Explanation : 

Chess Board Consecutive Squares Chosen

Given:
– Chess board has 2 diagonals with 8 squares each
– Need to select 6 consecutive squares
– These 6 squares must be consecutive along a diagonal

Step 1: Analyze one diagonal
– Diagonal has 8 squares
– Need to select 6 consecutive squares
– Think of 6 squares as one block
– This block can slide along diagonal

Step 2: Calculate possible positions on first diagonal
– 8 squares total, need 6 consecutive
– First position: squares 1-6
– Second position: squares 2-7
– Third position: squares 3-8
Total positions on first diagonal = 3

Step 3: Calculate possible positions on second diagonal
– Same calculation as first diagonal
– 3 possible positions
Total positions on second diagonal = 3

Step 4: Calculate total number of ways
– First diagonal: 3 ways
– Second diagonal: 3 ways
Total ways = 3 + 3 = 6

Explanation : 

Best Method- Check All option
15873 × 7 = 111111

Explanation : 

Price of an article is decreased & increased by 20, 25

Given:
– Original price = P
– First change: 20% decrease
– Second change: 25% increase

Step 1: Calculate price after 20% decrease
– Decrease amount = 20% of P = 0.2P
– New price = P – 0.2P
– New price = 0.8P

Step 2: Calculate final price after 25% increase
– Increase amount = 25% of 0.8P = 0.25(0.8P) = 0.2P
– Final price = 0.8P + 0.2P
– Final price = 1P

Step 3: Calculate net percentage change
– Original price = P
– Final price = P
– Net change = (Final price – Original price)/Original price × 100
– Net change = (P – P)/P × 100
– Net change = 0%

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.