1.07s Parametric and implicit differentiation

5 A curve is defined by the parametric equations $$x = 2 \cos \theta , \quad y = 3 \sin 2 \theta$$

1. 1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = a \sin \theta + b \operatorname { cosec } \theta$$ where \(a\) and \(b\) are integers.
   2. Find the gradient of the normal to the curve at the point where \(\theta = \frac { \pi } { 6 }\).
2. Show that the cartesian equation of the curve can be expressed as $$y ^ { 2 } = p x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where \(p\) is a rational number.

6 A curve is defined by the equation \(9 x ^ { 2 } - 6 x y + 4 y ^ { 2 } = 3\). Find the coordinates of the two stationary points of this curve.

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)

1 A curve is defined by the parametric equations \(x = \frac { t ^ { 2 } } { 2 } + 1 , y = \frac { 4 } { t } - 1\).

1. Find the gradient at the point on the curve where \(t = 2\).
2. Find a Cartesian equation of the curve.
   \includegraphics[max width=\textwidth, alt={}]{9f03a5f3-7fea-4fb7-b3bd-b4c0cdf662a2-02_1730_1709_977_153}

5 A curve is defined by the parametric equations \(x = \cos 2 t , y = \sin t\).
The point \(P\) on the curve is where \(t = \frac { \pi } { 6 }\).

1. Find the gradient at \(P\).
2. Find the equation of the normal to the curve at \(P\) in the form \(y = m x + c\).
3. The normal at \(P\) intersects the curve again at the point \(Q ( \cos 2 q , \sin q )\). Use the equation of the normal to form a quadratic equation in \(\sin q\) and hence find the \(x\)-coordinate of \(Q\).
   [0pt] [5 marks]

7 A curve has equation \(y ^ { 3 } + 2 \mathrm { e } ^ { - 3 x } y - x = k\), where \(k\) is a constant.
The point \(P \left( \ln 2 , \frac { 1 } { 2 } \right)\) lies on this curve.

1. Show that the exact value of \(k\) is \(q - \ln 2\), where \(q\) is a rational number.
2. Find the gradient of the curve at \(P\).

8. \begin{figure}[h] \end{figure} A table top, in the shape of a parallelogram, is made from two types of wood. The design is shown in Fig. 1. The area inside the ellipse is made from one type of wood, and the surrounding area is made from a second type of wood. The ellipse has parametric equations, $$x = 5 \cos \theta , \quad y = 4 \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The parallelogram consists of four line segments, which are tangents to the ellipse at the points where \(\theta = \alpha , \theta = - \alpha , \theta = \pi - \alpha , \theta = - \pi + \alpha\).

1. Find an equation of the tangent to the ellipse at ( \(5 \cos \alpha , 4 \sin \alpha\) ), and show that it can be written in the form $$5 y \sin \alpha + 4 x \cos \alpha = 20 .$$
2. Find by integration the area enclosed by the ellipse.
3. Hence show that the area enclosed between the ellipse and the parallelogram is $$\frac { 80 } { \sin 2 \alpha } - 20 \pi$$
4. Given that \(0 < \alpha < \frac { \pi } { 4 }\), find the value of \(\alpha\) for which the areas of two types of wood are equal.

1. A curve has the equation

$$x ^ { 2 } ( 2 + y ) - y ^ { 2 } = 0 .$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = \cos 2 t , \quad y = \operatorname { cosec } t , \quad 0 < t < \frac { \pi } { 2 } .$$ The point \(P\) on the curve has \(x\)-coordinate \(\frac { 1 } { 2 }\).

1. Find the value of the parameter \(t\) at \(P\).
2. Show that the tangent to the curve at \(P\) has the equation $$y = 2 x + 1$$ The shaded region is bounded by the curve, the coordinate axes and the line \(x = \frac { 1 } { 2 }\).
3. Show that the area of the shaded region is given by $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } k \cos t \mathrm {~d} t$$ where \(k\) is a positive integer to be found.
4. Hence find the exact area of the shaded region.
   7. continued
   7. continued

3. A curve has the equation $$4 x ^ { 2 } - 2 x y - y ^ { 2 } + 11 = 0$$ Find an equation for the normal to the curve at the point with coordinates \(( - 1 , - 3 )\). (8)
3. continued

7. \begin{figure}[h] \end{figure} Figure 2 shows the curve with parametric equations $$x = - 1 + 4 \cos \theta , \quad y = 2 \sqrt { 2 } \sin \theta , \quad 0 \leq \theta < 2 \pi$$ The point \(P\) on the curve has coordinates \(( 1 , \sqrt { 6 } )\).

1. Find the value of \(\theta\) at \(P\).
2. Show that the normal to the curve at \(P\) passes through the origin.
3. Find a cartesian equation for the curve.
   7. continued

3. A curve has the equation $$3 x ^ { 2 } - 2 x + x y + y ^ { 2 } - 11 = 0$$ The point \(P\) on the curve has coordinates \(( - 1,3 )\).

1. Show that the normal to the curve at \(P\) has the equation \(y = 2 - x\).
2. Find the coordinates of the point where the normal to the curve at \(P\) meets the curve again.
   3. continued

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve with parametric equations $$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$ The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the value of \(t\) at \(O\) and the value of \(t\) at \(A\). The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis.
2. Show that the volume of the solid formed is given by $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
3. Using the substitution \(u = \sin t\), or otherwise, evaluate this integral, giving your answer as an exact multiple of \(\pi\).
   6. continued

1. A curve has the equation

$$4 \cos x + 2 \sin y = 3$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
2. Find an equation for the tangent to the curve at the point \(\left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)\), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.

1. A curve has the equation

$$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).

1. A curve has the equation

$$x ^ { 3 } + 2 x y - y ^ { 2 } + 24 = 0$$ Show that the normal to the curve at the point \(( 2 , - 4 )\) has the equation \(y = 3 x - 10\). (8)

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve with equation \(y = \ln ( 2 + \cos x ) , 0 \leq x \leq \pi\).

1. Complete the table below for points on the curve, giving the \(y\) values to 4 decimal places.
2. Giving your answers to 3 decimal places, find estimates for the area of the region bounded by the curve and the coordinate axes using the trapezium rule with
   1. 1 strip,
   2. 2 strips,
   3. 4 strips.
3. Making your reasoning clear, suggest a value to 2 decimal places for the actual area of the region bounded by the curve and the coordinate axes.

   \(x\)	0	\(\frac { \pi } { 4 }\)	\(\frac { \pi } { 2 }\)	\(\frac { 3 \pi } { 4 }\)	\(\pi\)
   \(y\)	1.0986				0

   1. continued
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3056ad22-f87b-46c3-86cf-d46939927465-06_563_983_146_379} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows the curve with parametric equations $$x = \tan \theta , \quad y = \cos ^ { 2 } \theta , \quad - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$$ The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = - 1\) and \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

14
2 \end{array} \right) , $$ and\\ where \(a\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.\\ Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.\\ 4. continued\\ 5. A curve has the equation $$x ^ { 2 } - 4 x y + 2 y ^ { 2 } = 1$$
4. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in its simplest form in terms of \(x\) and \(y\).
5. Show that the tangent to the curve at the point \(P ( 1,2 )\) has the equation $$3 x - 2 y + 1 = 0$$ The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
6. Find the coordinates of \(Q\).\\ 5. continued\\ 6. The rate of increase in the number of bacteria in a culture, \(N\), at time \(t\) hours is proportional to \(N\).
7. Write down a differential equation connecting \(N\) and \(t\). Given that initially there are \(N _ { 0 }\) bacteria present in a culture,
8. Show that \(N = N _ { 0 } \mathrm { e } ^ { k t }\), where \(k\) is a positive constant. Given also that the number of bacteria present doubles every six hours,
9. find the value of \(k\),
10. find how long it takes for the number of bacteria to increase by a factor of ten, giving your answer to the nearest minute. of ten, giving your answer to the nearest minute.\\ 6. continued\\ 7. A curve has parametric equations $$x = \sec \theta + \tan \theta , \quad y = \operatorname { cosec } \theta + \cot \theta , \quad 0 < \theta < \frac { \pi } { 2 } .$$
11. Show that \(x + \frac { 1 } { x } = 2 \sec \theta\). Given that \(y + \frac { 1 } { y } = 2 \operatorname { cosec } \theta\),
12. find a cartesian equation for the curve.
13. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} \theta } = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right)\).
14. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
    7. continued
    7. continued

5 The position vector of a toy boat of mass 1.5 kg is modelled as \(\mathbf { r } = ( 2 + t ) \mathbf { i } + \left( 3 t - t ^ { 2 } \right) \mathbf { j }\) where lengths are in metres, \(t\) is the time in seconds, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal, perpendicular unit vectors and the origin is O .

1. Find the velocity of the boat when \(t = 4\).
2. Find the acceleration of the boat and the horizontal force acting on the boat.
3. Find the cartesian equation of the path of the boat referred to \(x\) - and \(y\)-axes in the directions of \(\mathbf { i }\) and \(\mathbf { j }\), respectively, with origin O . You are not required to simplify your answer.

12 The diagram shows the curve with parametric equations \(x = 2 \cosh t + \sinh t , y = \cosh t - 2 \sinh t\). \includegraphics[max width=\textwidth, alt={}, center]{83275e7c-7f5a-4f26-b81d-a041e67ac9a2-5_812_808_1283_246}

1. The curve crosses the positive \(x\)-axis at A .
   1. Determine the value of the parameter \(t\) at A , giving your answer in logarithmic form.
   2. Find the \(x\)-coordinate of A , giving your answer correct to \(\mathbf { 3 }\) significant figures.
2. The point B has parameter \(t = 0\). Determine the equation of the tangent to the curve at B .

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 \(m\).

1. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.

The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).

1. Use calculus to show that an equation of the tangent to \(P\) at \(A\) is $$y t = x + a t ^ { 2 }$$ The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
   The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
   Given that the area of the triangle \(B C D\) is 432
2. determine the coordinates of \(B\) and the coordinates of \(C\).

15 In this question you must show detailed reasoning. The equation of a curve is $$\ln y + x ^ { 3 } y = 8$$ Find the equation of the normal to the curve at the point where \(y = 1\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { c } = 0\), where \(a , b\) and \(c\) are constants to be found.