Given:

Distance traveled by the car = 60 km

Let the original speed of the car be 's' km/hr

Increased speed of the car = (s + 8) km/hr

Formula:

Time = Distance/Speed

Solution:

Let T_a & T_b be the time required to complete distance at original speed & increased speed respectively

⇒ 60s + 480 - 60s = 2s² + 16s

⇒ s² + 8s - 240 = 0

On solving s = -20 or s = 12 (∵ s is speed, neglecting negative value)

∴ The original speed of the car is 12 km/hr.

Given:

80 liters of milk solution has 60% milk

Calculation:

Let's denote the amount of pure milk to be added as x liters.

Initially, there are 80 liters of milk solution,

which contains 60% milk. So, there are (80 × 0.60),

⇒ 48 liters of milk in the 80 liters of milk solution.

After adding x liters of pure milk, the total volume of the milk solution will be (80 + x) liters.

We want this new solution to have 80% milk in it.

So, the equation will be:

(48 + x) / (80 + x) = 0.80

Now, let's solve for x:

⇒ 48 + x = 0.80 × (80 + x)

⇒ 48 + x = 64 + 0.80x

⇒ x - 0.80x = 64 - 48

⇒ 0.20x = 16

⇒ x = 16 / 0.20

⇒ x = 80

∴ Need to add 80 liters of pure milk to the milk solution to make it 80% milk.

Given

Total amount = Rs. 12,029

P's share = 5Q

R's share = P/3

Concept:

The total amount is the sum of the shares of P, Q, and R.

Solution:

Express all shares in terms of Q: P = 5Q, R = 5Q/3

Total = P + Q + R = 5Q + Q + 5Q/3 = 12,029

Solving the above equation we get Q = Rs. 1569

Therefore, R's share = 5Q/3 = 5 × 1569/3 = Rs. 2615

Therefore, the share of R is Rs. 2615.

Given:

Price raised by 45% and two successive discount of 15% is given

Formula used:

Calculation:

Let the cost price of an article is Rs100

As mentioned, the price is raised by 45% that means

New price = Old price + 45% of old price

New price = 145 rs

Since two successive discount of 15% has been given on new price,

After first discount,

Discounted Price =145 - 15% of 145

% change = 4.7625 %

The price of the article is increased by 4.7625 %.

Alternate Method

Let CP be 100

MP = 100 + (45 × 100/100)

⇒ 145

Net Dis = 15 + 15 - (15 × 15/100)

⇒ 27.75%

Final Price = 145 - (145 × 27.75%)

⇒ 104.7625

% change = 4.7625 %

The price of the article is increased by 4.7625 %.

Given:

If 6m61 is completely divisible by 11

Calculation:

For divisibility by 11,

The Sum of digits at even places minus the sum of digits at odd places should be 0 or divisible by 11

6 + 6 – M – 1 = 0 or a multiple of 11

∴ If the value of M is 0, then it is divisible by 11.

Given:

Length of train = 158.5 m

Length of bridge = 761.5 m

Time = 46 second

Formula used:

Speed = Distance/Time

Concept used:

1 m/s = (60 × 60)/1000 km/hour = 3600/1000 km/hour

To convert m/s to km/hour  we have to multiply by 18/5

To convert km/hour to m/s we have to multiply by 5/18

Calculation:

Total distance = 158.5 + 761.5 = 920 m

Speed = 920/46

⇒ 20 m/s

⇒ 20 × 18/5

⇒ 72 km/hour

∴ Speed of the train is 72 km/hour

Given:

r = 5

Concept used:

Surface Area = 4 x π x r x r

Solution:

SA(Sphere) = 4 x π x 5 x 5

= 100 π cm2

Hence, the correct option is 1.

Given:

The length of a diagonal of a square = (a + b)

Formula used:

Diagonal of a square = √2 Side

Area of a square = (Side)²

Calculation:

Side of the square = Diagonal of a square/√2 

⇒ Side of the square = (a + b)/√2 

The area of the square = [(a + b)/√2]²

= (a + b)²/2

= (a² + b² + 2ab)/2

= (a² + b²)/2 + ab

∴ The area of the square is 1/2 (a² + b²) + ab.

Given:

2x2 + 5x + 1 = 0

Concept used:

(a + b)² = a² + b² + 2ab

(a - b)² = a² + b² - 2ab

Calculation:

2x2 + 5x + 1 = 0

Now,

Formula Used:

​The formula for calculating compound interest with half-yearly compounding is:

A = P(1 + r%/2)²ⁿ

where

A is the final amount,

P is the principal amount,

r is the annual interest rate,

and n is the number of half-year periods.

Calculation:

The formula for calculating compound interest with half-yearly compounding is:

A = P(1 + r%/2)2n

In this case, P = ₹6,500, r = 10% per annum,

and n = 2 years x 2 half-year periods per year = 4 half-year periods.

So, plugging in these values, we get:

⇒ A = 6500(1 + 0.1/2)⁴

⇒ A = 6500(1.05)⁴

⇒ A = 7903.72

Therefore, the amount borrowed after 2 years with half-yearly compounding at 10% interest per annum is ₹7,900.