Explanation : 

Principal P becomes Q in 1 year. Compounded half-yearly at R% and annually at S%.

When interest is compounded half-yearly, it means that the interest is calculated twice in a year. This results in a higher amount at the end of the year compared to when the interest is compounded annually.

To understand this, let’s take an example. Suppose the principal amount is ₹1000, the annual interest rate is 10%, and the interest is compounded half-yearly.

After the first 6 months, the amount will be ₹1000 + ₹1000*(10/2)/100 = ₹1050.

After the next 6 months, the amount will be ₹1050 + ₹1050*(10/2)/100 = ₹1102.5.

So, at the end of the year, the amount is ₹1102.5 when the interest is compounded half-yearly.

Now, if the same principal amount is compounded annually, the amount at the end of the year will be ₹1000 + ₹1000*10/100 = ₹1100.

So, even though the annual interest rate is the same, the amount at the end of the year is higher when the interest is compounded half-yearly.

Therefore, for the same principal P to become Q in 1 year, the annual interest rate R% when compounded half-yearly has to be less than the annual interest rate S% when compounded annually. Hence, R<S.

Explanation : 

All the numbers in options (a), (b), and (c) are prime numbers. However, in option (d), 91 is not a prime number, so it is different from the others.

Explanation : 

Oldest Person Determination A,B,C,D,E,F

To determine the oldest person, we need to consider all the statements.

From statement 1, we know A is older than B.

Statement 2 tells us that C and D are of the same age.

Statement 3 tells us that E is the youngest, so E cannot be the oldest.

Statement 4 tells us that F is younger than D, so D could potentially be the oldest.

However, statement 5 tells us that F is older than A, which means F is also older than B (since A is older than B).
Therefore, to determine the oldest person, we need all five statements.

Explanation : 

Decoding ‘PLAYER’ based on pattern from ‘ZERO’ to ‘CHUR’

Each letter in the word “ZERO” is replaced by the letter that is three places ahead of it in the English alphabet.

Z -> C (Z is the 26th letter, C is the 3rd letter, 26+3 = 29, but since there are only 26 letters, we subtract 26 to get 3)
E -> H (E is the 5th letter, H is the 8th letter, 5+3 = 8)
R -> U (R is the 18th letter, U is the 21st letter, 18+3 = 21)
O -> R (O is the 15th letter, R is the 18th letter, 15+3 = 18)

So, using the same rule for “PLAYER”:

P -> S (P is the 16th letter, S is the 19th letter, 16+3 = 19)
L -> O (L is the 12th letter, O is the 15th letter, 12+3 = 15)
A -> D (A is the 1st letter, D is the 4th letter, 1+3 = 4)
Y -> B (Y is the 25th letter, B is the 2nd letter, 25+3 = 28, but since there are only 26 letters, we subtract 26 to get 2)
E -> H (E is the 5th letter, H is the 8th letter, 5+3 = 8)
R -> U (R is the 18th letter, U is the 21st letter, 18+3 = 21)

So “PLAYER” is written as “SODBHU”.

Explanation : 

Pattern Analysis

The pattern here is to add all the numbers and then add the digits of the resulting sum.

Let’s look at the examples:

1. 7@9@10 = 8
   Add all numbers: 7 + 9 + 10 = 26
   Add all digits: 2 + 6 = 8
2. 9@11@30 = 5
   Add all numbers: 9 + 11 + 30 = 50
   Add all digits: 5 + 0 = 5
3. 11@17@21 = 13
   Add all numbers: 11 + 17 + 21 = 49
   Add all digits: 4 + 9 = 13

Now, let’s apply this pattern to find the value of 23@4@15:

Add all numbers: 23 + 4 + 15 = 42
Add all digits: 4 + 2 = 6

So, 23@4@15 = 6, which corresponds to option (a).

Explanation : 

A, B, C work independently on alternate days

A, B, C working independently can do a piece of work in 8, 16, and 12 days respectively.

Let the total amount of work be LCM (8, 16, 12) = 48 units

So, Efficiency of A = 48/8 = 6 units/day
Efficiency of B = 48/16 = 3 units/day
Efficiency of C = 48/12 = 4 units/day

The amount of work done in 3 days (Monday + Tuesday + Wednesday) = 6 + 3 + 4 = 13 units
The cycle of work then repeats every 3 days.

To find out when the work will be finished, we need to find out how many full cycles of 3 days can be completed before the total work of 48 units is done.

48 units / 13 units per cycle = 3 cycles with a remainder of 9 units

This means that 3 full cycles of 3 days will be completed, which is 9 days, and 9 units of work will be remaining.

On the 10th day (Monday), A will work and do 6 units. This leaves 3 units of work remaining.
On the 11th day (Tuesday), B will work and do 3 units, completing the work.

So, the work will be finished on the 11th day, which is a Thursday.

Therefore, statement 1 is correct and statement 2 is incorrect.
Hence, the correct answer is option (a) 1 only.

Explanation : 

Greatest Value of Expression (p + q)(r+s)

The greatest value of (p + q)(r + s) will be achieved when p, q, r, and s are the largest distinct single digit positive numbers.

The largest distinct single digit positive numbers are 9, 8, 7 and 6.

So, the greatest value of (p + q)(r + s) is (9 + 8)(7 + 6) = 17 * 13 = 221.

However, if we rearrange the numbers as (9 + 7)(8 + 6), we get a larger value: 16 * 14 = 224.

But, the largest value is obtained when we rearrange the numbers as (9 + 6)(8 + 7) = 15 * 15 = 225.

So, the correct answer is (b) 225.

Explanation : 

7x + 96 is divisible by x

The statement “7x + 96 is divisible by x” implies that when you divide the expression “7x + 96” by “x”, you get a whole number (an integer) as a result. This means that “x” must be a factor of the constant term “96”, because the “7x” term will always be divisible by “x” (since it contains “x” as a factor).

To find all the possible values of “x”, we need to list all the factors of 96. The prime factorization of 96 is indeed (2^5 * 3). From this prime factorization, we can determine all the factors of 96 by taking various combinations of these prime factors.

Here are all the factors of 96:
1,2,4,8,16,32,3,6,12,24,48,96

These are the 12 factors of 96, which means that “x” can be any of these 12 values. If “x” is any of these values, then “7x + 96” will be divisible by “x”. Therefore, the statement is correct, and there are indeed 12 possible values for “x”.

Explanation : 

3-digit number ABC

The problem states that when a 3-digit number ABC is multiplied by D, it gives 37DD. Here, A, B, C, and D are different non-zero digits.

To solve this, we need to find a value for D such that 37DD is a 4-digit number. The only possible values for D are 1, 2, 3, 4, 5, 6, 7, 8, and 9.

If D = 1, then 37DD is not a 4-digit number. If D = 2, then 37DD is not a 4-digit number. If D = 3, then 37DD is not a 4-digit number.

If D = 4, then 37DD = 3744, which is a 4-digit number. Therefore, D = 4.

Now, we need to find a 3-digit number ABC such that ABC * 4 = 3744.

The only 3-digit number that satisfies this equation is 936. Therefore, A = 9, B = 3, and C = 6.

Finally, we need to find the value of A + B + C.

A + B + C = 9 + 3 + 6 = 18.

Therefore, the correct answer is (a) 18.

Explanation : 

Odd-even integers p, q, r, s, t

The sum or difference of two even numbers is always even, and the sum or difference of two odd numbers is always even. So, the sum or difference of any number of even numbers is always even.

For statement 1, since p, q, r, s, t are five integers where three are even and two are odd, we can group the even and odd numbers separately. The sum of three even numbers minus the sum of two odd numbers is even – even, which is definitely even.

For statement 2, the expression 2p + q + 2r – 2s + t can be rearranged as 2(p + r – s) + (q + t). The term 2(p + r – s) is definitely even as it is a product of 2 and some integer. However, (q + t) is the sum of two integers where one is even and the other is odd, which is definitely odd. Therefore, the sum of an even number and an odd number is odd. So, statement 2 is not definitely odd, it depends on the values of q and t.

Therefore, only statement 1 is correct.