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Given:

{x4 - (x - y)4}

Concept used:

x2 - y2 = (x + y)(x - y)

(x - y)2 = x2 + y2 - 2xy

Calculations:

x4 - (x - y)4

⇒ (x² - (x - y)²)(x² + (x - y)²)

⇒ (x - (x -y))(x + x - y)( x² + x² + y² - 2xy)

⇒ (y)(2x - y)(2x² + y² - 2xy)

∴ The factor of {x4 - (x - y)4} is y(2x - y) (2x2 + y2 - 2xy)

Given:

The sum of two numbers is 22 and the sum of their squares is 404.

Formula Used:

(x + y)² = x² + y² + 2xy

Calculations:

Let the two numbers are x and y 

According to Question

x + y = 22

x² + y² = 404

(x + y)2 = x2 + y2 + 2xy

⇒ (22)² = 404 + 2xy

⇒ 80 = 2xy

⇒ xy = 40

∴ Product of two numbers is 40

Given:

The sum of the digits of the minimum number by which 1800 is multiplied to get a perfect cube number

Calculations:

1800 = 2 × 3 × 3 × 10 × 10  

To get perfect cube 

3 × 5 

so, that the number will be perfect cube 

1800 ×  5 × 3 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5

1800 × 15 = \(\sqrt [3]{3 × 3 × 3 × 2 × 2× 2× 5 \times 5 \times 5}\)

So, the number multiplied by 1800 to get perfect square is 15

⇒ the sum of the digits of the number i.e 15 =  1  + 5 = 6

Given:

If a sum of money becomes \(\frac{41}{40}\) of itself in \(\frac{1}{4}\) year at a particular rate of simple interest.

Formula used:

SI(Simple Interest) = Amount(A) - Principle(P)

SI = P × R × T/100 , where R is the Rate of interest, T is the Time taken 

Calculations:

Let P = x Rs 

⇒ A = 41/40 × x Rs

T = 1/4 years

⇒ SI = 41/40 × x - x = x/40

Now, 

SI = P × R × T/100

⇒ R = SI × 100/ P × T 

⇒ R = x/40 × 100/ x × 1/4

⇒ R = 10%

∴ The annual rate of interest = 10%

Given:

Principle (P) = 2000

Amount  (A) = 2420

Rate of Interest (R) = 10%

Formula used:

A = P × (1 + R/100)ⁿ

Calculations:

⇒ 2420 = 2000 × (1 + 10/100)ⁿ

⇒ 2420/2000 = (11/10)ⁿ

⇒ 121/100 = (11/10)ⁿ

⇒ (11/10)² = (11/10)ⁿ

⇒ n = 2 

∴ After 2 years 2000 becomes 2420.

Given:

If the selling price of 7 articles is equal to the cost price of 6 articles

Formula used:

Loss = CP - SP

Profit = SP - CP 

Profit% = SP - CP/CP × 100

Loss% = CP - SP/SP × 100 

Calculations:

If the selling price of 7 articles is equal to the cost price of 6 articles

7SP = 6CP

⇒ SP/CP = 6/7

⇒ Selling price of an article is Rs. 6

⇒ Cost price of an article is Rs. 7

⇒ Loss% = CP - SP/SP × 100 

⇒ Loss% = 7 - 6/7 × 100

⇒ Loss% = 14\(\frac{2}{7}\)%

∴ option 2 is correct

Shortcut Trick

Ratio of Value = (Denomination × Ratio of Notes)

Value Ratio = (20 × 3) : (10 × 4) : (5 × 5) = 60 : 40 : 25 = 12 : 8 : 5

Total units = 12 + 8 + 5 = 25 units → ₹1000

1 unit = ₹40

Value of ₹5 notes = 5 × 40 = ₹200

Number of ₹5 notes = 200 ÷ 5 = 40

∴ The correct answer is 40.

Alternate Method

Given:

Ratio of ₹20, ₹10, and ₹5 notes = 3 : 4 : 5

Total amount in the bag = ₹1000

Calculations:

Let the number of notes be 3x, 4x, and 5x respectively.

⇒ Total Value = (20 × 3x) + (10 × 4x) + (5 × 5x) = 1000

⇒ 60x + 40x + 25x = 1000

⇒ 125x = 1000

⇒ x = 1000 ÷ 125

⇒ x = 8

⇒ Number of ₹5 notes = 5x = 5 × 8

⇒ Number of ₹5 notes = 40

∴ The correct answer is 40.

Additional Information

Value of Denominations

Total Value = Σ (Face Value of Note × Number of Notes). Always convert all units to either Rupees or Paise before solving.

Ratio and Proportion

If the ratio of notes is a:b:c, then the ratio of their values is (a×V₁):(b×V₂):(c×V₃), where V is the denomination value.

Given:

If the sum of the numerator and denominator of a fraction is 55 and the value of the fraction is 3/8

Calculations:

Let's take Numerator is 15 and Denominator is 55.

Then 15 + 40 = 55 (As sum of the numerator and denominator is 55)

And the fraction 15/40 = 3/8 (As the value of Fraction is 3/8)

⇒ The fraction is 15/40.

∴ option 4 is correct.

Explanations 

if a transversal intersects two parallel lines then each pair of corresponding angles are equal 

∴ option 3 is correct

Given:

A right-angled Triangle 

Concept used:

The side opposite to larger angle is greater

The Sum of the angle of the triangle is 180°

Calculations:

Let us take a right-angled triangle ABC 

⇒ ∠ABC + ∠BCA + ∠CAB = 180° 

⇒ ∠ABC = 90° 

⇒ ∠BCA + ∠CAB = 180° - 90° 

⇒ ∠BCA + ∠CAB =  90° 

⇒ ∠BCA < ∠ABC and  ∠CAB < ∠ABC

⇒ AB < AC and BC < AC 

∴ The longest side of the triangle is Hypotenuse.