwelling place of a bear.

Step 3: Recall the correct term.
While a bear might live in a cave, the more specific term for its den or home is a lair.

Answer: `Lair`

---

Problem-Solving Strategies

A systematic approach is paramount to achieving high accuracy under examination conditions. We recommend two primary strategies.

💡 The Sentence Method

This is the most reliable technique for solving analogy questions. Formulate a clear and precise sentence that defines the relationship between the first pair of words ($A$ and $B$). Then, apply the exact same sentence structure to the second pair ($C$ and $D$) to find the missing word.

Example: For `Surgeon : Scalpel`, the sentence is: "A surgeon is a professional who uses a scalpel as a tool."
Now apply this to `Author : ______`. The sentence becomes: "An author is a professional who uses a ______ as a tool." The answer would be `Pen` or `Keyboard`.

💡 Relationship Directionality

Pay close attention to the order of the words. The relationship from $A$ to $B$ is not the same as from $B$ to $A$.

For instance, `Chapter : Book` has a `Part : Whole` relationship.
However, `Book : Chapter` has a `Whole : Part` relationship.

If the question is `Chapter : Book :: Room : ______`, the answer is `House`.
If the question were `Book : Chapter :: House : ______`, the answer would be `Room`.
Always maintain the established direction of the relationship.

---

---

Common Mistakes

Even with a good vocabulary, certain conceptual errors can lead to incorrect answers. It is crucial to be aware of these pitfalls.

⚠️ Avoid These Errors
• ❌ Choosing a word that is merely associated, not analogous: For `Pen : Write :: Knife : ______`, the word `Sharp` is an attribute of a knife, but the relationship is `Tool : Action`. The correct answer must be an action, like `Cut`.
✅ Focus strictly on the identified relationship. The relationship type must be identical for both pairs.
• ❌ Reversing the relationship: For `Virus : Disease` (Cause : Effect), a common error is to match it with `Recovery : Medicine` (Effect : Cause).
✅ Maintain the exact order and direction of the relationship. The correct analogy would be `Medicine : Recovery` (Cause : Effect).
• ❌ Misinterpreting the primary meaning of a word: Some words have multiple meanings. The context provided by the first pair is essential to determine the intended meaning.
✅ Use the first pair to lock in the context before considering the second pair.

---

Practice Questions

:::question type="MCQ" question="Whisper : Shout :: Suggest : ______" options=["Demand","Request","Imply","Concur"] answer="Demand" hint="Consider the relationship of intensity between the first pair of words." solution="
Step 1: Analyze the relationship between `Whisper` and `Shout`.
`Shout` is a high-intensity form of `Whisper`. The relationship is one of increasing degree or intensity (`Low Intensity : High Intensity`).

Step 2: Apply this relationship to the word `Suggest`.
We need a word that represents a high-intensity form of suggesting.

Step 3: Evaluate the options.


• `Request` is similar in intensity to suggest.


• `Imply` is a less direct form of suggesting.


• `Concur` means to agree.


• `Demand` is a forceful and high-intensity way of putting forward a proposition, much like a shout is a high-intensity whisper.



Result: The correct analogous word is `Demand`.
Answer: \boxed{Demand}
"
:::

:::question type="NAT" question="In a specific coding, the relationship between shapes and numbers is defined by the number of vertices. If Square : 4 :: Octagon : ?" answer="8" hint="The relationship is 'Object : Count of its Vertices'." solution="
Step 1: Identify the relationship in `Square : 4`.
A square is a polygon with 4 vertices (or sides). The relationship is `Geometric Shape : Number of Vertices`.

Step 2: Apply this rule to `Octagon`.
An octagon is a polygon. We need to find its number of vertices.

Step 3: Recall the definition of an octagon.
By definition, an octagon is a polygon with 8 vertices.

Result: The correct number is 8.
Answer: \boxed{8}
"
:::

:::question type="MSQ" question="Which of the following pairs share the same logical relationship as `Ornithology : Birds`?" options=["Botany : Plants","Seismology : Earthquakes","Geology : Geese","Etymology : Insects"] answer="Botany : Plants,Seismology : Earthquakes" hint="First, identify the relationship in the given pair. Then, check each option to see if it matches that specific relationship type." solution="
Step 1: Determine the relationship in `Ornithology : Birds`.
Ornithology is the scientific study of birds. The relationship is `Field of Study : Subject of Study`.

Step 2: Evaluate each option against this relationship.


• A. Botany : Plants: Botany is the scientific study of plants. This matches the `Field of Study : Subject of Study` relationship. This is a correct option.


• B. Seismology : Earthquakes: Seismology is the scientific study of earthquakes. This also matches the relationship. This is a correct option.


• C. Geology : Geese: Geology is the study of the Earth, not geese. This is incorrect.


• D. Etymology : Insects: Etymology is the study of the origin of words. The study of insects is Entomology. This is incorrect.



Result: The correct pairs are `Botany : Plants` and `Seismology : Earthquakes`.
Answer: \boxed{Botany : Plants, Seismology : Earthquakes}
"
:::

:::question type="MCQ" question="Glove : Hand :: Sandal : ______" options=["Foot","Leather","Strap","Walk"] answer="Foot" hint="This is a 'Part of Body and its Covering' analogy." solution="
Step 1: Analyze the relationship in `Glove : Hand`.
A glove is a piece of clothing worn on the hand. The relationship is `Covering : Body Part`.

Step 2: Apply the same logic to `Sandal`.
A sandal is a piece of footwear worn on a specific body part.

Step 3: Identify the corresponding body part.
A sandal is worn on the foot.

Result: The correct answer is `Foot`.
Answer: \boxed{Foot}
"
:::

:::question type="MCQ" question="Indifference : Apathy :: Zeal : ______" options=["Passion","Laziness","Anxiety","Neutrality"] answer="Passion" hint="Determine if the first pair of words are synonyms or antonyms." solution="
Step 1: Establish the relationship between `Indifference` and `Apathy`.
`Indifference` is a lack of interest or concern. `Apathy` is also a lack of interest, enthusiasm, or concern. These words are synonyms.

Step 2: Apply the synonym relationship to `Zeal`.
`Zeal` means great energy or enthusiasm in pursuit of a cause or an objective. We need to find a synonym for it.

Step 3: Evaluate the options.


• `Passion` means strong and barely controllable emotion, which is a close synonym for zeal.


• `Laziness` is an antonym.


• `Anxiety` is a feeling of worry.


• `Neutrality` is the state of not supporting either side.



Result: The correct synonym is `Passion`.
Answer: \boxed{Passion}
"
:::

---

Summary

To excel in analogy questions, a disciplined approach is essential. The core principles can be distilled into a few key points for quick revision.

❗ Key Takeaways for GATE

• Relationship is Paramount: Your primary task is to identify the precise logical relationship in the given pair before considering the options. Do not be distracted by words that are merely associated.


• Employ the Sentence Method: Formulate a clear sentence connecting the first two words ($A$ and $B$). This crystallizes the relationship and provides a template to test the options for the second pair ($C$ and $D$).


• Recognize Common Patterns: Familiarize yourself with the major types of analogies, such as Tool-Action, Performer-Product, Part-Whole, Intensity, and Synonyms/Antonyms, as these frequently appear in the exam.


• Verify Directionality: The order of the words matters. The relationship $A:B$ is distinct from $B:A$. Ensure the direction of the relationship is consistent between both pairs.

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What's Next?

A strong foundation in analogy is beneficial for other areas of the GATE syllabus. The skills developed here are directly transferable.

💡 Continue Learning

This topic connects to:


• Classification (Odd One Out): In classification problems, you must identify an item that does not share a common property with others in a group. This requires the same relationship-identification skills used in analogy, but applied to a set rather than a pair.


• Verbal Reasoning & Vocabulary: Analogy questions are fundamentally a test of both vocabulary and reasoning. Expanding your vocabulary will directly improve your ability to discern the subtle relationships between words, which is crucial for all verbal sections of the exam.




Master these connections for a more holistic and robust preparation for the Analytical Aptitude section.

---

Chapter Summary

📖 Analogies and Relations - Key Takeaways

• Mastery of cardinal (North, South, East, West) and ordinal (North-East, South-West, etc.) directions is fundamental to solving problems of displacement and orientation. We must recognize that the effect of a 'left' or 'right' turn is always relative to the current direction of travel.

• For problems involving displacement, it is often most effective to resolve all movements into orthogonal components, typically along the North-South and East-West axes. The shortest distance between the initial and final points can then be determined using the Pythagorean theorem, $d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$.
  • In analyzing Blood Relations, the adoption of a systematic diagrammatic convention is not merely helpful but essential for accuracy. A clear visual representation of generational hierarchy, marital links, and sibling relationships prevents the misinterpretation of complex statements.
  • We must exercise precision in interpreting relational terminology. A clear distinction between paternal and maternal lineage (e.g., paternal uncle vs. maternal uncle) and a firm grasp of the definitions of terms like 'cousin', 'niece', and 'brother-in-law' are critical.
  • The core of solving an analogy lies in identifying the precise logical relationship that connects the first pair of terms. This relationship must be articulated as a clear rule before it can be applied to the second pair.
  • The types of relationships in analogies are diverse and may include, but are not limited to: cause and effect (e.g., Virus:Disease), part to whole (e.g., Blade:Fan), tool and object (e.g., Scalpel:Surgery), and synonym/antonym pairs.
  • Advanced aptitude questions frequently synthesize concepts. We may be required to use spatial reasoning derived from Direction Sense to map out the positions of individuals whose connections are described using the logic of Blood Relations.

---

Chapter Review Questions

:::question type="MCQ" question="Six members of a family—A, B, C, D, E, and F—are standing in a field. B is the son of C, but C is not his mother. A and C are a married couple. E is the brother of C. D is the daughter of A. F is the brother of B. The members are positioned such that D is at a reference point. C is 30 m to the north of D. F is 20 m to the east of D. A is 15 m to the south of F. Finally, E is 20 m to the west of A. In which direction is C with respect to E?" options=["North","South","East","West"] answer="A" hint="First, construct the family tree to establish all relationships. Then, plot the positions of each member on a 2D Cartesian plane to determine the final relative direction." solution="
Part 1: Decoding the Blood Relations

We are given the following relationships:


• B is the son of C, but C is not his mother. This implies C is the father of B. So, C is male.


• A and C are a married couple. Since C is male, A must be female (his wife).


• D is the daughter of A. Since A and C are married, D is their daughter.


• F is the brother of B. This means F is also a son of A and C.


• E is the brother of C. This makes E the paternal uncle of B, D, and F.



The family structure is:

• Parents: C (father) and A (mother)


• Children: B (son), D (daughter), F (son)


• Uncle: E (C's brother)



Part 2: Determining Positions on a Cartesian Plane

Let us set the reference point D at the origin $(0, 0)$.


• C is 30 m to the north of D. The coordinates of C are $(0, 30)$.


• F is 20 m to the east of D. The coordinates of F are $(20, 0)$.


• A is 15 m to the south of F. The coordinates of A are $(20, 0 - 15) = (20, -15)$.


• E is 20 m to the west of A. The coordinates of E are $(20 - 20, -15) = (0, -15)$.

Part 3: Finding the Relative Direction

We need to find the direction of C with respect to E.


• Coordinates of C: $C = (0, 30)$


• Coordinates of E: $E = (0, -15)$



The x-coordinates of C and E are the same ($x=0$), which means they lie on the same vertical line. The y-coordinate of C ($30$) is greater than the y-coordinate of E ($-15$). Therefore, C is directly to the North of E.

The correct option is A.
Answer: \boxed{North}
"
:::

:::question type="NAT" question="A robot follows a sequence of coded instructions. The code `(X, Y)` means 'move $X$ meters in direction $Y$'. The directions are analogous to a clock face: 12 represents North, 3 represents East, 6 represents South, and 9 represents West. The robot executes the following sequence from its starting point: (10, 12), (15, 3), (5, 6), (3, 9). What is the shortest distance (in meters) from its starting point to its final position?" answer="13" hint="Decode the analogical directions into standard North, South, East, and West components. Then, calculate the net displacement along the East-West and North-South axes before applying the Pythagorean theorem." solution="
The problem requires us to calculate the net displacement of the robot and then find the magnitude of the displacement vector.

Step 1: Decode the instructions

We translate the clock-face analogy into standard directions:


• (10, 12) $\implies$ 10 meters North


• (15, 3) $\implies$ 15 meters East


• (5, 6) $\implies$ 5 meters South


• (3, 9) $\implies$ 3 meters West



Step 2: Calculate net displacement along orthogonal axes

Let us calculate the net movement along the East-West axis ($\Delta x$) and the North-South axis ($\Delta y$). We can consider East and North as positive directions.

• Net East-West displacement ($\Delta x$):
The robot moves 15 m East and 3 m West.
$$\Delta x = 15 (\text{East}) - 3 (\text{West}) = +12 \text{ m}$$
This is a net displacement of 12 meters to the East.
• Net North-South displacement ($\Delta y$):
The robot moves 10 m North and 5 m South.
$$\Delta y = 10 (\text{North}) - 5 (\text{South}) = +5 \text{ m}$$
This is a net displacement of 5 meters to the North.

Step 3: Calculate the shortest distance

The shortest distance from the start to the end point is the magnitude of the net displacement vector. This can be found using the Pythagorean theorem.


$$d = \sqrt{(\Delta x)^2 + (\Delta y)^2}$$


$$d = \sqrt{(12)^2 + (5)^2}$$


$$d = \sqrt{144 + 25}$$


$$d = \sqrt{169}$$


$$d = 13$$

The shortest distance is 13 meters.
Answer: \boxed{13}
"
:::

:::question type="MCQ" question="In a symbolic language, 'P # Q' means 'P is the father of Q', 'P @ Q' means 'P is the sister of Q', 'P Q' means 'P is the mother of Q', and 'P Q' means 'P is the brother of Q'. Which of the following expressions definitively indicates that M is the paternal grandfather of N?" options=["M # K # N","K M # L N","M # K L @ N","N $ L @ K # M"] answer="A" hint="Translate each expression into a family tree. The target relationship requires M to be the father of a male, who in turn is the father of N." solution="
The objective is to find the expression where M is the father of N's father. Let us analyze each option.

A) M # K # N


• `M # K` means 'M is the father of K'. This establishes M as a male and K as his child.


• `K # N` means 'K is the father of N'. This establishes K as a male and N as his child.


• Combining these, M is the father of K, and K is the father of N. This makes M the paternal grandfather of N. This expression is correct.



B) K * M # L $ N

• `K * M` means 'K is the brother of M'.


• `M # L` means 'M is the father of L'.


• `L $ N` means 'L is the mother of N'.


• From this, L is N's mother, and M is L's father. This makes M the maternal grandfather of N, not the paternal grandfather. This expression is incorrect.



C) M # K * L @ N

• `M # K` means 'M is the father of K'.


• `K * L` means 'K is the brother of L'.


• `L @ N` means 'L is the sister of N'.


• Since K is the brother of L and L is the sister of N, K, L, and N are all siblings. M is the father of K, which means M is the father of all three siblings. Thus, M is the father of N, not the grandfather. This expression is incorrect.



D) N $ L @ K # M

• `N $ L` means 'N is the mother of L'.


• This statement already contradicts the requirement that N be the grandchild. Therefore, this expression is incorrect without further analysis.



Based on our analysis, only expression A correctly and definitively represents M as the paternal grandfather of N.
Answer: \boxed{M # K # N}
"
:::

:::question type="MCQ" question="A compass needle that is pointing North is rotated $135^{\circ}$ clockwise. It is then rotated $45^{\circ}$ counter-clockwise. The final direction is analogous to the initial direction in the same way that 'Up' is analogous to which of the following?" options=["Down","Right","Left","Itself"] answer="B" hint="First, determine the net angular displacement and the final direction of the compass needle. Then, identify the transformation rule (e.g., rotation) and apply the same rule to the word 'Up'." solution="
Step 1: Calculate the net rotation of the compass needle.

We define clockwise rotation as positive (+) and counter-clockwise rotation as negative (-).


• Initial rotation: $+135^{\circ}$


• Subsequent rotation: $-45^{\circ}$



The net angular displacement from the initial North direction is:

$$\text{Net Rotation} = 135^{\circ} - 45^{\circ} = 90^{\circ}$$

A net rotation of $+90^{\circ}$ signifies a $90^{\circ}$ clockwise rotation.

Step 2: Determine the final direction.

The initial direction is North. Rotating $90^{\circ}$ clockwise from North results in the direction East.


• Initial Direction: North


• Final Direction: East



Step 3: Formulate the analogy.

The analogy is based on the transformation from the initial state to the final state.
The relationship is: `North : East`
This transformation corresponds to a $90^{\circ}$ clockwise turn.

Step 4: Apply the analogy to the word 'Up'.

We must apply the same transformation rule (a $90^{\circ}$ clockwise turn) to the term 'Up'. If we visualize 'Up' as the primary direction on a 2D plane (equivalent to North), a $90^{\circ}$ clockwise turn would point to the 'Right'.

Therefore, 'Up' is analogous to 'Right'.
The correct option is B.
Answer: \boxed{Right}
"
:::

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What's Next?

💡 Continue Your GATE Journey
Having completed this chapter on Analogies and Relations, you have established a firm foundation for several key areas within the Analytical Aptitude section. The skills developed here are not isolated; rather, they are prerequisite for more complex reasoning tasks.
Key connections:
• The principles of spatial reasoning and relative positioning mastered in Direction Sense are directly applicable to the more advanced chapter on Spatial Aptitude, which includes topics like 2D & 3D visualization, paper folding, and cube problems. Your ability to mentally manipulate objects in space begins here.
• The logical deduction and rule-based analysis used to solve Blood Relations and Analogy problems are foundational for all structured reasoning puzzles. These skills will be called upon again in chapters like Seating Arrangements, where relational and positional constraints are often combined, and in Syllogisms, which require rigorous logical inference.