Explanation : 

Counting distinct non-zero digit combinations in the decimal form 0.XY

The number of numbers of the form 0.XY, where X and Y are distinct non-zero digits, can be found by counting the number of possibilities for X and Y, and then multiplying them together.

There are 9 choices for X (all digits except 0), and once X is chosen, there are 8 choices for Y (all digits except the chosen value of X).
Therefore, the total number of numbers of the form 0.XY is:
9 × 8 = 72

Therefore, the answer is (a) 72.

Explanation : 

Possibilities of three consecutive integers whose sum equals their product.

Let’s assume the three consecutive integers to be n-1, n, and n+1.
Their sum would be (n-1) + n + (n+1) = 3n
Their product would be (n-1)n(n+1) = n^3 – n
As per the question, their sum is equal to their product, so we can equate them:
3n = n^3 – n

Rearranging the equation:
n^3 – 4n = 0
n(n^2 – 4) = 0

n = 0 or n = ±2

Explanation : 

There are 9 cups placed on a table

Case 1: When we have 3 coffee cups in one row, 2 coffee cups in another row, and 1 coffee cup in the remaining row (3!).

For row having 3 coffee cups arrangement can be done in only one way.
For row having 2 coffee cups arrangement can be done in three ways.
For row having 1 coffee cup arrangement can be done in three ways.

Total arrangements for case 1 = 3! × 1 × 3 × 3 = 54.

Case 2: When we have 2 coffee cups in each row (1).
Each row has 2 coffee cups arrangement can be done in three ways.
Total arrangements for case 2 = 1 × 3 × 3 × 3 = 27

So, total arrangement for given conditions = 54 + 27 = 81.

Hence, the correct answer is an option(4) i.e., 81.

Explanation : 

To form a password, we need one non-zero digit, one vowel (in capital), and one consonant (in capital).

The password must start with a vowel and end with a consonant. Therefore, the password format will be Vowel-Digit-Consonant.

There are 5 vowels in the English alphabet (A, E, I, O, U).
There are 21 consonants in the English alphabet (the remaining letters after excluding the vowels).
There are 9 non-zero digits (1, 2, 3, 4, 5, 6, 7, 8, 9).

For each position in the password, we can use the number of available options for that specific position. So, to calculate the total number of possible passwords, we can multiply the available options for each position:

Total passwords = Number of vowels × Number of non-zero digits × Number of consonants
Total passwords = 5 × 9 × 21

Total passwords = 945

Thus, 945 such passwords can be generated (option c).

Explanation : 

Sum of all 3-digit numbers formed by distinct non-zero digits (A, B, C)

Let’s consider all possible 3-digit numbers formed by A, B and C without repetition: ABC, ACB, BAC, BCA, CAB and CBA. The sum of these numbers is x = ABC + ACB + BAC + BCA + CAB + CBA.

We can rewrite x as: x = 100A + 10B + C + 100A + 10C + B + 100B + 10A + C + 100B+10C+A+100C+10A+B+100C+10B+A.

By grouping the terms we get: x = (222)(A+B+C). Since A,B,C are distinct non-zero digits the minimum value for A+B+C is when A=1,B=2,C=3 so the minimum value for x is (222)(1+2+3)=1332. Therefore statement 1 is correct.

Statement 2: The 3-digit greatest value of x is 888: False, as the least value of x is 1332.

Explanation : 

A pie chart gives the expenditure

The total angle in a pie chart is 360°. The sum of the angles for items B, C, D and E is 90° + 50° + 45° + 75° = 260°. Therefore, the angle for item A is 360° – 260° = 100°.

Since the total angle in a pie chart represents 100%, each degree represents 100/360 = (5/18)%. Therefore, the percentage of expenditure on item A is (5/18) * 100 = (500/18)% ≈ (250/9)%.

So the correct answer is d. (250/9).

Explanation : 

P is a prime number and q is a composite number

Statement 1: p × q can be an odd number: Correct if p and q both are odd then their product is odd. e.g., p = 3 and q = 15 then p × q = 45, which is a odd number.

Statement 2: q / p can be a prime number: Correct if composite number q is made of two prime numbers and p is one of that prime numbers then q/p is an odd number. e.g., p = 3 and q = 15 then q/p = 5, which is a prime number.

Statement 3: p + q can be a prime number: Correct if p = 5 and q = 6, then p + q = 11 which is a prime number.

Hence, the correct answer is an option(4) i.e.,1, 2, and 3.

Explanation : 

Finding the smallest number greater than 1000 with specific conditions.

LCM of 6, 9, 12, 15 and 18 = 180
Smallest number greater than 1000 which is a multiple of 180 is 1080.

So, Required number = 1080 + 3 = 1083

Explanation : 

Difference between p and q

Let’s say that p = 10x + y and q = 10y + x where x and y are digits.

Then p × q = (10x + y)(10y + x) = 100xy + 10xy + 10xy + xy = 2430. Simplifying this equation gives us xy = 2430/121 = 20.

Since x and y are digits, the only possibility is that x=5 and y=4 or vice versa. Therefore p=54 and q=45 or vice versa. The difference between p and q is |54-45|=9.

Explanation : 

When 70% of a number x is added to another number y, the sum becomes 165% of the value of y.

We can represent this information as an equation:
y + 0.70x = 1.65y

When 60% of the number x is added to another number z, then the sum becomes 165% of the value of z.

We can represent this information as another equation:
z + 0.60x = 1.65z

Now we can solve the system of equations:

From the first equation, we can express x in terms of y:
0.70x = 1.65y – y
0.70x = 0.65y
x = 0.65y / 0.70
x = 13y/14

Now, substitute x in the second equation:
z + 0.60x = 1.65z
z + 0.60(13y/14) = 1.65z
z + 13y/23 = 1.65z

Now, let’s solve for z in terms of y:
13y/23 = 1.65z – z
13y/23 = 0.65z
z = 13y/23 / 0.65
z = 13y/15

Now we have:
x = 13y/14
z = 13y/15

Since 13/15 < 13/14, we can conclude that z < x.

Now let’s compare x and y. Since x = 13y/14, this means that x < y (because 13/14 < 1).

Thus, we have established the following relationship: z < x < y.

Therefore, the correct answer is (a) z < x < y.