In the following question, two statements (i) and (ii) are given. Choose the valid statement. i. 4x2 + kx − 7 is exactly divisible by x − 2 then the value of k is $\frac{9}{2}$. ii. (2x + 1) is a factor of 2x3 + x2 − 8x − 4.

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Only (ii) Explanation: i. Let f(x) = 4x2 + kx − 7 Let x − 2 = 0 x = 2 ∵ x − 2 is a factor of f(x). $\therefore$ f(2) = 0 $\Rightarrow$ 4(2)2 + k(2) − 7 = 0 $\Rightarrow$ 4(4) + 2k − 7 = 0 $\Rightarrow$ 16 + 2k − 7 = 0 $\Rightarrow$ 9 + 2k = 0 $\Rightarrow$ 2k = −9 $\Rightarrow$ k = -$\frac{9}{2}$ Therefore, statement (i) is invalid. ii. Let f(x) = 2x3 + x2 − 8x − 4 Let 2x + 1 = 0 x = -$\frac{1}{2}$ `f(-$\frac{1}{2}$) = 2(-$\frac{1}{2}$)^3 + (-$\frac{1}{2}$)^2 - 8(-$\frac{1}{2}$) - 4 = 2(-$\frac{1}{8}$) + ($\frac{1}{4}$) - (-4) - 4 = -$\frac{1}{4}$ + $\frac{1}{4}$ + 4 - 4 = 0 Since f(-$\frac{1}{2}$)` = 0, statement (ii) is valid.

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