Past Exam Questions

The parametric equations of a curve are

\(x = t - \tan t, \quad y = \ln(\cos t)\),

for \(-\frac{1}{2}\pi < t < \frac{1}{2}\pi\).

(i) Show that \(\frac{dy}{dx} = \cot t\).

(ii) Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures.

The parametric equations of a curve are

\(x = \\ln(2t + 3)\), \(y = \frac{3t + 2}{2t + 3}\).

Find the gradient of the curve at the point where it crosses the y-axis.

The parametric equations of a curve are

\(x = e^{-t} \cos t, \quad y = e^{-t} \sin t.\)

Show that \(\frac{dy}{dx} = \tan \left( t - \frac{1}{4} \pi \right).\)

The parametric equations of a curve are

\(x = \frac{4t}{2t + 3}\), \(y = 2 \ln(2t + 3)\).

1. Express \(\frac{dy}{dx}\) in terms of \(t\), simplifying your answer.
2. Find the gradient of the curve at the point for which \(x = 1\).

The parametric equations of a curve are

\(x = \sin 2\theta - \theta\), \(y = \cos 2\theta + 2 \sin \theta\).

Show that \(\frac{dy}{dx} = \frac{2 \cos \theta}{1 + 2 \sin \theta}\).

The diagram shows the curve with parametric equations

\(x = \\sin t + \\cos t, \quad y = \\sin^3 t + \\cos^3 t,\)

for \(\frac{1}{4}\pi < t < \frac{5}{4}\pi.\)

(i) Show that \(\frac{dy}{dx} = -3 \sin t \cos t.\)

(ii) Find the gradient of the curve at the origin.

(iii) Find the values of \(t\) for which the gradient of the curve is 1, giving your answers correct to 2 significant figures.

The parametric equations of a curve are

\(x = 3(1 + \\sin^2 t)\), \(y = 2 \\cos^3 t\).

Find \(\frac{dy}{dx}\) in terms of \(t\), simplifying your answer as far as possible.

The parametric equations of a curve are

\(x = \ln(\tan t)\), \(y = \sin^2 t\),

where \(0 < t < \frac{1}{2}\pi\).

(i) Express \(\frac{dy}{dx}\) in terms of \(t\).

(ii) Find the equation of the tangent to the curve at the point where \(x = 0\).

The parametric equations of a curve are

\(x = \frac{t}{2t + 3}\), \(y = e^{-2t}\).

Find the gradient of the curve at the point for which \(t = 0\).

The parametric equations of a curve are

\(x = \frac{\cos \theta}{2 - \sin \theta}\), \(y = \theta + 2 \cos \theta\).

Show that \(\frac{dy}{dx} = (2 - \sin \theta)^2\).

The parametric equations of a curve are

\(x = a \cos^3 t, \quad y = a \sin^3 t,\)

where \(a\) is a positive constant and \(0 < t < \frac{1}{2} \pi\).

(i) Express \(\frac{dy}{dx}\) in terms of \(t\).

(ii) Show that the equation of the tangent to the curve at the point with parameter \(t\) is

\(x \sin t + y \cos t = a \sin t \cos t.\)

(iii) Hence show that, if this tangent meets the \(x\)-axis at \(X\) and the \(y\)-axis at \(Y\), then the length of \(XY\) is always equal to \(a\).

The parametric equations of a curve are

\(x = a(2\theta - \sin 2\theta)\), \(y = a(1 - \cos 2\theta)\).

Show that \(\frac{dy}{dx} = \cot \theta\).

The parametric equations of a curve are

\(x = 2\theta + \sin 2\theta, \quad y = 1 - \cos 2\theta.\)

Show that \(\frac{dy}{dx} = \tan \theta.\)

The parametric equations of a curve are

\(x = te^{2t}\), \(y = t^2 + t + 3\).

(a) Show that \(\frac{dy}{dx} = e^{-2t}\).

(b) Hence show that the normal to the curve, where \(t = -1\), passes through the point \(\left( 0, 3 - \frac{1}{e^4} \right)\).

The parametric equations of a curve are

\(x = 2t - \tan t\), \(y = \ln(\sin 2t)\),

for \(0 < t < \frac{1}{2}\pi\).

Show that \(\frac{dy}{dx} = \cot t\).

The parametric equations of a curve are \(x = \frac{1}{\cos t}\), \(y = \ln \tan t\), where \(0 < t < \frac{1}{2}\pi\).

(a) Show that \(\frac{dy}{dx} = \frac{\cos t}{\sin^2 t}\).

(b) Find the equation of the tangent to the curve at the point where \(y = 0\).

The parametric equations of a curve are

\(x = 1 - \\cos \theta\),

\(y = \\cos \theta - \frac{1}{4} \\cos 2\theta\).

Show that \(\frac{dy}{dx} = -2 \\sin^2 \left( \frac{1}{2} \theta \right)\).

The parametric equations of a curve are

\(x = t + \ln(t + 2), \quad y = (t - 1)e^{-2t}\),

where \(t > -2\).

(a) Express \(\frac{dy}{dx}\) in terms of \(t\), simplifying your answer.

(b) Find the exact \(y\)-coordinate of the stationary point of the curve.

The parametric equations of a curve are

\(x = \ln(2 + 3t)\), \(y = \frac{t}{2 + 3t}\).

(a) Show that the gradient of the curve is always positive.

(b) Find the equation of the tangent to the curve at the point where it intersects the y-axis.

Find the exact coordinates of the stationary points of the curve \(y = \frac{e^{3x^2-1}}{1-x^2}\).