In the following question, two statements (i) and (ii) are given. Choose the valid statement. i. If (x + 1) and (x + 2) are the factors of x3 + px2 + qx − 2 then p = 0. ii. 2x3 + x2 + 17x + 46 is exactly divisible by x + 2.

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Only (ii) Explanation: i. Let f(x) = x3 + px2 + qx − 2 ∵ (x + 1) is a factor of f(x) ∴ f (−1) = 0 ⇒ −13 + p(−1)2 + q(−1) − 2 = 0 ⇒ −1 + p − q − 2 = 0 ⇒ p − q − 3 = 0 ⇒ p − q = 3 ...(i) (x + 2) is a factor of f(x) ∴ f (−2) = 0 ⇒ −23 + p(−2)2 + q(−2) − 2 = 0 ⇒ −8 + 4p − 2q − 2 = 0 ⇒ 4p − 2q − 10 = 0 Divide by 2: 2p − q = 5 ...(ii) Subtract (i) from (ii), (p − q) − (2p − q) = 5 − 3 2p − q − p + q = 2 p = 2 Therefore, statement (i) is invalid. ii. Let f(x) = 2x3 + x2 + 17x + 46 Let x + 2 = 0 x = −2 F(−2) = 2(−2)3 + (−2)2 + 17(−2) + 46 = 2(−8) + 4 − 34 + 46 = −16 + 4 − 34 + 46 = −12 − 34 + 46 = −46 + 46 = 0 Therefore, statement (ii) is valid.

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