Question

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

1. If x − a is a factor of 3x³ + x² − ax − 81, then a = 3.
2. x + 1 is a factor of 3x³ − 7x² + 4.

पर्याय

• Only (i)

• Only (ii)

• Both (i) and (ii)

• Neither (i) nor (ii)

Answer 1

Only (i)

Explanation:

i.

Let f(x) = 3x³ + x² − ax − 81

x − a = 0

x = a

f(a) = 0

⇒ 3(a)³ + a² − a(a) − 81 = 0

⇒ 3a³ − 81 = 0

⇒ a³ = `81/3`

⇒ a³ = 27

⇒ a = 3

Therefore, statement (i) is valid.

ii.

Let f(x) = 3x³ − 7x² + 4

Let x + 1 = 0

x = −1

f(−1) = 3(−1)³ − 7(−1)² + 4

= 3 − 7 + 4

= −6

Since f(x) ≠ 0

Therefore, g(x) is not a factor of f(x), so statement (ii) is false.