In the following question, two statements (i) and (ii) are given. Choose the valid statement. i. If x − a is a factor of 3x3 + x2 − ax − 81, then a = 3. ii. x + 1 is a factor of 3x3 − 7x2 + 4.

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Only (i) Explanation: i. Let f(x) = 3x3 + x2 − ax − 81 x − a = 0 x = a f(a) = 0 $\Rightarrow$ 3(a)3 + a2 − a(a) − 81 = 0 $\Rightarrow$ 3a3 − 81 = 0 $\Rightarrow$ a3 = $\frac{81}{3}$ $\Rightarrow$ a3 = 27 $\Rightarrow$ 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) $\neq$ 0 Therefore, g(x) is not a factor of f(x), so statement (ii) is false.

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