Applied context requiring Newton-Raphson

4 A curve has equation \(y = 2 \ln ( k - 3 x ) + x ^ { 2 } - 3 x\), where \(k\) is a positive constant.

1. Given that the curve has a point of inflection where \(x = 1\), show that \(k = 6\). It is also given that the curve intersects the \(x\)-axis at exactly one point.
2. Show by calculation that the \(x\)-coordinate of this point lies between 0.5 and 1.5 .
3. Use the Newton-Raphson method, with initial value \(x _ { 0 } = 1\), to find the \(x\)-coordinate of the point where the curve intersects the \(x\)-axis, giving your answer correct to 5 decimal places. Show the result of each iteration to 6 decimal places.
4. By choosing suitable bounds, verify that your answer to part (c) is correct to 5 decimal places.

4 The positive integers \(x , y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference - 4 . Also, \(x , \frac { 15 } { y }\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.

1. Show that \(y\) satisfies the equation \(y ^ { 4 } - 16 y ^ { 2 } - 225 = 0\).
2. Hence determine the sum to infinity of the geometric progression.

14 Fig. 14 shows a circle with centre O and radius \(r \mathrm {~cm}\). The chord AB is such that angle \(\mathrm { AOB } = x\) radians. The area of the shaded segment formed by AB is \(5 \%\) of the area of the circle. \begin{figure}[h] \end{figure}

1. Show that \(x - \sin x - \frac { 1 } { 10 } \pi = 0\). The Newton-Raphson method is to be used to find \(x\).
2. Write down the iterative formula to be used for the equation in part (a).
3. Use three iterations of the Newton-Raphson method with \(x _ { 0 } = 1.2\) to find the value of \(x\) to a suitable degree of accuracy.

7
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726} The diagram shows a particle \(P\) of mass 0.5 kg attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity \(6.86 \mathrm {~N} . P\) is released from rest at a point on the surface of the sphere where the acute angle \(A O P\) is at least 0.5 radians.

1. (a) For the case angle \(A O P = \alpha , P\) remains at rest. Show that \(\sin \alpha = 2.8 \alpha - 1.4\).
   (b) Use the iterative formula $$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$ with \(\alpha _ { 1 } = 0.8\), to find \(\alpha\) correct to 2 significant figures.
2. Given instead that angle \(A O P = 0.5\) radians when \(P\) is released, find the speed of \(P\) when angle \(A O P = 0.8\) radians, given that \(P\) is at all times in contact with the surface of the sphere. State whether the speed of \(P\) is increasing or decreasing when angle \(A O P = 0.8\) radians.