Past Exam Questions

The diagram shows the curve \(y = xe^{-\frac{1}{4}x^2}\), for \(x \geq 0\), and its maximum point \(M\).

(a) Find the exact coordinates of \(M\).

(b) Using the substitution \(x = \sqrt{u}\), or otherwise, find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 3\).

The diagram shows the curve \(y = \\sin^2 2x \\cos x\) for \(0 \leq x \leq \frac{1}{2} \pi\), and its maximum point \(M\).

(i) Find the \(x\)-coordinate of \(M\).

(ii) Using the substitution \(u = \\sin x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

The diagram shows the curve \(y = 5 \sin^3 x \cos^2 x\) for \(0 \leq x \leq \frac{1}{2} \pi\), and its maximum point \(M\).

(i) Find the \(x\)-coordinate of \(M\).

(ii) Using the substitution \(u = \cos x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis.

The diagram shows the curve \(y = x^2 \sqrt{1-x^2}\) for \(x \geq 0\) and its maximum point \(M\).

(i) Find the exact value of the \(x\)-coordinate of \(M\).

(ii) Show, by means of the substitution \(x = \sin \theta\), that the area \(A\) of the shaded region between the curve and the \(x\)-axis is given by

\(A = \frac{1}{4} \int_0^{\frac{\pi}{2}} \sin^2 2\theta \ d\theta.\)

(iii) Hence obtain the exact value of \(A\).

The diagram shows the curve \(y = (x + 5) \sqrt{3 - 2x}\) and its maximum point \(M\).

(a) Find the exact coordinates of \(M\).

(b) Using the substitution \(u = 3 - 2x\), find by integration the area of the shaded region bounded by the curve and the \(x\)-axis. Give your answer in the form \(a \sqrt{13}\), where \(a\) is a rational number.

The diagram shows part of the curve \(y = \\sin \\sqrt{x}\). This part of the curve intersects the x-axis at the point where \(x = a\).

(a) State the exact value of \(a\).

(b) Using the substitution \(u = \\sqrt{x}\), find the exact area of the shaded region in the first quadrant bounded by this part of the curve and the x-axis.

The diagram shows the curve \(y = \sin x \cos 2x\) for \(0 \leq x \leq \frac{1}{2}\pi\), and its maximum point \(M\).

(a) Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 significant figures.

(b) Using the substitution \(u = \cos x\), find the area of the shaded region enclosed by the curve and the \(x\)-axis in the first quadrant, giving your answer in a simplified exact form.

The diagram shows the curve \(y = \frac{x}{1 + 3x^4}\), for \(x \geq 0\), and its maximum point \(M\).

(a) Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

(b) Using the substitution \(u = \sqrt{3}x^2\), find by integration the exact area of the shaded region bounded by the curve, the \(x\)-axis and the line \(x = 1\).

The diagram shows the curve \(y = \\sin^3 x \\sqrt{\\cos x}\) for \(0 \leq x \leq \frac{1}{2} \pi\), and its maximum point \(M\).

(i) Using the substitution \(u = \\cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.

(ii) Showing all your working, find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

The diagram shows the curve \(y = 5 \sin^2 x \cos^3 x\) for \(0 \leq x \leq \frac{1}{2} \pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

(i) Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

(ii) Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\).

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

(i) Using the substitution \(u = \\cos x\), find by integration the exact area of the shaded region bounded by the curve and the \(x\)-axis.

(ii) Find the \(x\)-coordinate of \(M\). Give your answer correct to 2 decimal places.

The diagram shows the curve \(y = e^{2\sin x} \cos x\) for \(0 \leq x \leq \frac{1}{2}\pi\), and its maximum point \(M\).

(i) Using the substitution \(u = \sin x\), find the exact value of the area of the shaded region bounded by the curve and the axes.

(ii) Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places.

The diagram shows the curve \(y = x^{\frac{1}{2}} \ln x\). The shaded region between the curve, the x-axis and the line \(x = e\) is denoted by \(R\).

(i) Find the equation of the tangent to the curve at the point where \(x = 1\), giving your answer in the form \(y = mx + c\).

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

The diagram shows the curve \(y = e^{-\frac{1}{2}x} \sqrt{1 + 2x}\) and its maximum point \(M\). The shaded region between the curve and the axes is denoted by \(R\).

(i) Find the \(x\)-coordinate of \(M\).

(ii) Find by integration the volume of the solid obtained when \(R\) is rotated completely about the \(x\)-axis. Give your answer in terms of \(\pi\) and \(e\).

(a) Find the quotient and remainder when \(8x^3 + 4x^2 + 2x + 7\) is divided by \(4x^2 + 1\).

(b) Hence find the exact value of \(\int_0^{\frac{1}{2}} \frac{8x^3 + 4x^2 + 2x + 7}{4x^2 + 1} \, dx\).

Find the exact value of the constant k for which \(\int_{1}^{k} \frac{1}{2x-1} \, dx = 1\).

(i) Express \(\cos \theta + (\sqrt{3}) \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2} \pi\), giving the exact values of \(R\) and \(\alpha\).

(ii) Hence show that \(\int_{0}^{\frac{1}{2}\pi} \frac{1}{(\cos \theta + (\sqrt{3}) \sin \theta)^2} \, d\theta = \frac{1}{\sqrt{3}}\).

(i) By first expanding \(\cos(2x + x)\), show that \(\cos 3x \equiv 4 \cos^3 x - 3 \cos x\).

(ii) Hence solve the equation \(\cos 3x + 3 \cos x + 1 = 0\), for \(0 \leq x \leq \pi\).

(iii) Find the exact value of \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \cos^3 x \, dx\).

(i) Express \(\cos \theta + 2 \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). Give the exact values of \(R\) and \(\tan \alpha\).

(ii) Hence, showing all necessary working, show that \(\int_0^{\frac{1}{4}\pi} \frac{15}{(\cos \theta + 2 \sin \theta)^2} \, d\theta = 5.\)

(i) Prove that if \(y = \frac{1}{\cos \theta}\) then \(\frac{dy}{d\theta} = \sec \theta \tan \theta\).

(ii) Prove the identity \(\frac{1 + \sin \theta}{1 - \sin \theta} \equiv 2 \sec^2 \theta + 2 \sec \theta \tan \theta - 1\).

(iii) Hence find the exact value of \(\int_{0}^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} \, d\theta\).