If 2a-b=3 and 8a³-b³=999 , then the value of 4a²-b² .

If $2a-b=3$ and $8a^3-b^3=999$ , then the value of $4a^2-b^2$ .

Show answer & explanation

Correct answer: A

Explanation

The correct answer is option 1 i.e. 63.

x³ - y³ = (x - y)(x² + xy + y²)

Here, 2a - b = 3

⇒ (2a - b)² = 3² = 9

⇒ 4a² - 4ab + b² = 9 .....(1)

Also, 8a³ - b³ = 999

⇒ (2a - b)(4a² + 2ab + b²) = 999

⇒ 3(4a² + 2ab + b²) = 999

⇒ (4a² + 2ab + b²) = 999/3 = 333 .....(2)

Subtracting equation (1) from (2),

6ab = 324

⇒ ab = 54

Putting ab = 54 into the equation (1),

4a² - 4(54) + b² = 9

⇒ 4a² - 216 + b² = 9

⇒ 4a² + b² = 225

Now, (2a + b)² = 4a² + 4ab + b² = 225 + 4(54) = 225 + 216 = 441

⇒ 2a + b = √441 = 21

So, 4a² - b² = (2a - b)(2a + b) = 3 × 21 = 63

Therefore, the value of 4a² - b² is 63.

SSC CGL 2024 PYQ — Question 62