Explanation : 

Total Cost:
p + q + r = 50
Given q = 16 (the least),
p + r = 34

Statement I: p ≤ r

Possible pairs (p, r):
p = 16, r = 18
p = 17, r = 17
Conclusion: p can be 16 or 17. Statement I alone insufficient.

Statement II: r ≤ p

Possible pairs (p, r):
p = 17, r = 17
p = 18, r = 16
Conclusion: p can be 17 or 18. Statement II alone insufficient.

Using Both Statements Together:
p ≤ r and r ≤ p imply p = r
Thus, p = r = 17
Conclusion: p is uniquely determined as 17 using both statements.

Answer: c) The Question can be answered by the Statements using both together, but cannot be answered using either Statement alone.

Explanation : 

Objective: Determine if (x + y) is an integer.

Analysis:

1. Solving the System:
Let 2x + y = A (where A is an integer).
Let x + 2y = B (where B is an integer).

Multiply equation (1) by 2:
4x + 2y = 2A

Subtract equation (2):
3x = 2A – B
x = (2A – B)/3

Substitute x back:
y = A – 2x = A – 2(2A – B)/3 = (-A + 2B)/3

Therefore:
x + y = (2A – B)/3 + (-A + 2B)/3 = (A + B)/3

For (x + y) to be integer, A + B must be divisible by 3.

2. Evaluating Statements:
Using Statement I Alone: Knowing 2x + y is integer (A) doesn’t guarantee x + y is integer.

Using Statement II Alone: Knowing x + 2y is integer (B) doesn’t guarantee x + y is integer.

Using Both Statements Together: Even with both A and B known, x + y = (A + B)/3 is not guaranteed to be integer unless A + B is divisible by 3, which isn’t assured.

Conclusion:
Even with both statements, there’s insufficient information to determine whether (x + y) is an integer.

Answer: The Question cannot be answered even by using both the Statements together.

Explanation : 

Using Statement I Alone:

Primes less than 23: 2, 3, 5, 7, 11, 13, 17, 19.
Possible Combinations:
(2, 3, 5) -> Sum = 10 (Not prime)
(2, 3, 7) -> Sum = 12 (Not prime)
(3, 5, 11) -> Sum = 19 (Prime)
… (Other combinations either do not sum to a prime or exceed the sum limit)

Conclusion: The only valid combination under this statement is (3, 5, 11).

Using Statement II Alone:

Given: One of the numbers is 5.
Possible Combinations (without sum constraint):
(3, 5, 11) -> Sum = 19 (Prime)
(5, 7, 11) -> Sum = 23 (Prime)
(5, 7, 17) -> Sum = 29 (Prime)
(5, 13, 17) -> Sum = 35 (Not prime)
…and several others.

Conclusion: There are multiple valid combinations, such as (3, 5, 11), (5, 7, 11), (5, 7, 17), etc. Therefore, Statement II alone does not uniquely determine the answer.

Final Conclusion:
Statement I alone is sufficient to uniquely identify the three primes as (3, 5, 11).
Statement II alone is not sufficient as it allows for multiple valid combinations.

Answer: The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone.

Explanation : 

To determine the number of students in the class based on the given information, let’s analyze both statements individually and then consider them together.

Given:- Average marks = 60

Question:- What is the number of students in the class?

Analysis:

Statement I:
– Information Provided:
– Highest marks = 70
– Lowest marks = 50
– Implications:
– Without additional information about the distribution of marks or the total sum of marks, knowing just the highest and lowest marks doesn’t help in determining the number of students.
– Conclusion: Statement I alone is insufficient to determine the number of students.

Statement II:
– Information Provided:
– Removing the highest and lowest marks doesn’t change the average.
– Mathematical Implication:
– Let the number of students be n.
– Total sum of marks = Average × Number of students = 60n
– After removing highest and lowest marks:
– New sum = 60n – 70 – 50 = 60n – 120
– New number of students = n – 2
– New average = 60
– Setting up the equation:
60n – 120/n – 2 = 60
60n – 120 = 60(n – 2)
60n – 120 = 60n – 120
– The equation simplifies to 0 = 0, which is always true regardless of the value of n.
– Conclusion: Statement II alone is insufficient to determine the number of students.

Combining Statements I and II:

– Even when combining both statements:
– Highest Marks: 70
– Lowest Marks: 50
– Average remains 60 after exclusion.
– Attempts to solve for n with combined information still do not yield a unique solution for the number of students.
– Conclusion: Even together, the statements are insufficient to determine the exact number of students.

Final Answer: d) The Question cannot be answered even by using both the Statements together

Explanation : 

To determine who received the least amount among X, Y, and Z, let’s analyze the given statements:

Statement I: Given: X = 4/5(Y + Z)

Implication: The total amount distributed, T = X + Y + Z. Substituting from the statement:
T = 4/5(Y + Z) + Y + Z = 9/5(Y + Z)
Y + Z = 5/9T
X = 4/9T
Analysis: While we know X = 4/9T and Y + Z = 5/9T, we cannot determine the individual values of Y and Z. Without knowing how Y and Z are divided, we cannot conclusively identify who received the least.

Statement II: Given: Y = 2/7(X + Z)

Implication:
Let X + Z = a. Then, Y = 2/7a and T = a + 2/7a = 9/7a
Y = 2/9T
X + Z = 7/9T
Analysis: Even though we know Y = 2/9T, without the individual values of X and Z, it’s unclear whether Y is the least or if X or Z could be lesser based on their distribution.

Using Both Statements Together:

From Statement I: X = 4/9T
From Statement II: Y = 2/9T and X + Z = 7/9T
Substituting:
Y = 2/7(X + Z) = 2/7 × 7/9T = 2/9T
X = 4/9T
Z = 3/9T
Conclusion: The distribution ratios are X:Y:Z = 4:2:3. Clearly, Y receives the least amount.

Explanation : 

To determine the unique values of x and y (distinct natural numbers) that satisfy the given conditions, let’s analyze each statement individually and then consider them together.

Statement I: x/y is odd.

Interpretation: For x/y to be an odd integer, x must be an odd multiple of y. This implies that y must be a divisor of x, and the quotient x/y must be an odd number.

Limitations: Without additional information, there are multiple pairs (x, y) that can satisfy this condition. For example:
– x = 3, y = 1 (since 3/1 = 3 is odd)
– x = 9, y = 3 (since 9/3 = 3 is odd)

Conclusion: Statement I alone is insufficient to determine unique values of x and y.

Statement II: xy = 12.

Interpretation: The product of x and y is 12. Considering that x and y are distinct natural numbers, the possible pairs are:
– (1, 12)
– (2, 6)
– (3, 4)

Limitations: Multiple valid pairs satisfy this condition.

Conclusion: Statement II alone is insufficient to determine unique values of x and y.

Combining Statements I and II:

Now, let’s use both statements together.

1. From Statement II, the possible pairs are (1, 12), (2, 6), and (3, 4).

2. Applying Statement I (x/y is odd) to each pair:
– (1, 12): 1/12 is not an integer, so this pair is invalid.
– (2, 6): 2/6 = 1/3 is not an integer, so this pair is invalid.
– (3, 4): 3/4 is not an integer, so this pair appears invalid at first glance.

However, there’s a misunderstanding here. For x/y to be an odd integer, x must be a multiple of y, and the quotient must itself be odd.

Re-examining the possible pairs:
– (12, 1): 12/1 = 12 (Even)
– (6, 2): 6/2 = 3 (Odd)
– (4, 3): 4/3 is not an integer.

Valid Pair: (6, 2) since 6/2 = 3 is an odd integer.

Conclusion: Using both statements together, the unique values are x = 6 and y = 2.

Therefore, option c is the correct choice.

Explanation : 

Initial Movement:
Direction: West
Action: Walks 100 meters West.

First Turn:
Turn: Left
From West, turning left means facing South.
Action: Walks 100 meters South.

Second Turn:
Turn: 225° clockwise from South
Understanding the Turn:
– Starting Direction: South (which is 180° on a compass)
– Clockwise Turn: Adding 225° to the current direction.
– Calculation: 180° (South) + 225° = 405°
– Since 405° is equivalent to 45° (405° – 360°), the final direction is 45°, which corresponds to North-East on a compass.

Final Direction: North-East, Answer: d) North-East

Explanation : 

Let’s analyze the person’s movements step by step to determine the initial direction:

1. Initial Direction: Suppose the person starts by walking South.

2. First Movement:
– Walks 100 m South.
– Position: (0, -100).

3. First Turn (Left):
– From South, turning left means facing East.
– Walks 100 m East.
– Position: (100, -100).

4. Second Turn (Left):
– From East, turning left means facing North.
– Walks 300 m North.
– Position: (100, 200).

5. Third Turn (Right):
– From North, turning right means facing East again.
– Walks 100 m East.
– Final Position: (200, 200).

The final position (200, 200) places the office North-East of the house, which matches the given condition.

Answer: c) South

Explanation : 

Let p and q be positive integers

To determine the smallest value of k that does not uniquely determine the positive integers p and q (with p < q and p + q = k), let’s evaluate each option:

1. k = 3
– Possible pairs: (1, 2)
– Unique pair: Yes.

2. k = 4
– Possible pairs: (1, 3)
– Unique pair: Yes. (Note: (2, 2) is invalid since p < q.)

3. k = 5
– Possible pairs: (1, 4) and (2, 3)
– Unique pair: No. There are two distinct pairs.

4. k = 6
– Possible pairs: (1, 5) and (2, 4)
– Unique pair: No. There are two distinct pairs.

The smallest k where multiple pairs (p, q) satisfy the conditions is k = 5.

Explanation : 

32^5 + 2^27

To determine which of the given options divides 32^5 + 2^27, let’s simplify and analyze the expression step-by-step.

Simplify the Expression First, express everything in terms of powers of 2: 32 = 2^5 implies 32^5 = (2^5)^5 = 2^25

2^27 remains as is.

So, the expression becomes: 32^5 + 2^27 = 2^25 + 2^27 = 2^25(1 + 2^2) = 2^25 × 5

This shows that the expression is divisible by 2^25 and 5, and consequently by 2 × 5 = 10.