1.07m Tangents and normals: gradient and equations

5 The point \(P ( 1 , a )\), where \(a > 0\), lies on the curve \(y + 4 x = 5 x ^ { 2 } y ^ { 2 }\).

1. Show that \(a = 1\).
2. Find the gradient of the curve at \(P\).
3. Find an equation of the tangent to the curve at \(P\).

2 A curve is defined, for \(t \neq 0\), by the parametric equations $$x = 4 t + 3 , \quad y = \frac { 1 } { 2 t } - 1$$ At the point \(P\) on the curve, \(t = \frac { 1 } { 2 }\).

1. Find the gradient of the curve at the point \(P\).
2. Find an equation of the normal to the curve at the point \(P\).
3. Find a cartesian equation of the curve.

5. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve \(T\) with equation \(y = \cos 2 x\) and the circle \(C _ { 1 }\) that touches \(T\) at \(\left( \frac { \pi } { 4 } , 0 \right)\) and \(\left( \frac { 3 \pi } { 4 } , 0 \right)\) .

1. Find the radius of \(C _ { 1 }\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2a7c2530-a93c-4a26-bc37-c20c0f40c8f2-4_486_586_1199_841} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows a sketch of part of \(T\) and part of a circle \(C _ { 2 }\) that touches \(T\) at the point \(P\) with coordinates \(\left( \frac { \pi } { 2 } , - 1 \right)\) .For values of \(x\) close to \(\frac { \pi } { 2 }\) the curve \(T\) lies inside \(C _ { 2 }\) as shown in Figure 3.
2. Without doing any calculation,explain why the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(C _ { 2 }\) at \(P\) is less than the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) for \(T\) at \(P\) . The radius of \(C _ { 2 }\) is \(r\) .
3. Use the result from part(b)to find a value of \(k\) such that \(r > k\) . Given that \(C _ { 2 }\) cuts \(T\) at the point \(( 0,1 )\) ,
4. find the value of \(r\) .

8 A curve has equation $$y = x ^ { 3 } + p x ^ { 2 } + q x - 45$$ The curve passes through point \(R ( 2,3 )\) The gradient of the curve at \(R\) is 8
8

1. Find the value of \(p\) and the value of \(q\).
   8
2. Calculate the area enclosed between the normal to the curve at \(R\) and the coordinate 8 (b) axes. \(9 \quad\) A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = ( x - 2 ) ( x - 3 ) ^ { 2 }$$

12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive \(x\)-axis at the point \(P\).
12

1. Show that the gradient of the curve at \(P\) is \(- \frac { 3 } { 2 }\)

   12	Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   	[2 marks]
   	\(\_\_\_\_\)

5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where \(x = 0\)

4 A curve has equation $$y = \frac { x ^ { 2 } } { 8 } + 4 \sqrt { x }$$ 4

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 4
2. The point \(P\) with coordinates \(( 4,10 )\) lies on the curve.
   Find an equation of the tangent to the curve at the point \(P\) □
   4
3. Show that the curve has no stationary points.

8 In this question you must show detailed reasoning. A is the point \(( 1,0 ) , B\) is the point \(( 1,1 )\) and \(D\) is the point where the tangent to the curve \(y = x ^ { 3 }\) at B crosses the \(x\)-axis, as shown in Fig. 8. The tangent meets the \(y\)-axis at E. \begin{figure}[h] \end{figure}

1. Find the area of triangle ODE.
2. Find the area of the region bounded by the curve \(y = x ^ { 3 }\), the tangent at B and the \(y\)-axis.

9 In this question you must show detailed reasoning.
The curve \(x y + y ^ { 2 } = 8\) is shown in Fig. 9. \begin{figure}[h] \end{figure} Find the coordinates of the points on the curve at which the normal has gradient 2.

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\) Write down an equation of \(C _ { 2 }\) 14
2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\) 14
3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\) The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\) 14 (c) (i) Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
   [0pt] [2 marks]
   14 (c) (ii) Determine the translation vector.
   [0pt] [2 marks]
   14 (c) (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)

1. A curve \(C\) has equation \(y = \mathrm { f } ( x )\).

The point \(P\) with \(x\) coordinate 3 lies on \(C\) \section*{Given}

• \(\mathrm { f } ^ { \prime } ( x ) = 4 x ^ { 2 } + k x + 3\) where \(k\) is a constant
• the normal to \(C\) at \(P\) has equation \(y = - \frac { 1 } { 24 } x + 5\)
  1. show that \(k = - 5\)
  2. Hence find \(\mathrm { f } ( x )\).

9. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 6 x ^ { 2 } + 4 x - 2 } { 2 x + 1 } \quad x > - \frac { 1 } { 2 }$$

1. Find \(\mathrm { f } ^ { \prime } ( x )\), giving the answer in simplest form. The line \(l\) is the normal to \(C\) at the point \(P ( 2,6 )\)
2. Show that an equation for \(l\) is $$16 y + 5 x = 106$$
3. Write \(\mathrm { f } ( x )\) in the form \(A x + B + \frac { D } { 2 x + 1 }\) where \(A , B\) and \(D\) are constants. The region \(R\), shown shaded in Figure 5, is bounded by \(C , l\) and the \(x\)-axis.
4. Use algebraic integration to find the exact area of \(R\), giving your answer in the form \(P + Q \ln 3\), where \(P\) and \(Q\) are rational constants.
   (Solutions based entirely on calculator technology are not acceptable.)

6 The curve \(y = x ^ { 3 } + a x ^ { 2 } + b x + 1\) has a gradient of 11 at the point \(( 1,7 )\). Find the values of \(a\) and \(b\).

12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).

1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-3_648_679_342_733}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

5 The curve \(C\) has equation \(y = \frac { 12 ( x + 1 ) } { ( x - 2 ) ^ { 2 } }\).

1. Determine the coordinates of any stationary points of \(C\).
2. Sketch \(C\).

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}

1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.

9 Find the equations of all the horizontal tangents to the curve with equation \(y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36\).

10 The diagram shows the curve with equation $$x = ( y - 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(y\)-axis at \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{afc8561d-94ae-42c0-bc6c-e9b091938368-3_588_780_1087_680}

1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
2. Find the exact gradient of the curve at each of the points \(A\) and \(B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{7895dcbc-2ae0-498f-8770-7b738feed7c9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).

1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
2. Calculate the exact area of the region bounded by the curve and the tangent.

2

1. Express \(2 x ^ { 2 } + 6 x + 5\) in the form \(p ( x + q ) ^ { 2 } + r\).
2. State the equation of the line of symmetry of the curve \(y = 2 x ^ { 2 } + 6 x + 5\).
3. Find the value of the constant \(k\) for which the line \(y = k - 2 x\) is a tangent to the curve \(y = 2 x ^ { 2 } + 6 x + 5\).

7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.

1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
2. Show that this normal does not meet the curve again.

5 \includegraphics[max width=\textwidth, alt={}, center]{69214874-18a7-495d-892d-2a0a7019cbe9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).

1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
2. Calculate the exact area of the region bounded by the curve and the tangent.

3 Show that the equation of the tangent to the curve \(y = \ln \left( x ^ { 2 } + 3 \right)\) at the point \(( 1 , \ln 4 )\) is $$2 y - x = \ln ( 16 ) - 1$$