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Given:

Perimeter of Rhombus = 52 m

Shorter diagonal = 10 m

Calculation:

We know that perimeter of the rhombus = 4a

where a is side of Rhombus

4a = 52

a = 13

Therefore each side of Rhombus is 13 m

One of the diagonal = 10m

The length of other diagonal is

= 2 × (13² - (10/2)² )^1/2

= 2 × (169 - 25)^1/2

= 2 × (144^1/2)

= 2 × 12

= 24 m

Hence, option 4 is correct.

Given:

In a 100 m race, A runs at 8 kmph.

A gives B a start of 4 m and still beats him by 15 seconds.

Calculation:

Time taken by A to cover 100 m = (60 × 60/8000) × 100 sec = 45 sec

B covers (100 - 4) = 96 m in (45 + 15) sec = 60 sec

B's speed = (96 × 60 × 60) / (60 × 1000) kmph = 5.76 kmph

Hence, option 2 is correct.

Alternate MethodGiven:

A's speed = 8 km/h = 8 × (5/18) = 20/9 m/sec.

A gives B a start of 4 m and still beats him by 15 seconds

Formula used:

Time = distance/speed

Calculation:

Time is taken by A to cover a 100 m race = 100/(20/9)

⇒ 45

Distance covered by B = 100 - 4 = 96 m

Time taken by B = 45 + 15 = 60 sec.

The speed of B = 96/ 60

⇒ 96/60 m/sec

⇒ (96/60) × (18/5)

⇒ 5.76 km/h

∴ The speed of B is 5.76 km/h

Time taken by A to cover 100 m = (60 x 60 / 8000) x 100 sec = 45 sec. B covers (100 - 4) m = 96 m in (45 + 15) sec = 60 sec. B's speed = (96 x 60 x 60)/ (60 x 1000) km/hr = 5.76 km/hr.Read more on Sarthaks.com - https://www.sarthaks.com/450230/in-100-race-runs-at-km-per-hour-if-gives-start-of-and-still-him-by-15-seconds-what-is-the-speed-of

Given:

A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours.

Calculation:

Time from 8 a.m. on a day to 1 p.m. on the following day = 29 hours. 

24 hours 10 min. of this clock = 24 hours of the correct clock. 

145/6 hrs of this clock = 24 hours of the correct clock. 

29 hours of this clock = 24 × 6/145 × 29 hours of the correct clock = 28 hours 48 minutes fo the correct clock

Therefore the correct time is 24 hours 48 minutes

So correct time is 29 hours - 28 hours 48 minutes before 1 pm or 12 minute before 1 pm or 48 minute past 12

Hence, option 2 is correct.

Calculation:

x = \(2-\frac{1}{2-\frac{1}{2-\frac{1}{\cdots}}}\)

This expression can be written as:

x = 2 - (1/x)

x = (2x - 1) / x

x² = 2x - 1

x² - 2x + 1 = 0

(x -1)² = 0

x = 1

Hence, option 1 is correct.

Calculation:

Numbers between 200 and 600 which are divisible by 4, 5 and 6.

LCM (4, 5 and 6) = 60

Multiple of 60 between 200 and 600,

240, 300, 360, 420, 480, 540

There are 6 numbers between 200 and 600 which are divisible by 60.

Hence, option 3 is correct.

Calculation:

Let the value of n be 1.

(3⁴ⁿ - 4³ⁿ) = (3^{4 × 1} - 4^{3 × 1})

(3⁴ⁿ - 4³ⁿ) = 81 - 64 = 17

Let the value of n be 2.

(3⁴ⁿ - 4³ⁿ) = (3^{4 × 2} - 4^{3 × 2})

(3⁴ⁿ - 4³ⁿ) = 6651 - 4096 = 2465

(3⁴ⁿ - 4³ⁿ) = 2465 = 17 × 145

Hence, 17 is a multiple of (3⁴ⁿ - 4³ⁿ)

Hence, option 2 is correct.

Calculation:

Let a and b be the two numbers.

a² - b² = 256000  ----(1.)

a + b = 1000   -------(2.)

b  = 1000 - a    --------(3.)

Put equation (3.) in equation (1.)

a² - b² = 256000 

a ² - (1000 - a)² = 256000

a ²  - (1000² + a ² - 2000 a) = 256000

a ²  - 1000000 - a ² + 2000 a= 256000

2000 a = 1000000 + 256000

a = 1256000 / 2000 = 628

b = 1000 - 628 = 372

Hence, option 3 is correct.

Calculation:

log_√2(32)

It can be written as:

log_√2(32) = (log 32) / (log 2^1/2 )

log_√2(32) = (log 2⁵) / (log 2^1/2 )

log_√2(32) = (5 log 2) / (1/2 (log 2))                    [ log xⁿ = n log x]

log_√2(32) = 5 / (1/2)

log_√2(32) = 10

Hence, option 3 is correct.

Shortcut Trick

Use the formula: (Quantity₁ × Price₁%) ÷ Value₁ = (Quantity₂ × Price₂%) ÷ Value₂

Where Price% is (100 − Loss%) or (100 + Gain%).

(45 × 80) ÷ 40 = (n × 120) ÷ 24

90 = 5n ⇒ n = 18

∴ The correct answer is 18.

Alternate Method

Given: Selling Price (SP₁) of 45 lemons = ₹40, Loss = 20%, Target Gain = 20%, Target Revenue = ₹24. Formulas: CP = (SP × 100) ÷ (100 − Loss%), New SP = CP × (100 + Gain%) ÷ 100.

⇒ Selling Price of 1 lemon = 40 ÷ 45 = ₹8/9

⇒ Cost Price (CP) of 1 lemon = (8/9 × 100) ÷ 80 = ₹10/9

⇒ Required SP for 20% gain = (10/9) × 120 ÷ 100 = ₹4/3 per lemon

⇒ Number of lemons for ₹24 = 24 ÷ (4/3) = 24 × 3 ÷ 4 = 18

∴ The correct answer is 18.

Additional Information

Price and Quantity Relationship

When total expenditure or revenue is constant, the Price per article is inversely proportional to the Quantity sold. Ratio of Price % × Quantity = Constant.

Profit and Loss on SP

Standard Profit/Loss is calculated on Cost Price unless specified. Formula: SP = CP(1 ± R/100).

Given:

Length of the train = 110 m

Speed of the train = 58 kmph

Speed of the passerby = 4 kmph

Concept used:

Time travel to pass a passerby =  Length of the train / Relative Speed

Calculation:

Time to pass a passerby =  Length of the train / Relative Speed

Relative speed = 58 - 4 = 54 kmph = 54 × (5/18) = 15 m/sec

Time to pass the passerby = 110/15 = \(7\frac{1}{3}\) seconds

Hence, option 3 is correct.