Mathematical Reasoning

Step 1: Understand the Concept
This problem is a classic Direction Sense Test. To solve these, it is helpful to visualize a standard compass where North is up, South is down, East is right, and West is left. When the person "turns right" or "turns left," their new direction depends on which way they were currently facing. We track the movement from a fixed starting point (O) to determine the final relative position.

Step 2: Analysis of Movement
Let's trace the path step-by-step from the starting point (O):
* Initial Move: The man walks 5 km South. He is now at Point A, facing South.
* First Turn: While facing South, a "Right" turn points him toward the West. He walks 3 km in this direction to reach Point B.
* Second Turn: While facing West, a "Left" turn points him back toward the South. He walks 5 km further to reach Point C.

Step 3: Conclusion
Now, look at the position of Point C relative to the original starting point O:
* Point C is further South than the starting point (5 km + 5 km = 10 km South).
* Point C is also to the West of the starting point (3 km West).
Combining these two, Point C is located in the South-West direction from the starting point O.

Final Answer: (C)

Step 1: Concept
Blood relation analysis.
Step 2: Analysis
- "My father's son" refers to the man himself (since he has no siblings). - The statement then becomes: "That man's father is ME."
Step 3: Conclusion
If the man is the father of the person in the photograph, the photograph belongs to his son.
Final Answer: (B)

Step 1: Analyze the inequality.
We are given the inequality . We need to determine for which values of this inequality holds true.

Step 2: Check small values of .
- For : So, the inequality does not hold for . - For : So, the inequality holds for . - For : So, the inequality holds for . - For : So, the inequality holds for .

Step 3: General observation.
As increases, grows much faster than . Hence, the inequality holds for .

Step 4: Conclusion.
Thus, the inequality holds true for , corresponding to option (C).

Step 1: Identify the original statement.
The original statement is: This is a logical OR statement.

Step 2: Apply negation to the OR statement.
The negation of an OR statement is given by De Morgan's law. De Morgan's law states that: Thus, the negation of the statement “Ram is in Class X or Rashmi is in Class XII” is: This simplifies to:

Step 3: Conclusion.
Thus, the negation of the original statement is “Ram is not in Class X and Rashmi is not in Class XII,” which does not match any of the given options exactly. Therefore, the correct answer is option (D), “None of these.”

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Equality when
Step 3: Final Answer

Concept: Evaluate logical expression using truth table.

Step 1: Understand components.


Step 2: Check cases.
Expression is sometimes true and sometimes false.

Step 3: Conclusion from truth table.
Not always true (not tautology), not always false (not contradiction). Conclusion:
Answer = neither tautology nor contradiction