Explanation : 

The difference between 2 digit number and the number obtained is 54

Given:
– Original number = 10x + y (where x is tens digit, y is units digit)
– New number after interchanging = 10y + x
– Difference between these numbers = 54

Step 1: Write equation for the difference
(10x + y) – (10y + x) = 54

Step 2: Simplify left side
10x + y – 10y – x = 54
9x – 9y = 54

Step 3: Further simplify
9(x – y) = 54
x – y = 6

Step 4: Analyze what we know
– Difference between tens and units digit is 6
– Since it’s a 2-digit number:
* x must be from 1 to 9
* y must be from 0 to 9
* x must be greater than y by 6

Step 5: Find possible digit pairs
Since x – y = 6:
– If y = 0, x = 6
– If y = 1, x = 7
– If y = 2, x = 8
– If y = 3, x = 9

Analysis:

1. Statement 1 (about sum of digits) is incorrect
– Multiple pairs of digits satisfy x – y = 6
– Each pair would give different sums
– Examples: 6+0=6, 7+1=8, 8+2=10, 9+3=12

2. Statement 2 (about difference of digits) is correct
– We proved x – y = 6
– This is a definite value

Answer: Statement 1 is incorrect and Statement 2 is correct.

Explanation : 

Pie Chart Analysis

A pie chart represents 100% of the data as a full circle with 360 degrees. In this case, proteins correspond to 16% and water corresponds to 70%, so the other dry elements correspond to 100% – 16% – 70% = 14%. Since p represents the sum of proteins and other dry elements, p = 16% + 14% = 30%.

Since p represents 30% of the data, the central angle of the sector representing p on the pie diagram is (30/100) * 360 = 108 degrees.

Explanation : 

Biology Class at High School Population Growth Model

Given:
– Initial population (2021) = 50 animals
– Population doubles every 12 years
– Need formula for population P after n years

Step 1: Identify Growth Type
– Not linear growth (population doesn’t increase by fixed amount)
– Population multiplies by 2 at fixed intervals
– This indicates exponential growth

Step 2: Basic Exponential Growth Formula
P = Initial Value × (growth factor)^(time/interval)

Step 3: Identify Components
– Initial Value = 50
– Growth factor = 2 (doubles)
– Time = n years
– Interval = 12 years

Step 4: Analyze Options
a) P = 12 + 50n
Incorrect: Shows linear growth

b) P = 50 + 12n
Incorrect: Shows linear growth

c) P = 50(2)^12n
Incorrect: Time interval ratio is wrong

d) P = 50(2)^(n/12)
Correct: Shows proper exponential growth

Step 5: Verify Option D
After 0 years: P = 50(2)^0 = 50 (correct initial population)
After 12 years: P = 50(2)^1 = 100 (population doubles)
After 24 years: P = 50(2)^2 = 200 (doubles again)

Answer: The correct formula is P = 50(2)^(n/12)
This formula accurately represents population doubling every 12 years from an initial value of 50 animals.

Explanation : 

Problem: Finding when petrol and diesel prices are equal ?

Given:
– 2021 is non leap year
– Petrol price formula: 80 + 0.1m (where m = 1 to 100, constant thereafter)
– Diesel price formula: 69 + 0.15n (for any value of n)

Step 1: Set up equation for equal prices
80 + 0.1m = 69 + 0.15n

Step 2: Solve for n in terms of m
80 + 0.1m = 69 + 0.15n
0.1m – 0.15n = -11
-0.15n = -11 – 0.1m
n = (2m + 220)/3

Step 3: Find n when m = 100
n = (2(100) + 220)/3
n = (200 + 220)/3
n = 420/3
n = 140

Answer: The prices of petrol and diesel are equal on the 140th day of 2021, which is May 20th.

Explanation : 

Problem: Find the digit P in 3798125P369 so that the number is divisible by 7

Step 1: Divide into Three-Digit Blocks (from right to left)
– Write as: 037, 981, 25P, 369
– This splits the number into blocks of three: 037|981|25P|369

Step 2: Apply Alternating Addition and Subtraction Rule
– Start from rightmost block: 369
– Then alternate: 369 – 25P + 981 – 037
– Note: This is based on the divisibility rule of 7 where we alternate adding and subtracting blocks of 3 digits

Step 3: Simplify the Expression
369 – (250 + P) + 981 – 37
= 369 – 250 – P + 981 – 37
= 1063 – P

Step 4: Solve for P Using Modulo 7
– For the number to be divisible by 7:
– 1063 – P must be divisible by 7
– 1063 ÷ 7 = 151 remainder 6
– So, 1063 ≡ 6 (mod 7)
– Therefore: 6 – P ≡ 0 (mod 7)
– This means P = 6

Answer: P = 6

Explanation : 

Problem: Find the remainder when 3^2019 is divided by 10

Let’s solve this step by step:

1) To find 3^2019 mod 10, examine the pattern of remainders when successive powers of 3 are divided by 10:
3^1 = 3
3^2 = 9
3^3 = 27 ≡ 7 (mod 10)
3^4 = 81 ≡ 1 (mod 10)
3^5 = 243 ≡ 3 (mod 10)
3^6 = 729 ≡ 9 (mod 10)
3^7 ≡ 7 (mod 10)
3^8 ≡ 1 (mod 10)

2) The pattern of remainders is: 3, 9, 7, 1, 3, 9, 7, 1, …
This forms a cycle of length 4

3) For 3^2019:
2019 = 504 × 4 + 3
Thus 3^2019 will have the same remainder as 3^3

4) Since 3^3 ≡ 7 (mod 10)

Therefore, when 3^2019 is divided by 10, the remainder is 7.

The answer is (c) 7.

Explanation : 

Integers are listed from 700 to 1000

The sum of the digits of an integer is 10 if the digits add up to 10. For example, the number 703 has a digit sum of 7 + 0 + 3 = 10. We can find all integers between 700 and 1000 that have a digit sum of 10 by listing them out:

703, 712, 721, 730, 802, 811, 820, 901, 910.

There are 9 integers between 700 and 1000 that have a digit sum of 10. So the correct answer is (d) 9.

Explanation : 

A boy plays with a ball

Starting height = 1.5 meters
Bounce ratio = 4/5 (ball reaches 4/5th of previous height)
Stop condition = height < 0.5 meters (50 cm)

Height Calculations:

1. Initial Drop: 1.5 meters

2. First Bounce:
Height = 1.5 × 4/5 = 1.2 meters

3. Second Bounce:
Height = 1.2 × 4/5 = 0.96 meters

4. Third Bounce:
Height = 0.96 × 4/5 = 0.768 meters

5. Fourth Bounce:
Height = 0.768 × 4/5 = 0.6144 meters

6. Fifth Bounce:
Height = 0.6144 × 4/5 = 0.49152 meters
(This is less than 50 cm, so ball stops)

Result:
– Ball hits ground 5 times before stopping
– Final height (0.49152 m) is below stopping condition (0.5 m)
– Answer: 5 bounces

Explanation : 

A student appeared in 6 papers

– Total marks per subject = 100
– Total subjects = 6
– Total possible marks = 600
– Overall score = 60% = 360 marks
– Marks distribution pattern: 5x, 6x, 7x, 8x, 9x, 10x

Step 1: Form equation
5x + 6x + 7x + 8x + 9x + 10x = 360
45x = 360

Step 2: Solve for x
x = 360/45 = 8

Step 3: Calculate marks in each subject
Subject 1: 5x = 5 × 8 = 40 marks
Subject 2: 6x = 6 × 8 = 48 marks
Subject 3: 7x = 7 × 8 = 56 marks
Subject 4: 8x = 8 × 8 = 64 marks
Subject 5: 9x = 9 × 8 = 72 marks
Subject 6: 10x = 10 × 8 = 80 marks

Step 4: Count subjects below 60%
60% of 100 = 60 marks
Subjects below 60 marks:
1. 40 marks
2. 48 marks
3. 56 marks

Therefore: Student scored less than 60% in 3 subjects.

Explanation :

There are two classes a and b

Class A (Initial):
– Total students: 25
– Highest score: 21
– Lowest score: 17
– Four students will be shifted out

Class B (Initial):
– Total students: 30
– Highest score: 30
– Lowest score: 22
– Will receive four students from Class A

Analysis:

1. Effect on Class B’s Average:
– Initial students: 30
– New total: 34 students
– All incoming students have scores ≤ 21
– All original students have scores ≥ 22
– Therefore, adding any four students from Class A will definitely lower Class B’s average

2. Effect on Class A’s Average:
– Initial students: 25
– New total: 21 students
– Cannot determine if average will increase because:

• If highest scoring students (around 21) are shifted out → average will decrease
• If lowest scoring students (around 17) are shifted out → average will increase
• Without knowing which students are shifted, change in average is uncertain

Conclusion:
– Class B’s average will definitely decrease
– Class A’s average could increase or decrease depending on which students are shifted