Past Exam Questions

(i) By sketching a suitable pair of graphs, show that the equation \(\cot x = 1 + x^2\), where \(x\) is in radians, has only one root in the interval \(0 < x < \frac{1}{2}\pi\).

(ii) Verify by calculation that this root lies between 0.5 and 0.8.

(iii) Use the iterative formula \(x_{n+1} = \arctan\left( \frac{1}{1 + x_n^2} \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(\sqrt{x} = e^x - 3\) has only one root.

(b) Show by calculation that this root lies between 1 and 2.

(c) Show that, if a sequence of values given by the iterative formula \(x_{n+1} = \ln(3 + \sqrt{x_n})\) converges, then it converges to the root of the equation in (a).

(d) Use the iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(i) By sketching suitable graphs, show that the equation \(4x^2 - 1 = \cot x\) has only one root in the interval \(0 < x < \frac{1}{2}\pi\).

(ii) Verify by calculation that this root lies between 0.6 and 1.

(iii) Use the iterative formula \(x_{n+1} = \frac{1}{2}\sqrt{1 + \cot x_n}\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The constant a is such that \(\int_{0}^{a} xe^{\frac{1}{2}x} \, dx = 6\).

(i) Show that a satisfies the equation \(x = 2 + e^{-\frac{1}{2}x}\).

(ii) By sketching a suitable pair of graphs, show that this equation has only one root.

(iii) Verify by calculation that this root lies between 2 and 2.5.

(iv) Use an iterative formula based on the equation in part (i) to calculate the value of a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(i) By sketching a suitable pair of graphs, show that the equation \(2 - x = \ln x\) has only one root.

(ii) Verify by calculation that this root lies between 1.4 and 1.7.

(iii) Show that this root also satisfies the equation \(x = \frac{1}{3}(4 + x - 2 \ln x)\).

(iv) Use the iterative formula \(x_{n+1} = \frac{1}{3}(4 + x_n - 2 \ln x_n)\), with initial value \(x_1 = 1.5\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(i) By sketching a suitable pair of graphs, show that the equation \(2 \cot x = 1 + e^x\), where \(x\) is in radians, has only one root in the interval \(0 < x < \frac{1}{2} \pi\).

(ii) Verify by calculation that this root lies between 0.5 and 1.0.

(iii) Show that this root also satisfies the equation \(x = \arctan\left(\frac{2}{1 + e^x}\right)\).

(iv) Use the iterative formula \(x_{n+1} = \arctan\left(\frac{2}{1 + e^{x_n}}\right)\), with initial value \(x_1 = 0.7\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(i) By sketching a suitable pair of graphs, show that the equation \(\csc x = \frac{1}{2}x + 1\), where \(x\) is in radians, has a root in the interval \(0 < x < \frac{1}{2}\pi\).

(ii) Verify, by calculation, that this root lies between 0.5 and 1.

(iii) Show that this root also satisfies the equation \(x = \sin^{-1} \left( \frac{2}{x+2} \right)\).

(iv) Use the iterative formula \(x_{n+1} = \sin^{-1} \left( \frac{2}{x_n+2} \right)\), with initial value \(x_1 = 0.75\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The diagram shows a sector OAB of a circle with centre O and radius r. The angle AOB is \(\alpha\) radians, where \(0 < \alpha < \frac{1}{2}\pi\). The point N on OA is such that BN is perpendicular to OA. The area of the triangle ONB is half the area of the sector OAB.

1. Show that \(\alpha\) satisfies the equation \(\sin 2x = x\).
2. By sketching a suitable pair of graphs, show that this equation has exactly one root in the interval \(0 < x < \frac{1}{2}\pi\).
3. Use the iterative formula \(x_{n+1} = \sin(2x_n)\), with initial value \(x_1 = 1\), to find \(\alpha\) correct to 2 decimal places, showing the result of each iteration.

(i) By sketching suitable graphs, show that the equation \(\sec x = 3 - x^2\) has exactly one root in the interval \(0 < x < \frac{1}{2}\pi\).

(ii) Show that, if a sequence of values given by the iterative formula \(x_{n+1} = \cos^{-1} \left( \frac{1}{3-x_n^2} \right)\) converges, then it converges to a root of the equation given in part (i).

(iii) Use this iterative formula, with initial value \(x_1 = 1\), to determine the root in the interval \(0 < x < \frac{1}{2}\pi\) correct to 2 decimal places, showing the result of each iteration.

(a) By sketching a suitable pair of graphs, show that the equation \(\ln x = 3x - x^2\) has one real root.

(b) Verify by calculation that the root lies between 2 and 2.8.

(c) Use the iterative formula \(x_{n+1} = \sqrt{3x_n - \ln x_n}\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(4 - x^2 = \sec \frac{1}{2}x\) has exactly one root in the interval \(0 \leq x < \pi\).

(b) Verify by calculation that this root lies between 1 and 2.

(c) Use the iterative formula \(x_{n+1} = \sqrt{4 - \sec \frac{1}{2}x_n}\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(\cot \frac{1}{2}x = 1 + e^{-x}\) has exactly one root in the interval \(0 < x \leq \pi\).

(b) Verify by calculation that this root lies between 1 and 1.5.

(c) Use the iterative formula \(x_{n+1} = 2 \arctan \left( \frac{1}{1 + e^{-x_n}} \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(\csc x = 1 + e^{-\frac{1}{2}x}\) has exactly two roots in the interval \(0 < x < \pi\).

(b) The sequence of values given by the iterative formula \(x_{n+1} = \pi - \sin^{-1}\left( \frac{1}{e^{-\frac{1}{2}x_n} + 1} \right)\), with initial value \(x_1 = 2\), converges to one of these roots. Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(x^5 = 2 + x\) has exactly one real root.

(b) Show that if a sequence of values given by the iterative formula \(x_{n+1} = \frac{4x_n^5 + 2}{5x_n^4 - 1}\) converges, then it converges to the root of the equation in part (a).

(c) Use the iterative formula with initial value \(x_1 = 1.5\) to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

(a) By sketching a suitable pair of graphs, show that the equation \(\sec x = 2 - \frac{1}{2}x\) has exactly one root in the interval \(0 \leq x < \frac{1}{2}\pi\).

(b) Verify by calculation that this root lies between 0.8 and 1.

(c) Use the iterative formula \(x_{n+1} = \cos^{-1}\left(\frac{2}{4-x_n}\right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(i) By sketching a suitable pair of graphs, show that the equation \(\ln(x+2) = 4e^{-x}\) has exactly one real root.

(ii) Show by calculation that this root lies between \(x = 1\) and \(x = 1.5\).

(iii) Use the iterative formula \(x_{n+1} = \ln\left( \frac{4}{\ln(x_n + 2)} \right)\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.