1.03g Parametric equations: of curves and conversion to cartesian

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
2. Hence, using algebraic integration, find the exact area of \(R\). [3]

A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).

1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
2. Find the cartesian equation of the curve. [3]
3. State the set of values that \(x\) can take and hence sketch the curve. [3]

The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}

1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]

In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]

The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$

1. Find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [2]

The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).

1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]

The line \(l\) intersects the curve \(C\) again at the point \(Q\).

1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]

In this question you must show detailed reasoning. \includegraphics{figure_12} The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the exact \(y\)-coordinate of \(P\). [1]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).

1. Determine the exact coordinates of \(Y\). [4]

The curve \(C\) and the line segment \(PY\) are rotated \(2\pi\) radians about the \(y\)-axis.

1. Determine the exact volume of the solid generated. Give your answer in the form \(\pi(\ln p + q)\), where \(p\) and \(q\) are rational numbers. [8]

[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\)]

The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [2]

The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).

1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]

The line \(l\) intersects the curve \(C\) again at the point \(Q\).

1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]

A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t^2, \quad y = t^3.$$

1. Show that the equation of the tangent at the point with parameter \(t\) is $$2y = 3tx - t^3.$$ [4]

1. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A\left(\frac{19}{2}, -\frac{15}{8}\right)\) and it meets the \(x\)-axis at the point \(B\). Find the area of triangle \(OAB\), where \(O\) is the origin. [7]

\includegraphics{figure_6} The diagram shows the curve with parametric equations \(x = \ln(t^2 - 4)\), \(y = \frac{4}{t}\), where \(t > 2\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = \ln 5\) and \(x = \ln 12\).

1. Show that the area of \(R\) is given by $$\int_a^b \frac{8}{t(t^2 - 4)} dt,$$ where \(a\) and \(b\) are constants to be determined. [4]
2. In this question you must show detailed reasoning. Hence find the exact area of \(R\), giving your answer in the form \(\ln k\), where \(k\) is a constant to be determined. [8]
3. Find a cartesian equation of the curve in the form \(y = \text{f}(x)\). [3]

The parametric equations of a curve are given by \(x = 2\cos\theta\) and \(y = 3\sin\theta\) for \(0 \leq \theta < 2\pi\).

1. Find \(\frac{dy}{dx}\) in terms of \(\theta\). [2]

The tangents to the curve at the points P and Q pass through the point (2, 6).

1. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2\sin\theta + \cos\theta = 1\). [4]
2. Find the values of \(\theta\) at the points P and Q. [5]

A curve is given by \(x = t^2 - 2\ln t\), \(y = 4t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\). Given that \(A = k\pi\), where \(k\) is an integer, write down an integral which gives \(A\) and find the value of \(k\). [4]

The parametric equations of a curve are \(x = \frac{1}{1 + t^2}\) and \(y = \frac{t}{1 + t^2}\), \(t \in \mathbb{R}\).

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [5]
2. Hence find the coordinates of the stationary points of the curve. [2]

% \includegraphics{figure_2} - Shows a circular tower with center T at (0,1), a goat at point G attached to the base at O, with string along arc OA then tangent AG A circular tower stands in a large horizontal field of grass. A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower. Taking the point \(O\) as the origin \((0, 0)\), the centre of the base of the tower is at the point \(T(0, 1)\). The radius of the base of the tower is 1. The string has length \(\pi\) and you may ignore the size of the goat. The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.

1. Write down the equation of \(C\) for \(y < 0\). [1] When the goat is at the point \(G(x, y)\), with \(x > 0\) and \(y > 0\), as shown in Figure 2, the string lies along \(OAG\) where \(OA\) is an arc of the circle with angle \(OTA = \theta\) radians and \(AG\) is a tangent to the circle at \(A\).
2. With the aid of a suitable diagram show that $$x = \sin \theta + (\pi - \theta) \cos \theta$$ $$y = 1 - \cos \theta + (\pi - \theta) \sin \theta$$ [5]
3. By considering \(\int y \frac{dx}{d\theta} d\theta\), show that the area between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form $$\int_0^{\pi} u \sin u \, du + \int_0^{\pi} u^2 \sin^2 u \, du + \int_0^{\pi} u \sin u \cos u \, du$$ [5]
4. Show that \(\int_0^{\pi} u^2 \sin^2 u \, du = \frac{\pi^3}{6} + \int_0^{\pi} u \sin u \cos u \, du\) [4]
5. Hence find the area of grass that can be reached by the goat. [8]

The curve \(C\) has parametric equations $$x = \cos^2 t$$ $$y = \cos t \sin t$$ where \(0 \leq t < \pi\)

1. Show that \(C\) is a circle and find its centre and its radius. [5]

% Figure 1 shows a sketch of C with point P, rectangle R with diagonal OP \includegraphics{figure_1} Figure 1 Figure 1 shows a sketch of \(C\). The point \(P\), with coordinates \((\cos^2 \alpha, \cos\alpha \sin \alpha)\), \(0 < \alpha < \frac{\pi}{2}\), lies on \(C\). The rectangle \(R\) has one side on the \(x\)-axis, one side on the \(y\)-axis and \(OP\) as a diagonal, where \(O\) is the origin.

1. Show that the area of \(R\) is \(\sin\alpha \cos^3 \alpha\) [1]
2. Find the maximum area of \(R\), as \(\alpha\) varies. [7]

[Total 13 marks]