Find parameter from gradient condition

7 A curve has parametric equations $$x = \frac { 2 t + 3 } { t + 2 } , \quad y = t ^ { 2 } + a t + 1$$ where \(a\) is a constant. It is given that, at the point \(P\) on the curve, the gradient is 1 .

1. Show that the value of \(t\) at \(P\) satisfies the equation $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1 = 0$$
2. It is given that \(( t + 1 )\) is a factor of $$2 t ^ { 3 } + ( a + 8 ) t ^ { 2 } + ( 4 a + 8 ) t + 4 a - 1$$ Find the value of \(a\).
3. Hence show that \(P\) is the only point on the curve at which the gradient is 1 .
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7
\includegraphics[max width=\textwidth, alt={}, center]{ce0d5faa-9428-4afd-829d-7634c5bd150d-12_446_613_274_760} The diagram shows the curve with parametric equations $$x = k \tan t , \quad y = 3 \sin 2 t - 4 \sin t ,$$ for \(0 < t < \frac { 1 } { 2 } \pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).

1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction.
2. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(k\) and \(\cos t\).
3. Given that the normal to the curve at \(P\) has gradient \(\frac { 9 } { 10 }\), find the value of \(k\), giving your answer as an exact fraction.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7 A curve is defined by the parametric equations $$x = 3 t - 2 \sin t , \quad y = 5 t + 4 \cos t$$ where \(0 \leqslant t \leqslant 2 \pi\). At each of the points \(P\) and \(Q\) on the curve, the gradient of the curve is \(\frac { 5 } { 2 }\).

1. Show that the values of \(t\) at \(P\) and \(Q\) satisfy the equation \(10 \cos t - 8 \sin t = 5\).
2. Express \(10 \cos t - 8 \sin t\) in the form \(R \cos ( t + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and the value of \(\alpha\) correct to 3 significant figures.
3. Hence find the values of \(t\) at the points \(P\) and \(Q\).

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\includegraphics[max width=\textwidth, alt={}, center]{68f4b2dc-a05d-4061-aaf0-de15cfe186a9-10_657_792_269_664} The diagram shows the curve with parametric equations $$x = 3 \cos 2 \theta , \quad y = 4 \sin \theta ,$$ for \(\pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). Points \(P\) and \(Q\) lie on the curve. The gradient of the curve at \(P\) is 2 . The straight line \(3 x + y = 0\) meets the curve at \(Q\).

1. Find the value of \(\theta\) at \(P\), giving your answer correct to 3 significant figures.
2. Find the gradient of the curve at \(Q\), giving your answer correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The parametric equations of a curve are $$x = 2 \sin \theta + \sin 2 \theta , \quad y = 2 \cos \theta + \cos 2 \theta$$ where \(0 < \theta < \pi\).

1. Obtain an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\).
2. Hence find the exact coordinates of the point on the curve at which the tangent is parallel to the \(y\)-axis.

7 The parametric equations of a curve are $$x = 2 t - \sin 2 t , \quad y = 5 t + \cos 2 t$$ for \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). At the point \(P\) on the curve, the gradient of the curve is 2 .

1. Show that the value of the parameter at \(P\) satisfies the equation \(2 \sin 2 t - 4 \cos 2 t = 1\).
2. By first expressing \(2 \sin 2 t - 4 \cos 2 t\) in the form \(R \sin ( 2 t - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\), find the coordinates of \(P\). Give each coordinate correct to 3 significant figures.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The parametric equations of a curve are $$x = \sqrt { t } + 3 , \quad y = \ln t$$ for \(t > 0\).

1. Obtain a simplified expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
2. Hence find the exact coordinates of the point on the curve at which the gradient of the normal is - 2 .

8. \begin{figure}[h] \end{figure} The curve shown in Figure 3 has parametric equations $$x = t - 4 \sin t , y = 1 - 2 \cos t , \quad - \frac { 2 \pi } { 3 } \leqslant t \leqslant \frac { 2 \pi } { 3 }$$ The point \(A\), with coordinates ( \(k , 1\) ), lies on the curve. Given that \(k > 0\)

1. find the exact value of \(k\),
2. find the gradient of the curve at the point \(A\). There is one point on the curve where the gradient is equal to \(- \frac { 1 } { 2 }\)
3. Find the value of \(t\) at this point, showing each step in your working and giving your answer to 4 decimal places.
   [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]

6 A curve has parametric equations $$x = 9 t - \ln ( 9 t ) , \quad y = t ^ { 3 } - \ln \left( t ^ { 3 } \right)$$ Show that there is only one value of \(t\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and state that value.

5 The parametric equations of a curve are $$x = 2 + 3 \sin \theta \text { and } y = 1 - 2 \cos \theta \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$$

1. Find the coordinates of the point on the curve where the gradient is \(\frac { 1 } { 2 }\).
2. Find the cartesian equation of the curve.

9 A particle moves in the \(x - y\) plane so that at time \(t\) seconds, where \(t \geqslant 0\), its coordinates are given by \(x = \mathrm { e } ^ { 2 t } - 4 \mathrm { e } ^ { t } + 3 , y = 2 \mathrm { e } ^ { - 3 t }\).

1. Explain why the path of the particle never crosses the \(x\)-axis.
2. Determine the exact values of \(t\) when the path of the particle intersects the \(y\)-axis.
3. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 2 \mathrm { e } ^ { 4 t } - \mathrm { e } ^ { 5 t } }\).
4. Hence find the coordinates of the particle when its path is parallel to the \(y\)-axis.

1 A family of curves is given by the parametric equations
\(x ( t ) = \cos ( t ) - \frac { \cos ( ( m + 1 ) t ) } { m + 1 }\) and \(y ( t ) = \sin ( t ) - \frac { \sin ( ( m + 1 ) t ) } { m + 1 }\)
where \(0 \leqslant t < 2 \pi\) and \(m\) is a positive integer.

1. 1. Sketch the curves in the cases \(m = 3 , m = 4\) and \(m = 5\) on separate axes in the Printed Answer Booklet.
   2. State one common feature of these three curves.
   3. State a feature for the case \(m = 4\) which is absent in the cases \(m = 3\) and \(m = 5\).
2. 1. Determine, in terms of \(m\), the values of \(t\) for which \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) but \(\frac { \mathrm { d } y } { \mathrm {~d} t } \neq 0\).
   2. Describe the tangent to the curve at the points corresponding to such values of \(t\).
3. 1. Show that the curve lies between the circle centred at the origin with radius $$1 - \frac { 1 } { m + 1 }$$ and the circle centred at the origin with radius $$1 + \frac { 1 } { m + 1 }$$
   2. Hence, or otherwise, show that the area \(A\) bounded by the curve satisfies $$\frac { m ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } } < A < \frac { ( m + 2 ) ^ { 2 } \pi } { ( m + 1 ) ^ { 2 } }$$
   3. Find the limit of the area bounded by the curve as \(m\) tends to infinity.
4. The arc length of a curve defined by parametric equations \(x ( t )\) and \(y ( t )\) between points corresponding to \(t = c\) and \(t = d\), where \(c < d\), is $$\int _ { c } ^ { d } \sqrt { \left( \frac { \mathrm {~d} x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } } \mathrm {~d} t$$ Use this to show that the length of the curve is independent of \(m\).

10 In this question you must show detailed reasoning. Fig. 10 shows the curve given parametrically by the equations \(\mathrm { x } = \frac { 1 } { \mathrm { t } ^ { 2 } } , \mathrm { y } = \frac { 1 } { \mathrm { t } ^ { 3 } } - \frac { 1 } { \mathrm { t } }\), for \(t > 0\). \begin{figure}[h] \end{figure}

1. Show that \(\frac { d y } { d x } = \frac { 3 - t ^ { 2 } } { 2 t }\).
2. Find the coordinates of the point on the curve at which the tangent to the curve is parallel to the line \(4 \mathrm { y } + \mathrm { x } = 1\).
3. Find the cartesian equation of the curve. Give your answer in factorised form.

5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where \(k\) is a positive constant.

1. For the case \(k = 1\), use your graphical calculator to sketch the curve. Describe its main features.
2. Sketch the curve for a value of \(k\) between 0 and 1 . Describe briefly how the main features differ from those for the case \(k = 1\).
3. For the case \(k = 2\) :
   (A) sketch the curve;
   (B) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\);
   (C) show that the width of each loop, measured parallel to the \(x\)-axis, is $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
4. Use your calculator to find, correct to one decimal place, the value of \(k\) for which successive loops just touch each other.

12 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 \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\). The tangents to the curve at the points P and Q pass through the point \(( 2,6 )\).
2. Show that the values of \(\theta\) at the points P and Q satisfy the equation \(2 \sin \theta + \cos \theta = 1\).
3. Find the values of \(\theta\) at the points \(P\) and \(Q\).

6 A design for a surfboard is shown in Figure 1. \begin{figure}[h] \end{figure} The curve of the top half of the surfboard can be modelled by the parametric equations $$\begin{aligned} & x = - 2 t ^ { 2 } \\ & y = 9 t - 0.7 t ^ { 2 } \end{aligned}$$ for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres. \begin{figure}[h] \end{figure} 6

1. Find the length of the surfboard.
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2. 1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
      6
3. (ii) Hence, show that the width of the surfboard is approximately one third of its length.
   \(7 \quad\) A planet takes \(T\) days to complete one orbit of the Sun.
   \(T\) is known to be related to the planet's average distance \(d\), in millions of kilometres, from the Sun. A graph of \(\log _ { 10 } T\) against \(\log _ { 10 } d\) is shown with data for Mercury and Uranus labelled.
   \includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-08_752_1447_580_296}

9 A water slide is the shape of a curve \(P Q\) as shown in Figure 1 below. \begin{figure}[h] \end{figure} The curve can be modelled by the parametric equations $$\begin{aligned} & x = t - \frac { 1 } { t } + 4.8 \\ & y = t + \frac { 2 } { t } \end{aligned}$$ where \(0.2 \leq t \leq 3\) The horizontal distance from \(O\) is \(x\) metres. The vertical distance above the point \(O\) at ground level is \(y\) metres.
\(P\) is the point where \(t = 0.2\) and \(Q\) is the point where \(t = 3\) 9

1. To make sure speeds are safe at \(Q\), the difference in height between \(P\) and \(Q\) must be less than 7 metres. Show that the slide meets this safety requirement.
   9
2. 1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\)
      9
3. (ii) A vertical support, \(R S\), is to be added between the ground and the lowest point on the slide as shown in Figure 2 below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{6fba7e53-de46-460b-9bef-f1a6962f2e7d-16_590_1278_475_470}
   \end{figure} Find the length of \(R S\)
   9
4. (iii) Find the acute angle the slide makes with the horizontal at \(Q\) Give your answer to the nearest degree.
   \section*{END OF SECTION A}