Explanation : 

In how many ways a batsman can score 25 runs ?

Let x = singles, y = fours, z = sixes
We need to find integer solutions to: x + 4y + 6z = 25

Approach: Systematically consider possible values for z (sixes) from maximum to minimum.

Solutions:
z = 4: (1,0,4) → 1 way (four sixes and one single)
z = 3: (7,0,3), (3,1,3) → 2 ways
z = 2: (13,0,2), (9,1,2), (5,2,2), (1,3,2) → 4 ways
z = 1: (19,0,1), (15,1,1), (11,2,1), (7,3,1), (3,4,1) → 5 ways
z = 0: (25,0,0), (21,1,0), (17,2,0), (13,3,0), (9,4,0), (5,5,0), (1,6,0) → 7 ways

Each tuple represents (singles, fours, sixes).

Total number of ways: 1 + 2 + 4 + 5 + 7 = 19

Therefore, there are 19 different ways for a batsman to score exactly 25 runs using combinations of singles, fours, and sixes.

Explanation : 

1. 1000 litres = 1 m³
This is correct. One cubic meter (m³) is equal to 1000 liters.

2. 1 metric ton = 1000 kg
This is correct. A metric ton, also known as a tonne, is defined as 1000 kilograms.

3. 1 hectare = 10000 m²
This is correct. A hectare is a unit of area equal to 10,000 square meters.

Since all three statements are correct, the answer is: d) 1, 2 and 3

Explanation : 

Father Son Age Problem

Let’s approach this step-by-step:

1. Let’s define some variables:
F = Father’s present age
S = Son’s present age
n = Number of years mentioned in the problem

2. From the given information:
F – n = S (Father’s age n years ago was equal to son’s present age)
F = 4(S – n) (Father’s present age is 4 times son’s age n years ago)

3. We’re also told that the sum of their present ages is 130:
F + S = 130

4. Let’s solve the equations:
From the first equation: S = F – n
Substituting this into the second equation:
F = 4(F – n – n) = 4F – 8n

3F = 8n
F = 8n/3

5. Now, let’s use the third equation:
F + S = 130
(8n/3) + (8n/3 – n) = 130
16n/3 – n = 130
16n/3 – 3n/3 = 130
13n/3 = 130
13n = 390
n = 30

6. Now that we know n, we can find F and S:
F = 8n/3 = 8(30)/3 = 80
S = F – n = 80 – 30 = 50

7. The difference in their ages is:
80 – 50 = 30

Therefore, the difference in their ages is 30 years.

The correct answer is a) 30 years.

Explanation : 

In an examination, 80% of students passed in English

Let’s approach this step-by-step:

1) Let’s define our variables:
Let x be the total number of students.

2) We know:
– 80% passed English, so 20% failed English
– 70% passed Hindi, so 30% failed Hindi
– 15% failed both subjects

3) Let’s create a Venn diagram in our mind:
– Let A be the set of students who failed English
– Let B be the set of students who failed Hindi
– The intersection of A and B (A ∩ B) is 15% of x

4) We need to find:
(Students who failed English only) + (Students who failed Hindi only)

5) Students who failed English only:
20% – 15% = 5% of x

6) Students who failed Hindi only:
30% – 15% = 15% of x

7) Total percentage of students who failed in only one subject:
5% + 15% = 20%

Therefore, the correct answer is b) 20%.

Explanation : 

R’s salary more than P’s salary

Let’s approach this step-by-step to verify the correct answer:

1) Let’s assume R’s salary is 100 units (for easy calculation).

2) Q’s salary is 20% lower than R’s:
Q’s salary = 100 – (20% of 100) = 100 – 20 = 80 units

3) P’s salary is 20% lower than Q’s:
P’s salary = 80 – (20% of 80) = 80 – 16 = 64 units

4) Now, we need to calculate how much higher R’s salary is compared to P’s:
Difference = R’s salary – P’s salary = 100 – 64 = 36 units

5) To express this as a percentage increase from P to R:
Percentage increase = (Difference / P’s salary) × 100
= (36 / 64) × 100
= 0.5625 × 100
= 56.25%

Therefore, the correct answer is indeed b) 56.25%.

R’s salary is 56.25% more than P’s salary.

Explanation : 

Number is mistakenly divided by 4 instead of multiplying by 4

1) Our original number is 100.

2) The correct operation should be: 100 * 4 = 400

3) The mistaken operation is: 100 ÷ 4 = 25

4) Now, let’s calculate the percentage change:
(Mistaken result – Correct result) / Correct result * 100
= (25 – 400) / 400 * 100
= -375 / 400 * 100
= -0.9375 * 100
= -93.75%

5) The negative sign indicates a decrease. The question asks for the percentage change, which is the absolute value.

Therefore, the percentage change is 93.75%.

Explanation : 

Two persons P and Q

Let’s approach this problem step by step:

1) Let’s say Q’s investment is x.
Then, P’s investment is x + 14,000.

2) We know that profit is proportional to both the amount invested and the time of investment.
So, we can set up an equation:

P’s share : Q’s share = (x + 14,000) * 8 : x * 10

3) We’re told that P’s share is ₹400 more than Q’s share out of a total profit of ₹2,000.
So, P’s share = 1200 and Q’s share = 800.

4) Now we can set up the proportion:

1200 : 800 = (x + 14,000) * 8 : x * 10

5) Cross multiply:

1200 * 10x = 800 * (8x + 112,000)
12000x = 6400x + 89,600,000

6) Solve for x:

5600x = 89,600,000
x = 16,000

7) Remember, x is Q’s investment. We need to find P’s investment:

P’s investment = x + 14,000 = 16,000 + 14,000 = 30,000

Therefore, the capital contributed by P is ₹30,000.

The correct answer is a) ₹30,000.

Explanation : 

Solution Steps

1. Let’s split the sequence in sets of 5 character: A_BCD_BBCDABC_DABC_D Splits into: A_BCD / _BBCD / ABC_D / ABC_D
2. Analyzing this grouping, we can identify the pattern:
   • Each group consists of 5 characters
   • The pattern involves repeating ABCD in various arrangements, sometimes with repetitions
3. The correct grouping of the sequence by Hit & Trail Method is: A A BCD A BBCD ABC C D ABC D D (Hit & Trail)
4. Observations from this pattern:
   • The first group has two A’s followed by BCD
   • The second group starts with A, then has two B’s, followed by CD
   • The third group has ABC, then C, then D
   • The fourth group has ABC, then two D’s
5. The last group in our given sequence is incomplete: ABC_D
6. To complete it according to the established pattern, we need to add another D, making it: ABC D D
7. Therefore, the missing letters are A, A, C, D