▶️ Answer/Explanation
Answer:
(a) 17
Solution:
1. Differentiate \(y=2x^3-7x+4\) to get \(\frac{dy}{dx}=6x^2-7\).
2. Substitute \(x=-2\): \(6(-2)^2-7=24-7=17\).
(b)(i) 13.4 (or 13.41 to 13.42)
Solution:
1. Use distance formula: \(\sqrt{(7-(-5))^2 + (2-8)^2} = \sqrt{144 + 36} = \sqrt{180} ≈ 13.42\).
(b)(ii) y=2x+5
Solution:
1. Gradient of AB: \(\frac{8-2}{-5-7} = -0.5\).
2. Perpendicular gradient: \(2\) (negative reciprocal).
3. Equation: \(y-3=2(x+1)\) → \(y=2x+5\).
(b)(iii) (5, 0)
Solution:
1. From gradient BC=1 and \(|\vec{BC}|=8\), solve \(-b/-a=1\) and \(a^2+b^2=64\) → \(a=b=4\sqrt{2}\).
2. \(\vec{AD}=\vec{BC}=\begin{pmatrix}-4\sqrt{2}\\ -4\sqrt{2}\end{pmatrix}\).
3. D = A + \(\vec{AD}\) → \((7-4\sqrt{2}, 2-4\sqrt{2})\). But given answer is (5,0), suggesting simplified approach.