Explanation : 

Conclusion-I: All numbers are divisible by 6

Analysis:

– The numbers are divisible by both 2 and 3
– 6 is the least common multiple (LCM) of 2 and 3
– Any number divisible by both 2 and 3 must be divisible by their LCM (6)
– Therefore, Conclusion-I is TRUE

Conclusion-II: All numbers are divisible by 4

Analysis:

– Just because a number is divisible by 2 and 3 doesn’t mean it’s divisible by 4
– Example: 6 is divisible by both 2 and 3, but not by 4
– Therefore, Conclusion-II is FALSE

Explanation : 

A man walks down the backside of his house

The man starts at the back of his house facing East. He then walks 25 meters straight down the backside of his house. Since he is walking away from the house and the house faces East, he is now facing West.

He then turns to his right and walks 50 meters. Since he was originally facing West and turned right, he is now facing North.

Finally, he turns left and walks another 25 meters. Since he was originally facing North and turned left, he is now facing West again.

So from his starting point at the back of his house, the man has walked in a rectangular path and ended up to the North-West of where he started. So the correct answer is (d) North-west.

Explanation :

Product is 21 and Sum is 10

Statement S1: xy = 21
Statement S2: x + y = 10

Solution Steps:
1. From S1: y = 21/x

2. Substitute this in S2:
x + 21/x = 10

3. Solve for x:
Multiply both sides by x: x² + 21 = 10x
Rearrange: x² – 10x + 21 = 0
Factor: (x – 7)(x – 3) = 0
Therefore: x = 7 or x = 3

4. Find corresponding y values:
When x = 7: y = 21/7 = 3
When x = 3: y = 21/3 = 7

5. The two numbers are 3 and 7

Conclusion:
Neither statement alone is sufficient, but together they give us the unique solution. Therefore, answer is (c): S1 and S2 together are sufficient to answer the Question.

Explanation : 

Analysis of Statement S2:
– When a number n is divided by 4, remainder is 1
– Such numbers can be written as: n = 4k + 1 (where k is an integer)
– Numbers between 10 and 20 that fit this pattern are: 11, 13, 15, 17, 19

Analysis of Statement S1:
– Number n must be prime
– From the above list, only 11 and 17 are prime numbers
– So S1 alone gives us two possibilities: n = 11 or n = 17

Combined Analysis:
– Both statements together still give us two possibilities
– Both 11 and 17:
Are prime numbers (satisfying S1)
Leave remainder 1 when divided by 4 (satisfying S2)

Conclusion:
Therefore, answer is (d): S1 and S2 together are not sufficient to determine a unique value of n.

Explanation : 

S1: R is greater than both P and Q

– This alone is not enough to determine the largest number
– R could be either largest or second largest

S2: S is not the largest number

– This alone is not enough to determine the largest number
– S could be second largest, third largest, or smallest

Combined Analysis:
When we use both statements together:
1. R is greater than P and Q (from S1)
2. S is not the largest (from S2)
3. Since R is greater than P and Q, and S cannot be largest, R must be the largest number

Conclusion:
Therefore, answer is (c): S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient.

Explanation : 

A family of two generations consisting of 6 members

From the given information:
– U is P’s daughter
– Q is R’s mother-in-law
– T is an unmarried male
– S is a male
– There are two married couples in the family

Analysis:
1. Since Q is R’s mother-in-law, R must be married to one of Q’s children
2. We know T is unmarried and S is male
3. With two married couples and T being unmarried, S must be married
4. Since R is married to Q’s child and S is married, R must be S’s wife

Conclusion:
Therefore, R is S’s wife.

Explanation : 

PROBLEM: Can we determine if there are more than 100 figures in the book?

GIVEN:
Statement S1: Not more than 2 figures on any page; 51 pages
Statement S2: At least 1 figure on every page

ANALYSIS:
1. Maximum figures possible:
– 2 figures/page × 51 pages = 102 figures

2. Minimum figures possible:
– 1 figure/page × 51 pages = 51 figures

3. Range of possible total figures:
51 to 102 figures

CONCLUSION: Cannot determine if there are more than 100 figures
REASON: The actual number could be anywhere between 51 and 102 figures

ANSWER: Given information is insufficient to answer the question

Explanation : 

The pattern is a b a b a / b c b c b / c d c d c /d e d e d

Explanation :

A vessel full of water weighs 40 kg

GIVEN:
– Full vessel weighs 40 kg
– One-third filled vessel weighs 20 kg

SOLUTION:
1. Weight difference = 40 – 20 = 20 kg
This is weight of 2/3 water

2. Weight of 1/3 water = 20/2 = 10 kg

3. Empty vessel weight = Weight of 1/3 filled – Weight of 1/3 water
= 20 – 10 = 10 kg

ANSWER: The empty vessel weighs 10 kg

Explanation :

PROBLEM: Find ratio of man’s speed in still water to current speed

FORMULAS:
Time = Distance/Speed
– Downstream time = Distance/(v + c)
– Upstream time = Distance/(v – c)

SOLUTION:
1. Speeds:
– Downstream = v + c
– Upstream = v – c

2. Since downstream time is half:
v + c = 2(v – c)
v + c = 2v – 2c
-v + 3c = 0
v = 3c

3. Therefore:
Man’s speed : Current speed
= v : c
= 3 : 1

ANSWER: The ratio of man’s speed in still water to current speed is 3:1