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Given:

8a3 - (pb)3 = (2a - 3b) (4a2 + 6ab + 9b2)

Formula used:

a³ - b³ = (a -b)(a² + ab + b2)

Calculation:

(2a - 3b) (4a2 + 6ab + 9b2)

⇒ (2a - 3b) [(2a)2 + 2a × 3b + (3b)2]

⇒ (2a)³ - (3b)³

⇒ 8a3 - (3b)3 =  8a3 - (pb)³

So, p = 3

∴ The correct answer is option (2).

The correct answer is Anthracite.

Key Points

• Coal is a fossil fuel that mainly contains carbon.
• It is formed by the decay of vegetation that existed millions of years ago.
• The best quality of coal is Anthracite.
• “Anthracite is the highest rank of coal.”
• It is a hard/brittle/black lustrous coal, also known as hard coal.
• It has the highest percentage of carbon i.e. 92%-98%

 and a low percentage of volatile matter.

• It ignites slowly with a blue flame.

Additional Information

• Bituminous- It is the most widely available coal in India. It is of medium quality. It contains about 40 to 75 per cent carbon. In India, much of the coal comes under this category.
• Lignite- It is of low quality. It is brown. It contains about 30 to 40 per cent coal.
• Peat- It is the middle stage in the process of coal formation. It contains less than 40 per cent carbon. It has more impurities. The parts of the wood are remarkably present in it.

Given:

5 years ago, the ratio of ages of A and B was 6 ∶ 7.

After 5 years, the ratio of their ages will be 8 ∶ 9.

Concept:

Let the ages of A and B 5 years ago be 6x and 7x respectively.

Hence, the present age of A is (6x + 5) and B is (7x + 5).

Formula Used:

After 5 years, their ages will be (6x + 10) and (7x + 10) respectively.

The new ratio given is 8 ∶ 9.

Calculation:

We have,

⇒ (6x + 10)/(7x + 10) = 8/9

⇒ 9(6x + 10) = 8(7x + 10)

⇒ 54x + 90 = 56x + 80

⇒ 56x - 54x = 90 - 80

⇒ 2x = 10

⇒ x = 5

Hence, the present age of A is:

⇒ 6x + 5 = 6(5) + 5 = 30 + 5 = 35

∴ The present age of A is 35 years, which corresponds to option 2.

Given:

In a joint business, the capitals of A, B, and C are in the ratio \(\dfrac{1}{4}\) ∶ \(\dfrac{1}{3}\) ∶ \(\dfrac{1}{2}\).

At the end of the year, they make a profit of Rs. 11700.

Concept used:

Ratio of investments is the same as the ratio of profits or losses.

Calculation:

The capitals of A, B, and C are in the ratio  \(\dfrac{1}{4}\) ∶ \(\dfrac{1}{3}\) ∶ \(\dfrac{1}{2}\) = 3 : 4 : 6.

∴ The ratio of profit = 3 : 4 : 6

Let the profits of A, B, and C be 3x, 4x and 6x respectively.

Total profit = Rs. 11700

∴ Profit of B = \(11700\times \dfrac{4x}{13x} \) = Rs. 3600

Hence, the profit of B is Rs. 3600.

The given word is: MEDITERRANEAN

Now, let's check each option,

1) READER → Can be formed.

2. ARRANGE → Cannot be formed (Because there is no 'G' in MEDITERRANEAN)

3. DINNER → Can be formed.

4. MEDIATE → Can be formed.

Hence, the correct answer is "ARRANGE".

Given:

10 cows or 7 buffaloes can gaze a field in 60 days working 8 hours a day.

We need to find the number of days required for 25 cows and 14 buffaloes working 12 hours a day to do 9 times their work.

Concept:

Work done is directly proportional to the number of animals and the number of hours worked.

If W is the work done by one cow in one hour, then 10 cows work in 60 days for 8 hours each day is:

Formula Used:

Total Work = Number of Animals × Number of Days × Number of Hours

Calculation:

We have,

⇒ 10 cows × 60 days × 8 hours = 4800 cow-hours

Similarly,

⇒ 7 buffaloes × 60 days × 8 hours = 3360 buffalo-hours

Assuming 1 buffalo = (4800 cow-hours) / (3360 buffalo-hours) = 1.428 cows

Now, for 25 cows and 14 buffaloes working 12 hours a day to do 9 times the work:

Total equivalent cows = 25 cows + 14 buffaloes × 1.428 = 25 + 20 = 45 cows

Total work required = 9 × 4800 cow-hours = 43200 cow-hours

Let the number of days required be D. Then,

⇒ 45 cows × D days × 12 hours = 43200 cow-hours

⇒ 540D = 43200

⇒ D = 43200 / 540

⇒ D = 80

∴ The number of days required is 80.

Hence, the correct answer is option 1.

The correct answer is ​Regelation

Key Points

• Regelation is defined as the phenomenon in which the ice melts to the water below 0°C on the application of pressure and refreezes back to ice on the removal of pressure.
• Regelation demonstrates the idea of compressing the ice and turning it into water under pressure and when the pressure is removed, it solidifies again.

Other Examples

• Glacier acts as a source of a river due to regelation. The mass of the glacier exerts pressure on the lower surface lowering the melting point of the ice at its base. This results in the melting of ice and propels the glacier to slide over the liquid. Under appropriate conditions, liquid water flows from the base of the glacier to lower altitudes when the temperature of the air is above the freezing point of water.

The logic followed here is:

Logic: The resultant is the product of the sum of digits of both numbers.

324 × 150 = 54

→ (3 + 2 + 4) = 9

→ (1 + 5 + 0) = 6

Resultant = 9 × 6 = 54

251 × 402 = 48

→ (2 + 5 + 1) = 8

→ (4 + 0 + 2) = 6

Resultant = 8 × 6 = 48

Similarly,

735 × 866 = ?

→ (7 + 3 + 5) = 15

→ (8 + 6 + 6) = 20

Resultant = 15 × 20 = 300

Hence, the correct answer is "300".

Given:

Product of HCF and LCM of two numbers = 24

Subtraction of the two numbers = 2

Concept:

If two numbers are a and b, then HCF(a, b) × LCM(a, b) = a × b

Given, a - b = 2

Calculation:

⇒ Let the two numbers be a and b, and a > b

⇒ a - b = 2

⇒ HCF(a, b) × LCM(a, b) = a × b

⇒ 24 = a × b

⇒ Since a - b = 2, let b = x and a = x + 2

⇒ (x + 2) × x = 24

⇒ x² + 2x = 24

⇒ x² + 2x - 24 = 0

⇒ Solving the quadratic equation:

⇒ x = [-2 ± √(2² - 4×1×(-24))] / 2×1

⇒ x = [-2 ± √(4 + 96)] / 2

⇒ x = [-2 ± √100] / 2

⇒ x = [-2 ± 10] / 2

⇒ x = 4 (since x must be positive)

⇒ Therefore, b = 4 and a = 4 + 2 = 6

∴ The larger number is 6.

Calculation:

(0.5)² = 0.25

\(\sqrt {0 \cdot 49} \) = \(\sqrt {0.7\times0.7} \) = 0.7

\(\sqrt[3]{{0 \cdot 008}}\) = \(\sqrt[3]{0.2\times0.2\times0.2}\) = 0.2

0.23

Here, 0.2 is the least value among all of them.

Thus, \(\sqrt[3]{{0 \cdot 008}}\) is the least value.

∴ The correct answer is option(4).