Explanation : 

Remainder Calculation

The factorization of 100 is 4 x 5^2.

If we look at the given numbers, we can see that 85, 95 each contain a factor of 5, and 96 contain a factor of 4.

Therefore, we can say that the product of these numbers contains at least 4 x 5^2 = 100 as a factor.

This means that when the product of these numbers is divided by 100, the remainder will be 0.

So, the correct answer is (a) 0.

Explanation : 

The first statement is incorrect.

If we have four letters and four envelopes, and we place one letter correctly, then we have three letters and three envelopes left.

If we now place another letter correctly, we would have two letters and two envelopes left, which means either both of these would be correct or both would be incorrect. If we place one of these two letters incorrectly, the last letter would also have to go into the incorrect envelope because there’s only one choice left. Therefore, it’s not possible to have exactly one letter in an incorrect envelope. The possible scenarios are 0, 2, or 4 letters in the correct envelopes.

The second statement is correct.

There are indeed six ways in which only two letters can go into the correct envelopes.

This can be calculated using the combination formula 4C2 (which stands for “”4 choose 2″”), which gives us the number of ways to choose 2 items from a set of 4. The formula is 4! / (2!(4-2)!) = 6.
So, the correct answer is (b) 2 only.

Explanation : 

Since E = 0 and F = 8, the product of AB and CD must be a number in the form of 08X, where X is a digit from 1 to 9.

The only way to get a product of 08X is if one of the numbers (AB or CD) is a multiple of 4 and the other number ends with 2 or 7.

The possible pairs of numbers are (12, 34), (17, 24), (24, 17), (34, 12).

However, since the sum of DEF and GHI is 975, the sum of the digits D, G, H, I must be 9 (since E = 0 and F = 8).

The only pair of numbers that satisfies this condition is (12, 34), which gives a product of 408.

Therefore, A = 1, B = 2, C = 3, D = 4.

So, A + B + C = 1 + 2 + 3 = 6.

Therefore, the correct answer is (a) 6.

Explanation : 

The powers of 2 cycle in a pattern of 4 when divided by 6.

Specifically, the remainders are 2, 4, 2, 0, 2, 4, 2, 0, and so on. This pattern repeats every 4 powers.

Since 192 is a multiple of 4, the remainder when 2^192 is divided by 6 is the same as the remainder when 2^4 is divided by 6, which is 4.

So, the answer is (d) 4.

Explanation : 

Flag with 4 horizontal stripes

The first stripe can be any one of the 3 colours, so there are 3 options for the first stripe.

For the second stripe, it cannot be the same colour as the first stripe, so there are only 2 options.

Similarly, for the third and fourth stripes, they cannot be the same colour as the stripe before them, so there are 2 options for each of them.

Therefore, the total number of different ways to design the flag is 3 * 2 * 2 * 2 = 24.

Explanation : 

From the given question, we need to find whether (p+q-r) is greater than (p-q+r) or not.

Is (p+q-r) greater than (p-q+r), where p, q and r are integers ?

If we simplify this, we get 2q > 2r or q > r. So, we need to determine whether q is greater than r or not.

Statement-1: (p-q) is positive. This means p > q. But this statement doesn’t give any information about r. So, we cannot answer the question using this statement alone.

Statement-2: (p-r) is negative. This means p < r. But this statement doesn’t give any information about q. So, we cannot answer the question using this statement alone.

Using both the statements together, we know that p > q and p < r. This implies that r > p > q or r > q. So, q is not greater than r. Therefore, (p+q-r) is not greater than (p-q+r).

So, the correct answer is (c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone.

Explanation : 

Question – Is p greater than q?

Statement-1: p x q is greater than zero. This statement implies that both p and q are either positive or negative. However, it does not provide information about whether p is greater than q.

Statement-2: p^2 is greater than q^2. This statement implies that the absolute value of p is greater than the absolute value of q. However, it does not provide information about whether p is greater than q, because both p and q could be negative.

Even if we use both statements together, we cannot definitively answer whether p is greater than q. For example, if p and q are both positive and p^2 is greater than q^2, then p is greater than q. But if p and q are both negative and p^2 is greater than q^2, then p is actually less than q.

Therefore, the correct answer is (d) The Question cannot be answered even by using both the Statements together.

Explanation : 

4-Digit Numbers Sum

The 4-digit numbers less than 2000 formed by the digits 1, 2, 3, and 4 where none of the digits is repeated are 1234, 1243, 1324, 1342, 1423, and 1432.

Adding these numbers together gives the sum:

1234 + 1243 + 1324 + 1342 + 1423 + 1432 = 7998

So, the correct answer is (a) 7998.