Question

In the following question, two statements (i) and (ii) are given. Choose the valid statement.

1. 4x² + kx − 7 is exactly divisible by x − 2 then the value of k is `9/2`.
2. (2x + 1) is a factor of 2x³ + x² − 8x − 4.

विकल्प

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (ii)

Explanation:

i.

Let  f(x) = 4x² + kx − 7

Let x − 2 = 0

x = 2

∵ x − 2 is a factor of f(x).

∴ f(2) = 0

⇒ 4(2)² + k(2) − 7 = 0

⇒ 4(4) + 2k − 7 = 0

⇒ 16 + 2k − 7 = 0

⇒ 9 + 2k = 0

⇒ 2k = −9

⇒ k = `-9/2`

Therefore, statement (i) is invalid.

ii.

Let f(x) = 2x³ + x² − 8x − 4

Let 2x + 1 = 0

x = `-1/2`

`f(-1/2) = 2(-1/2)^3 + (-1/2)^2 - 8(-1/2) - 4`

= `2(-1/8) + (1/4) - (-4) - 4`

= `-1/4 + 1/4 + 4 - 4`

= 0

Since `f(-1/2)` = 0, statement (ii) is valid.