Explanation : 

Given main statement: Pradeep becomes either a Director or a Producer

Let’s define:
D = Pradeep is a Director
P = Pradeep is a Producer
R = Not Director
S = Not Producer

Analyzing statement pairs:

1. S implies P:
– If S is true (Not Producer)
– From main statement (must be Director or Producer)
– Then D must be true (Director)
This is valid

2. R implies Q:
– If R is true (Not Director)
– From main statement (must be Director or Producer)
– Then P must be true (Producer)
This is valid

Both pairs are logically valid based on the main statement that Pradeep must be either Director or Producer.

Answer: c) Both SP and RQ are valid implications.

Explanation : 

Given:
– P and Q are two-digit ages
– P = 10a + b
– Q = 10b + a where a and b are digits (1-9)

Statement I: P is greater than Q

– This means 10a + b > 10b + a
– Simplifies to a > b
– Multiple values possible
– Not sufficient alone

Statement II: Sum of ages is 11/6 times their difference

– (P + Q) = 11/6 × (P – Q)
– (10a + b) + (10b + a) = 11/6 × [(10a + b) – (10b + a)]
– 11a + 11b = 11/6 × (9a – 9b)
– 66(a + b) = 99(a – b)
– 5b = a

Since a and b must be single digits:
– Only possible solution is a = 5, b = 1
– Therefore P = 51 and Q = 15
Statement II alone is sufficient.

Answer: a) The question can be answered using Statement II alone, but cannot be answered using Statement I alone.

Explanation : 

Statement I Alone:

– Knowing only the size (12 MB) isn’t enough to find download time
– Need transfer rate to calculate time

Statement II Alone:

– Knowing only transfer rate (2.4 KB/s) isn’t enough
– Need file size to calculate time

Both Statements Together:
1. Convert 12 megabytes to kilobytes:
12 MB = 12 × 1024 = 12,288 KB

2. Calculate time:
Time = Size ÷ Rate
= 12,288 KB ÷ 2.4 KB/s
= 5,120 seconds
= 1 hour, 25 minutes, 20 seconds

Explanation : 

Statement I Alone:

1. Rearranging m + n > mn gives (m-1)(n-1) < 1
2. Since m and n are natural numbers, and m > n:
– n must equal 1
– m can be any natural number > 1
This alone doesn’t give unique values.

Statement II Alone:

1. Factors of 24 give these pairs:
(1,24), (2,12), (3,8), (4,6), (6,4), (8,3), (12,2), (24,1)
This alone doesn’t give unique values.

Both Statements Together:
1. From Statement I, n must be 1
2. From Statement II, mn = 24
3. Substituting n = 1:
m × 1 = 24
Therefore m = 24

The unique solution is m = 24 and n = 1.

Answer: c) The question can be answered using both statements together, but cannot be answered using either statement alone.

Explanation : 

Conclusion-I: India is the world’s largest exporter of milk.
Analysis: Being largest producer doesn’t mean largest exporter. Production and export are different metrics. India might consume most milk domestically rather than export.

Conclusion-II: India does not import milk.
Analysis: Being largest producer doesn’t mean no imports. A country can both produce and import based on various factors like regional demand and quality requirements.

Answer: d) Neither Conclusion-I nor Conclusion-II follows

Explanation : 

Analyzing Statement I Alone:

P + Q = R + S
This only shows sum of P and Q equals sum of R and S.
Examples:
P = 7, Q = 5, R = 4, S = 8 → P > Q
P = 6, Q = 6, R = 5, S = 7 → P = Q
P = 5, Q = 7, R = 4, S = 8 → P < Q
Statement I alone insufficient.

Analyzing Statement II Alone:

P + S > Q + R
Shows P and S together score more than Q and R.
Examples:
P = 7, S = 5, Q = 5, R = 4 → P > Q
P = 3, S = 6, Q = 4, R = 3 → P < Q
Statement II alone insufficient.

Combining Statements:

From I: P + Q = R + S → P = R + S – Q
From II: P + S > Q + R → S > Q
Substituting: P = R + (S – Q)
Examples:
If R = 35, S = 60, Q = 50 → P = 45 → P < Q
If R = 41, S = 60, Q = 50 → P = 51 → P > Q
Combined statements still insufficient.

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

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