Past Exam Questions

It is given that \(\int_0^p 4xe^{-\frac{1}{2}x} \, dx = 9\), where \(p\) is a positive constant.

(i) Show that \(p = 2 \ln \left( \frac{8p + 16}{7} \right)\).

(ii) Use an iterative process based on the equation in part (i) to find the value of \(p\) correct to 3 significant figures. Use a starting value of 3.5 and give the result of each iteration to 5 significant figures.

The diagram shows part of the curve \(y = \\cos(\sqrt{x})\) for \(x \geq 0\), where \(x\) is in radians. The shaded region between the curve, the axes and the line \(x = p^2\), where \(p > 0\), is denoted by \(R\). The area of \(R\) is equal to 1.

(i) Use the substitution \(x = u^2\) to find \(\int_0^{p^2} \cos(\sqrt{x}) \, dx\). Hence show that \(\sin p = \frac{3 - 2 \cos p}{2p}\).

(ii) Use the iterative formula \(p_{n+1} = \sin^{-1} \left( \frac{3 - 2 \cos p_n}{2p_n} \right)\), with initial value \(p_1 = 1\), to find the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

It is given that \(\int_1^a x \ln x \, dx = 22\), where \(a\) is a constant greater than 1.

(i) Show that \(a = \sqrt{\frac{87}{2 \ln a - 1}}\).

(ii) Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 6 and give the result of each iteration to 4 decimal places.

(i) Given that \(\int_1^a \frac{\ln x}{x^2} \, dx = \frac{2}{5}\), show that \(a = \frac{5}{3}(1 + \ln a)\).

(ii) Use an iteration formula based on the equation \(a = \frac{5}{3}(1 + \ln a)\) to find the value of \(a\) correct to 2 decimal places. Use an initial value of 4 and give the result of each iteration to 4 decimal places.

The diagram shows the curve \(y = x \cos 2x\) for \(0 \leq x \leq \frac{1}{4} \pi\). The point \(M\) is a maximum point.

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2x \tan 2x\).
2. The equation in part (i) can be rearranged in the form \(x = \frac{1}{2} \arctan \left( \frac{1}{2x} \right)\). Use the iterative formula \(x_{n+1} = \frac{1}{2} \arctan \left( \frac{1}{2x_n} \right)\), with initial value \(x_1 = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
3. Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from \(0\) to \(\frac{1}{4} \pi\).

The constant \(a\) is such that \(\int_{1}^{a} x^2 \ln x \, dx = 4\).

(a) Show that \(a = \left( \frac{35}{3 \ln a - 1} \right)^{\frac{1}{3}}\).

(b) Verify by calculation that \(a\) lies between 2.4 and 2.8.

(c) Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The constant \(a\) is such that \(\int_1^a \frac{\ln x}{\sqrt{x}} \, dx = 6\).

(a) Show that \(a = \exp \left( \frac{1}{\sqrt{a}} + 2 \right)\).

[\(\exp(x)\) is an alternative notation for \(e^x\).]

(b) Verify by calculation that \(a\) lies between 9 and 11.

(c) Use an iterative formula based on the equation in part (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The diagram shows the curve \(y = \sqrt{x} \cos x\), for \(0 \leq x \leq \frac{3}{2}\pi\), and its minimum point \(M\), where \(x = a\). The shaded region between the curve and the x-axis is denoted by \(R\).

(a) Show that \(a\) satisfies the equation \(\tan a = \frac{1}{2a}\).

(b) The sequence of values given by the iterative formula \(a_{n+1} = \pi + \arctan\left(\frac{1}{2a_n}\right)\), with initial value \(x_1 = 3\), converges to \(a\). Use this formula to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(c) Find the volume of the solid obtained when the region \(R\) is rotated completely about the x-axis. Give your answer in terms of \(\pi\).

It is given that \(\int_0^a x \cos \frac{1}{3}x \, dx = 3\), where the constant \(a\) is such that \(0 < a < \frac{3}{2}\pi\).

(i) Show that \(a\) satisfies the equation \(a = \frac{4 - 3 \cos \frac{1}{3}a}{\sin \frac{1}{3}a}.\)

(ii) Verify by calculation that \(a\) lies between 2.5 and 3.

(iii) Use an iterative formula based on the equation in part (i) to calculate \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

The positive constant \(a\) is such that \(\int_0^a x e^{-\frac{1}{2}x} \, dx = 2\).

(i) Show that \(a\) satisfies the equation \(a = 2 \ln(a + 2)\).

(ii) Verify by calculation that \(a\) lies between 3 and 3.5.

(iii) Use an iteration based on the equation in part (i) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

It is given that \(\int_{1}^{a} x^{-2} \ln x \, dx = 2\), where \(a > 1\).

(i) Show that \(a^{\frac{3}{2}} = \frac{7 + 2a^{\frac{3}{2}}}{3 \ln a}\).

(ii) Show by calculation that \(a\) lies between 2 and 4.

(iii) Use the iterative formula \(a_{n+1} = \left( \frac{7 + 2a_n^{\frac{3}{2}}}{3 \ln a_n} \right)^{\frac{2}{3}}\) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

The diagram shows the curve \(y = x^2 \cos 2x\) for \(0 \leq x \leq \frac{1}{4}\pi\). The curve has a maximum point at \(M\) where \(x = p\).

(i) Show that \(p\) satisfies the equation \(p = \frac{1}{2} \arctan \left( \frac{1}{p} \right)\).

(ii) Use the iterative formula \(p_{n+1} = \frac{1}{2} \arctan \left( \frac{1}{p_n} \right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

(iii) Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis.

It is given that \(\int_0^a x \cos x \, dx = 0.5\), where \(0 < a < \frac{1}{2} \pi\).

(i) Show that \(a\) satisfies the equation \(\sin a = \frac{1.5 - \cos a}{a}\).

(ii) Verify by calculation that \(a\) is greater than 1.

(iii) Use the iterative formula \(a_{n+1} = \sin^{-1} \left( \frac{1.5 - \cos a_n}{a_n} \right)\) to determine the value of \(a\) correct to 4 decimal places, giving the result of each iteration to 6 decimal places.

The diagram shows a circle with centre O and radius r. The angle of the minor sector AOB of the circle is x radians. The area of the major sector of the circle is 3 times the area of the shaded region.

\((a) Show that x = \frac{3}{4} \sin x + \frac{1}{2} \pi.\)

(b) Show by calculation that the root of the equation in (a) lies between 2 and 2.5.

(c) Use an iterative formula based on the equation in (a) to calculate this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

In the diagram, A is a point on the circumference of a circle with centre O and radius r. A circular arc with centre A meets the circumference at B and C. The angle OAB is θ radians. The shaded region is bounded by the circumference of the circle and the arc with centre A joining B and C. The area of the shaded region is equal to half the area of the circle.

(i) Show that \(\cos 2θ = \frac{2 \sin 2θ - π}{4θ}\).

(ii) Use the iterative formula \(θ_{n+1} = \frac{1}{2} \cos^{-1} \left( \frac{2 \sin 2θ_n - π}{4θ_n} \right)\), with initial value \(θ_1 = 1\), to determine \(θ\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places.

In the diagram, ABC is a triangle in which angle ABC is a right angle and BC = a. A circular arc, with centre C and radius a, joins B and the point M on AC. The angle ACB is b8 radians. The area of the sector CMB is equal to one third of the area of the triangle ABC.

(i) Show that b8 satisfies the equation

\(\tan \theta = 3\theta\).

(ii) This equation has one root in the interval \(0 < \theta < \frac{1}{2}\pi\). Use the iterative formula

\(\theta_{n+1} = \arctan(3\theta_n)\)

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The diagram shows a semicircle ACB with centre O and radius r. The tangent at C meets AB produced at T. The angle BOC is x radians. The area of the shaded region is equal to the area of the semicircle.

(i) Show that x satisfies the equation \(\tan x = x + \pi\).

(ii) Use the iterative formula \(x_{n+1} = \arctan(x_n + \pi)\) to determine x correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

The diagram shows a circle with centre O and radius 10 cm. The chord AB divides the circle into two regions whose areas are in the ratio 1 : 4 and it is required to find the length of AB. The angle AOB is \(\theta\) radians.

(i) Show that \(\theta = \frac{2}{5}\pi + \sin \theta\).

(ii) Showing all your working, use an iterative formula, based on the equation in part (i), with an initial value of 2.1, to find \(\theta\) correct to 2 decimal places. Hence find the length of AB in centimetres correct to 1 decimal place.

The diagram shows a semicircle ACB with centre O and radius r. The angle BOC is x radians. The area of the shaded segment is a quarter of the area of the semicircle.

(i) Show that x satisfies the equation

\(x = \frac{3}{4}\pi - \sin x\).

(ii) This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.

(iii) Use the iterative formula

\(x_{n+1} = \frac{3}{4}\pi - \sin x_n\)

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

In the diagram, ABCD is a rectangle with AB = 3a and AD = a. A circular arc, with centre A and radius r, joins points M and N on AB and CD respectively. The angle MAN is x radians. The perimeter of the sector AMN is equal to half the perimeter of the rectangle.

1. Show that x satisfies the equation \(\sin x = \frac{1}{4}(2 + x)\).
2. This equation has only one root in the interval \(0 < x < \frac{1}{2}\pi\). Use the iterative formula \(x_{n+1} = \sin^{-1}\left(\frac{2 + x_n}{4}\right)\), with initial value \(x_1 = 0.8\), to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.