Convert to Cartesian (exponential/logarithmic)

2 A curve has parametric equations \(x = \mathrm { e } ^ { 3 t } , y = t \mathrm { e } ^ { 2 t }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\).
2. Find the cartesian equation of the curve in the form \(y = a x ^ { b } \ln x\), where \(a\) and \(b\) are constants to be determined.

3 Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \begin{figure}[h] \end{figure}

1. Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
3. Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\). An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
4. Find the volume of the object.

2 Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \begin{figure}[h] \end{figure}

1. Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
3. Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\). An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
4. Find the volume of the object.

6 A curve has parametric equations $$x = \mathrm { e } ^ { 2 t } , \quad y = \frac { 2 t } { 1 + t }$$

1. Find the gradient of the curve at the point where \(t = 0\).
2. Find \(y\) in terms of \(x\).

3 A curve has parametric equations $$x = \mathrm { e } ^ { 2 t } , \quad y = \frac { 2 t } { 1 + t }$$

1. Find the gradient of the curve at the point where \(t = 0\).
2. Find \(y\) in terms of \(x\).

7 Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u , y = u + \frac { 1 } { u } , \quad 1 \leqslant u \leqslant 10 .$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \begin{figure}[h] \end{figure}

1. Find the lengths \(\mathrm { OA } , \mathrm { OB }\) and AC .
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(u\). Hence find the angle \(\theta\).
3. Show that the cartesian equation of the curve is \(y = \mathrm { e } ^ { \frac { 1 } { 5 } x } + \mathrm { e } ^ { - \frac { 1 } { 5 } x }\). An object is formed by rotating the region OACB through \(360 ^ { \circ }\) about \(\mathrm { O } x\).
4. Find the volume of the object. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

5 A curve has parametric equations \(x = \mathrm { e } ^ { 3 t } , y = t \mathrm { e } ^ { 2 t }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Hence find the exact gradient of the curve at the point with parameter \(t = 1\).
2. Find the cartesian equation of the curve in the form \(y = a x ^ { b } \ln x\), where \(a\) and \(b\) are constants to be determined.

1. The curve \(C\) has parametric equations

$$x = t ^ { 2 } + 6 t - 16 \quad y = 6 \ln ( t + 3 ) \quad t > - 3$$

1. Show that a Cartesian equation for \(C\) is $$y = A \ln ( x + B ) \quad x > - B$$ where \(A\) and \(B\) are integers to be found. The curve \(C\) cuts the \(y\)-axis at the point \(P\)
2. Show that the equation of the tangent to \(C\) at \(P\) can be written in the form $$a x + b y = c \ln 5$$ where \(a\), \(b\) and \(c\) are integers to be found.

4 A curve is defined by the parametric equations $$x = 3 \mathrm { e } ^ { t } , \quad y = \mathrm { e } ^ { 2 t } - \mathrm { e } ^ { - 2 t }$$

1. 1. Find the gradient of the curve at the point where \(t = 0\).
   2. Find an equation of the tangent to the curve at the point where \(t = 0\).
2. Show that the cartesian equation of the curve can be written in the form $$y = \frac { x ^ { 2 } } { k } - \frac { k } { x ^ { 2 } }$$ where \(k\) is an integer.

4 A curve is defined by the parametric equations \(x = 8 \mathrm { e } ^ { - 2 t } - 4 , y = 2 \mathrm { e } ^ { 2 t } + 4\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. The point \(P\), where \(t = \ln 2\), lies on the curve.
   1. Find the gradient of the curve at \(P\).
   2. Find the coordinates of \(P\).
   3. The normal at \(P\) crosses the \(x\)-axis at the point \(Q\). Find the coordinates of \(Q\).
3. Find the Cartesian equation of the curve in the form \(x y + 4 y - 4 x = k\), where \(k\) is an integer.
   (3 marks)

6

1. A curve is in the first quadrant. It has parametric equations \(x = \cosh t + \sinh t , y = \cosh t - \sinh t\) where \(t \in \mathbb { R }\). Show that the cartesian equation of the curve is \(x y = 1\). Fig. 6 shows the curve from part (i). P is a point on the curve. O is the origin. Point A lies on the \(x\)-axis, point B lies on the \(y\)-axis and OAPB is a rectangle. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d39832-5519-463d-ac7a-5d406ffd7be0-3_931_1050_598_479} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Find the smallest possible value of the perimeter of rectangle OAPB. Justify your answer.

5 A curve is defined by the parametric equations $$\begin{aligned} & x = 4 \times 2 ^ { - t } + 3 \\ & y = 3 \times 2 ^ { t } - 5 \end{aligned}$$ 5

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 } \times 2 ^ { 2 t }\)
   5
2. Find the Cartesian equation of the curve in the form \(x y + a x + b y = c\), where \(a , b\) and \(c\) are integers.